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# Maxima 5.41.0 Manual

Maxima is a computer algebra system, implemented in Lisp.

Maxima is derived from the Macsyma system, developed at MIT in the years 1968 through 1982 as part of Project MAC. MIT turned over a copy of the Macsyma source code to the Department of Energy in 1982; that version is now known as DOE Macsyma. A copy of DOE Macsyma was maintained by Professor William F. Schelter of the University of Texas from 1982 until his death in 2001. In 1998, Schelter obtained permission from the Department of Energy to release the DOE Macsyma source code under the GNU Public License, and in 2000 he initiated the Maxima project at SourceForge to maintain and develop DOE Macsyma, now called Maxima.

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# 1. Introduction to Maxima

Start Maxima with the command "maxima". Maxima will display version information and a prompt. End each Maxima command with a semicolon. End the session with the command "quit();". Here's a sample session:

    [wfs@chromium]$ maxima
    Maxima 5.9.1 http://maxima.sourceforge.net
    Using Lisp CMU Common Lisp 19a
    Distributed under the GNU Public License. See the file COPYING.
    Dedicated to the memory of William Schelter.
    This is a development version of Maxima. The function bug_report()
    provides bug reporting information.
    (%i1) factor(10!);
                                8  4  2
    (%o1)                      2  3  5  7
    (%i2) expand ((x + y)^6);
           6        5       2  4       3  3       4  2      5      6
    (%o2) y  + 6 x y  + 15 x  y  + 20 x  y  + 15 x  y  + 6 x  y + x
    (%i3) factor (x^6 - 1);
                                  2            2
    (%o3)       (x - 1) (x + 1) (x  - x + 1) (x  + x + 1)
    (%i4) quit();
    [wfs@chromium]$

Maxima can search the info pages. Use the `describe` command to show information about the command or all the commands and variables containing a string. The question mark `?` (exact search) and double question mark `??` (inexact search) are abbreviations for `describe`:

    (%i1) ?? integ
     0: Functions and Variables for Elliptic Integrals
     1: Functions and Variables for Integration
     2: Introduction to Elliptic Functions and Integrals
     3: Introduction to Integration
     4: askinteger  (Functions and Variables for Simplification)
     5: integerp  (Functions and Variables for Miscellaneous Options)
     6: integer_partitions  (Functions and Variables for Sets)
     7: integrate  (Functions and Variables for Integration)
     8: integrate_use_rootsof  (Functions and Variables for Integration)
     9: integration_constant_counter  (Functions and Variables for
        Integration)
     10: nonnegintegerp  (Functions and Variables for linearalgebra)
    Enter space-separated numbers, `all' or `none': 5 4

     -- Function: integerp (<expr>)
         Returns `true' if <expr> is a literal numeric integer, otherwise
         `false'.

         `integerp' returns false if its argument is a symbol, even if the
         argument is declared integer.

         Examples:

              (%i1) integerp (0);
              (%o1)                         true
              (%i2) integerp (1);
              (%o2)                         true
              (%i3) integerp (-17);
              (%o3)                         true
              (%i4) integerp (0.0);
              (%o4)                         false
              (%i5) integerp (1.0);
              (%o5)                         false
              (%i6) integerp (%pi);
              (%o6)                         false
              (%i7) integerp (n);
              (%o7)                         false
              (%i8) declare (n, integer);
              (%o8)                         done
              (%i9) integerp (n);
              (%o9)                         false

     -- Function: askinteger (<expr>, integer)
     -- Function: askinteger (<expr>)
     -- Function: askinteger (<expr>, even)
     -- Function: askinteger (<expr>, odd)
         `askinteger (<expr>, integer)' attempts to determine from the
         `assume' database whether <expr> is an integer.  `askinteger'
         prompts the user if it cannot tell otherwise, and attempt to
         install the information in the database if possible.  `askinteger
         (<expr>)' is equivalent to `askinteger (<expr>, integer)'.

         `askinteger (<expr>, even)' and `askinteger (<expr>, odd)'
         likewise attempt to determine if <expr> is an even integer or odd
         integer, respectively.

    (%o1)                                true

To use a result in later calculations, you can assign it to a variable or refer to it by its automatically supplied label. In addition, `%` refers to the most recent calculated result:

    (%i1) u: expand ((x + y)^6);
           6        5       2  4       3  3       4  2      5      6
    (%o1) y  + 6 x y  + 15 x  y  + 20 x  y  + 15 x  y  + 6 x  y + x
    (%i2) diff (u, x);
             5         4       2  3       3  2       4        5
    (%o2) 6 y  + 30 x y  + 60 x  y  + 60 x  y  + 30 x  y + 6 x
    (%i3) factor (%o2);
                                        5
    (%o3)                      6 (y + x)

Maxima knows about complex numbers and numerical constants:

    (%i1) cos(%pi);
    (%o1)                          - 1
    (%i2) exp(%i*%pi);
    (%o2)                          - 1

Maxima can do differential and integral calculus:

    (%i1) u: expand ((x + y)^6);
           6        5       2  4       3  3       4  2      5      6
    (%o1) y  + 6 x y  + 15 x  y  + 20 x  y  + 15 x  y  + 6 x  y + x
    (%i2) diff (%, x);
             5         4       2  3       3  2       4        5
    (%o2) 6 y  + 30 x y  + 60 x  y  + 60 x  y  + 30 x  y + 6 x
    (%i3) integrate (1/(1 + x^3), x);
                                      2 x - 1
                    2            atan(-------)
               log(x  - x + 1)        sqrt(3)    log(x + 1)
    (%o3)    - --------------- + ------------- + ----------
                      6             sqrt(3)          3

Maxima can solve linear systems and cubic equations:

    (%i1) linsolve ([3*x + 4*y = 7, 2*x + a*y = 13], [x, y]);
                            7 a - 52        25
    (%o1)              [x = --------, y = -------]
                            3 a - 8       3 a - 8
    (%i2) solve (x^3 - 3*x^2 + 5*x = 15, x);
    (%o2)       [x = - sqrt(5) %i, x = sqrt(5) %i, x = 3]

Maxima can solve nonlinear sets of equations. Note that if you don't want a result printed, you can finish your command with `$` instead of `;`.

    (%i1) eq_1: x^2 + 3*x*y + y^2 = 0$
    (%i2) eq_2: 3*x + y = 1$
    (%i3) solve ([eq_1, eq_2]);
                  3 sqrt(5) + 7      sqrt(5) + 3
    (%o3) [[y = - -------------, x = -----------],
                        2                 2

                                   3 sqrt(5) - 7        sqrt(5) - 3
                              [y = -------------, x = - -----------]]
                                         2                   2

Maxima can generate plots of one or more functions:

    (%i1) plot2d (sin(x)/x, [x, -20, 20])$

    (%i2) plot2d ([atan(x), erf(x), tanh(x)], [x, -5, 5], [y, -1.5, 2])$

    (%i3) plot3d (sin(sqrt(x^2 + y^2))/sqrt(x^2 + y^2),
             [x, -12, 12], [y, -12, 12])$

Categories: Help

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# 2. Bug Detection and Reporting

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## 2.1 Functions and Variables for Bug Detection and Reporting

Function: run_testsuite ([options])

  Run the Maxima test suite. Tests producing the desired answer are considered "passes," as are tests that do not produce the desired answer, but are marked as known bugs.

  `run_testsuite` takes the following optional keyword arguments

  `display_all`

    Display all tests. Normally, the tests are not displayed, unless the test fails. (Defaults to `false`).

  `display_known_bugs`

    Displays tests that are marked as known bugs. (Default is `false`).

  `tests`

    This is a single test or a list of tests that should be run. Each test can be specified by either a string or a symbol. By default, all tests are run. The complete set of tests is specified by `testsuite_files`.

  `time`

    Display time information. If `true`, the time taken for each test file is displayed. If `all`, the time for each individual test is shown if `display_all` is `true`. The default is `false`, so no timing information is shown.

  `share_tests`

    Load additional tests for the `share` directory. If `true`, these additional tests are run as a part of the testsuite. If `false`, no tests from the `share` directory are run. If `only`, only the tests from the `share` directory are run. Of course, the actual set of test that are run can be controlled by the `tests` option. The default is `false`.

  For example `run_testsuite(display_known_bugs = true, tests=[rtest5])` runs just test `rtest5` and displays the test that are marked as known bugs.

  `run_testsuite(display_all = true, tests=["rtest1", rtest1a])` will run tests `rtest1` and `rtest2`, and displays each test.

  `run_testsuite` changes the Maxima environment. Typically a test script executes `kill` to establish a known environment (namely one without user-defined functions and variables) and then defines functions and variables appropriate to the test.

  `run_testsuite` returns `done`.

  Categories: Debugging

Option variable: testsuite_files

  `testsuite_files` is the set of tests to be run by `run_testsuite`. It is a list of names of the files containing the tests to run. If some of the tests in a file are known to fail, then instead of listing the name of the file, a list containing the file name and the test numbers that fail is used.

  For example, this is a part of the default set of tests:

       ["rtest13s", ["rtest14", 57, 63]]

  This specifies the testsuite consists of the files "rtest13s" and "rtest14", but "rtest14" contains two tests that are known to fail: 57 and 63.

  Categories: Debugging · Global variables

Option variable: share_testsuite_files

  `share_testsuite_files` is the set of tests from the `share` directory that is run as a part of the test suite by `run_testsuite`..

  Categories: Debugging · Global variables

Function: bug_report ()

  Prints out Maxima and Lisp version numbers, and gives a link to the Maxima project bug report web page. The version information is the same as reported by `build_info`.

  When a bug is reported, it is helpful to copy the Maxima and Lisp version information into the bug report.

  `bug_report` returns an empty string `""`.

  Categories: Debugging

Function: build_info ()

  Returns a summary of the parameters of the Maxima build, as a Maxima structure (defined by `defstruct`). The fields of the structure are: `version`, `timestamp`, `host`, `lisp_name`, and `lisp_version`. When the pretty-printer is enabled (via `display2d`), the structure is displayed as a short table.

  See also `bug_report`.

  Examples:

      (%i1) build_info ();
      (%o1)
      Maxima version: "5.36.1"
      Maxima build date: "2015-06-02 11:26:48"
      Host type: "x86_64-unknown-linux-gnu"
      Lisp implementation type: "GNU Common Lisp (GCL)"
      Lisp implementation version: "GCL 2.6.12"
      (%i2) x : build_info ()$
      (%i3) x@version;
      (%o3)                               5.36.1
      (%i4) x@timestamp;
      (%o4)                         2015-06-02 11:26:48
      (%i5) x@host;
      (%o5)                      x86_64-unknown-linux-gnu
      (%i6) x@lisp_name;
      (%o6)                        GNU Common Lisp (GCL)
      (%i7) x@lisp_version;
      (%o7)                             GCL 2.6.12
      (%i8) x;
      (%o8)
      Maxima version: "5.36.1"
      Maxima build date: "2015-06-02 11:26:48"
      Host type: "x86_64-unknown-linux-gnu"
      Lisp implementation type: "GNU Common Lisp (GCL)"
      Lisp implementation version: "GCL 2.6.12"

  The Maxima version string can (here 5.36.1) can look very different:

      (%i1) build_info();
      (%o1)
      Maxima version: "branch_5_37_base_331_g8322940_dirty"
      Maxima build date: "2016-01-01 15:37:35"
      Host type: "x86_64-unknown-linux-gnu"
      Lisp implementation type: "CLISP"
      Lisp implementation version: "2.49 (2010-07-07) (built 3605577779) (memory 3660647857)"

  In that case, Maxima was not build from a released sourcecode, but directly from the GIT-checkout of the sourcecode. In the example, the checkout is 331 commits after the latest GIT tag (usually a Maxima (major) release (5.37 in our example)) and the abbreviated commit hash of the last commit was "8322940".

  Categories: Debugging

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# 3. Help

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## 3.1 Documentation

The Maxima on-line user's manual can be viewed in different forms. From the Maxima interactive prompt, the user's manual is viewed as plain text by the `?` command (i.e., the `describe` function). The user's manual is viewed as `info` hypertext by the `info` viewer program and as a web page by any ordinary web browser.

`example` displays examples for many Maxima functions. For example,

    (%i1) example (integrate);

yields

    (%i2) test(f):=block([u],u:integrate(f,x),ratsimp(f-diff(u,x)))
    (%o2) test(f) := block([u], u : integrate(f, x),
                                             ratsimp(f - diff(u, x)))
    (%i3) test(sin(x))
    (%o3)                           0
    (%i4) test(1/(x+1))
    (%o4)                           0
    (%i5) test(1/(x^2+1))
    (%o5)                           0

and additional output.

Categories: Console interaction

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## 3.2 Functions and Variables for Help

Function: apropos (string)

  Searches for Maxima names which have string appearing anywhere within them. Thus, `apropos (exp)` returns a list of all the flags and functions which have `exp` as part of their names, such as `expand`, `exp`, and `exponentialize`. Thus if you can only remember part of the name of something you can use this command to find the rest of the name. Similarly, you could say `apropos (tr_)` to find a list of many of the switches relating to the translator, most of which begin with `tr_`.

  `apropos("")` returns a list with all Maxima names.

  `apropos` returns the empty list `[]`, if no name is found.

  Example:

  Show all Maxima symbols which have `"gamma"` in the name:

      (%i1) apropos("gamma");
      (%o1) [%gamma, gamma, gammalim, gamma_expand, gamma_incomplete_lower,
      gamma_incomplete, gamma_incomplete_generalized,
      gamma_incomplete_regularized, Gamma, log_gamma, makegamma,
      prefer_gamma_incomplete, gamma-incomplete,
      gamma_incomplete_generalized_regularized]

  Categories: Help

Function: demo (filename)

  Evaluates Maxima expressions in filename and displays the results. `demo` pauses after evaluating each expression and continues after the user enters a carriage return. (If running in Xmaxima, `demo` may need to see a semicolon `;` followed by a carriage return.)

  `demo` searches the list of directories `file_search_demo` to find `filename`. If the file has the suffix `dem`, the suffix may be omitted. See also `file_search`.

  `demo` evaluates its argument. `demo` returns the name of the demonstration file.

  Example:

      (%i1) demo ("disol");

      batching /home/wfs/maxima/share/simplification/disol.dem
       At the _ prompt, type ';' followed by enter to get next demo
      (%i2)                      load(disol)

      _
      (%i3)           exp1 : a (e (g + f) + b (d + c))
      (%o3)               a (e (g + f) + b (d + c))

      _
      (%i4)                disolate(exp1, a, b, e)
      (%t4)                         d + c

      (%t5)                         g + f

      (%o5)                   a (%t5 e + %t4 b)

      _

  Categories: Help · Console interaction · File input

Function: describe
`describe` (string)
`describe` (string, exact)
`describe` (string, inexact)

  `describe(string)` is equivalent to `describe(string, exact)`.

  `describe(string, exact)` finds an item with title equal (case-insensitive) to string, if there is any such item.

  `describe(string, inexact)` finds all documented items which contain string in their titles. If there is more than one such item, Maxima asks the user to select an item or items to display.

  At the interactive prompt, `? foo` (with a space between `?` and `foo`) is equivalent to `describe("foo", exact)`, and `?? foo` is equivalent to `describe("foo", inexact)`.

  `describe("", inexact)` yields a list of all topics documented in the on-line manual.

  `describe` quotes its argument. `describe` returns `true` if some documentation is found, otherwise `false`.

  See also Documentation.

  Example:

      (%i1) ?? integ
       0: Functions and Variables for Elliptic Integrals
       1: Functions and Variables for Integration
       2: Introduction to Elliptic Functions and Integrals
       3: Introduction to Integration
       4: askinteger  (Functions and Variables for Simplification)
       5: integerp  (Functions and Variables for Miscellaneous Options)
       6: integer_partitions  (Functions and Variables for Sets)
       7: integrate  (Functions and Variables for Integration)
       8: integrate_use_rootsof  (Functions and Variables for
          Integration)
       9: integration_constant_counter  (Functions and Variables for
          Integration)
       10: nonnegintegerp  (Functions and Variables for linearalgebra)
      Enter space-separated numbers, `all' or `none': 7 8

       -- Function: integrate (<expr>, <x>)
       -- Function: integrate (<expr>, <x>, <a>, <b>)
           Attempts to symbolically compute the integral of <expr> with
           respect to <x>.  `integrate (<expr>, <x>)' is an indefinite
           integral, while `integrate (<expr>, <x>, <a>, <b>)' is a
           definite integral, [...]

       -- Option variable: integrate_use_rootsof
           Default value: `false'

           When `integrate_use_rootsof' is `true' and the denominator of
           a rational function cannot be factored, `integrate' returns
           the integral in a form which is a sum over the roots (not yet
           known) of the denominator.
           [...]

  In this example, items 7 and 8 were selected (output is shortened as indicated by `[...]`). All or none of the items could have been selected by entering `all` or `none`, which can be abbreviated `a` or `n`, respectively.

  Categories: Help · Console interaction

Function: example
`example` (topic)
`example` ()

  `example (topic)` displays some examples of topic, which is a symbol or a string. To get examples for operators like `if`, `do`, or `lambda` the argument must be a string, e.g. `example ("do")`. `example` is not case sensitive. Most topics are function names.

  `example ()` returns the list of all recognized topics.

  The name of the file containing the examples is given by the global option variable `manual_demo`, which defaults to `"manual.demo"`.

  `example` quotes its argument. `example` returns `done` unless no examples are found or there is no argument, in which case `example` returns the list of all recognized topics.

  Examples:

      (%i1) example(append);
      (%i2) append([y+x,0,-3.2],[2.5e+20,x])
      (%o2)             [y + x, 0, - 3.2, 2.5e+20, x]
      (%o2)                         done
      (%i3) example("lambda");
      (%i4) lambda([x,y,z],x^2+y^2+z^2)
                                          2    2    2
      (%o4)            lambda([x, y, z], x  + y  + z )
      (%i5) %(1,2,a)
                                    2
      (%o5)                        a  + 5
      (%i6) 1+2+a
      (%o6)                         a + 3
      (%o6)                         done

  Categories: Help · Console interaction

Option variable: manual_demo

  Default value: `"manual.demo"`

  `manual_demo` specifies the name of the file containing the examples for the function `example`. See `example`.

  Categories: Help · Global variables

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# 4. Command Line

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## 4.1 Introduction to Command Line

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## 4.2 Functions and Variables for Command Line

System variable: __

  `__` is the input expression currently being evaluated. That is, while an input expression expr is being evaluated, `__` is expr.

  `__` is assigned the input expression before the input is simplified or evaluated. However, the value of `__` is simplified (but not evaluated) when it is displayed.

  `__` is recognized by `batch` and `load`. In a file processed by `batch`, `__` has the same meaning as at the interactive prompt. In a file processed by `load`, `__` is bound to the input expression most recently entered at the interactive prompt or in a batch file; `__` is not bound to the input expressions in the file being processed. In particular, when `load (filename)` is called from the interactive prompt, `__` is bound to `load (filename)` while the file is being processed.

  See also `_` and `%`.

  Examples:

      (%i1) print ("I was called as", __);
      I was called as print(I was called as, __)
      (%o1)              print(I was called as, __)
      (%i2) foo (__);
      (%o2)                     foo(foo(__))
      (%i3) g (x) := (print ("Current input expression =", __), 0);
      (%o3) g(x) := (print("Current input expression =", __), 0)
      (%i4) [aa : 1, bb : 2, cc : 3];
      (%o4)                       [1, 2, 3]
      (%i5) (aa + bb + cc)/(dd + ee + g(x));
                                  cc + bb + aa
      Current input expression = --------------
                                 g(x) + ee + dd
                                      6
      (%o5)                        -------
                                   ee + dd

  Categories: Global variables

System variable: _

  `_` is the most recent input expression (e.g., `%i1`, `%i2`, `%i3`, …).

  `_` is assigned the input expression before the input is simplified or evaluated. However, the value of `_` is simplified (but not evaluated) when it is displayed.

  `_` is recognized by `batch` and `load`. In a file processed by `batch`, `_` has the same meaning as at the interactive prompt. In a file processed by `load`, `_` is bound to the input expression most recently evaluated at the interactive prompt or in a batch file; `_` is not bound to the input expressions in the file being processed.

  See also `__` and `%`.

  Examples:

      (%i1) 13 + 29;
      (%o1)                          42
      (%i2) :lisp $_
      ((MPLUS) 13 29)
      (%i2) _;
      (%o2)                          42
      (%i3) sin (%pi/2);
      (%o3)                           1
      (%i4) :lisp $_
      ((%SIN) ((MQUOTIENT) $%PI 2))
      (%i4) _;
      (%o4)                           1
      (%i5) a: 13$
      (%i6) b: 29$
      (%i7) a + b;
      (%o7)                          42
      (%i8) :lisp $_
      ((MPLUS) $A $B)
      (%i8) _;
      (%o8)                         b + a
      (%i9) a + b;
      (%o9)                          42
      (%i10) ev (_);
      (%o10)                         42

  Categories: Console interaction · Global variables

System variable: %

  `%` is the output expression (e.g., `%o1`, `%o2`, `%o3`, …) most recently computed by Maxima, whether or not it was displayed.

  `%` is recognized by `batch` and `load`. In a file processed by `batch`, `%` has the same meaning as at the interactive prompt. In a file processed by `load`, `%` is bound to the output expression most recently computed at the interactive prompt or in a batch file; `%` is not bound to output expressions in the file being processed.

  See also `_`, `%%`, and `%th`.

  Categories: Console interaction · Global variables

System variable: %%

  In compound statements, namely `block`, `lambda`, or `(s_1, ..., s_n)`, `%%` is the value of the previous statement.

  At the first statement in a compound statement, or outside of a compound statement, `%%` is undefined.

  `%%` is recognized by `batch` and `load`, and it has the same meaning as at the interactive prompt.

  See also `%`.

  Examples:

  The following two examples yield the same result.

      (%i1) block (integrate (x^5, x), ev (%%, x=2) - ev (%%, x=1));
                                     21
      (%o1)                          --
                                     2
      (%i2) block ([prev], prev: integrate (x^5, x),
                     ev (prev, x=2) - ev (prev, x=1));
                                     21
      (%o2)                          --
                                     2

  A compound statement may comprise other compound statements. Whether a statement be simple or compound, `%%` is the value of the previous statement.

      (%i3) block (block (a^n, %%*42), %%/6);
                                       n
      (%o3)                         7 a

  Within a compound statement, the value of `%%` may be inspected at a break prompt, which is opened by executing the `break` function. For example, entering `%%;` in the following example yields `42`.

      (%i4) block (a: 42, break ())$

      Entering a Maxima break point. Type 'exit;' to resume.
      _%%;
      42
      _

  Categories: Global variables

Function: %th (i)

  The value of the i'th previous output expression. That is, if the next expression to be computed is the n'th output, `%th (m)` is the (n - m)'th output.

  `%th` is recognized by `batch` and `load`. In a file processed by `batch`, `%th` has the same meaning as at the interactive prompt. In a file processed by `load`, `%th` refers to output expressions most recently computed at the interactive prompt or in a batch file; `%th` does not refer to output expressions in the file being processed.

  See also `%` and `%%`.

  Example:

  `%th` is useful in `batch` files or for referring to a group of output expressions. This example sets `s` to the sum of the last five output expressions.

      (%i1) 1;2;3;4;5;
      (%o1)                           1
      (%o2)                           2
      (%o3)                           3
      (%o4)                           4
      (%o5)                           5
      (%i6) block (s: 0, for i:1 thru 5 do s: s + %th(i), s);
      (%o6)                          15

  Categories: Console interaction

Special symbol: ?

  As prefix to a function or variable name, `?` signifies that the name is a Lisp name, not a Maxima name. For example, `?round` signifies the Lisp function `ROUND`. See Lisp and Maxima for more on this point.

  The notation `? word` (a question mark followed a word, separated by whitespace) is equivalent to `describe("word")`. The question mark must occur at the beginning of an input line; otherwise it is not recognized as a request for documentation. See also `describe`.

  Categories: Help · Console interaction

Special symbol: ??

  The notation `?? word` (`??` followed a word, separated by whitespace) is equivalent to `describe("word", inexact)`. The question mark must occur at the beginning of an input line; otherwise it is not recognized as a request for documentation. See also `describe`.

  Categories: Help · Console interaction

Input terminator: $

  The dollar sign `$` terminates an input expression, and the most recent output `%` and an output label, e.g. `%o1`, are assigned the result, but the result is not displayed.

  See also `;`.

  Example:

      (%i1) 1 + 2 + 3 $
      (%i2) %;
      (%o2)                           6
      (%i3) %o1;
      (%o3)                           6

Input terminator: ;

  The semicolon `;` terminates an input expression, and the resulting output is displayed.

  See also `$`.

  Example:

      (%i1) 1 + 2 + 3;
      (%o1)                           6

Option variable: inchar

  Default value: `%i`

  `inchar` is the prefix of the labels of expressions entered by the user. Maxima automatically constructs a label for each input expression by concatenating `inchar` and `linenum`.

  `inchar` may be assigned any string or symbol, not necessarily a single character. Because Maxima internally takes into account only the first char of the prefix, the prefixes `inchar`, `outchar`, and `linechar` should have a different first char. Otherwise some commands like `kill(inlabels)` do not work as expected.

  See also `labels`.

  Example:

      (%i1) inchar: "input";
      (%o1)                         input
      (input2) expand((a+b)^3);
                           3        2      2      3
      (%o2)               b  + 3 a b  + 3 a  b + a

  Categories: Display flags and variables

System variable: infolists

  Default value: `[]`

  `infolists` is a list of the names of all of the information lists in Maxima. These are:

  `labels`

    All bound `%i`, `%o`, and `%t` labels.

  `values`

    All bound atoms which are user variables, not Maxima options or switches, created by `:` or `::` or functional binding.

  `functions`

    All user-defined functions, created by `:=` or `define`.

  `arrays`

    All declared and undeclared arrays, created by `:`, `::`, or `:=`.

  `macros`

    All user-defined macro functions, created by `::=`.

  `myoptions`

    All options ever reset by the user (whether or not they are later reset to their default values).

  `rules`

    All user-defined pattern matching and simplification rules, created by `tellsimp`, `tellsimpafter`, `defmatch`, or `defrule`.

  `aliases`

    All atoms which have a user-defined alias, created by the `alias`, `ordergreat`, `orderless` functions or by declaring the atom as a `noun` with `declare`.

  `dependencies`

    All atoms which have functional dependencies, created by the `depends`, `dependencies`, or `gradef` functions.

  `gradefs`

    All functions which have user-defined derivatives, created by the `gradef` function.

  `props`

    All atoms which have any property other than those mentioned above, such as properties established by `atvalue` or `matchdeclare`, etc., as well as properties established in the `declare` function.

  `let_rule_packages`

    All user-defined `let` rule packages plus the special package `default_let_rule_package`. (`default_let_rule_package` is the name of the rule package used when one is not explicitly set by the user.)

  Categories: Declarations and inferences · Global variables

Function: kill
`kill` (a_1, …, a_n)
`kill` (labels)
`kill` (inlabels, outlabels, linelabels)
`kill` (n)
`kill` ([m, n])
`kill` (values, functions, arrays, …)
`kill` (all)
`kill` (allbut (a_1, …, a_n))

  Removes all bindings (value, function, array, or rule) from the arguments a_1, …, a_n. An argument a_k may be a symbol or a single array element. When a_k is a single array element, `kill` unbinds that element without affecting any other elements of the array.

  Several special arguments are recognized. Different kinds of arguments may be combined, e.g., `kill (inlabels, functions, allbut (foo, bar))`.

  `kill (labels)` unbinds all input, output, and intermediate expression labels created so far. `kill (inlabels)` unbinds only input labels which begin with the current value of `inchar`. Likewise, `kill (outlabels)` unbinds only output labels which begin with the current value of `outchar`, and `kill (linelabels)` unbinds only intermediate expression labels which begin with the current value of `linechar`.

  `kill (n)`, where n is an integer, unbinds the n most recent input and output labels.

  `kill ([m, n])` unbinds input and output labels m through n.

  `kill (infolist)`, where infolist is any item in `infolists` (such as `values`, `functions`, or `arrays`) unbinds all items in infolist. See also `infolists`.

  `kill (all)` unbinds all items on all infolists. `kill (all)` does not reset global variables to their default values; see `reset` on this point.

  `kill (allbut (a_1, ..., a_n))` unbinds all items on all infolists except for a_1, …, a_n. `kill (allbut (infolist))` unbinds all items except for the ones on infolist, where infolist is `values`, `functions`, `arrays`, etc.

  The memory taken up by a bound property is not released until all symbols are unbound from it. In particular, to release the memory taken up by the value of a symbol, one unbinds the output label which shows the bound value, as well as unbinding the symbol itself.

  `kill` quotes its arguments. The quote-quote operator `''` defeats quotation.

  `kill (symbol)` unbinds all properties of symbol. In contrast, the functions `remvalue`, `remfunction`, `remarray`, and `remrule` unbind a specific property.

  `kill` always returns `done`, even if an argument has no binding.

  Categories: Evaluation · Console interaction · Session management

Function: labels (symbol) Returns the list of input, output, or intermediate expression labels which begin

  with symbol. Typically symbol is the value of `inchar`, `outchar`, or `linechar`. If no labels begin with symbol, `labels` returns an empty list.

  By default, Maxima displays the result of each user input expression, giving the result an output label. The output display is suppressed by terminating the input with `$` (dollar sign) instead of `;` (semicolon). An output label is constructed and bound to the result, but not displayed, and the label may be referenced in the same way as displayed output labels. See also `%`, `%%`, and `%th`.

  Intermediate expression labels can be generated by some functions. The option variable `programmode` controls whether `solve` and some other functions generate intermediate expression labels instead of returning a list of expressions. Some other functions, such as `ldisplay`, always generate intermediate expression labels.

  See also `inchar`, `outchar`, `linechar`, and `infolists`.

  Categories: Display functions · Console interaction

System variable: labels

  The variable `labels` is the list of input, output, and intermediate expression labels, including all previous labels if `inchar`, `outchar`, or `linechar` were redefined.

  Categories: Display flags and variables · Console interaction

Option variable: linechar

  Default value: `%t`

  `linechar` is the prefix of the labels of intermediate expressions generated by Maxima. Maxima constructs a label for each intermediate expression (if displayed) by concatenating `linechar` and `linenum`.

  `linechar` may be assigned any string or symbol, not necessarily a single character. Because Maxima internally takes into account only the first char of the prefix, the prefixes `inchar`, `outchar`, and `linechar` should have a different first char. Otherwise some commands like `kill(inlabels)` do not work as expected.

  Intermediate expressions might or might not be displayed. See `programmode` and `labels`.

  Categories: Display flags and variables

System variable: linenum

  The line number of the current pair of input and output expressions.

  Categories: Display flags and variables · Console interaction

System variable: myoptions

  Default value: `[]`

  `myoptions` is the list of all options ever reset by the user, whether or not they get reset to their default value.

  Categories: Global variables · Session management · Console interaction

Option variable: nolabels

  Default value: `false`

  When `nolabels` is `true`, input and output result labels (`%i` and `%o`, respectively) are displayed, but the labels are not bound to results, and the labels are not appended to the `labels` list. Since labels are not bound to results, garbage collection can recover the memory taken up by the results.

  Otherwise input and output result labels are bound to results, and the labels are appended to the `labels` list.

  Intermediate expression labels (`%t`) are not affected by `nolabels`; whether `nolabels` is `true` or `false`, intermediate expression labels are bound and appended to the `labels` list.

  See also `batch`, `load`, and `labels`.

  Categories: Global flags · Session management

Option variable: optionset

  Default value: `false`

  When `optionset` is `true`, Maxima prints out a message whenever a Maxima option is reset. This is useful if the user is doubtful of the spelling of some option and wants to make sure that the variable he assigned a value to was truly an option variable.

  Example:

      (%i1) optionset:true;
      assignment: assigning to option optionset
      (%o1)                         true
      (%i2) gamma_expand:true;
      assignment: assigning to option gamma_expand
      (%o2)                         true

  Categories: Global flags · Session management · Console interaction

Option variable: outchar

  Default value: `%o`

  `outchar` is the prefix of the labels of expressions computed by Maxima. Maxima automatically constructs a label for each computed expression by concatenating `outchar` and `linenum`.

  `outchar` may be assigned any string or symbol, not necessarily a single character. Because Maxima internally takes into account only the first char of the prefix, the prefixes `inchar`, `outchar` and `linechar` should have a different first char. Otherwise some commands like `kill(inlabels)` do not work as expected.

  See also `labels`.

  Example:

      (%i1) outchar: "output";
      (output1)                    output
      (%i2) expand((a+b)^3);
                           3        2      2      3
      (output2)           b  + 3 a b  + 3 a  b + a

  Categories: Display flags and variables

Function: playback
`playback` ()
`playback` (n)
`playback` ([m, n])
`playback` ([m])
`playback` (input)
`playback` (slow)
`playback` (time)
`playback` (grind)

  Displays input, output, and intermediate expressions, without recomputing them. `playback` only displays the expressions bound to labels; any other output (such as text printed by `print` or `describe`, or error messages) is not displayed. See also `labels`.

  `playback` quotes its arguments. The quote-quote operator `''` defeats quotation. `playback` always returns `done`.

  `playback ()` (with no arguments) displays all input, output, and intermediate expressions generated so far. An output expression is displayed even if it was suppressed by the `$` terminator when it was originally computed.

  `playback (n)` displays the most recent n expressions. Each input, output, and intermediate expression counts as one.

  `playback ([m, n])` displays input, output, and intermediate expressions with numbers from m through n, inclusive.

  `playback ([m])` is equivalent to `playback ([m, m])`; this usually prints one pair of input and output expressions.

  `playback (input)` displays all input expressions generated so far.

  `playback (slow)` pauses between expressions and waits for the user to press `enter`. This behavior is similar to `demo`. `playback (slow)` is useful in conjunction with `save` or `stringout` when creating a secondary-storage file in order to pick out useful expressions.

  `playback (time)` displays the computation time for each expression.

  `playback (grind)` displays input expressions in the same format as the `grind` function. Output expressions are not affected by the `grind` option. See `grind`.

  Arguments may be combined, e.g., `playback ([5, 10], grind, time, slow)`.

  Categories: Display functions · Console interaction

Option variable: prompt

  Default value: `_`

  `prompt` is the prompt symbol of the `demo` function, `playback (slow)` mode, and the Maxima break loop (as invoked by `break`).

  Categories: Global variables · Console interaction

Function: quit ()

  Terminates the Maxima session. Note that the function must be invoked as `quit();` or `quit()$`, not `quit` by itself.

  To stop a lengthy computation, type `control-C`. The default action is to return to the Maxima prompt. If `*debugger-hook*` is `nil`, `control-C` opens the Lisp debugger. See also Debugging.

  Categories: Console interaction

Function: read (expr_1, …, expr_n)

  Prints expr_1, …, expr_n, then reads one expression from the console and returns the evaluated expression. The expression is terminated with a semicolon `;` or dollar sign `$`.

  See also `readonly`

  Example:

      (%i1) foo: 42$
      (%i2) foo: read ("foo is", foo, " -- enter new value.")$
      foo is 42  -- enter new value.
      (a+b)^3;
      (%i3) foo;
                                           3
      (%o3)                         (b + a)

  Categories: Console interaction

Function: readonly (expr_1, …, expr_n)

  Prints expr_1, …, expr_n, then reads one expression from the console and returns the expression (without evaluation). The expression is terminated with a `;` (semicolon) or `$` (dollar sign).

  See also `read`.

  Examples:

      (%i1) aa: 7$
      (%i2) foo: readonly ("Enter an expression:");
      Enter an expression:
      2^aa;
                                        aa
      (%o2)                            2
      (%i3) foo: read ("Enter an expression:");
      Enter an expression:
      2^aa;
      (%o3)                            128

  Categories: Console interaction

Function: reset ()

  Resets many global variables and options, and some other variables, to their default values.

  `reset` processes the variables on the Lisp list `*variable-initial-values*`. The Lisp macro `defmvar` puts variables on this list (among other actions). Many, but not all, global variables and options are defined by `defmvar`, and some variables defined by `defmvar` are not global variables or options.

  Categories: Session management

Option variable: showtime

  Default value: `false`

  When `showtime` is `true`, the computation time and elapsed time is printed with each output expression.

  The computation time is always recorded, so `time` and `playback` can display the computation time even when `showtime` is `false`.

  See also `timer`.

  Categories: Display flags and variables · Debugging

Function: to_lisp ()

  Enters the Lisp system under Maxima. `(to-maxima)` returns to Maxima.

  Example:

  Define a function and enter the Lisp system under Maxima. The definition is inspected on the property list, then the function definition is extracted, factored and stored in the variable $result. The variable can be used in Maxima after returning to Maxima.

      (%i1) f(x):=x^2+x;
                                        2
      (%o1)                    f(x) := x  + x
      (%i2) to_lisp();
      Type (to-maxima) to restart, ($quit) to quit Maxima.
      MAXIMA> (symbol-plist '$f)
      (MPROPS (NIL MEXPR ((LAMBDA) ((MLIST) $X)
                                   ((MPLUS) ((MEXPT) $X 2) $X))))
      MAXIMA> (setq $result ($factor (caddr (mget '$f 'mexpr))))
      ((MTIMES SIMP FACTORED) $X ((MPLUS SIMP IRREDUCIBLE) 1 $X))
      MAXIMA> (to-maxima)
      Returning to Maxima
      (%o2)                         true
      (%i3) result;
      (%o3)                       x (x + 1)

  Categories: Console interaction

System variable: values

  Initial value: `[]`

  `values` is a list of all bound user variables (not Maxima options or switches). The list comprises symbols bound by `:`, or `::`.

  If the value of a variable is removed with the commands `kill`, `remove`, or `remvalue` the variable is deleted from `values`.

  See `functions` for a list of user defined functions.

  Examples:

  First, `values` shows the symbols `a`, `b`, and `c`, but not `d`, it is not bound to a value, and not the user function `f`. The values are removed from the variables. `values` is the empty list.

      (%i1) [a:99, b:: a-90, c:a-b, d, f(x):=x^2];
                                                 2
      (%o1)              [99, 9, 90, d, f(x) := x ]
      (%i2) values;
      (%o2)                       [a, b, c]
      (%i3) [kill(a), remove(b,value), remvalue(c)];
      (%o3)                   [done, done, [c]]
      (%i4) values;
      (%o4)                          []

  Categories: Evaluation · Global variables

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## 4.3 Functions and Variables for Display

Option variable: %edispflag

  Default value: `false`

  When `%edispflag` is `true`, Maxima displays `%e` to a negative exponent as a quotient. For example, `%e^-x` is displayed as `1/%e^x`. See also `exptdispflag`.

  Example:

      (%i1) %e^-10;
                                     - 10
      (%o1)                        %e
      (%i2) %edispflag:true$
      (%i3) %e^-10;
                                     1
      (%o3)                         ----
                                      10
                                    %e

  Categories: Exponential and logarithm functions · Display flags and variables

Option variable: absboxchar

  Default value: `!`

  `absboxchar` is the character used to draw absolute value signs around expressions which are more than one line tall.

  Example:

      (%i1) abs((x^3+1));
                                  ! 3    !
      (%o1)                       !x  + 1!

  Categories: Display flags and variables

Function: disp (expr_1, expr_2, …)

  is like `display` but only the value of the arguments are displayed rather than equations. This is useful for complicated arguments which don't have names or where only the value of the argument is of interest and not the name.

  See also `ldisp` and `print`.

  Example:

      (%i1) b[1,2]:x-x^2$
      (%i2) x:123$
      (%i3) disp(x, b[1,2], sin(1.0));
                                     123

                                        2
                                   x - x

                             0.8414709848078965

      (%o3)                         done

  Categories: Display functions

Function: display (expr_1, expr_2, …)

  Displays equations whose left side is expr_i unevaluated, and whose right side is the value of the expression centered on the line. This function is useful in blocks and `for` statements in order to have intermediate results displayed. The arguments to `display` are usually atoms, subscripted variables, or function calls.

  See also `ldisplay`, `disp`, and `ldisp`.

  Example:

      (%i1) b[1,2]:x-x^2$
      (%i2) x:123$
      (%i3) display(x, b[1,2], sin(1.0));
                                   x = 123

                                            2
                               b     = x - x
                                1, 2

                        sin(1.0) = 0.8414709848078965

      (%o3)                         done

  Categories: Display functions

Option variable: display2d

  Default value: `true`

  When `display2d` is `false`, the console display is a string (1-dimensional) form rather than a display (2-dimensional) form.

  See also `leftjust` to switch between a left justified and a centered display of equations.

  Example:

      (%i1) x/(x^2+1);
                                     x
      (%o1)                        ------
                                    2
                                   x  + 1
      (%i2) display2d:false$
      (%i3) x/(x^2+1);
      (%o3) x/(x^2+1)

  Categories: Display flags and variables

Option variable: display_format_internal

  Default value: `false`

  When `display_format_internal` is `true`, expressions are displayed without being transformed in ways that hide the internal mathematical representation. The display then corresponds to what `inpart` returns rather than `part`.

  Examples:

      User     part       inpart
      a-b;      a - b     a + (- 1) b

                 a            - 1
      a/b;       -         a b
                 b
                             1/2
      sqrt(x);   sqrt(x)    x

                4 X        4
      X*4/3;    ---        - X
                 3         3

  Categories: Display flags and variables

Function: dispterms (expr)

  Displays expr in parts one below the other. That is, first the operator of expr is displayed, then each term in a sum, or factor in a product, or part of a more general expression is displayed separately. This is useful if expr is too large to be otherwise displayed. For example if `P1`, `P2`, … are very large expressions then the display program may run out of storage space in trying to display `P1 + P2 + ...` all at once. However, `dispterms (P1 + P2 + ...)` displays `P1`, then below it `P2`, etc. When not using `dispterms`, if an exponential expression is too wide to be displayed as `A^B` it appears as `expt (A, B)` (or as `ncexpt (A, B)` in the case of `A^^B`).

  Example:

      (%i1) dispterms(2*a*sin(x)+%e^x);

      +

      2 a sin(x)

        x
      %e

      (%o1)                         done

  Categories: Display functions

Special symbol: expt (a, b)
Special symbol: ncexpt (a, b)

  If an exponential expression is too wide to be displayed as `a^b` it appears as `expt (a, b)` (or as `ncexpt (a, b)` in the case of `a^^b`).

  `expt` and `ncexpt` are not recognized in input.

Option variable: exptdispflag

  Default value: `true`

  When `exptdispflag` is `true`, Maxima displays expressions with negative exponents using quotients. See also `%edispflag`.

  Example:

      (%i1) exptdispflag:true;
      (%o1)                         true
      (%i2) 10^-x;
                                      1
      (%o2)                          ---
                                       x
                                     10
      (%i3) exptdispflag:false;
      (%o3)                         false
      (%i4) 10^-x;
                                      - x
      (%o4)                         10

  Categories: Expressions · Display flags and variables

Function: grind (expr)

  The function `grind` prints expr to the console in a form suitable for input to Maxima. `grind` always returns `done`.

  When expr is the name of a function or macro, `grind` prints the function or macro definition instead of just the name.

  See also `string`, which returns a string instead of printing its output. `grind` attempts to print the expression in a manner which makes it slightly easier to read than the output of `string`.

  `grind` evaluates its argument.

  Examples:

      (%i1) aa + 1729;
      (%o1)                       aa + 1729
      (%i2) grind (%);
      aa+1729$
      (%o2)                         done
      (%i3) [aa, 1729, aa + 1729];
      (%o3)                 [aa, 1729, aa + 1729]
      (%i4) grind (%);
      [aa,1729,aa+1729]$
      (%o4)                         done
      (%i5) matrix ([aa, 17], [29, bb]);
                                 [ aa  17 ]
      (%o5)                      [        ]
                                 [ 29  bb ]
      (%i6) grind (%);
      matrix([aa,17],[29,bb])$
      (%o6)                         done
      (%i7) set (aa, 17, 29, bb);
      (%o7)                   {17, 29, aa, bb}
      (%i8) grind (%);
      {17,29,aa,bb}$
      (%o8)                         done
      (%i9) exp (aa / (bb + 17)^29);
                                      aa
                                  -----------
                                           29
                                  (bb + 17)
      (%o9)                     %e
      (%i10) grind (%);
      %e^(aa/(bb+17)^29)$
      (%o10)                        done
      (%i11) expr: expand ((aa + bb)^10);
               10           9        2   8         3   7         4   6
      (%o11) bb   + 10 aa bb  + 45 aa  bb  + 120 aa  bb  + 210 aa  bb
               5   5         6   4         7   3        8   2
       + 252 aa  bb  + 210 aa  bb  + 120 aa  bb  + 45 aa  bb
              9        10
       + 10 aa  bb + aa
      (%i12) grind (expr);
      bb^10+10*aa*bb^9+45*aa^2*bb^8+120*aa^3*bb^7+210*aa^4*bb^6
           +252*aa^5*bb^5+210*aa^6*bb^4+120*aa^7*bb^3+45*aa^8*bb^2
           +10*aa^9*bb+aa^10$
      (%o12)                        done
      (%i13) string (expr);
      (%o13) bb^10+10*aa*bb^9+45*aa^2*bb^8+120*aa^3*bb^7+210*aa^4*bb^6\
      +252*aa^5*bb^5+210*aa^6*bb^4+120*aa^7*bb^3+45*aa^8*bb^2+10*aa^9*\
      bb+aa^10
      (%i14) cholesky (A):= block ([n : length (A), L : copymatrix (A),
        p : makelist (0, i, 1, length (A))],
        for i thru n do for j : i thru n do
        (x : L[i, j], x : x - sum (L[j, k] * L[i, k], k, 1, i - 1),
        if i = j then p[i] : 1 / sqrt(x) else L[j, i] : x * p[i]),
        for i thru n do L[i, i] : 1 / p[i],
        for i thru n do for j : i + 1 thru n do L[i, j] : 0, L)$
      define: warning: redefining the built-in function cholesky
      (%i15) grind (cholesky);
      cholesky(A):=block(
               [n:length(A),L:copymatrix(A),
                p:makelist(0,i,1,length(A))],
               for i thru n do
                   (for j from i thru n do
                        (x:L[i,j],x:x-sum(L[j,k]*L[i,k],k,1,i-1),
                         if i = j then p[i]:1/sqrt(x)
                             else L[j,i]:x*p[i])),
               for i thru n do L[i,i]:1/p[i],
               for i thru n do (for j from i+1 thru n do L[i,j]:0),L)$
      (%o15)                        done
      (%i16) string (fundef (cholesky));
      (%o16) cholesky(A):=block([n:length(A),L:copymatrix(A),p:makelis\
      t(0,i,1,length(A))],for i thru n do (for j from i thru n do (x:L\
      [i,j],x:x-sum(L[j,k]*L[i,k],k,1,i-1),if i = j then p[i]:1/sqrt(x\
      ) else L[j,i]:x*p[i])),for i thru n do L[i,i]:1/p[i],for i thru \
      n do (for j from i+1 thru n do L[i,j]:0),L)

  Categories: Display functions

Option variable: grind

  When the variable `grind` is `true`, the output of `string` and `stringout` has the same format as that of `grind`; otherwise no attempt is made to specially format the output of those functions. The default value of the variable `grind` is `false`.

  `grind` can also be specified as an argument of `playback`. When `grind` is present, `playback` prints input expressions in the same format as the `grind` function. Otherwise, no attempt is made to specially format input expressions.

  Categories: Display flags and variables

Option variable: ibase

  Default value: `10`

  `ibase` is the base for integers read by Maxima.

  `ibase` may be assigned any integer between 2 and 36 (decimal), inclusive. When `ibase` is greater than 10, the numerals comprise the decimal numerals 0 through 9 plus letters of the alphabet `A`, `B`, `C`, …, as needed to make `ibase` digits in all. Letters are interpreted as digits only if the first digit is 0 through 9.

  Uppercase and lowercase letters are not distinguished. The numerals for base 36, the largest acceptable base, comprise 0 through 9 and `A` through `Z`.

  Whatever the value of `ibase`, when an integer is terminated by a decimal point, it is interpreted in base 10.

  See also `obase`.

  Examples:

  `ibase` less than 10 (for example binary numbers).

      (%i1) ibase : 2 $
      (%i2) obase;
      (%o2)                          10
      (%i3) 1111111111111111;
      (%o3)                         65535

  `ibase` greater than 10. Letters are interpreted as digits only if the first digit is 0 through 9 which means that hexadecimal numbers might need to be prepended by a 0.

      (%i1) ibase : 16 $
      (%i2) obase;
      (%o2)                          10
      (%i3) 1000;
      (%o3)                         4096
      (%i4) abcd;
      (%o4)                         abcd
      (%i5) symbolp (abcd);
      (%o5)                         true
      (%i6) 0abcd;
      (%o6)                         43981
      (%i7) symbolp (0abcd);
      (%o7)                         false

  When an integer is terminated by a decimal point, it is interpreted in base 10.

      (%i1) ibase : 36 $
      (%i2) obase;
      (%o2)                          10
      (%i3) 1234;
      (%o3)                         49360
      (%i4) 1234.;
      (%o4)                         1234

  Categories: Console interaction

Function: ldisp (expr_1, …, expr_n)

  Displays expressions expr_1, …, expr_n to the console as printed output. `ldisp` assigns an intermediate expression label to each argument and returns the list of labels.

  See also `disp`, `display`, and `ldisplay`.

  Examples:

      (%i1) e: (a+b)^3;
                                         3
      (%o1)                       (b + a)
      (%i2) f: expand (e);
                           3        2      2      3
      (%o2)               b  + 3 a b  + 3 a  b + a
      (%i3) ldisp (e, f);
                                         3
      (%t3)                       (b + a)

                           3        2      2      3
      (%t4)               b  + 3 a b  + 3 a  b + a

      (%o4)                      [%t3, %t4]
      (%i4) %t3;
                                         3
      (%o4)                       (b + a)
      (%i5) %t4;
                           3        2      2      3
      (%o5)               b  + 3 a b  + 3 a  b + a

  Categories: Display functions

Function: ldisplay (expr_1, …, expr_n)

  Displays expressions expr_1, …, expr_n to the console as printed output. Each expression is printed as an equation of the form `lhs = rhs` in which `lhs` is one of the arguments of `ldisplay` and `rhs` is its value. Typically each argument is a variable. `ldisp` assigns an intermediate expression label to each equation and returns the list of labels.

  See also `display`, `disp`, and `ldisp`.

  Examples:

      (%i1) e: (a+b)^3;
                                         3
      (%o1)                       (b + a)
      (%i2) f: expand (e);
                           3        2      2      3
      (%o2)               b  + 3 a b  + 3 a  b + a
      (%i3) ldisplay (e, f);
                                           3
      (%t3)                     e = (b + a)

                             3        2      2      3
      (%t4)             f = b  + 3 a b  + 3 a  b + a

      (%o4)                      [%t3, %t4]
      (%i4) %t3;
                                           3
      (%o4)                     e = (b + a)
      (%i5) %t4;
                             3        2      2      3
      (%o5)             f = b  + 3 a b  + 3 a  b + a

  Categories: Display functions

Option variable: leftjust

  Default value: `false`

  When `leftjust` is `true`, equations in 2D-display are drawn left justified rather than centered.

  See also `display2d` to switch between 1D- and 2D-display.

  Example:

      (%i1) expand((x+1)^3);
                              3      2
      (%o1)                  x  + 3 x  + 3 x + 1
      (%i2) leftjust:true$
      (%i3) expand((x+1)^3);
             3      2
      (%o3) x  + 3 x  + 3 x + 1

  Categories: Display flags and variables

Option variable: linel

  Default value: `79`

  `linel` is the assumed width (in characters) of the console display for the purpose of displaying expressions. `linel` may be assigned any value by the user, although very small or very large values may be impractical. Text printed by built-in Maxima functions, such as error messages and the output of `describe`, is not affected by `linel`.

  Categories: Display flags and variables

Option variable: lispdisp

  Default value: `false`

  When `lispdisp` is `true`, Lisp symbols are displayed with a leading question mark `?`. Otherwise, Lisp symbols are displayed with no leading mark. This has the same effect for 1-d and 2-d display.

  Examples:

      (%i1) lispdisp: false$
      (%i2) ?foo + ?bar;
      (%o2)                       foo + bar
      (%i3) lispdisp: true$
      (%i4) ?foo + ?bar;
      (%o4)                      ?foo + ?bar

  Categories: Display flags and variables

Option variable: negsumdispflag

  Default value: `true`

  When `negsumdispflag` is `true`, `x - y` displays as `x - y` instead of as `- y + x`. Setting it to `false` causes the special check in display for the difference of two expressions to not be done. One application is that thus `a + %i*b` and `a - %i*b` may both be displayed the same way.

  Categories: Display flags and variables

Option variable: obase

  Default value: `10`

  `obase` is the base for integers displayed by Maxima.

  `obase` may be assigned any integer between 2 and 36 (decimal), inclusive. When `obase` is greater than 10, the numerals comprise the decimal numerals 0 through 9 plus capital letters of the alphabet A, B, C, …, as needed. A leading 0 digit is displayed if the leading digit is otherwise a letter. The numerals for base 36, the largest acceptable base, comprise 0 through 9, and A through Z.

  See also `ibase`.

  Examples:

      (%i1) obase : 2;
      (%o1)                          10
      (%i10) 2^8 - 1;
      (%o10)                      11111111
      (%i11) obase : 8;
      (%o3)                          10
      (%i4) 8^8 - 1;
      (%o4)                       77777777
      (%i5) obase : 16;
      (%o5)                          10
      (%i6) 16^8 - 1;
      (%o6)                       0FFFFFFFF
      (%i7) obase : 36;
      (%o7)                          10
      (%i8) 36^8 - 1;
      (%o8)                       0ZZZZZZZZ

  Categories: Display flags and variables · Console interaction

Option variable: pfeformat

  Default value: `false`

  When `pfeformat` is `true`, a ratio of integers is displayed with the solidus (forward slash) character, and an integer denominator `n` is displayed as a leading multiplicative term `1/n`.

  Examples:

      (%i1) pfeformat: false$
      (%i2) 2^16/7^3;
                                    65536
      (%o2)                         -----
                                     343
      (%i3) (a+b)/8;
                                    b + a
      (%o3)                         -----
                                      8
      (%i4) pfeformat: true$
      (%i5) 2^16/7^3;
      (%o5)                       65536/343
      (%i6) (a+b)/8;
      (%o6)                      1/8 (b + a)

  Categories: Display flags and variables

Option variable: powerdisp

  Default value: `false`

  When `powerdisp` is `true`, a sum is displayed with its terms in order of increasing power. Thus a polynomial is displayed as a truncated power series, with the constant term first and the highest power last.

  By default, terms of a sum are displayed in order of decreasing power.

  Example:

      (%i1) powerdisp:true;
      (%o1)                         true
      (%i2) x^2+x^3+x^4;
                                 2    3    4
      (%o2)                     x  + x  + x
      (%i3) powerdisp:false;
      (%o3)                         false
      (%i4) x^2+x^3+x^4;
                                 4    3    2
      (%o4)                     x  + x  + x

  Categories: Display flags and variables

Function: print (expr_1, …, expr_n)

  Evaluates and displays expr_1, …, expr_n one after another, from left to right, starting at the left edge of the console display.

  The value returned by `print` is the value of its last argument. `print` does not generate intermediate expression labels.

  See also `display`, `disp`, `ldisplay`, and `ldisp`. Those functions display one expression per line, while `print` attempts to display two or more expressions per line.

  To display the contents of a file, see `printfile`.

  Examples:

      (%i1) r: print ("(a+b)^3 is", expand ((a+b)^3), "log (a^10/b) is",
            radcan (log (a^10/b)))$
                  3        2      2      3
      (a+b)^3 is b  + 3 a b  + 3 a  b + a  log (a^10/b) is

                                                    10 log(a) - log(b)
      (%i2) r;
      (%o2)                  10 log(a) - log(b)
      (%i3) disp ("(a+b)^3 is", expand ((a+b)^3), "log (a^10/b) is",
            radcan (log (a^10/b)))$
                                 (a+b)^3 is

                           3        2      2      3
                          b  + 3 a b  + 3 a  b + a

                               log (a^10/b) is

                             10 log(a) - log(b)

  Categories: Display functions

Option variable: sqrtdispflag

  Default value: `true`

  When `sqrtdispflag` is `false`, causes `sqrt` to display with exponent 1/2.

  Categories: Mathematical functions · Display flags and variables

Option variable: stardisp

  Default value: `false`

  When `stardisp` is `true`, multiplication is displayed with an asterisk `*` between operands.

  Categories: Display flags and variables

Option variable: ttyoff

  Default value: `false`

  When `ttyoff` is `true`, output expressions are not displayed. Output expressions are still computed and assigned labels. See `labels`.

  Text printed by built-in Maxima functions, such as error messages and the output of `describe`, is not affected by `ttyoff`.

  Categories: Display flags and variables

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# 5. Data Types and Structures

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## 5.1 Numbers

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### 5.1.1 Introduction to Numbers

### Complex numbers

A complex expression is specified in Maxima by adding the real part of the expression to `%i` times the imaginary part. Thus the roots of the equation `x^2 - 4*x + 13 = 0` are `2 + 3*%i` and `2 - 3*%i`. Note that simplification of products of complex expressions can be effected by expanding the product. Simplification of quotients, roots, and other functions of complex expressions can usually be accomplished by using the `realpart`, `imagpart`, `rectform`, `polarform`, `abs`, `carg` functions.

Categories: Complex variables

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### 5.1.2 Functions and Variables for Numbers

Function: bfloat (expr)

  Converts all numbers and functions of numbers in expr to bigfloat numbers. The number of significant digits in the resulting bigfloats is specified by the global variable `fpprec`.

  When `float2bf` is `false` a warning message is printed when a floating point number is converted into a bigfloat number (since this may lead to loss of precision).

  Categories: Numerical evaluation

Function: bfloatp (expr)

  Returns `true` if expr is a bigfloat number, otherwise `false`.

  Categories: Numerical evaluation · Predicate functions

Option variable: bftorat

  Default value: `false`

  `bftorat` controls the conversion of bfloats to rational numbers. When `bftorat` is `false`, `ratepsilon` will be used to control the conversion (this results in relatively small rational numbers). When `bftorat` is `true`, the rational number generated will accurately represent the bfloat.

  Note: `bftorat` has no effect on the transformation to rational numbers with the function `rationalize`.

  Example:

      (%i1) ratepsilon:1e-4;
      (%o1)                         1.e-4
      (%i2) rat(bfloat(11111/111111)), bftorat:false;
      `rat' replaced 9.99990999991B-2 by 1/10 = 1.0B-1
                                     1
      (%o2)/R/                       --
                                     10
      (%i3) rat(bfloat(11111/111111)), bftorat:true;
      `rat' replaced 9.99990999991B-2 by 11111/111111 = 9.99990999991B-2
                                   11111
      (%o3)/R/                     ------
                                   111111

  Categories: Numerical evaluation

Option variable: bftrunc

  Default value: `true`

  `bftrunc` causes trailing zeroes in non-zero bigfloat numbers not to be displayed. Thus, if `bftrunc` is `false`, `bfloat (1)` displays as `1.000000000000000B0`. Otherwise, this is displayed as `1.0B0`.

  Categories: Numerical evaluation

Function: evenp (expr)

  Returns `true` if expr is a literal even integer, otherwise `false`.

  `evenp` returns `false` if expr is a symbol, even if expr is declared `even`.

  Categories: Predicate functions

Function: float (expr)

  Converts integers, rational numbers and bigfloats in expr to floating point numbers. It is also an `evflag`, `float` causes non-integral rational numbers and bigfloat numbers to be converted to floating point.

  Categories: Numerical evaluation · Evaluation flags

Option variable: float2bf

  Default value: `true`

  When `float2bf` is `false`, a warning message is printed when a floating point number is converted into a bigfloat number (since this may lead to loss of precision).

  Categories: Numerical evaluation

Function: floatnump (expr)

  Returns `true` if expr is a floating point number, otherwise `false`.

  Categories: Numerical evaluation · Predicate functions

Option variable: fpprec

  Default value: 16

  `fpprec` is the number of significant digits for arithmetic on bigfloat numbers. `fpprec` does not affect computations on ordinary floating point numbers.

  See also `bfloat` and `fpprintprec`.

  Categories: Numerical evaluation

Option variable: fpprintprec

  Default value: 0

  `fpprintprec` is the number of digits to print when printing an ordinary float or bigfloat number.

  For ordinary floating point numbers, when `fpprintprec` has a value between 2 and 16 (inclusive), the number of digits printed is equal to `fpprintprec`. Otherwise, `fpprintprec` is 0, or greater than 16, and the number of digits printed is 16.

  For bigfloat numbers, when `fpprintprec` has a value between 2 and `fpprec` (inclusive), the number of digits printed is equal to `fpprintprec`. Otherwise, `fpprintprec` is 0, or greater than `fpprec`, and the number of digits printed is equal to `fpprec`.

  For both ordinary floats and bigfloats, trailing zero digits are suppressed. The actual number of digits printed is less than `fpprintprec` if there are trailing zero digits.

  `fpprintprec` cannot be 1.

  Categories: Numerical evaluation · Display flags and variables

Function: integerp (expr)

  Returns `true` if expr is a literal numeric integer, otherwise `false`.

  `integerp` returns `false` if expr is a symbol, even if expr is declared `integer`.

  Examples:

      (%i1) integerp (0);
      (%o1)                         true
      (%i2) integerp (1);
      (%o2)                         true
      (%i3) integerp (-17);
      (%o3)                         true
      (%i4) integerp (0.0);
      (%o4)                         false
      (%i5) integerp (1.0);
      (%o5)                         false
      (%i6) integerp (%pi);
      (%o6)                         false
      (%i7) integerp (n);
      (%o7)                         false
      (%i8) declare (n, integer);
      (%o8)                         done
      (%i9) integerp (n);
      (%o9)                         false

  Categories: Predicate functions

Option variable: m1pbranch

  Default value: `false`

  `m1pbranch` is the principal branch for `-1` to a power. Quantities such as `(-1)^(1/3)` (that is, an "odd" rational exponent) and `(-1)^(1/4)` (that is, an "even" rational exponent) are handled as follows:

                    domain:real

      (-1)^(1/3):      -1
      (-1)^(1/4):   (-1)^(1/4)

                   domain:complex
      m1pbranch:false          m1pbranch:true
      (-1)^(1/3)               1/2+%i*sqrt(3)/2
      (-1)^(1/4)              sqrt(2)/2+%i*sqrt(2)/2

  Categories: Expressions · Global flags

Function: nonnegintegerp (n)

  Return `true` if and only if `n >= 0` and n is an integer.

  Categories: Package linearalgebra · Predicate functions

Function: numberp (expr)

  Returns `true` if expr is a literal integer, rational number, floating point number, or bigfloat, otherwise `false`.

  `numberp` returns `false` if expr is a symbol, even if expr is a symbolic number such as `%pi` or `%i`, or declared to be `even`, `odd`, `integer`, `rational`, `irrational`, `real`, `imaginary`, or `complex`.

  Examples:

      (%i1) numberp (42);
      (%o1)                         true
      (%i2) numberp (-13/19);
      (%o2)                         true
      (%i3) numberp (3.14159);
      (%o3)                         true
      (%i4) numberp (-1729b-4);
      (%o4)                         true
      (%i5) map (numberp, [%e, %pi, %i, %phi, inf, minf]);
      (%o5)      [false, false, false, false, false, false]
      (%i6) declare (a, even, b, odd, c, integer, d, rational,
           e, irrational, f, real, g, imaginary, h, complex);
      (%o6)                         done
      (%i7) map (numberp, [a, b, c, d, e, f, g, h]);
      (%o7) [false, false, false, false, false, false, false, false]

  Categories: Predicate functions

Option variable: numer

  `numer` causes some mathematical functions (including exponentiation) with numerical arguments to be evaluated in floating point. It causes variables in `expr` which have been given numerals to be replaced by their values. It also sets the `float` switch on.

  See also `%enumer`.

  Examples:

      (%i1) [sqrt(2), sin(1), 1/(1+sqrt(3))];
                                              1
      (%o1)            [sqrt(2), sin(1), -----------]
                                         sqrt(3) + 1
      (%i2) [sqrt(2), sin(1), 1/(1+sqrt(3))],numer;
      (%o2) [1.414213562373095, 0.8414709848078965, 0.3660254037844387]

  Categories: Numerical evaluation · Evaluation flags

Option variable: numer_pbranch

  Default value: `false`

  The option variable `numer_pbranch` controls the numerical evaluation of the power of a negative integer, rational, or floating point number. When `numer_pbranch` is `true` and the exponent is a floating point number or the option variable `numer` is `true` too, Maxima evaluates the numerical result using the principal branch. Otherwise a simplified, but not an evaluated result is returned.

  Examples:

      (%i1) (-2)^0.75;
                                       0.75
      (%o1)                       (- 2)
      (%i2) (-2)^0.75,numer_pbranch:true;
      (%o2)       1.189207115002721 %i - 1.189207115002721
      (%i3) (-2)^(3/4);
                                     3/4  3/4
      (%o3)                     (- 1)    2
      (%i4) (-2)^(3/4),numer;
                                                0.75
      (%o4)              1.681792830507429 (- 1)
      (%i5) (-2)^(3/4),numer,numer_pbranch:true;
      (%o5)       1.189207115002721 %i - 1.189207115002721

  Categories: Numerical evaluation

Function: numerval (x_1, expr_1, …, var_n, expr_n)

  Declares the variables `x_1`, …, x_n to have numeric values equal to `expr_1`, …, `expr_n`. The numeric value is evaluated and substituted for the variable in any expressions in which the variable occurs if the `numer` flag is `true`. See also `ev`.

  The expressions `expr_1`, …, `expr_n` can be any expressions, not necessarily numeric.

  Categories: Declarations and inferences · Numerical evaluation

Function: oddp (expr)

  Returns `true` if expr is a literal odd integer, otherwise `false`.

  `oddp` returns `false` if expr is a symbol, even if expr is declared `odd`.

  Categories: Predicate functions

Option variable: ratepsilon

  Default value: `2.0e-15`

  `ratepsilon` is the tolerance used in the conversion of floating point numbers to rational numbers, when the option variable `bftorat` has the value `false`. See `bftorat` for an example.

  Categories: Numerical evaluation · Rational expressions

Function: rationalize (expr)

  Convert all double floats and big floats in the Maxima expression expr to their exact rational equivalents. If you are not familiar with the binary representation of floating point numbers, you might be surprised that `rationalize (0.1)` does not equal 1/10. This behavior isn't special to Maxima - the number 1/10 has a repeating, not a terminating, binary representation.

      (%i1) rationalize (0.5);
                                      1
      (%o1)                           -
                                      2
      (%i2) rationalize (0.1);
                              3602879701896397
      (%o2)                   -----------------
                              36028797018963968
      (%i3) fpprec : 5$
      (%i4) rationalize (0.1b0);
                                   209715
      (%o4)                        -------
                                   2097152
      (%i5) fpprec : 20$
      (%i6) rationalize (0.1b0);
                           236118324143482260685
      (%o6)                ----------------------
                           2361183241434822606848
      (%i7) rationalize (sin (0.1*x + 5.6));
                     3602879701896397 x   3152519739159347
      (%o7)      sin(------------------ + ----------------)
                     36028797018963968    562949953421312

  Categories: Numerical evaluation

Function: ratnump (expr)

  Returns `true` if expr is a literal integer or ratio of literal integers, otherwise `false`.

  Categories: Predicate functions · Rational expressions

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## 5.2 Strings

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### 5.2.1 Introduction to Strings

Strings (quoted character sequences) are enclosed in double quote marks `"` for input, and displayed with or without the quote marks, depending on the global variable `stringdisp`.

Strings may contain any characters, including embedded tab, newline, and carriage return characters. The sequence `\"` is recognized as a literal double quote, and `\\` as a literal backslash. When backslash appears at the end of a line, the backslash and the line termination (either newline or carriage return and newline) are ignored, so that the string continues with the next line. No other special combinations of backslash with another character are recognized; when backslash appears before any character other than `"`, `\`, or a line termination, the backslash is ignored. There is no way to represent a special character (such as tab, newline, or carriage return) except by embedding the literal character in the string.

There is no character type in Maxima; a single character is represented as a one-character string.

The `stringproc` add-on package contains many functions for working with strings.

Examples:

    (%i1) s_1 : "This is a string.";
    (%o1)                   This is a string.
    (%i2) s_2 : "Embedded \"double quotes\" and backslash \\ characters.";
    (%o2) Embedded "double quotes" and backslash \ characters.
    (%i3) s_3 : "Embedded line termination
    (%o3) Embedded line termination
    in this string.
    (%i4) in this string.";
    (%o4) Ignore the line termination characters in this string.
    (%i5) s_4 : "Ignore the \
    (%o5)                         false
    (%i6) line termination \
    (%o6)                   This is a string.
    (%i7) characters in \
    (%o7)                         true
    (%i8) this string.";
    (%o8)                  "This is a string."
    (%i9) stringdisp : false;

Categories: Syntax

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### 5.2.2 Functions and Variables for Strings

Function: concat (arg_1, arg_2, …)

  Concatenates its arguments. The arguments must evaluate to atoms. The return value is a symbol if the first argument is a symbol and a string otherwise.

  `concat` evaluates its arguments. The single quote `'` prevents evaluation.

  See also `sconcat` and `eval_string`.

      (%i1) y: 7$
      (%i2) z: 88$
      (%i3) concat (y, z/2);
      (%o3)                          744
      (%i4) concat ('y, z/2);
      (%o4)                          y44

  A symbol constructed by `concat` may be assigned a value and appear in expressions. The `::` (double colon) assignment operator evaluates its left-hand side.

      (%i5) a: concat ('y, z/2);
      (%o5)                          y44
      (%i6) a:: 123;
      (%o6)                          123
      (%i7) y44;
      (%o7)                          123
      (%i8) b^a;
                                     y44
      (%o8)                         b
      (%i9) %, numer;
                                     123
      (%o9)                         b

  Note that although `concat (1, 2)` looks like a number, it is a string.

      (%i10) concat (1, 2) + 3;
      (%o10)                       12 + 3

  Categories: Expressions · Strings

Function: sconcat (arg_1, arg_2, …)

  Concatenates its arguments into a string. Unlike `concat`, the arguments do not need to be atoms.

  See also `concat` and `eval_string`.

      (%i1) sconcat ("xx[", 3, "]:", expand ((x+y)^3));
      (%o1)               xx[3]:y^3+3*x*y^2+3*x^2*y+x^3

  Categories: Expressions · Strings

Function: string (expr)

  Converts `expr` to Maxima's linear notation just as if it had been typed in.

  The return value of `string` is a string, and thus it cannot be used in a computation.

  Categories: Strings

Option variable: stringdisp

  Default value: `false`

  When `stringdisp` is `true`, strings are displayed enclosed in double quote marks. Otherwise, quote marks are not displayed.

  `stringdisp` is always `true` when displaying a function definition.

  Examples:

      (%i1) stringdisp: false$
      (%i2) "This is an example string.";
      (%o2)              This is an example string.
      (%i3) foo () :=
            print ("This is a string in a function definition.");
      (%o3) foo() :=
                    print("This is a string in a function definition.")
      (%i4) stringdisp: true$
      (%i5) "This is an example string.";
      (%o5)             "This is an example string."

  Categories: Display flags and variables

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## 5.3 Constants

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### 5.3.1 Functions and Variables for Constants

Constant: %e

  `%e` represents the base of the natural logarithm, also known as Euler's number. The numeric value of `%e` is the double-precision floating-point value 2.718281828459045d0.

  Categories: Constants

Constant: %i

  `%i` represents the imaginary unit, sqrt(- 1).

  Categories: Constants

Constant: false

  `false` represents the Boolean constant of the same name. Maxima implements `false` by the value `NIL` in Lisp.

  Categories: Constants

Constant: %gamma

  The Euler-Mascheroni constant, 0.5772156649015329 ....

  Categories: Constants

Constant: ind

  `ind` represents a bounded, indefinite result.

  See also `limit`.

  Example:

      (%i1) limit (sin(1/x), x, 0);
      (%o1)                          ind

  Categories: Constants

Constant: inf

  `inf` represents real positive infinity.

  Categories: Constants

Constant: infinity

  `infinity` represents complex infinity.

  Categories: Constants

Constant: minf

  `minf` represents real minus (i.e., negative) infinity.

  Categories: Constants

Constant: %phi

  `%phi` represents the so-called golden mean, (1 + sqrt(5))/2. The numeric value of `%phi` is the double-precision floating-point value 1.618033988749895d0.

  `fibtophi` expresses Fibonacci numbers `fib(n)` in terms of `%phi`.

  By default, Maxima does not know the algebraic properties of `%phi`. After evaluating `tellrat(%phi^2 - %phi - 1)` and `algebraic: true`, `ratsimp` can simplify some expressions containing `%phi`.

  Examples:

  `fibtophi` expresses Fibonacci numbers `fib(n)` in terms of `%phi`.

      (%i1) fibtophi (fib (n));
                                 n             n
                             %phi  - (1 - %phi)
      (%o1)                  -------------------
                                 2 %phi - 1
      (%i2) fib (n-1) + fib (n) - fib (n+1);
      (%o2)          - fib(n + 1) + fib(n) + fib(n - 1)
      (%i3) fibtophi (%);
                  n + 1             n + 1       n             n
              %phi      - (1 - %phi)        %phi  - (1 - %phi)
      (%o3) - --------------------------- + -------------------
                      2 %phi - 1                2 %phi - 1
                                                n - 1             n - 1
                                            %phi      - (1 - %phi)
                                          + ---------------------------
                                                    2 %phi - 1
      (%i4) ratsimp (%);
      (%o4)                           0

  By default, Maxima does not know the algebraic properties of `%phi`. After evaluating `tellrat (%phi^2 - %phi - 1)` and `algebraic: true`, `ratsimp` can simplify some expressions containing `%phi`.

      (%i1) e : expand ((%phi^2 - %phi - 1) * (A + 1));
                       2                      2
      (%o1)        %phi  A - %phi A - A + %phi  - %phi - 1
      (%i2) ratsimp (e);
                        2                     2
      (%o2)        (%phi  - %phi - 1) A + %phi  - %phi - 1
      (%i3) tellrat (%phi^2 - %phi - 1);
                                  2
      (%o3)                  [%phi  - %phi - 1]
      (%i4) algebraic : true;
      (%o4)                         true
      (%i5) ratsimp (e);
      (%o5)                           0

  Categories: Constants

Constant: %pi

  `%pi` represents the ratio of the perimeter of a circle to its diameter. The numeric value of `%pi` is the double-precision floating-point value 3.141592653589793d0.

  Categories: Constants

Constant: true

  `true` represents the Boolean constant of the same name. Maxima implements `true` by the value `T` in Lisp.

  Categories: Constants

Constant: und

  `und` represents an undefined result.

  See also `limit`.

  Example:

      (%i1) limit (x*sin(x), x, inf);
      (%o1)                          und

  Categories: Constants

Constant: zeroa

  `zeroa` represents an infinitesimal above zero. `zeroa` can be used in expressions. `limit` simplifies expressions which contain infinitesimals.

  See also `zerob` and `limit`.

  Example:

  `limit` simplifies expressions which contain infinitesimals:

      (%i1) limit(zeroa);
      (%o1)                           0
      (%i2) limit(x+zeroa);
      (%o2)                           x

  Categories: Constants

Constant: zerob

  `zerob` represents an infinitesimal below zero. `zerob` can be used in expressions. `limit` simplifies expressions which contain infinitesimals.

  See also `zeroa` and `limit`.

  Categories: Constants

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## 5.4 Lists

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### 5.4.1 Introduction to Lists

Lists are the basic building block for Maxima and Lisp. All data types other than arrays, hash tables and numbers are represented as Lisp lists, These Lisp lists have the form

    ((MPLUS) $A 2)

to indicate an expression `a+2`. At Maxima level one would see the infix notation `a+2`. Maxima also has lists which are printed as

    [1, 2, 7, x+y]

for a list with 4 elements. Internally this corresponds to a Lisp list of the form

    ((MLIST) 1 2 7 ((MPLUS) $X $Y))

The flag which denotes the type field of the Maxima expression is a list itself, since after it has been through the simplifier the list would become

    ((MLIST SIMP) 1 2 7 ((MPLUS SIMP) $X $Y))

[ < ] | [ > ] |  | [ << ] | [ Up ] | [ >> ] |  |  |  |  | [Top] | [Contents] | [Index] | [ ? ]

### 5.4.2 Functions and Variables for Lists

Operator: [
Operator: ]

  `[` and `]` mark the beginning and end, respectively, of a list.

  `[` and `]` also enclose the subscripts of a list, array, hash array, or array function. Note that other than for arrays accessing the `n`th element of a list may need an amount of time that is roughly proportional to `n`, See section Performance considerations for Lists.

  Examples:

      (%i1) x: [a, b, c];
      (%o1)                       [a, b, c]
      (%i2) x[3];
      (%o2)                           c
      (%i3) array (y, fixnum, 3);
      (%o3)                           y
      (%i4) y[2]: %pi;
      (%o4)                          %pi
      (%i5) y[2];
      (%o5)                          %pi
      (%i6) z['foo]: 'bar;
      (%o6)                          bar
      (%i7) z['foo];
      (%o7)                          bar
      (%i8) g[k] := 1/(k^2+1);
                                        1
      (%o8)                     g  := ------
                                 k     2
                                      k  + 1
      (%i9) g[10];
                                      1
      (%o9)                          ---
                                     101

  Categories: Lists · Operators

Function: append (list_1, …, list_n)

  Returns a single list of the elements of list_1 followed by the elements of list_2, … `append` also works on general expressions, e.g. `append (f(a,b), f(c,d,e));` yields `f(a,b,c,d,e)`.

  See also `addrow`, `addcol` and `join`.

  Do `example(append);` for an example.

  Categories: Lists · Expressions

Function: assoc
`assoc` (key, list, default)
`assoc` (key, list)

  This function searches for key in the left hand side of the input list. The list argument should be a list, each of whose elements is an expression with exactly two parts. Most usually, the elements of list are themselves lists, each with two elements.

  The `assoc` function iterates along list, checking the first part of each element for equality with key. If an element is found where the comparison is true, `assoc` returns the second part of that element. If there is no such element in the list, `assoc` returns either `false` or default, if given.

  For example, in the expression `assoc (y, [[x,1], [y,2], [z,3]])`, the `assoc` function searches for `x` in the left hand side of the list `[[y,1],[x,2]]` and finds it at the second term, returning `2`. In `assoc (z, [[x,1], [z,2], [z,3]])`, the search stops at the first term starting with `z` and returns `2`. In `assoc(x, [[y,1]])`, there is no matching element, so `assoc` returns `false`.

      (%i1) assoc (y, [[x,1], [y,2],[z,3]]);
      (%o1)                           2
      (%i2) assoc (z, [[x,1], [z,2], [z,3]]);
      (%o2)                           2
      (%i3) assoc (x, [[y,1]]);
      (%o3)                         false

  Categories: Lists · Expressions

Function: cons
`cons` (expr, list)
`cons` (expr_1, expr_2)

  `cons (expr, list)` returns a new list constructed of the element expr as its first element, followed by the elements of list. This is analogous to the Lisp language construction operation "cons".

  The Maxima function `cons` can also be used where the second argument is other than a list and this might be useful. In this case, `cons (expr_1, expr_2)` returns an expression with same operator as expr_2 but with argument `cons(expr_1, args(expr_2))`. Examples:

      (%i1) cons(a,[b,c,d]);
      (%o1)                     [a, b, c, d]
      (%i2) cons(a,f(b,c,d));
      (%o2)                     f(a, b, c, d)

  In general, `cons` applied to a nonlist doesn't make sense. For instance, `cons(a,b^c)` results in an illegal expression, since '^' cannot take three arguments.

  When `inflag` is true, `cons` operates on the internal structure of an expression, otherwise `cons` operates on the displayed form. Especially when `inflag` is true, `cons` applied to a nonlist sometimes gives a surprising result; for example

      (%i1) cons(a,-a), inflag : true;
                                       2
      (%o1)                         - a
      (%i2) cons(a,-a), inflag : false;
      (%o2)                           0

  Categories: Lists · Expressions

Function: copylist (list)

  Returns a copy of the list list.

  Categories: Lists

Function: create_list (form, x_1, list_1, …, x_n, list_n)

  Create a list by evaluating form with x_1 bound to each element of list_1, and for each such binding bind x_2 to each element of list_2, … The number of elements in the result will be the product of the number of elements in each list. Each variable x_i must actually be a symbol - it will not be evaluated. The list arguments will be evaluated once at the beginning of the iteration.

      (%i1) create_list (x^i, i, [1, 3, 7]);
                                      3   7
      (%o1)                      [x, x , x ]

  With a double iteration:

      (%i1) create_list ([i, j], i, [a, b], j, [e, f, h]);
      (%o1)   [[a, e], [a, f], [a, h], [b, e], [b, f], [b, h]]

  Instead of list_i two args may be supplied each of which should evaluate to a number. These will be the inclusive lower and upper bounds for the iteration.

      (%i1) create_list ([i, j], i, [1, 2, 3], j, 1, i);
      (%o1)   [[1, 1], [2, 1], [2, 2], [3, 1], [3, 2], [3, 3]]

  Note that the limits or list for the `j` variable can depend on the current value of `i`.

  Categories: Lists

Function: delete
`delete` (expr_1, expr_2)
`delete` (expr_1, expr_2, n)

  `delete(expr_1, expr_2)` removes from expr_2 any arguments of its top-level operator which are the same (as determined by "=") as expr_1. Note that "=" tests for formal equality, not equivalence. Note also that arguments of subexpressions are not affected.

  expr_1 may be an atom or a non-atomic expression. expr_2 may be any non-atomic expression. `delete` returns a new expression; it does not modify expr_2.

  `delete(expr_1, expr_2, n)` removes from expr_2 the first n arguments of the top-level operator which are the same as expr_1. If there are fewer than n such arguments, then all such arguments are removed.

  Examples:

  Removing elements from a list.

      (%i1) delete (y, [w, x, y, z, z, y, x, w]);
      (%o1)                  [w, x, z, z, x, w]

  Removing terms from a sum.

      (%i1) delete (sin(x), x + sin(x) + y);
      (%o1)                         y + x

  Removing factors from a product.

      (%i1) delete (u - x, (u - w)*(u - x)*(u - y)*(u - z));
      (%o1)                (u - w) (u - y) (u - z)

  Removing arguments from an arbitrary expression.

      (%i1) delete (a, foo (a, b, c, d, a));
      (%o1)                     foo(b, c, d)

  Limit the number of removed arguments.

      (%i1) delete (a, foo (a, b, a, c, d, a), 2);
      (%o1)                    foo(b, c, d, a)

  Whether arguments are the same as expr_1 is determined by "=". Arguments which are `equal` but not "=" are not removed.

      (%i1) [is (equal (0, 0)), is (equal (0, 0.0)), is (equal (0, 0b0))];
      (%o1)                  [true, true, true]
      (%i2) [is (0 = 0), is (0 = 0.0), is (0 = 0b0)];
      (%o2)                 [true, false, false]
      (%i3) delete (0, [0, 0.0, 0b0]);
      (%o3)                     [0.0, 0.0b0]
      (%i4) is (equal ((x + y)*(x - y), x^2 - y^2));
      (%o4)                         true
      (%i5) is ((x + y)*(x - y) = x^2 - y^2);
      (%o5)                         false
      (%i6) delete ((x + y)*(x - y), [(x + y)*(x - y), x^2 - y^2]);
                                    2    2
      (%o6)                       [x  - y ]

  Categories: Lists · Expressions

Function: eighth (expr)

  Returns the 8'th item of expression or list expr. See `first` for more details.

  Categories: Lists · Expressions

Function: endcons
`endcons` (expr, list)
`endcons` (expr_1, expr_2)

  `endcons (expr, list)` returns a new list constructed of the elements of list followed by expr. The Maxima function `endcons` can also be used where the second argument is other than a list and this might be useful. In this case, `endcons (expr_1, expr_2)` returns an expression with same operator as expr_2 but with argument `endcons(expr_1, args(expr_2))`. Examples:

      (%i1) endcons(a,[b,c,d]);
      (%o1)                     [b, c, d, a]
      (%i2) endcons(a,f(b,c,d));
      (%o2)                     f(b, c, d, a)

  In general, `endcons` applied to a nonlist doesn't make sense. For instance, `endcons(a,b^c)` results in an illegal expression, since '^' cannot take three arguments.

  When `inflag` is true, `endcons` operates on the internal structure of an expression, otherwise `endcons` operates on the displayed form. Especially when `inflag` is true, `endcons` applied to a nonlist sometimes gives a surprising result; for example

      (%i1) endcons(a,-a), inflag : true;
                                       2
      (%o1)                         - a
      (%i2) endcons(a,-a), inflag : false;
      (%o2)                           0

  Categories: Lists · Expressions

Function: fifth (expr)

  Returns the 5'th item of expression or list expr. See `first` for more details.

  Categories: Lists · Expressions

Function: first (expr)

  Returns the first part of expr which may result in the first element of a list, the first row of a matrix, the first term of a sum, etc.:

      (%i1) matrix([1,2],[3,4]);
                                         [ 1  2 ]
      (%o1)                              [      ]
                                         [ 3  4 ]
      (%i2) first(%);
      (%o2)                              [1,2]
      (%i3) first(%);
      (%o3)                              1
      (%i4) first(a*b/c+d+e/x);
                                         a b
      (%o4)                              ---
                                          c
      (%i5) first(a=b/c+d+e/x);
      (%o5)                              a

  Note that `first` and its related functions, `rest` and `last`, work on the form of expr which is displayed not the form which is typed on input. If the variable `inflag` is set to `true` however, these functions will look at the internal form of expr. One reason why this may make a difference is that the simplifier re-orders expressions:

      (%i1) x+y;
      (%o1)                              y+1
      (%i2) first(x+y),inflag : true;
      (%o2)                              x
      (%i3) first(x+y),inflag : false;
      (%o3)                              y

  The functions `second` … `tenth` yield the second through the tenth part of their input argument.

  See also `firstn` and `part`.

  Categories: Lists · Expressions

Function: firstn (expr, count)

  Returns the first count arguments of expr, if expr has at least count arguments. Returns expr if expr has less than count arguments.

  expr may be any nonatomic expression. When expr is something other than a list, `firstn` returns an expression which has the same operator as expr. count must be a nonnegative integer.

  `firstn` honors the global flag `inflag`, which governs whether the internal form of an expression is processed (when `inflag` is true) or the displayed form (when `inflag` is false).

  Note that `firstn(expr, 1)`, which returns a nonatomic expression containing the first argument, is not the same as `first(expr)`, which returns the first argument by itself.

  See also `lastn` and `rest`.

  Examples:

  `firstn` returns the first count elements of expr, if expr has at least count elements.

      (%i1) mylist : [1, a, 2, b, 3, x, 4 - y, 2*z + sin(u)];
      (%o1)        [1, a, 2, b, 3, x, 4 - y, 2 z + sin(u)]
      (%i2) firstn (mylist, 0);
      (%o2)                          []
      (%i3) firstn (mylist, 1);
      (%o3)                          [1]
      (%i4) firstn (mylist, 2);
      (%o4)                        [1, a]
      (%i5) firstn (mylist, 7);
      (%o5)               [1, a, 2, b, 3, x, 4 - y]

  `firstn` returns expr if expr has less than count elements.

      (%i1) mylist : [1, a, 2, b, 3, x, 4 - y, 2*z + sin(u)];
      (%o1)        [1, a, 2, b, 3, x, 4 - y, 2 z + sin(u)]
      (%i2) firstn (mylist, 100);
      (%o2)        [1, a, 2, b, 3, x, 4 - y, 2 z + sin(u)]

  expr may be any nonatomic expression.

      (%i1) myfoo : foo(1, a, 2, b, 3, x, 4 - y, 2*z + sin(u));
      (%o1)      foo(1, a, 2, b, 3, x, 4 - y, 2 z + sin(u))
      (%i2) firstn (myfoo, 4);
      (%o2)                    foo(1, a, 2, b)
      (%i3) mybar : bar[m, n](1, a, 2, b, 3, x, 4 - y, 2*z + sin(u));
      (%o3)    bar    (1, a, 2, b, 3, x, 4 - y, 2 z + sin(u))
                  m, n
      (%i4) firstn (mybar, 4);
      (%o4)                  bar    (1, a, 2, b)
                                m, n
      (%i5) mymatrix : genmatrix (lambda ([i, j], 10*i + j), 10, 4) $
      (%i6) firstn (mymatrix, 3);
                             [ 11  12  13  14 ]
                             [                ]
      (%o6)                  [ 21  22  23  24 ]
                             [                ]
                             [ 31  32  33  34 ]

  `firstn` honors the global flag `inflag`.

      (%i1) myexpr : a + b + c + d + e;
      (%o1)                   e + d + c + b + a
      (%i2) firstn (myexpr, 3), inflag=true;
      (%o2)                       c + b + a
      (%i3) firstn (myexpr, 3), inflag=false;
      (%o3)                       e + d + c

  Note that `firstn(expr, 1)` is not the same as `first(expr)`.

      (%i1) firstn ([w, x, y, z], 1);
      (%o1)                          [w]
      (%i2) first ([w, x, y, z]);
      (%o2)                           w

  Categories: Lists · Expressions

Function: fourth (expr)

  Returns the 4'th item of expression or list expr. See `first` for more details.

  Categories: Lists · Expressions

Function: join (l, m)

  Creates a new list containing the elements of lists l and m, interspersed. The result has elements `[l[1], m[1], l[2], m[2], ...]`. The lists l and m may contain any type of elements.

  If the lists are different lengths, `join` ignores elements of the longer list.

  Maxima complains if l or m is not a list.

  See also `append`.

  Examples:

      (%i1) L1: [a, sin(b), c!, d - 1];
      (%o1)                [a, sin(b), c!, d - 1]
      (%i2) join (L1, [1, 2, 3, 4]);
      (%o2)          [a, 1, sin(b), 2, c!, 3, d - 1, 4]
      (%i3) join (L1, [aa, bb, cc, dd, ee, ff]);
      (%o3)        [a, aa, sin(b), bb, c!, cc, d - 1, dd]

  Categories: Lists

Function: last (expr)

  Returns the last part (term, row, element, etc.) of the expr.

  See also `lastn`.

  Categories: Lists · Expressions

Function: lastn (expr, count)

  Returns the last count arguments of expr, if expr has at least count arguments. Returns expr if expr has less than count arguments.

  expr may be any nonatomic expression. When expr is something other than a list, `lastn` returns an expression which has the same operator as expr. count must be a nonnegative integer.

  `lastn` honors the global flag `inflag`, which governs whether the internal form of an expression is processed (when `inflag` is true) or the displayed form (when `inflag` is false).

  Note that `lastn(expr, 1)`, which returns a nonatomic expression containing the last argument, is not the same as `last(expr)`, which returns the last argument by itself.

  See also `firstn` and `rest`.

  Examples:

  `lastn` returns the last count elements of expr, if expr has at least count elements.

      (%i1) mylist : [1, a, 2, b, 3, x, 4 - y, 2*z + sin(u)];
      (%o1)        [1, a, 2, b, 3, x, 4 - y, 2 z + sin(u)]
      (%i2) lastn (mylist, 0);
      (%o2)                          []
      (%i3) lastn (mylist, 1);
      (%o3)                    [2 z + sin(u)]
      (%i4) lastn (mylist, 2);
      (%o4)                 [4 - y, 2 z + sin(u)]
      (%i5) lastn (mylist, 7);
      (%o5)         [a, 2, b, 3, x, 4 - y, 2 z + sin(u)]

  `lastn` returns expr if expr has less than count elements.

      (%i1) mylist : [1, a, 2, b, 3, x, 4 - y, 2*z + sin(u)];
      (%o1)        [1, a, 2, b, 3, x, 4 - y, 2 z + sin(u)]
      (%i2) lastn (mylist, 100);
      (%o2)        [1, a, 2, b, 3, x, 4 - y, 2 z + sin(u)]

  expr may be any nonatomic expression.

      (%i1) myfoo : foo(1, a, 2, b, 3, x, 4 - y, 2*z + sin(u));
      (%o1)      foo(1, a, 2, b, 3, x, 4 - y, 2 z + sin(u))
      (%i2) lastn (myfoo, 4);
      (%o2)            foo(3, x, 4 - y, 2 z + sin(u))
      (%i3) mybar : bar[m, n](1, a, 2, b, 3, x, 4 - y, 2*z + sin(u));
      (%o3)    bar    (1, a, 2, b, 3, x, 4 - y, 2 z + sin(u))
                  m, n
      (%i4) lastn (mybar, 4);
      (%o4)          bar    (3, x, 4 - y, 2 z + sin(u))
                        m, n
      (%i5) mymatrix : genmatrix (lambda ([i, j], 10*i + j), 10, 4) $
      (%i6) lastn (mymatrix, 3);
                           [ 81   82   83   84  ]
                           [                    ]
      (%o6)                [ 91   92   93   94  ]
                           [                    ]
                           [ 101  102  103  104 ]

  `lastn` honors the global flag `inflag`.

      (%i1) myexpr : a + b + c + d + e;
      (%o1)                   e + d + c + b + a
      (%i2) lastn (myexpr, 3), inflag=true;
      (%o2)                       e + d + c
      (%i3) lastn (myexpr, 3), inflag=false;
      (%o3)                       c + b + a

  Note that `lastn(expr, 1)` is not the same as `last(expr)`.

      (%i1) lastn ([w, x, y, z], 1);
      (%o1)                          [z]
      (%i2) last ([w, x, y, z]);
      (%o2)                           z

  Categories: Lists · Expressions

Function: length (expr)

  Returns (by default) the number of parts in the external (displayed) form of expr. For lists this is the number of elements, for matrices it is the number of rows, and for sums it is the number of terms (see `dispform`).

  The `length` command is affected by the `inflag` switch. So, e.g. `length(a/(b*c));` gives 2 if `inflag` is `false` (Assuming `exptdispflag` is `true`), but 3 if `inflag` is `true` (the internal representation is essentially `a*b^-1*c^-1`).

  Determining a list's length typically needs an amount of time proportional to the number of elements in the list. If the length of a list is used inside a loop it therefore might drastically increase the performance if the length is calculated outside the loop instead.

  Categories: Lists · Expressions

Option variable: listarith

  Default value: `true`

  If `false` causes any arithmetic operations with lists to be suppressed; when `true`, list-matrix operations are contagious causing lists to be converted to matrices yielding a result which is always a matrix. However, list-list operations should return lists.

  Categories: Lists · Global flags

Function: listp (expr)

  Returns `true` if expr is a list else `false`.

  Categories: Lists · Predicate functions

Function: lreduce
`lreduce` (F, s)
`lreduce` (F, s, s_0)

  Extends the binary function F to an n-ary function by composition, where s is a list.

  `lreduce(F, s)` returns `F(... F(F(s_1, s_2), s_3), ... s_n)`. When the optional argument s_0 is present, the result is equivalent to `lreduce(F, cons(s_0, s))`.

  The function F is first applied to the leftmost list elements, thus the name "lreduce".

  See also `rreduce`, `xreduce`, and `tree_reduce`.

  Examples:

  `lreduce` without the optional argument.

      (%i1) lreduce (f, [1, 2, 3]);
      (%o1)                     f(f(1, 2), 3)
      (%i2) lreduce (f, [1, 2, 3, 4]);
      (%o2)                  f(f(f(1, 2), 3), 4)

  `lreduce` with the optional argument.

      (%i1) lreduce (f, [1, 2, 3], 4);
      (%o1)                  f(f(f(4, 1), 2), 3)

  `lreduce` applied to built-in binary operators. `/` is the division operator.

      (%i1) lreduce ("^", args ({a, b, c, d}));
                                     b c d
      (%o1)                       ((a ) )
      (%i2) lreduce ("/", args ({a, b, c, d}));
                                      a
      (%o2)                         -----
                                    b c d

  Categories: Lists

Function: makelist
`makelist` ()
`makelist` (expr, n)
`makelist` (expr, i, i_max)
`makelist` (expr, i, i_0, i_max)
`makelist` (expr, i, i_0, i_max, step)
`makelist` (expr, x, list)

  The first form, `makelist ()`, creates an empty list. The second form, `makelist (expr)`, creates a list with expr as its single element. `makelist (expr, n)` creates a list of n elements generated from expr.

  The most general form, `makelist (expr, i, i_0, i_max, step)`, returns the list of elements obtained when `ev (expr, i=j)` is applied to the elements j of the sequence: i_0, i_0 + step, i_0 + 2*step, ..., with |j| less than or equal to |i_max|.

  The increment step can be a number (positive or negative) or an expression. If it is omitted, the default value 1 will be used. If both i_0 and step are omitted, they will both have a default value of 1.

  `makelist (expr, x, list)` returns a list, the `j`'th element of which is equal to `ev (expr, x=list[j])` for `j` equal to 1 through `length (list)`.

  Examples:

      (%i1) makelist (concat (x,i), i, 6);
      (%o1)               [x1, x2, x3, x4, x5, x6]
      (%i2) makelist (x=y, y, [a, b, c]);
      (%o2)                 [x = a, x = b, x = c]
      (%i3) makelist (x^2, x, 3, 2*%pi, 2);
      (%o3)                        [9, 25]
      (%i4) makelist (random(6), 4);
      (%o4)                     [2, 0, 2, 5]
      (%i5) flatten (makelist (makelist (i^2, 3), i, 4));
      (%o5)        [1, 1, 1, 4, 4, 4, 9, 9, 9, 16, 16, 16]
      (%i6) flatten (makelist (makelist (i^2, i, 3), 4));
      (%o6)         [1, 4, 9, 1, 4, 9, 1, 4, 9, 1, 4, 9]

  Categories: Lists

Function: member (expr_1, expr_2)

  Returns `true` if `is(expr_1 = a)` for some element a in `args(expr_2)`, otherwise returns `false`.

  `expr_2` is typically a list, in which case `args(expr_2) = expr_2` and `is(expr_1 = a)` for some element a in `expr_2` is the test.

  `member` does not inspect parts of the arguments of `expr_2`, so it may return `false` even if `expr_1` is a part of some argument of `expr_2`.

  See also `elementp`.

  Examples:

      (%i1) member (8, [8, 8.0, 8b0]);
      (%o1)                         true
      (%i2) member (8, [8.0, 8b0]);
      (%o2)                         false
      (%i3) member (b, [a, b, c]);
      (%o3)                         true
      (%i4) member (b, [[a, b], [b, c]]);
      (%o4)                         false
      (%i5) member ([b, c], [[a, b], [b, c]]);
      (%o5)                         true
      (%i6) F (1, 1/2, 1/4, 1/8);
                                     1  1  1
      (%o6)                     F(1, -, -, -)
                                     2  4  8
      (%i7) member (1/8, %);
      (%o7)                         true
      (%i8) member ("ab", ["aa", "ab", sin(1), a + b]);
      (%o8)                         true

  Categories: Lists · Expressions · Predicate functions

Function: ninth (expr)

  Returns the 9'th item of expression or list expr. See `first` for more details.

  Categories: Lists · Expressions

Function: pop (list)

  `pop` removes and returns the first element from the list list. The argument list must be a mapatom that is bound to a nonempty list. If the argument list is not bound to a nonempty list, Maxima signals an error. For examples, see `push`.

  Categories: Lists · Expressions

Function: push (item, list)

  `push` prepends the item item to the list list and returns a copy of the new list. The second argument list must be a mapatom that is bound to a list. The first argument item can be any Maxima symbol or expression. If the argument list is not bound to a list, Maxima signals an error.

  To remove the first item from a list, see `pop`.

  Examples:

      (%i1) ll: [];
      (%o1)                          []
      (%i2) push (x, ll);
      (%o2)                          [x]
      (%i3) push (x^2+y, ll);
                                       2
      (%o3)                      [y + x , x]
      (%i4) a: push ("string", ll);
                                           2
      (%o4)                  [string, y + x , x]
      (%i5) pop (ll);
      (%o5)                        string
      (%i6) pop (ll);
                                        2
      (%o6)                        y + x
      (%i7) pop (ll);
      (%o7)                           x
      (%i8) ll;
      (%o8)                          []
      (%i9) a;
                                           2
      (%o9)                  [string, y + x , x]

  Categories: Lists · Expressions

Function: rest
`rest` (expr, n)
`rest` (expr)

  Returns expr with its first n elements removed if n is positive and its last `- n` elements removed if n is negative. If n is 1 it may be omitted. The first argument expr may be a list, matrix, or other expression. When expr is an atom, `rest` signals an error; when expr is an empty list and `partswitch` is false, `rest` signals an error. When expr is an empty list and `partswitch` is true, `rest` returns `end`.

  Applying `rest` to expression such as `f(a,b,c)` returns `f(b,c)`. In general, applying `rest` to an nonlist doesn't make sense. For example, because '^' requires two arguments, `rest(a^b)` results in an error message. The functions `args` and `op` may be useful as well, since `args(a^b)` returns `[a,b]` and `op(a^b)` returns ^.

  See also `firstn` and `lastn`.

      (%i1) rest(a+b+c);
      (%o1) b+a
      (%i2) rest(a+b+c,2);
      (%o2) a
      (%i3) rest(a+b+c,-2);
      (%o3) c

  Categories: Lists · Expressions

Function: reverse (list)

  Reverses the order of the members of the list (not the members themselves). `reverse` also works on general expressions, e.g. `reverse(a=b);` gives `b=a`.

  Categories: Lists · Expressions

Function: rreduce
`rreduce` (F, s)
`rreduce` (F, s, s_{n + 1})

  Extends the binary function F to an n-ary function by composition, where s is a list.

  `rreduce(F, s)` returns `F(s_1, ... F(s_{n - 2}, F(s_{n - 1}, s_n)))`. When the optional argument s_{n + 1} is present, the result is equivalent to `rreduce(F, endcons(s_{n + 1}, s))`.

  The function F is first applied to the rightmost list elements, thus the name "rreduce".

  See also `lreduce`, `tree_reduce`, and `xreduce`.

  Examples:

  `rreduce` without the optional argument.

      (%i1) rreduce (f, [1, 2, 3]);
      (%o1)                     f(1, f(2, 3))
      (%i2) rreduce (f, [1, 2, 3, 4]);
      (%o2)                  f(1, f(2, f(3, 4)))

  `rreduce` with the optional argument.

      (%i1) rreduce (f, [1, 2, 3], 4);
      (%o1)                  f(1, f(2, f(3, 4)))

  `rreduce` applied to built-in binary operators. `/` is the division operator.

      (%i1) rreduce ("^", args ({a, b, c, d}));
                                       d
                                      c
                                     b
      (%o1)                         a
      (%i2) rreduce ("/", args ({a, b, c, d}));
                                     a c
      (%o2)                          ---
                                     b d

  Categories: Lists

Function: second (expr)

  Returns the 2'nd item of expression or list expr. See `first` for more details.

  Categories: Lists · Expressions

Function: seventh (expr)

  Returns the 7'th item of expression or list expr. See `first` for more details.

  Categories: Lists · Expressions

Function: sixth (expr)

  Returns the 6'th item of expression or list expr. See `first` for more details.

  Categories: Lists · Expressions

Function: sort
`sort` (L, P)
`sort` (L)

  `sort(L, P)` sorts a list L according to a predicate `P` of two arguments which defines a strict weak order on the elements of L. If `P(a, b)` is `true`, then `a` appears before `b` in the result. If neither `P(a, b)` nor `P(b, a)` are `true`, then `a` and `b` are equivalent, and appear in the result in the same order as in the input. That is, `sort` is a stable sort.

  If `P(a, b)` and `P(b, a)` are both `true` for some elements of L, then P is not a valid sort predicate, and the result is undefined. If `P(a, b)` is something other than `true` or `false`, `sort` signals an error.

  The predicate may be specified as the name of a function or binary infix operator, or as a `lambda` expression. If specified as the name of an operator, the name must be enclosed in double quotes.

  The sorted list is returned as a new object; the argument L is not modified.

  `sort(L)` is equivalent to `sort(L, orderlessp)`.

  The default sorting order is ascending, as determined by `orderlessp`. The predicate `ordergreatp` sorts a list in descending order.

  All Maxima atoms and expressions are comparable under `orderlessp` and `ordergreatp`.

  Operators `<` and `>` order numbers, constants, and constant expressions by magnitude. Note that `orderlessp` and `ordergreatp` do not order numbers, constants, and constant expressions by magnitude.

  `ordermagnitudep` orders numbers, constants, and constant expressions the same as `<`, and all other elements the same as `orderlessp`.

  Examples:

  `sort` sorts a list according to a predicate of two arguments which defines a strict weak order on the elements of the list.

      (%i1) sort ([1, a, b, 2, 3, c], 'orderlessp);
      (%o1)                  [1, 2, 3, a, b, c]
      (%i2) sort ([1, a, b, 2, 3, c], 'ordergreatp);
      (%o2)                  [c, b, a, 3, 2, 1]

  The predicate may be specified as the name of a function or binary infix operator, or as a `lambda` expression. If specified as the name of an operator, the name must be enclosed in double quotes.

      (%i1) L : [[1, x], [3, y], [4, w], [2, z]];
      (%o1)           [[1, x], [3, y], [4, w], [2, z]]
      (%i2) foo (a, b) := a[1] > b[1];
      (%o2)                 foo(a, b) := a  > b
                                          1    1
      (%i3) sort (L, 'foo);
      (%o3)           [[4, w], [3, y], [2, z], [1, x]]
      (%i4) infix (">>");
      (%o4)                          >>
      (%i5) a >> b := a[1] > b[1];
      (%o5)                  (a >> b) := a  > b
                                          1    1
      (%i6) sort (L, ">>");
      (%o6)           [[4, w], [3, y], [2, z], [1, x]]
      (%i7) sort (L, lambda ([a, b], a[1] > b[1]));
      (%o7)           [[4, w], [3, y], [2, z], [1, x]]

  `sort(L)` is equivalent to `sort(L, orderlessp)`.

      (%i1) L : [a, 2*b, -5, 7, 1 + %e, %pi];
      (%o1)             [a, 2 b, - 5, 7, %e + 1, %pi]
      (%i2) sort (L);
      (%o2)             [- 5, 7, %e + 1, %pi, a, 2 b]
      (%i3) sort (L, 'orderlessp);
      (%o3)             [- 5, 7, %e + 1, %pi, a, 2 b]

  The default sorting order is ascending, as determined by `orderlessp`. The predicate `ordergreatp` sorts a list in descending order.

      (%i1) L : [a, 2*b, -5, 7, 1 + %e, %pi];
      (%o1)             [a, 2 b, - 5, 7, %e + 1, %pi]
      (%i2) sort (L);
      (%o2)             [- 5, 7, %e + 1, %pi, a, 2 b]
      (%i3) sort (L, 'ordergreatp);
      (%o3)             [2 b, a, %pi, %e + 1, 7, - 5]

  All Maxima atoms and expressions are comparable under `orderlessp` and `ordergreatp`.

      (%i1) L : [11, -17, 29b0, 9*c, 7.55, foo(x, y), -5/2, b + a];
                                                       5
      (%o1)  [11, - 17, 2.9b1, 9 c, 7.55, foo(x, y), - -, b + a]
                                                       2
      (%i2) sort (L, orderlessp);
                      5
      (%o2)  [- 17, - -, 7.55, 11, 2.9b1, b + a, 9 c, foo(x, y)]
                      2
      (%i3) sort (L, ordergreatp);
                                                        5
      (%o3)  [foo(x, y), 9 c, b + a, 2.9b1, 11, 7.55, - -, - 17]
                                                        2

  Operators `<` and `>` order numbers, constants, and constant expressions by magnitude. Note that `orderlessp` and `ordergreatp` do not order numbers, constants, and constant expressions by magnitude.

      (%i1) L : [%pi, 3, 4, %e, %gamma];
      (%o1)                [%pi, 3, 4, %e, %gamma]
      (%i2) sort (L, ">");
      (%o2)                [4, %pi, 3, %e, %gamma]
      (%i3) sort (L, ordergreatp);
      (%o3)                [%pi, %gamma, %e, 4, 3]

  `ordermagnitudep` orders numbers, constants, and constant expressions the same as `<`, and all other elements the same as `orderlessp`.

      (%i1) L : [%i, 1+%i, 2*x, minf, inf, %e, sin(1), 0, 1, 2, 3, 1.0, 1.0b0];
      (%o1) [%i, %i + 1, 2 x, minf, inf, %e, sin(1), 0, 1, 2, 3, 1.0,
                                                                 1.0b0]
      (%i2) sort (L, ordermagnitudep);
      (%o2) [minf, 0, sin(1), 1, 1.0, 1.0b0, 2, %e, 3, inf, %i,
                                                           %i + 1, 2 x]
      (%i3) sort (L, orderlessp);
      (%o3) [0, 1, 1.0, 2, 3, sin(1), 1.0b0, %e, %i, %i + 1, inf,
                                                             minf, 2 x]

  Categories: Lists

Function: sublist (list, p)

  Returns the list of elements of list for which the predicate `p` returns `true`.

  Example:

      (%i1) L: [1, 2, 3, 4, 5, 6];
      (%o1)                  [1, 2, 3, 4, 5, 6]
      (%i2) sublist (L, evenp);
      (%o2)                       [2, 4, 6]

  Categories: Lists

Function: sublist_indices (L, P)

  Returns the indices of the elements `x` of the list L for which the predicate `maybe(P(x))` returns `true`; this excludes `unknown` as well as `false`. P may be the name of a function or a lambda expression. L must be a literal list.

  Examples:

      (%i1) sublist_indices ('[a, b, b, c, 1, 2, b, 3, b],
                             lambda ([x], x='b));
      (%o1)                     [2, 3, 7, 9]
      (%i2) sublist_indices ('[a, b, b, c, 1, 2, b, 3, b], symbolp);
      (%o2)                  [1, 2, 3, 4, 7, 9]
      (%i3) sublist_indices ([1 > 0, 1 < 0, 2 < 1, 2 > 1, 2 > 0],
                             identity);
      (%o3)                       [1, 4, 5]
      (%i4) assume (x < -1);
      (%o4)                       [x < - 1]
      (%i5) map (maybe, [x > 0, x < 0, x < -2]);
      (%o5)                [false, true, unknown]
      (%i6) sublist_indices ([x > 0, x < 0, x < -2], identity);
      (%o6)                          [2]

  Categories: Lists

Function: tenth (expr)

  Returns the 10'th item of expression or list expr. See `first` for more details.

  Categories: Lists · Expressions

Function: third (expr)

  Returns the 3'rd item of expression or list expr. See `first` for more details.

  Categories: Lists · Expressions

Function: tree_reduce
`tree_reduce` (F, s)
`tree_reduce` (F, s, s_0)

  Extends the binary function F to an n-ary function by composition, where s is a set or list.

  `tree_reduce` is equivalent to the following: Apply F to successive pairs of elements to form a new list `[F(s_1, s_2), F(s_3, s_4), ...]`, carrying the final element unchanged if there are an odd number of elements. Then repeat until the list is reduced to a single element, which is the return value.

  When the optional argument s_0 is present, the result is equivalent `tree_reduce(F, cons(s_0, s))`.

  For addition of floating point numbers, `tree_reduce` may return a sum that has a smaller rounding error than either `rreduce` or `lreduce`.

  The elements of s and the partial results may be arranged in a minimum-depth binary tree, thus the name "tree_reduce".

  Examples:

  `tree_reduce` applied to a list with an even number of elements.

      (%i1) tree_reduce (f, [a, b, c, d]);
      (%o1)                  f(f(a, b), f(c, d))

  `tree_reduce` applied to a list with an odd number of elements.

      (%i1) tree_reduce (f, [a, b, c, d, e]);
      (%o1)               f(f(f(a, b), f(c, d)), e)

  Categories: Sets · Lists

Function: unique (L)

  Returns the unique elements of the list L.

  When all the elements of L are unique, `unique` returns a shallow copy of L, not L itself.

  If L is not a list, `unique` returns L.

  Example:

      (%i1) unique ([1, %pi, a + b, 2, 1, %e, %pi, a + b, [1]]);
      (%o1)              [1, 2, %e, %pi, [1], b + a]

Function: xreduce
`xreduce` (F, s)
`xreduce` (F, s, s_0)

  Extends the function F to an n-ary function by composition, or, if F is already n-ary, applies F to s. When F is not n-ary, `xreduce` is the same as `lreduce`. The argument s is a list.

  Functions known to be n-ary include addition `+`, multiplication `*`, `and`, `or`, `max`, `min`, and `append`. Functions may also be declared n-ary by `declare(F, nary)`. For these functions, `xreduce` is expected to be faster than either `rreduce` or `lreduce`.

  When the optional argument s_0 is present, the result is equivalent to `xreduce(s, cons(s_0, s))`.

  Floating point addition is not exactly associative; be that as it may, `xreduce` applies Maxima's n-ary addition when s contains floating point numbers.

  Examples:

  `xreduce` applied to a function known to be n-ary. `F` is called once, with all arguments.

      (%i1) declare (F, nary);
      (%o1)                         done
      (%i2) F ([L]) := L;
      (%o2)                      F([L]) := L
      (%i3) xreduce (F, [a, b, c, d, e]);
      (%o3)                    [a, b, c, d, e]

  `xreduce` applied to a function not known to be n-ary. `G` is called several times, with two arguments each time.

      (%i1) G ([L]) := L;
      (%o1)                      G([L]) := L
      (%i2) xreduce (G, [a, b, c, d, e]);
      (%o2)                 [[[[a, b], c], d], e]
      (%i3) lreduce (G, [a, b, c, d, e]);
      (%o3)                 [[[[a, b], c], d], e]

  Categories: Sets · Lists

[ < ] | [ > ] |  | [ << ] | [ Up ] | [ >> ] |  |  |  |  | [Top] | [Contents] | [Index] | [ ? ]

### 5.4.3 Performance considerations for Lists

Lists provide efficient ways of appending and removing elements. They can be created without knowing their final dimensions. Lisp provides efficient means of copying and handling lists. Also nested lists do not need to be strictly rectangular. These advantages over declared arrays come with the drawback that the amount of time needed for accessing a random element within a list may be roughly proportional to the element's distance from its beginning. Efficient traversal of lists is still possible, though, by using the list as a stack or a fifo:

    (%i1) l:[Test,1,2,3,4];
    (%o1)                  [Test, 1, 2, 3, 4]
    (%i2) while l # [] do
       disp(pop(l));
                                  Test

                                    1

                                    2

                                    3

                                    4

    (%o2)                         done

Another even faster example would be:

    (%i1) l:[Test,1,2,3,4];
    (%o1)                  [Test, 1, 2, 3, 4]
    (%i2) for i in l do
       disp(pop(l));
                                  Test

                                    1

                                    2

                                    3

                                    4

    (%o2)                         done

Beginning traversal with the last element of a list is possible after reversing the list using `reverse ()`. If the elements of a long list need to be processed in a different order performance might be increased by converting the list into a declared array first.

It is also to note that the ending condition of `for` loops is tested for every iteration which means that the result of a `length` should be cached if it is used in the ending condition:

    (%i1) l:makelist(i,i,1,100000)$
    (%i2) lngth:length(l);
    (%o2)                        100000
    (%i3) x:1;
    (%o3)                           1
    (%i4) for i:1 thru lngth do
        x:x+1$
    (%i5) x;
    (%o5)                        100001

[ < ] | [ > ] |  | [ << ] | [ Up ] | [ >> ] |  |  |  |  | [Top] | [Contents] | [Index] | [ ? ]

## 5.5 Arrays

[ < ] | [ > ] |  | [ << ] | [ Up ] | [ >> ] |  |  |  |  | [Top] | [Contents] | [Index] | [ ? ]

### 5.5.1 Functions and Variables for Arrays

Function: array
`array` (name, dim_1, …, dim_n)
`array` (name, type, dim_1, …, dim_n)
`array` ([name_1, …, name_m], dim_1, …, dim_n)

  Creates an n-dimensional array. n may be less than or equal to 5. The subscripts for the i'th dimension are the integers running from 0 to dim_i.

  `array (name, dim_1, ..., dim_n)` creates a general array.

  `array (name, type, dim_1, ..., dim_n)` creates an array, with elements of a specified type. type can be `fixnum` for integers of limited size or `flonum` for floating-point numbers.

  `array ([name_1, ..., name_m], dim_1, ..., dim_n)` creates m arrays, all of the same dimensions.

  If the user assigns to a subscripted variable before declaring the corresponding array, an undeclared array is created. Undeclared arrays, otherwise known as hashed arrays (because hash coding is done on the subscripts), are more general than declared arrays. The user does not declare their maximum size, and they grow dynamically by hashing as more elements are assigned values. The subscripts of undeclared arrays need not even be numbers. However, unless an array is rather sparse, it is probably more efficient to declare it when possible than to leave it undeclared. The `array` function can be used to transform an undeclared array into a declared array.

  Categories: Arrays

Function: arrayapply (A, [i_1, …, i_n])

  Evaluates `A [i_1, ..., i_n]`, where A is an array and i_1, …, i_n are integers.

  This is reminiscent of `apply`, except the first argument is an array instead of a function.

  Categories: Expressions · Arrays

Function: arrayinfo (A)

  Returns information about the array A. The argument A may be a declared array, an undeclared (hashed) array, an array function, or a subscripted function.

  For declared arrays, `arrayinfo` returns a list comprising the atom `declared`, the number of dimensions, and the size of each dimension. The elements of the array, both bound and unbound, are returned by `listarray`.

  For undeclared arrays (hashed arrays), `arrayinfo` returns a list comprising the atom `hashed`, the number of subscripts, and the subscripts of every element which has a value. The values are returned by `listarray`.

  For array functions, `arrayinfo` returns a list comprising the atom `hashed`, the number of subscripts, and any subscript values for which there are stored function values. The stored function values are returned by `listarray`.

  For subscripted functions, `arrayinfo` returns a list comprising the atom `hashed`, the number of subscripts, and any subscript values for which there are lambda expressions. The lambda expressions are returned by `listarray`.

  See also `listarray`.

  Examples:

  `arrayinfo` and `listarray` applied to a declared array.

      (%i1) array (aa, 2, 3);
      (%o1)                          aa
      (%i2) aa [2, 3] : %pi;
      (%o2)                          %pi
      (%i3) aa [1, 2] : %e;
      (%o3)                          %e
      (%i4) arrayinfo (aa);
      (%o4)                 [declared, 2, [2, 3]]
      (%i5) listarray (aa);
      (%o5) [#####, #####, #####, #####, #####, #####, %e, #####,
                                              #####, #####, #####, %pi]

  `arrayinfo` and `listarray` applied to an undeclared (hashed) array.

      (%i1) bb [FOO] : (a + b)^2;
                                         2
      (%o1)                       (b + a)
      (%i2) bb [BAR] : (c - d)^3;
                                         3
      (%o2)                       (c - d)
      (%i3) arrayinfo (bb);
      (%o3)               [hashed, 1, [BAR], [FOO]]
      (%i4) listarray (bb);
                                    3         2
      (%o4)                 [(c - d) , (b + a) ]

  `arrayinfo` and `listarray` applied to an array function.

      (%i1) cc [x, y] := y / x;
                                           y
      (%o1)                      cc     := -
                                   x, y    x
      (%i2) cc [u, v];
                                      v
      (%o2)                           -
                                      u
      (%i3) cc [4, z];
                                      z
      (%o3)                           -
                                      4
      (%i4) arrayinfo (cc);
      (%o4)              [hashed, 2, [4, z], [u, v]]
      (%i5) listarray (cc);
                                    z  v
      (%o5)                        [-, -]
                                    4  u

  `arrayinfo` and `listarray` applied to a subscripted function.

      (%i1) dd [x] (y) := y ^ x;
                                           x
      (%o1)                     dd (y) := y
                                  x
      (%i2) dd [a + b];
                                          b + a
      (%o2)                  lambda([y], y     )
      (%i3) dd [v - u];
                                          v - u
      (%o3)                  lambda([y], y     )
      (%i4) arrayinfo (dd);
      (%o4)             [hashed, 1, [b + a], [v - u]]
      (%i5) listarray (dd);
                               b + a                v - u
      (%o5)      [lambda([y], y     ), lambda([y], y     )]

  Categories: Arrays

Function: arraymake (A, [i_1, …, i_n])

  Returns the expression `A[i_1, ..., i_n]`. The result is an unevaluated array reference.

  `arraymake` is reminiscent of `funmake`, except the return value is an unevaluated array reference instead of an unevaluated function call.

  Examples:

      (%i1) arraymake (A, [1]);
      (%o1)                          A
                                      1
      (%i2) arraymake (A, [k]);
      (%o2)                          A
                                      k
      (%i3) arraymake (A, [i, j, 3]);
      (%o3)                       A
                                   i, j, 3
      (%i4) array (A, fixnum, 10);
      (%o4)                           A
      (%i5) fillarray (A, makelist (i^2, i, 1, 11));
      (%o5)                           A
      (%i6) arraymake (A, [5]);
      (%o6)                          A
                                      5
      (%i7) ''%;
      (%o7)                          36
      (%i8) L : [a, b, c, d, e];
      (%o8)                    [a, b, c, d, e]
      (%i9) arraymake ('L, [n]);
      (%o9)                          L
                                      n
      (%i10) ''%, n = 3;
      (%o10)                          c
      (%i11) A2 : make_array (fixnum, 10);
      (%o11)          {Array:  #(0 0 0 0 0 0 0 0 0 0)}
      (%i12) fillarray (A2, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]);
      (%o12)          {Array:  #(1 2 3 4 5 6 7 8 9 10)}
      (%i13) arraymake ('A2, [8]);
      (%o13)                         A2
                                       8
      (%i14) ''%;
      (%o14)                          9

  Categories: Expressions · Arrays

System variable: arrays

  Default value: `[]`

  `arrays` is a list of arrays that have been allocated. These comprise arrays declared by `array`, hashed arrays constructed by implicit definition (assigning something to an array element), and array functions defined by `:=` and `define`. Arrays defined by `make_array` are not included.

  See also `array`, `arrayapply`, `arrayinfo`, `arraymake`, `fillarray`, `listarray`, and `rearray`.

  Examples:

      (%i1) array (aa, 5, 7);
      (%o1)                          aa
      (%i2) bb [FOO] : (a + b)^2;
                                         2
      (%o2)                       (b + a)
      (%i3) cc [x] := x/100;
                                         x
      (%o3)                      cc  := ---
                                   x    100
      (%i4) dd : make_array ('any, 7);
      (%o4)       {Array:  #(NIL NIL NIL NIL NIL NIL NIL)}
      (%i5) arrays;
      (%o5)                     [aa, bb, cc]

  Categories: Arrays · Global variables

Function: arraysetapply (A, [i_1, …, i_n], x)

  Assigns x to `A[i_1, ..., i_n]`, where A is an array and i_1, …, i_n are integers.

  `arraysetapply` evaluates its arguments.

  Categories: Expressions · Arrays

Function: fillarray (A, B)

  Fills array A from B, which is a list or an array.

  If a specific type was declared for A when it was created, it can only be filled with elements of that same type; it is an error if an attempt is made to copy an element of a different type.

  If the dimensions of the arrays A and B are different, A is filled in row-major order. If there are not enough elements in B the last element is used to fill out the rest of A. If there are too many, the remaining ones are ignored.

  `fillarray` returns its first argument.

  Examples:

  Create an array of 9 elements and fill it from a list.

      (%i1) array (a1, fixnum, 8);
      (%o1)                          a1
      (%i2) listarray (a1);
      (%o2)              [0, 0, 0, 0, 0, 0, 0, 0, 0]
      (%i3) fillarray (a1, [1, 2, 3, 4, 5, 6, 7, 8, 9]);
      (%o3)                          a1
      (%i4) listarray (a1);
      (%o4)              [1, 2, 3, 4, 5, 6, 7, 8, 9]

  When there are too few elements to fill the array, the last element is repeated. When there are too many elements, the extra elements are ignored.

      (%i1) a2 : make_array (fixnum, 8);
      (%o1)             {Array:  #(0 0 0 0 0 0 0 0)}
      (%i2) fillarray (a2, [1, 2, 3, 4, 5]);
      (%o2)             {Array:  #(1 2 3 4 5 5 5 5)}
      (%i3) fillarray (a2, [4]);
      (%o3)             {Array:  #(4 4 4 4 4 4 4 4)}
      (%i4) fillarray (a2, makelist (i, i, 1, 100));
      (%o4)             {Array:  #(1 2 3 4 5 6 7 8)}

  Multple-dimension arrays are filled in row-major order.

      (%i1) a3 : make_array (fixnum, 2, 5);
      (%o1)        {Array:  #2A((0 0 0 0 0) (0 0 0 0 0))}
      (%i2) fillarray (a3, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]);
      (%o2)        {Array:  #2A((1 2 3 4 5) (6 7 8 9 10))}
      (%i3) a4 : make_array (fixnum, 5, 2);
      (%o3)     {Array:  #2A((0 0) (0 0) (0 0) (0 0) (0 0))}
      (%i4) fillarray (a4, a3);
      (%o4)     {Array:  #2A((1 2) (3 4) (5 6) (7 8) (9 10))}

  Categories: Arrays

Function: listarray (A)

  Returns a list of the elements of the array A. The argument A may be a declared array, an undeclared (hashed) array, an array function, or a subscripted function.

  Elements are listed in row-major order. That is, elements are sorted according to the first index, then according to the second index, and so on. The sorting order of index values is the same as the order established by `orderless`.

  For undeclared arrays, array functions, and subscripted functions, the elements correspond to the index values returned by `arrayinfo`.

  Unbound elements of declared general arrays (that is, not `fixnum` and not `flonum`) are returned as `#####`. Unbound elements of declared `fixnum` or `flonum` arrays are returned as 0 or 0.0, respectively. Unbound elements of undeclared arrays, array functions, and subscripted functions are not returned.

  Examples:

  `listarray` and `arrayinfo` applied to a declared array.

      (%i1) array (aa, 2, 3);
      (%o1)                          aa
      (%i2) aa [2, 3] : %pi;
      (%o2)                          %pi
      (%i3) aa [1, 2] : %e;
      (%o3)                          %e
      (%i4) listarray (aa);
      (%o4) [#####, #####, #####, #####, #####, #####, %e, #####,
                                              #####, #####, #####, %pi]
      (%i5) arrayinfo (aa);
      (%o5)                 [declared, 2, [2, 3]]

  `listarray` and `arrayinfo` applied to an undeclared (hashed) array.

      (%i1) bb [FOO] : (a + b)^2;
                                         2
      (%o1)                       (b + a)
      (%i2) bb [BAR] : (c - d)^3;
                                         3
      (%o2)                       (c - d)
      (%i3) listarray (bb);
                                    3         2
      (%o3)                 [(c - d) , (b + a) ]
      (%i4) arrayinfo (bb);
      (%o4)               [hashed, 1, [BAR], [FOO]]

  `listarray` and `arrayinfo` applied to an array function.

      (%i1) cc [x, y] := y / x;
                                           y
      (%o1)                      cc     := -
                                   x, y    x
      (%i2) cc [u, v];
                                      v
      (%o2)                           -
                                      u
      (%i3) cc [4, z];
                                      z
      (%o3)                           -
                                      4
      (%i4) listarray (cc);
                                    z  v
      (%o4)                        [-, -]
                                    4  u
      (%i5) arrayinfo (cc);
      (%o5)              [hashed, 2, [4, z], [u, v]]

  `listarray` and `arrayinfo` applied to a subscripted function.

      (%i1) dd [x] (y) := y ^ x;
                                           x
      (%o1)                     dd (y) := y
                                  x
      (%i2) dd [a + b];
                                          b + a
      (%o2)                  lambda([y], y     )
      (%i3) dd [v - u];
                                          v - u
      (%o3)                  lambda([y], y     )
      (%i4) listarray (dd);
                               b + a                v - u
      (%o4)      [lambda([y], y     ), lambda([y], y     )]
      (%i5) arrayinfo (dd);
      (%o5)             [hashed, 1, [b + a], [v - u]]

  Categories: Arrays

Function: make_array (type, dim_1, …, dim_n)

  Creates and returns a Lisp array. type may be `any`, `flonum`, `fixnum`, `hashed` or `functional`. There are n indices, and the i'th index runs from 0 to dim_i - 1.

  The advantage of `make_array` over `array` is that the return value doesn't have a name, and once a pointer to it goes away, it will also go away. For example, if `y: make_array (...)` then `y` points to an object which takes up space, but after `y: false`, `y` no longer points to that object, so the object can be garbage collected.

  Examples:

      (%i1) A1 : make_array (fixnum, 10);
      (%o1)           {Array:  #(0 0 0 0 0 0 0 0 0 0)}
      (%i2) A1 [8] : 1729;
      (%o2)                         1729
      (%i3) A1;
      (%o3)          {Array:  #(0 0 0 0 0 0 0 0 1729 0)}
      (%i4) A2 : make_array (flonum, 10);
      (%o4) {Array:  #(0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0)}
      (%i5) A2 [2] : 2.718281828;
      (%o5)                      2.718281828
      (%i6) A2;
      (%o6)
           {Array:  #(0.0 0.0 2.718281828 0.0 0.0 0.0 0.0 0.0 0.0 0.0)}
      (%i7) A3 : make_array (any, 10);
      (%o7) {Array:  #(NIL NIL NIL NIL NIL NIL NIL NIL NIL NIL)}
      (%i8) A3 [4] : x - y - z;
      (%o8)                      - z - y + x
      (%i9) A3;
      (%o9) {Array:  #(NIL NIL NIL NIL ((MPLUS SIMP) $X ((MTIMES SIMP)\
       -1 $Y) ((MTIMES SIMP) -1 $Z))
        NIL NIL NIL NIL NIL)}
      (%i10) A4 : make_array (fixnum, 2, 3, 5);
      (%o10) {Array:  #3A(((0 0 0 0 0) (0 0 0 0 0) (0 0 0 0 0)) ((0 0 \
      0 0 0) (0 0 0 0 0) (0 0 0 0 0)))}
      (%i11) fillarray (A4, makelist (i, i, 1, 2*3*5));
      (%o11) {Array:  #3A(((1 2 3 4 5) (6 7 8 9 10) (11 12 13 14 15))
          ((16 17 18 19 20) (21 22 23 24 25) (26 27 28 29 30)))}
      (%i12) A4 [0, 2, 1];
      (%o12)                         12

  Categories: Arrays

Function: rearray (A, dim_1, …, dim_n)

  Changes the dimensions of an array. The new array will be filled with the elements of the old one in row-major order. If the old array was too small, the remaining elements are filled with `false`, `0.0` or `0`, depending on the type of the array. The type of the array cannot be changed.

  Categories: Arrays

Function: remarray
`remarray` (A_1, …, A_n)
`remarray` (all)

  Removes arrays and array associated functions and frees the storage occupied. The arguments may be declared arrays, undeclared (hashed) arrays, array functions, and subscripted functions.

  `remarray (all)` removes all items in the global list `arrays`.

  It may be necessary to use this function if it is desired to redefine the values in a hashed array.

  `remarray` returns the list of arrays removed.

  `remarray` quotes its arguments.

  Categories: Arrays

Function: subvar (x, i)

  Evaluates the subscripted expression `x[i]`.

  `subvar` evaluates its arguments.

  `arraymake (x, [i])` constructs the expression `x[i]`, but does not evaluate it.

  Examples:

      (%i1) x : foo $
      (%i2) i : 3 $
      (%i3) subvar (x, i);
      (%o3)                         foo
                                       3
      (%i4) foo : [aa, bb, cc, dd, ee]$
      (%i5) subvar (x, i);
      (%o5)                          cc
      (%i6) arraymake (x, [i]);
      (%o6)                         foo
                                       3
      (%i7) ''%;
      (%o7)                          cc

  Categories: Expressions · Arrays

Function: subvarp (expr)

  Returns `true` if expr is a subscripted variable, for example `a[i]`.

  Categories: Predicate functions

Option variable: use_fast_arrays

  Default value: `false`

  When `use_fast_arrays` is `true`, undeclared arrays and arrays declared by `array` are values instead of properties, and undeclared arrays are implemented as Lisp hash tables.

  When `use_fast_arrays` is `false`, undeclared arrays and arrays declared by `array` are properties, and undeclared arrays are implemented with Maxima's own hash table implementation.

  Arrays created by `make_array` are not affected by `use_fast_arrays`.

  See also `translate_fast_arrays`.

  Categories: Arrays · Global flags

Option variable: translate_fast_arrays

  Default value: `false`

  When `translate_fast_arrays` is `true`, the Maxima-to-Lisp translator generates code that assumes arrays are values instead of properties, as if `use_fast_arrays` were `true`.

  When `translate_fast_arrays` is `false`, the Maxima-to-Lisp translator generates code that assumes arrays are properties, as if `use_fast_arrays` were `false`.

  Categories: Arrays · Translation flags and variables

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## 5.6 Structures

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### 5.6.1 Introduction to Structures

Maxima provides a simple data aggregate called a structure. A structure is an expression in which arguments are identified by name (the field name) and the expression as a whole is identified by its operator (the structure name). A field value can be any expression.

A structure is defined by the `defstruct` function; the global variable `structures` is the list of user-defined structures. The function `new` creates instances of structures. The `@` operator refers to fields. `kill(S)` removes the structure definition S, and `kill(x@ a)` unbinds the field a of the structure instance x.

In the pretty-printing console display (with `display2d` equal to `true`), structure instances are displayed with the value of each field represented as an equation, with the field name on the left-hand side and the value on the right-hand side. (The equation is only a display construct; only the value is actually stored.) In 1-dimensional display (via `grind` or with `display2d` equal to `false`), structure instances are displayed without the field names.

There is no way to use a field name as a function name, although a field value can be a lambda expression. Nor can the values of fields be restricted to certain types; any field can be assigned any kind of expression. There is no way to make some fields accessible or inaccessible in different contexts; all fields are always visible.

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### 5.6.2 Functions and Variables for Structures

Global variable: structures

  `structures` is the list of user-defined structures defined by `defstruct`.

  Categories: Structures · Global variables

Function: defstruct
`defstruct` (S(a_1, …, a_n))
`defstruct` (S(a_1 = v_1, …, a_n = v_n))

  Define a structure, which is a list of named fields a_1, …, a_n associated with a symbol S. An instance of a structure is just an expression which has operator S and exactly `n` arguments. `new(S)` creates a new instance of structure S.

  An argument which is just a symbol a specifies the name of a field. An argument which is an equation `a = v` specifies the field name a and its default value v. The default value can be any expression.

  `defstruct` puts S on the list of user-defined structures, `structures`.

  `kill(S)` removes S from the list of user-defined structures, and removes the structure definition.

  Examples:

      (%i1) defstruct (foo (a, b, c));
      (%o1)                    [foo(a, b, c)]
      (%i2) structures;
      (%o2)                    [foo(a, b, c)]
      (%i3) new (foo);
      (%o3)                     foo(a, b, c)
      (%i4) defstruct (bar (v, w, x = 123, y = %pi));
      (%o4)             [bar(v, w, x = 123, y = %pi)]
      (%i5) structures;
      (%o5)      [foo(a, b, c), bar(v, w, x = 123, y = %pi)]
      (%i6) new (bar);
      (%o6)              bar(v, w, x = 123, y = %pi)
      (%i7) kill (foo);
      (%o7)                         done
      (%i8) structures;
      (%o8)             [bar(v, w, x = 123, y = %pi)]

  Categories: Structures

Function: new
`new` (S)
`new` (S (v_1, …, v_n))

  `new` creates new instances of structures.

  `new(S)` creates a new instance of structure S in which each field is assigned its default value, if any, or no value at all if no default was specified in the structure definition.

  `new(S(v_1, ..., v_n))` creates a new instance of S in which fields are assigned the values v_1, …, v_n.

  Examples:

      (%i1) defstruct (foo (w, x = %e, y = 42, z));
      (%o1)              [foo(w, x = %e, y = 42, z)]
      (%i2) new (foo);
      (%o2)               foo(w, x = %e, y = 42, z)
      (%i3) new (foo (1, 2, 4, 8));
      (%o3)            foo(w = 1, x = 2, y = 4, z = 8)

  Categories: Structures

Operator: @

  `@` is the structure field access operator. The expression `x@ a` refers to the value of field a of the structure instance x. The field name is not evaluated.

  If the field a in x has not been assigned a value, `x@ a` evaluates to itself.

  `kill(x@ a)` removes the value of field a in x.

  Examples:

      (%i1) defstruct (foo (x, y, z));
      (%o1)                    [foo(x, y, z)]
      (%i2) u : new (foo (123, a - b, %pi));
      (%o2)           foo(x = 123, y = a - b, z = %pi)
      (%i3) u@z;
      (%o3)                          %pi
      (%i4) u@z : %e;
      (%o4)                          %e
      (%i5) u;
      (%o5)            foo(x = 123, y = a - b, z = %e)
      (%i6) kill (u@z);
      (%o6)                         done
      (%i7) u;
      (%o7)              foo(x = 123, y = a - b, z)
      (%i8) u@z;
      (%o8)                          u@z

  The field name is not evaluated.

      (%i1) defstruct (bar (g, h));
      (%o1)                      [bar(g, h)]
      (%i2) x : new (bar);
      (%o2)                       bar(g, h)
      (%i3) x@h : 42;
      (%o3)                          42
      (%i4) h : 123;
      (%o4)                          123
      (%i5) x@h;
      (%o5)                          42
      (%i6) x@h : 19;
      (%o6)                          19
      (%i7) x;
      (%o7)                    bar(g, h = 19)
      (%i8) h;
      (%o8)                          123

  Categories: Structures · Operators

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# 6. Expressions

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## 6.1 Introduction to Expressions

There are a number of reserved words which should not be used as variable names. Their use would cause a possibly cryptic syntax error.

    integrate            next           from                 diff
    in                   at             limit                sum
    for                  and            elseif               then
    else                 do             or                   if
    unless               product        while                thru
    step

Most things in Maxima are expressions. A sequence of expressions can be made into an expression by separating them by commas and putting parentheses around them. This is similar to the C comma expression.

    (%i1) x: 3$
    (%i2) (x: x+1, x: x^2);
    (%o2)                          16
    (%i3) (if (x > 17) then 2 else 4);
    (%o3)                           4
    (%i4) (if (x > 17) then x: 2 else y: 4, y+x);
    (%o4)                          20

Even loops in Maxima are expressions, although the value they return is the not too useful `done`.

    (%i1) y: (x: 1, for i from 1 thru 10 do (x: x*i))$
    (%i2) y;
    (%o2)                         done

Whereas what you really want is probably to include a third term in the comma expression which actually gives back the value.

    (%i3) y: (x: 1, for i from 1 thru 10 do (x: x*i), x)$
    (%i4) y;
    (%o4)                        3628800

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## 6.2 Nouns and Verbs

Maxima distinguishes between operators which are "nouns" and operators which are "verbs". A verb is an operator which can be executed. A noun is an operator which appears as a symbol in an expression, without being executed. By default, function names are verbs. A verb can be changed into a noun by quoting the function name or applying the `nounify` function. A noun can be changed into a verb by applying the `verbify` function. The evaluation flag `nouns` causes `ev` to evaluate nouns in an expression.

The verb form is distinguished by a leading dollar sign `$` on the corresponding Lisp symbol. In contrast, the noun form is distinguished by a leading percent sign `%` on the corresponding Lisp symbol. Some nouns have special display properties, such as `'integrate` and `'derivative` (returned by `diff`), but most do not. By default, the noun and verb forms of a function are identical when displayed. The global flag `noundisp` causes Maxima to display nouns with a leading quote mark `'`.

See also `noun`, `nouns`, `nounify`, and `verbify`.

Examples:

    (%i1) foo (x) := x^2;
                                         2
    (%o1)                     foo(x) := x
    (%i2) foo (42);
    (%o2)                         1764
    (%i3) 'foo (42);
    (%o3)                        foo(42)
    (%i4) 'foo (42), nouns;
    (%o4)                         1764
    (%i5) declare (bar, noun);
    (%o5)                         done
    (%i6) bar (x) := x/17;
                                        x
    (%o6)                     bar(x) := --
                                        17
    (%i7) bar (52);
    (%o7)                        bar(52)
    (%i8) bar (52), nouns;
    (%o8)                        bar(52)
    (%i9) integrate (1/x, x, 1, 42);
    (%o9)                        log(42)
    (%i10) 'integrate (1/x, x, 1, 42);
                                 42
                                /
                                [   1
    (%o10)                      I   - dx
                                ]   x
                                /
                                 1
    (%i11) ev (%, nouns);
    (%o11)                       log(42)

Categories: Evaluation · Nouns and verbs

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## 6.3 Identifiers

Maxima identifiers may comprise alphabetic characters, plus the numerals 0 through 9, plus any special character preceded by the backslash `\` character.

A numeral may be the first character of an identifier if it is preceded by a backslash. Numerals which are the second or later characters need not be preceded by a backslash.

Characters may be declared alphabetic by the `declare` function. If so declared, they need not be preceded by a backslash in an identifier. The alphabetic characters are initially `A` through `Z`, `a` through `z`, `%`, and `_`.

Maxima is case-sensitive. The identifiers `foo`, `FOO`, and `Foo` are distinct. See Lisp and Maxima for more on this point.

A Maxima identifier is a Lisp symbol which begins with a dollar sign `$`. Any other Lisp symbol is preceded by a question mark `?` when it appears in Maxima. See Lisp and Maxima for more on this point.

Examples:

    (%i1) %an_ordinary_identifier42;
    (%o1)               %an_ordinary_identifier42
    (%i2) embedded\ spaces\ in\ an\ identifier;
    (%o2)           embedded spaces in an identifier
    (%i3) symbolp (%);
    (%o3)                         true
    (%i4) [foo+bar, foo\+bar];
    (%o4)                 [foo + bar, foo+bar]
    (%i5) [1729, \1729];
    (%o5)                     [1729, 1729]
    (%i6) [symbolp (foo\+bar), symbolp (\1729)];
    (%o6)                     [true, true]
    (%i7) [is (foo\+bar = foo+bar), is (\1729 = 1729)];
    (%o7)                    [false, false]
    (%i8) baz\~quux;
    (%o8)                       baz~quux
    (%i9) declare ("~", alphabetic);
    (%o9)                         done
    (%i10) baz~quux;
    (%o10)                      baz~quux
    (%i11) [is (foo = FOO), is (FOO = Foo), is (Foo = foo)];
    (%o11)                [false, false, false]
    (%i12) :lisp (defvar *my-lisp-variable* '$foo)
    *MY-LISP-VARIABLE*
    (%i12) ?\*my\-lisp\-variable\*;
    (%o12)                         foo

Categories: Syntax

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## 6.4 Inequality

Maxima has the inequality operators `<`, `<=`, `>=`, `>`, `#`, and `notequal`. See `if` for a description of conditional expressions.

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## 6.5 Functions and Variables for Expressions

Function: alias (new_name_1, old_name_1, …, new_name_n, old_name_n)

  provides an alternate name for a (user or system) function, variable, array, etc. Any even number of arguments may be used.

  Categories: Declarations and inferences

System variable: aliases

  Default value: `[]`

  `aliases` is the list of atoms which have a user defined alias (set up by the `alias`, `ordergreat`, `orderless` functions or by declaring the atom a `noun` with `declare`.)

  Categories: Declarations and inferences · Global variables

Keyword: allbut

  works with the `part` commands (i.e. `part`, `inpart`, `substpart`, `substinpart`, `dpart`, and `lpart`). For example,

      (%i1) expr : e + d + c + b + a;
      (%o1)                   e + d + c + b + a
      (%i2) part (expr, [2, 5]);
      (%o2)                         d + a

  while

      (%i1) expr : e + d + c + b + a;
      (%o1)                   e + d + c + b + a
      (%i2) part (expr, allbut (2, 5));
      (%o2)                       e + c + b

  `allbut` is also recognized by `kill`.

      (%i1) [aa : 11, bb : 22, cc : 33, dd : 44, ee : 55];
      (%o1)                 [11, 22, 33, 44, 55]
      (%i2) kill (allbut (cc, dd));
      (%o0)                         done
      (%i1) [aa, bb, cc, dd];
      (%o1)                   [aa, bb, 33, 44]

  `kill(allbut(a_1, a_2, ...))` has the effect of `kill(all)` except that it does not kill the symbols a_1, a_2, …

Function: args (expr)

  Returns the list of arguments of `expr`, which may be any kind of expression other than an atom. Only the arguments of the top-level operator are extracted; subexpressions of `expr` appear as elements or subexpressions of elements of the list of arguments.

  The order of the items in the list may depend on the global flag `inflag`.

  `args (expr)` is equivalent to `substpart ("[", expr, 0)`. See also `substpart`, `apply`, `funmake`, and `op`.

  How to convert a matrix to a nested list:

      (%i1) M:matrix([1,2],[3,4]);
                                  [ 1  2 ]
      (%o1)                       [      ]
                                  [ 3  4 ]
      (%i2) args(M);
      (%o2)                   [[1, 2], [3, 4]]

  Since maxima internally treats a sum of `n` terms as a summation command with `n` arguments args() can extract the list of terms in a sum:

      (%i1) a+b+c;
      (%o1)                       c + b + a
      (%i2) args(%);
      (%o2)                       [c, b, a]

  Categories: Expressions

Function: atom (expr)

  Returns `true` if expr is atomic (i.e. a number, name or string) else `false`. Thus `atom(5)` is `true` while `atom(a[1])` and `atom(sin(x))` are `false` (assuming `a[1]` and `x` are unbound).

  Categories: Expressions · Predicate functions

Function: box
`box` (expr)
`box` (expr, a)

  Returns expr enclosed in a box. The return value is an expression with `box` as the operator and expr as the argument. A box is drawn on the display when `display2d` is `true`.

  `box (expr, a)` encloses expr in a box labelled by the symbol a. The label is truncated if it is longer than the width of the box.

  `box` evaluates its argument. However, a boxed expression does not evaluate to its content, so boxed expressions are effectively excluded from computations. `rembox` removes the box again.

  `boxchar` is the character used to draw the box in `box` and in the `dpart` and `lpart` functions.

  See also `rembox`, `dpart` and `lpart`.

  Examples:

      (%i1) box (a^2 + b^2);
                                  """""""""
                                  " 2    2"
      (%o1)                       "b  + a "
                                  """""""""
      (%i2) a : 1234;
      (%o2)                         1234
      (%i3) b : c - d;
      (%o3)                         c - d
      (%i4) box (a^2 + b^2);
                            """"""""""""""""""""
                            "       2          "
      (%o4)                 "(c - d)  + 1522756"
                            """"""""""""""""""""
      (%i5) box (a^2 + b^2, term_1);
                            term_1""""""""""""""
                            "       2          "
      (%o5)                 "(c - d)  + 1522756"
                            """"""""""""""""""""
      (%i6) 1729 - box (1729);
                                       """"""
      (%o6)                     1729 - "1729"
                                       """"""
      (%i7) boxchar: "-";
      (%o7)                           -
      (%i8) box (sin(x) + cos(y));
                              -----------------
      (%o8)                   -cos(y) + sin(x)-
                              -----------------

  Categories: Expressions

Option variable: boxchar

  Default value: `"`

  `boxchar` is the character used to draw the box in the `box` and in the `dpart` and `lpart` functions.

  All boxes in an expression are drawn with the current value of `boxchar`; the drawing character is not stored with the box expression.

  Categories: Expressions

Function: collapse (expr)

  Collapses expr by causing all of its common (i.e., equal) subexpressions to share (i.e., use the same cells), thereby saving space. (`collapse` is a subroutine used by the `optimize` command.) Thus, calling `collapse` may be useful after loading in a `save` file. You can collapse several expressions together by using `collapse ([expr_1, ..., expr_n])`. Similarly, you can collapse the elements of the array `A` by doing `collapse (listarray ('A))`.

  Categories: Expressions

Function: copy (e)

  Return a copy of the Maxima expression e. Although e can be any Maxima expression, the copy function is the most useful when e is either a list or a matrix; consider:

      (%i1) m : [1,[2,3]]$
      (%i2) mm : m$
      (%i3) mm[2][1] : x$
      (%i4) m;
      (%o4)                      [1, [x, 3]]
      (%i5) mm;
      (%o5)                      [1, [x, 3]]

  Let's try the same experiment, but this time let mm be a copy of m

      (%i1) m : [1,[2,3]]$
      (%i2) mm : copy(m)$
      (%i3) mm[2][1] : x$
      (%i4) m;
      (%o4)                      [1, [2, 3]]
      (%i5) mm;
      (%o5)                      [1, [x, 3]]

  This time, the assignment to mm does not change the value of m.

  Categories: Expressions

Function: disolate (expr, x_1, …, x_n)

  is similar to `isolate``(expr, x)` except that it enables the user to isolate more than one variable simultaneously. This might be useful, for example, if one were attempting to change variables in a multiple integration, and that variable change involved two or more of the integration variables. This function is autoloaded from ``simplification/disol.mac'`. A demo is available by `demo("disol")$`.

  Categories: Expressions

Function: dispform
`dispform` (expr)
`dispform` (expr, all)

  Returns the external representation of expr.

  `dispform(expr)` returns the external representation with respect to the main (top-level) operator. `dispform(expr, all)` returns the external representation with respect to all operators in expr.

  See also `part`, `inpart`, and `inflag`.

  Examples:

  The internal representation of `- x` is "negative one times `x`" while the external representation is "minus `x`".

      (%i1) - x;
      (%o1)                          - x
      (%i2) ?format (true, "~S~%", %);
      ((MTIMES SIMP) -1 $X)
      (%o2)                         false
      (%i3) dispform (- x);
      (%o3)                          - x
      (%i4) ?format (true, "~S~%", %);
      ((MMINUS SIMP) $X)
      (%o4)                         false

  The internal representation of `sqrt(x)` is "`x` to the power 1/2" while the external representation is "square root of `x`".

      (%i1) sqrt (x);
      (%o1)                        sqrt(x)
      (%i2) ?format (true, "~S~%", %);
      ((MEXPT SIMP) $X ((RAT SIMP) 1 2))
      (%o2)                         false
      (%i3) dispform (sqrt (x));
      (%o3)                        sqrt(x)
      (%i4) ?format (true, "~S~%", %);
      ((%SQRT SIMP) $X)
      (%o4)                         false

  Use of the optional argument `all`.

      (%i1) expr : sin (sqrt (x));
      (%o1)                     sin(sqrt(x))
      (%i2) freeof (sqrt, expr);
      (%o2)                         true
      (%i3) freeof (sqrt, dispform (expr));
      (%o3)                         true
      (%i4) freeof (sqrt, dispform (expr, all));
      (%o4)                         false

  Categories: Expressions

Function: dpart (expr, n_1, …, n_k)

  Selects the same subexpression as `part`, but instead of just returning that subexpression as its value, it returns the whole expression with the selected subexpression displayed inside a box. The box is actually part of the expression.

      (%i1) dpart (x+y/z^2, 1, 2, 1);
                                   y
      (%o1)                       ---- + x
                                     2
                                  """
                                  "z"
                                  """

  Categories: Expressions

Option variable: exptisolate

  Default value: `false`

  `exptisolate`, when `true`, causes `isolate (expr, var)` to examine exponents of atoms (such as `%e`) which contain `var`.

  Categories: Expressions

Option variable: exptsubst

  Default value: `false`

  `exptsubst`, when `true`, permits substitutions such as `y` for `%e^x` in `%e^(a x)`.

      (%i1) %e^(a*x);
                                      a x
      (%o1)                         %e
      (%i2) exptsubst;
      (%o2)                         false
      (%i3) subst(y, %e^x, %e^(a*x));
                                      a x
      (%o3)                         %e
      (%i4) exptsubst: not exptsubst;
      (%o4)                         true
      (%i5) subst(y, %e^x, %e^(a*x));
                                      a
      (%o5)                          y

  Categories: Exponential and logarithm functions · Expressions

Function: freeof (x_1, …, x_n, expr)

  `freeof (x_1, expr)` returns `true` if no subexpression of expr is equal to x_1 or if x_1 occurs only as a dummy variable in expr, or if x_1 is neither the noun nor verb form of any operator in expr, and returns `false` otherwise.

  `freeof (x_1, ..., x_n, expr)` is equivalent to `freeof (x_1, expr) and ... and freeof (x_n, expr)`.

  The arguments x_1, …, x_n may be names of functions and variables, subscripted names, operators (enclosed in double quotes), or general expressions. `freeof` evaluates its arguments.

  `freeof` operates only on expr as it stands (after simplification and evaluation) and does not attempt to determine if some equivalent expression would give a different result. In particular, simplification may yield an equivalent but different expression which comprises some different elements than the original form of expr.

  A variable is a dummy variable in an expression if it has no binding outside of the expression. Dummy variables recognized by `freeof` are the index of a sum or product, the limit variable in `limit`, the integration variable in the definite integral form of `integrate`, the original variable in `laplace`, formal variables in `at` expressions, and arguments in `lambda` expressions.

  The indefinite form of `integrate` is not free of its variable of integration.

  Examples:

  Arguments are names of functions, variables, subscripted names, operators, and expressions. `freeof (a, b, expr)` is equivalent to `freeof (a, expr) and freeof (b, expr)`.

      (%i1) expr: z^3 * cos (a[1]) * b^(c+d);
                                       d + c  3
      (%o1)                   cos(a ) b      z
                                   1
      (%i2) freeof (z, expr);
      (%o2)                         false
      (%i3) freeof (cos, expr);
      (%o3)                         false
      (%i4) freeof (a[1], expr);
      (%o4)                         false
      (%i5) freeof (cos (a[1]), expr);
      (%o5)                         false
      (%i6) freeof (b^(c+d), expr);
      (%o6)                         false
      (%i7) freeof ("^", expr);
      (%o7)                         false
      (%i8) freeof (w, sin, a[2], sin (a[2]), b*(c+d), expr);
      (%o8)                         true

  `freeof` evaluates its arguments.

      (%i1) expr: (a+b)^5$
      (%i2) c: a$
      (%i3) freeof (c, expr);
      (%o3)                         false

  `freeof` does not consider equivalent expressions. Simplification may yield an equivalent but different expression.

      (%i1) expr: (a+b)^5$
      (%i2) expand (expr);
                5        4       2  3       3  2      4      5
      (%o2)    b  + 5 a b  + 10 a  b  + 10 a  b  + 5 a  b + a
      (%i3) freeof (a+b, %);
      (%o3)                         true
      (%i4) freeof (a+b, expr);
      (%o4)                         false
      (%i5) exp (x);
                                       x
      (%o5)                          %e
      (%i6) freeof (exp, exp (x));
      (%o6)                         true

  A summation or definite integral is free of its dummy variable. An indefinite integral is not free of its variable of integration.

      (%i1) freeof (i, 'sum (f(i), i, 0, n));
      (%o1)                         true
      (%i2) freeof (x, 'integrate (x^2, x, 0, 1));
      (%o2)                         true
      (%i3) freeof (x, 'integrate (x^2, x));
      (%o3)                         false

  Categories: Expressions

Option variable: inflag

  Default value: `false`

  When `inflag` is `true`, functions for part extraction inspect the internal form of `expr`.

  Note that the simplifier re-orders expressions. Thus `first (x + y)` returns `x` if `inflag` is `true` and `y` if `inflag` is `false`. (`first (y + x)` gives the same results.)

  Also, setting `inflag` to `true` and calling `part` or `substpart` is the same as calling `inpart` or `substinpart`.

  Functions affected by the setting of `inflag` are: `part`, `substpart`, `first`, `rest`, `last`, `length`, the `for` … `in` construct, `map`, `fullmap`, `maplist`, `reveal` and `pickapart`.

  Categories: Expressions

Function: inpart (expr, n_1, …, n_k)

  is similar to `part` but works on the internal representation of the expression rather than the displayed form and thus may be faster since no formatting is done. Care should be taken with respect to the order of subexpressions in sums and products (since the order of variables in the internal form is often different from that in the displayed form) and in dealing with unary minus, subtraction, and division (since these operators are removed from the expression). `part (x+y, 0)` or `inpart (x+y, 0)` yield `+`, though in order to refer to the operator it must be enclosed in "s. For example `... if inpart (%o9,0) = "+" then ...`.

  Examples:

      (%i1) x + y + w*z;
      (%o1)                      w z + y + x
      (%i2) inpart (%, 3, 2);
      (%o2)                           z
      (%i3) part (%th (2), 1, 2);
      (%o3)                           z
      (%i4) 'limit (f(x)^g(x+1), x, 0, minus);
                                        g(x + 1)
      (%o4)                 limit   f(x)
                            x -> 0-
      (%i5) inpart (%, 1, 2);
      (%o5)                       g(x + 1)

  Categories: Expressions

Function: isolate (expr, x)

  Returns expr with subexpressions which are sums and which do not contain var replaced by intermediate expression labels (these being atomic symbols like `%t1`, `%t2`, …). This is often useful to avoid unnecessary expansion of subexpressions which don't contain the variable of interest. Since the intermediate labels are bound to the subexpressions they can all be substituted back by evaluating the expression in which they occur.

  `exptisolate` (default value: `false`) if `true` will cause `isolate` to examine exponents of atoms (like `%e`) which contain var.

  `isolate_wrt_times` if `true`, then `isolate` will also isolate with respect to products. See `isolate_wrt_times`. See also `disolate`.

  Do `example (isolate)` for examples.

  Categories: Expressions

Option variable: isolate_wrt_times

  Default value: `false`

  When `isolate_wrt_times` is `true`, `isolate` will also isolate with respect to products. E.g. compare both settings of the switch on

      (%i1) isolate_wrt_times: true$
      (%i2) isolate (expand ((a+b+c)^2), c);

      (%t2)                          2 a

      (%t3)                          2 b

                                2            2
      (%t4)                    b  + 2 a b + a

                           2
      (%o4)               c  + %t3 c + %t2 c + %t4
      (%i4) isolate_wrt_times: false$
      (%i5) isolate (expand ((a+b+c)^2), c);
                           2
      (%o5)               c  + 2 b c + 2 a c + %t4

  Categories: Expressions

Option variable: listconstvars

  Default value: `false`

  When `listconstvars` is `true` the list returned by `listofvars` contains constant variables, such as `%e`, `%pi`, `%i` or any variables declared as constant that occur in expr. A variable is declared as `constant` type via `declare`, and `constantp` returns `true` for all variables declared as `constant`. The default is to omit constant variables from `listofvars` return value.

  Categories: Expressions

Option variable: listdummyvars

  Default value: `true`

  When `listdummyvars` is `false`, "dummy variables" in the expression will not be included in the list returned by `listofvars`. (The meaning of "dummy variables" is as given in `freeof`. "Dummy variables" are mathematical things like the index of a sum or product, the limit variable, and the definite integration variable.)

  Example:

      (%i1) listdummyvars: true$
      (%i2) listofvars ('sum(f(i), i, 0, n));
      (%o2)                        [i, n]
      (%i3) listdummyvars: false$
      (%i4) listofvars ('sum(f(i), i, 0, n));
      (%o4)                          [n]

  Categories: Expressions

Function: listofvars (expr)

  Returns a list of the variables in expr.

  `listconstvars` if `true` causes `listofvars` to include `%e`, `%pi`, `%i`, and any variables declared constant in the list it returns if they appear in expr. The default is to omit these.

  See also the option variable `listdummyvars` to exclude or include "dummy variables" in the list of variables.

      (%i1) listofvars (f (x[1]+y) / g^(2+a));
      (%o1)                     [g, a, x , y]
                                        1

  Categories: Expressions

Function: lfreeof (list, expr)

  For each member m of list, calls `freeof (m, expr)`. It returns `false` if any call to `freeof` does and `true` otherwise.

  Example:

      (%i1) lfreeof ([ a, x], x^2+b);
      (%o1)                         false
      (%i2) lfreeof ([ b, x], x^2+b);
      (%o2)                         false
      (%i3) lfreeof ([ a, y], x^2+b);
      (%o3)                         true

  Categories: Expressions

Function: lpart (label, expr, n_1, …, n_k)

  is similar to `dpart` but uses a labelled box. A labelled box is similar to the one produced by `dpart` but it has a name in the top line.

  Categories: Expressions

Property: mainvar

  You may declare variables to be `mainvar`. The ordering scale for atoms is essentially: numbers < constants (e.g., `%e`, `%pi`) < scalars < other variables < mainvars. E.g., compare `expand ((X+Y)^4)` with `(declare (x, mainvar), expand ((x+y)^4))`. (Note: Care should be taken if you elect to use the above feature. E.g., if you subtract an expression in which `x` is a `mainvar` from one in which `x` isn't a `mainvar`, resimplification e.g. with `ev (expr, simp)` may be necessary if cancellation is to occur. Also, if you save an expression in which `x` is a `mainvar`, you probably should also save `x`.)

  Categories: Declarations and inferences · Expressions

Property: noun

  `noun` is one of the options of the `declare` command. It makes a function so declared a "noun", meaning that it won't be evaluated automatically.

  Example:

      (%i1) factor (12345678);
                                   2
      (%o1)                     2 3  47 14593
      (%i2) declare (factor, noun);
      (%o2)                         done
      (%i3) factor (12345678);
      (%o3)                   factor(12345678)
      (%i4) ''%, nouns;
                                   2
      (%o4)                     2 3  47 14593

  Categories: Nouns and verbs

Option variable: noundisp

  Default value: `false`

  When `noundisp` is `true`, nouns display with a single quote. This switch is always `true` when displaying function definitions.

  Categories: Display flags and variables · Nouns and verbs

Function: nounify (f)

  Returns the noun form of the function name f. This is needed if one wishes to refer to the name of a verb function as if it were a noun. Note that some verb functions will return their noun forms if they can't be evaluated for certain arguments. This is also the form returned if a function call is preceded by a quote.

  See also `verbify`.

  Categories: Nouns and verbs

Function: nterms (expr)

  Returns the number of terms that expr would have if it were fully expanded out and no cancellations or combination of terms occurred. Note that expressions like `sin (expr)`, `sqrt (expr)`, `exp (expr)`, etc. count as just one term regardless of how many terms expr has (if it is a sum).

  Categories: Expressions

Function: op (expr)

  Returns the main operator of the expression expr. `op (expr)` is equivalent to `part (expr, 0)`.

  `op` returns a string if the main operator is a built-in or user-defined prefix, binary or n-ary infix, postfix, matchfix, or nofix operator. Otherwise, if expr is a subscripted function expression, `op` returns the subscripted function; in this case the return value is not an atom. Otherwise, expr is an array function or ordinary function expression, and `op` returns a symbol.

  `op` observes the value of the global flag `inflag`.

  `op` evaluates it argument.

  See also `args`.

  Examples:

      (%i1) stringdisp: true$
      (%i2) op (a * b * c);
      (%o2)                          "*"
      (%i3) op (a * b + c);
      (%o3)                          "+"
      (%i4) op ('sin (a + b));
      (%o4)                          sin
      (%i5) op (a!);
      (%o5)                          "!"
      (%i6) op (-a);
      (%o6)                          "-"
      (%i7) op ([a, b, c]);
      (%o7)                          "["
      (%i8) op ('(if a > b then c else d));
      (%o8)                         "if"
      (%i9) op ('foo (a));
      (%o9)                          foo
      (%i10) prefix (foo);
      (%o10)                        "foo"
      (%i11) op (foo a);
      (%o11)                        "foo"
      (%i12) op (F [x, y] (a, b, c));
      (%o12)                        F
                                     x, y
      (%i13) op (G [u, v, w]);
      (%o13)                          G

  Categories: Expressions · Operators

Function: operatorp
`operatorp` (expr, op)
`operatorp` (expr, [op_1, …, op_n])

  `operatorp (expr, op)` returns `true` if op is equal to the operator of expr.

  `operatorp (expr, [op_1, ..., op_n])` returns `true` if some element op_1, …, op_n is equal to the operator of expr.

  Categories: Operators · Predicate functions

Option variable: opsubst

  Default value: `true`

  When `opsubst` is `false`, `subst` does not attempt to substitute into the operator of an expression. E.g., `(opsubst: false, subst (x^2, r, r+r[0]))` will work.

      (%i1) r+r[0];
      (%o1)                        r + r
                                        0
      (%i2) opsubst;
      (%o2)                         true
      (%i3) subst (x^2, r, r+r[0]);
                                  2     2
      (%o3)                      x  + (x )
                                          0
      (%i4) opsubst: not opsubst;
      (%o4)                         false
      (%i5) subst (x^2, r, r+r[0]);
                                    2
      (%o5)                        x  + r
                                         0

  Categories: Expressions

Function: optimize (expr)

  Returns an expression that produces the same value and side effects as expr but does so more efficiently by avoiding the recomputation of common subexpressions. `optimize` also has the side effect of "collapsing" its argument so that all common subexpressions are shared. Do `example (optimize)` for examples.

  Categories: Expressions

Option variable: optimprefix

  Default value: `%`

  `optimprefix` is the prefix used for generated symbols by the `optimize` command.

  Categories: Expressions

Function: ordergreat (v_1, …, v_n)
Function: orderless (v_1, …, v_n)

  `ordergreat` changes the canonical ordering of Maxima expressions such that v_1 succeeds v_2 succeeds … succeeds v_n, and v_n succeeds any other symbol not mentioned as an argument.

  `orderless` changes the canonical ordering of Maxima expressions such that v_1 precedes v_2 precedes … precedes v_n, and v_n precedes any other variable not mentioned as an argument.

  The order established by `ordergreat` and `orderless` is dissolved by `unorder`. `ordergreat` and `orderless` can be called only once each, unless `unorder` is called; only the last call to `ordergreat` and `orderless` has any effect.

  See also `ordergreatp`.

  Categories: Expressions

Function: ordergreatp (expr_1, expr_2)
Function: orderlessp (expr_1, expr_2)

  `ordergreatp` returns `true` if expr_1 succeeds expr_2 in the canonical ordering of Maxima expressions, and `false` otherwise.

  `orderlessp` returns `true` if expr_1 precedes expr_2 in the canonical ordering of Maxima expressions, and `false` otherwise.

  All Maxima atoms and expressions are comparable under `ordergreatp` and `orderlessp`, although there are isolated examples of expressions for which these predicates are not transitive; that is a bug.

  The canonical ordering of atoms (symbols, literal numbers, and strings) is the following.

  (integers and floats) precede (bigfloats) precede (declared constants) precede (strings) precede (declared scalars) precede (first argument to `orderless`) precedes … precedes (last argument to `orderless`) precedes (other symbols) precede (last argument to `ordergreat`) precedes … precedes (first argument to `ordergreat`) precedes (declared main variables)

  For non-atomic expressions, the canonical ordering is derived from the ordering for atoms. For the built-in `+` `*` and `^` operators, the ordering is not easily summarized. For other built-in operators and all other functions and operators, expressions are ordered by their arguments (beginning with the first argument), then by the name of the operator or function. In the case of subscripted expressions, the subscripted symbol is considered the operator and the subscript is considered an argument.

  The canonical ordering of expressions is modified by the functions `ordergreat` and `orderless`, and the `mainvar`, `constant`, and `scalar` declarations.

  See also `sort`.

  Examples:

  Ordering ordinary symbols and constants. Note that `%pi` is not ordered according to its numerical value.

      (%i1) stringdisp : true;
      (%o1)                         true
      (%i2) sort ([%pi, 3b0, 3.0, x, X, "foo", 3, a, 4, "bar", 4.0, 4b0]);
      (%o2) [3, 3.0, 4, 4.0, 3.0b0, 4.0b0, %pi, "bar", "foo", X, a, x]

  Effect of `ordergreat` and `orderless` functions.

      (%i1) sort ([M, H, K, T, E, W, G, A, P, J, S]);
      (%o1)           [A, E, G, H, J, K, M, P, S, T, W]
      (%i2) ordergreat (S, J);
      (%o2)                         done
      (%i3) orderless (M, H);
      (%o3)                         done
      (%i4) sort ([M, H, K, T, E, W, G, A, P, J, S]);
      (%o4)           [M, H, A, E, G, K, P, T, W, J, S]

  Effect of `mainvar`, `constant`, and `scalar` declarations.

      (%i1) sort ([aa, foo, bar, bb, baz, quux, cc, dd, A1, B1, C1]);
      (%o1)   [A1, B1, C1, aa, bar, baz, bb, cc, dd, foo, quux]
      (%i2) declare (aa, mainvar);
      (%o2)                         done
      (%i3) declare ([baz, quux], constant);
      (%o3)                         done
      (%i4) declare ([A1, B1], scalar);
      (%o4)                         done
      (%i5) sort ([aa, foo, bar, bb, baz, quux, cc, dd, A1, B1, C1]);
      (%o5)   [baz, quux, A1, B1, C1, bar, bb, cc, dd, foo, aa]

  Ordering non-atomic expressions.

      (%i1) sort ([1, 2, n, f(1), f(2), f(2, 1), g(1), g(1, 2), g(n),
             f(n, 1)]);
      (%o1) [1, 2, f(1), g(1), g(1, 2), f(2), f(2, 1), n, g(n),
                                                               f(n, 1)]
      (%i2) sort ([foo(1), X[1], X[k], foo(k), 1, k]);
      (%o2)            [1, X , foo(1), k, X , foo(k)]
                            1              k

  Categories: Expressions · Predicate functions

Function: part (expr, n_1, …, n_k)

  Returns parts of the displayed form of `expr`. It obtains the part of `expr` as specified by the indices n_1, …, n_k. First part n_1 of `expr` is obtained, then part n_2 of that, etc. The result is part n_k of … part n_2 of part n_1 of `expr`. If no indices are specified `expr` is returned.

  `part` can be used to obtain an element of a list, a row of a matrix, etc.

  If the last argument to a `part` function is a list of indices then several subexpressions are picked out, each one corresponding to an index of the list. Thus `part (x + y + z, [1, 3])` is `z+x`.

  `piece` holds the last expression selected when using the `part` functions. It is set during the execution of the function and thus may be referred to in the function itself as shown below.

  If `partswitch` is set to `true` then `end` is returned when a selected part of an expression doesn't exist, otherwise an error message is given.

  See also `inpart`, `substpart`, `substinpart`, `dpart`, and `lpart`.

  Examples:

      (%i1) part(z+2*y+a,2);
      (%o1)                          2 y
      (%i2) part(z+2*y+a,[1,3]);
      (%o2)                         z + a
      (%i3) part(z+2*y+a,2,1);
      (%o3)                           2

  `example (part)` displays additional examples.

  Categories: Expressions

Function: partition (expr, x)

  Returns a list of two expressions. They are (1) the factors of expr (if it is a product), the terms of expr (if it is a sum), or the list (if it is a list) which don't contain x and, (2) the factors, terms, or list which do.

  Examples:

      (%i1) partition (2*a*x*f(x), x);
      (%o1)                     [2 a, x f(x)]
      (%i2) partition (a+b, x);
      (%o2)                      [b + a, 0]
      (%i3) partition ([a, b, f(a), c], a);
      (%o3)                  [[b, c], [a, f(a)]]

  Categories: Expressions

Option variable: partswitch

  Default value: `false`

  When `partswitch` is `true`, `end` is returned when a selected part of an expression doesn't exist, otherwise an error message is given.

  Categories: Expressions

Function: pickapart (expr, n)

  Assigns intermediate expression labels to subexpressions of expr at depth n, an integer. Subexpressions at greater or lesser depths are not assigned labels. `pickapart` returns an expression in terms of intermediate expressions equivalent to the original expression expr.

  See also `part`, `dpart`, `lpart`, `inpart`, and `reveal`.

  Examples:

      (%i1) expr: (a+b)/2 + sin (x^2)/3 - log (1 + sqrt(x+1));
                                                2
                                           sin(x )   b + a
      (%o1)       - log(sqrt(x + 1) + 1) + ------- + -----
                                              3        2
      (%i2) pickapart (expr, 0);
                                                2
                                           sin(x )   b + a
      (%t2)       - log(sqrt(x + 1) + 1) + ------- + -----
                                              3        2
      (%o2)                          %t2
      (%i3) pickapart (expr, 1);

      (%t3)                - log(sqrt(x + 1) + 1)

                                        2
                                   sin(x )
      (%t4)                        -------
                                      3

                                    b + a
      (%t5)                         -----
                                      2

      (%o5)                    %t5 + %t4 + %t3
      (%i5) pickapart (expr, 2);

      (%t6)                 log(sqrt(x + 1) + 1)

                                        2
      (%t7)                        sin(x )

      (%t8)                         b + a

                               %t8   %t7
      (%o8)                    --- + --- - %t6
                                2     3
      (%i8) pickapart (expr, 3);

      (%t9)                    sqrt(x + 1) + 1

                                      2
      (%t10)                         x

                        b + a              sin(%t10)
      (%o10)            ----- - log(%t9) + ---------
                          2                    3
      (%i10) pickapart (expr, 4);

      (%t11)                     sqrt(x + 1)
                            2
                       sin(x )   b + a
      (%o11)           ------- + ----- - log(%t11 + 1)
                          3        2
      (%i11) pickapart (expr, 5);

      (%t12)                        x + 1

                         2
                    sin(x )   b + a
      (%o12)        ------- + ----- - log(sqrt(%t12) + 1)
                       3        2
      (%i12) pickapart (expr, 6);
                        2
                   sin(x )   b + a
      (%o12)       ------- + ----- - log(sqrt(x + 1) + 1)
                      3        2

  Categories: Expressions

System variable: piece

  Holds the last expression selected when using the `part` functions. It is set during the execution of the function and thus may be referred to in the function itself.

  Categories: Expressions

Function: psubst
`psubst` (list, expr)
`psubst` (a, b, expr)

  `psubst(a, b, expr)` is simliar to `subst`. See `subst`.

  In distinction from `subst` the function `psubst` makes parallel substitutions, if the first argument list is a list of equations.

  See also `sublis` for making parallel substitutions.

  Example:

  The first example shows parallel substitution with `psubst`. The second example shows the result for the function `subst`, which does a serial substitution.

      (%i1) psubst ([a^2=b, b=a], sin(a^2) + sin(b));
      (%o1)                    sin(b) + sin(a)
      (%i2) subst ([a^2=b, b=a], sin(a^2) + sin(b));
      (%o2)                       2 sin(a)

  Categories: Expressions

Function: rembox
`rembox` (expr, unlabelled)
`rembox` (expr, label)
`rembox` (expr)

  Removes boxes from expr.

  `rembox (expr, unlabelled)` removes all unlabelled boxes from expr.

  `rembox (expr, label)` removes only boxes bearing label.

  `rembox (expr)` removes all boxes, labelled and unlabelled.

  Boxes are drawn by the `box`, `dpart`, and `lpart` functions.

  Examples:

      (%i1) expr: (a*d - b*c)/h^2 + sin(%pi*x);
                                        a d - b c
      (%o1)                sin(%pi x) + ---------
                                            2
                                           h
      (%i2) dpart (dpart (expr, 1, 1), 2, 2);
      dpart: fell off the end.
       -- an error. To debug this try: debugmode(true);
      (%i3) expr2: lpart (BAR, lpart (FOO, %, 1), 2);
                                        BAR""""""""
                         FOO"""""""""   "a d - b c"
      (%o3)              "sin(%pi x)" + "---------"
                         """"""""""""   "    2    "
                                        "   h     "
                                        """""""""""
      (%i4) rembox (expr2, unlabelled);
                                        BAR""""""""
                         FOO"""""""""   "a d - b c"
      (%o4)              "sin(%pi x)" + "---------"
                         """"""""""""   "    2    "
                                        "   h     "
                                        """""""""""
      (%i5) rembox (expr2, FOO);
                                       BAR""""""""
                                       "a d - b c"
      (%o5)               sin(%pi x) + "---------"
                                       "    2    "
                                       "   h     "
                                       """""""""""
      (%i6) rembox (expr2, BAR);
                          FOO"""""""""   a d - b c
      (%o6)               "sin(%pi x)" + ---------
                          """"""""""""       2
                                            h
      (%i7) rembox (expr2);
                                        a d - b c
      (%o7)                sin(%pi x) + ---------
                                            2
                                           h

  Categories: Expressions

Function: reveal (expr, depth)

  Replaces parts of expr at the specified integer depth with descriptive summaries.

  - Sums and differences are replaced by `Sum(n)` where n is the number of operands of the sum.
  - Products are replaced by `Product(n)` where n is the number of operands of the product.
  - Exponentials are replaced by `Expt`.
  - Quotients are replaced by `Quotient`.
  - Unary negation is replaced by `Negterm`.
  - Lists are replaced by `List(n)` where n ist the number of elements of the list.

  When depth is greater than or equal to the maximum depth of expr, `reveal (expr, depth)` returns expr unmodified.

  `reveal` evaluates its arguments. `reveal` returns the summarized expression.

  Example:

      (%i1) e: expand ((a - b)^2)/expand ((exp(a) + exp(b))^2);
                                2            2
                               b  - 2 a b + a
      (%o1)               -------------------------
                              b + a     2 b     2 a
                          2 %e      + %e    + %e
      (%i2) reveal (e, 1);
      (%o2)                       Quotient
      (%i3) reveal (e, 2);
                                   Sum(3)
      (%o3)                        ------
                                   Sum(3)
      (%i4) reveal (e, 3);
                           Expt + Negterm + Expt
      (%o4)               ------------------------
                          Product(2) + Expt + Expt
      (%i5) reveal (e, 4);
                             2                 2
                            b  - Product(3) + a
      (%o5)         ------------------------------------
                               Product(2)     Product(2)
                    2 Expt + %e           + %e
      (%i6) reveal (e, 5);
                               2            2
                              b  - 2 a b + a
      (%o6)              --------------------------
                             Sum(2)     2 b     2 a
                         2 %e       + %e    + %e
      (%i7) reveal (e, 6);
                                2            2
                               b  - 2 a b + a
      (%o7)               -------------------------
                              b + a     2 b     2 a
                          2 %e      + %e    + %e

  Categories: Expressions · Display functions

Function: sublis (list, expr)

  Makes multiple parallel substitutions into an expression. list is a list of equations. The left hand side of the equations must be an atom.

  The variable `sublis_apply_lambda` controls simplification after `sublis`.

  See also `psubst` for making parallel substitutions.

  Example:

      (%i1) sublis ([a=b, b=a], sin(a) + cos(b));
      (%o1)                    sin(b) + cos(a)

  Categories: Expressions

Option variable: sublis_apply_lambda

  Default value: `true`

  Controls whether `lambda`'s substituted are applied in simplification after `sublis` is used or whether you have to do an `ev` to get things to apply. `true` means do the application.

  Categories: Expressions

Option variable: subnumsimp

  Default value: `false`

  If `true` then the functions `subst` and `psubst` can substitute a subscripted variable `f[x]` with a number, when only the symbol `f` is given.

  See also `subst`.

      (%i1) subst(100,g,g[x]+2);

      subst: cannot substitute 100 for operator g in expression g
                                                                 x
       -- an error. To debug this try: debugmode(true);

      (%i2) subst(100,g,g[x]+2),subnumsimp:true;
      (%o2)                          102

  Categories: Expressions

Function: subst (a, b, c)

  Substitutes a for b in c. b must be an atom or a complete subexpression of c. For example, `x+y+z` is a complete subexpression of `2*(x+y+z)/w` while `x+y` is not. When b does not have these characteristics, one may sometimes use `substpart` or `ratsubst` (see below). Alternatively, if b is of the form `e/f` then one could use `subst (a*f, e, c)` while if b is of the form `e^(1/f)` then one could use `subst (a^f, e, c)`. The `subst` command also discerns the `x^y` in `x^-y` so that `subst (a, sqrt(x), 1/sqrt(x))` yields `1/a`. a and b may also be operators of an expression enclosed in double-quotes `"` or they may be function names. If one wishes to substitute for the independent variable in derivative forms then the `at` function (see below) should be used.

  `subst` is an alias for `substitute`.

  The commands `subst (eq_1, expr)` or `subst ([eq_1, ..., eq_k], expr)` are other permissible forms. The eq_i are equations indicating substitutions to be made. For each equation, the right side will be substituted for the left in the expression expr. The equations are substituted in serial from left to right in expr. See the functions `sublis` and `psubst` for making parallel substitutions.

  `exptsubst` if `true` permits substitutions like `y` for `%e^x` in `%e^(a*x)` to take place.

  When `opsubst` is `false`, `subst` will not attempt to substitute into the operator of an expression. E.g. `(opsubst: false, subst (x^2, r, r+r[0]))` will work.

  See also `at`, `ev` and `psubst`.

  Examples:

      (%i1) subst (a, x+y, x + (x+y)^2 + y);
                                          2
      (%o1)                      y + x + a
      (%i2) subst (-%i, %i, a + b*%i);
      (%o2)                       a - %i b

  The substitution is done in serial for a list of equations. Compare this with a parallel substitution:

      (%i1) subst([a=b, b=c], a+b);
      (%o1)                          2 c
      (%i2) sublis([a=b, b=c], a+b);
      (%o2)                         c + b

  Single-character Operators like `+` and `-` have to be quoted in order to be replaced by subst. It is to note, though, that `a+b-c` might be expressed as `a+b+(-1*c)` internally.

      (%i3) subst(["+"="-"],a+b-c);
      (%o3)                                 c-b+a

  The difference between `subst` and `at` can be seen in the following example:

      (%i1) g1:y(t)=a*x(t)+b*diff(x(t),t);
                                  d
      (%o1)             y(t) = b (-- (x(t))) + a x(t)
                                  dt
      (%i2) subst('diff(x(t),t)=1,g1);
      (%o2)                   y(t) = a x(t) + b
      (%i3) at(g1,'diff(x(t),t)=1);
                                    !
                           d        !
      (%o3)      y(t) = b (-- (x(t))!             ) + a x(t)
                           dt       !d
                                    !-- (x(t)) = 1
                                     dt

  For further examples, do `example (subst)`.

  Categories: Expressions

Function: substinpart (x, expr, n_1, …, n_k)

  Similar to `substpart`, but `substinpart` works on the internal representation of expr.

  Examples:

      (%i1) x . 'diff (f(x), x, 2);
                                    2
                                   d
      (%o1)                   x . (--- (f(x)))
                                     2
                                   dx
      (%i2) substinpart (d^2, %, 2);
                                        2
      (%o2)                        x . d
      (%i3) substinpart (f1, f[1](x + 1), 0);
      (%o3)                       f1(x + 1)

  If the last argument to a `part` function is a list of indices then several subexpressions are picked out, each one corresponding to an index of the list. Thus

      (%i1) part (x + y + z, [1, 3]);
      (%o1)                         z + x

  `piece` holds the value of the last expression selected when using the `part` functions. It is set during the execution of the function and thus may be referred to in the function itself as shown below. If `partswitch` is set to `true` then `end` is returned when a selected part of an expression doesn't exist, otherwise an error message is given.

      (%i1) expr: 27*y^3 + 54*x*y^2 + 36*x^2*y + y + 8*x^3 + x + 1;
                    3         2       2            3
      (%o1)     27 y  + 54 x y  + 36 x  y + y + 8 x  + x + 1
      (%i2) part (expr, 2, [1, 3]);
                                        2
      (%o2)                         54 y
      (%i3) sqrt (piece/54);
      (%o3)                        abs(y)
      (%i4) substpart (factor (piece), expr, [1, 2, 3, 5]);
                                     3
      (%o4)               (3 y + 2 x)  + y + x + 1
      (%i5) expr: 1/x + y/x - 1/z;
                                   1    y   1
      (%o5)                     (- -) + - + -
                                   z    x   x
      (%i6) substpart (xthru (piece), expr, [2, 3]);
                                  y + 1   1
      (%o6)                       ----- - -
                                    x     z

  Also, setting the option `inflag` to `true` and calling `part` or `substpart` is the same as calling `inpart` or `substinpart`.

  Categories: Expressions

Function: substpart (x, expr, n_1, …, n_k)

  Substitutes x for the subexpression picked out by the rest of the arguments as in `part`. It returns the new value of expr. x may be some operator to be substituted for an operator of expr. In some cases x needs to be enclosed in double-quotes `"` (e.g. `substpart ("+", a*b, 0)` yields `b + a`).

  Example:

      (%i1) 1/(x^2 + 2);
                                     1
      (%o1)                        ------
                                    2
                                   x  + 2
      (%i2) substpart (3/2, %, 2, 1, 2);
                                     1
      (%o2)                       --------
                                   3/2
                                  x    + 2
      (%i3) a*x + f(b, y);
      (%o3)                     a x + f(b, y)
      (%i4) substpart ("+", %, 1, 0);
      (%o4)                    x + f(b, y) + a

  Also, setting the option `inflag` to `true` and calling `part` or `substpart` is the same as calling `inpart` or `substinpart`.

  Categories: Expressions

Function: symbolp (expr)

  Returns `true` if expr is a symbol, else `false`.

  See also Identifiers.

  Categories: Predicate functions

Function: unorder ()

  Disables the aliasing created by the last use of the ordering commands `ordergreat` and `orderless`. `ordergreat` and `orderless` may not be used more than one time each without calling `unorder`. `unorder` does not substitute back in expressions the original symbols for the aliases introduced by `ordergreat` and `orderless`. Therefore, after execution of `unorder` the aliases appear in previous expressions.

  See also `ordergreat` and `orderless`.

  Examples:

  `ordergreat(a)` introduces an alias for the symbol `a`. Therefore, the difference of `%o2` and `%o4` does not vanish. `unorder` does not substitute back the symbol `a` and the alias appears in the output `%o7`.

      (%i1) unorder();
      (%o1)                          []
      (%i2) b*x + a^2;
                                         2
      (%o2)                       b x + a
      (%i3) ordergreat (a);
      (%o3)                         done
      (%i4) b*x + a^2;
       %th(1) - %th(3);
                                   2
      (%o4)                       a  + b x
      (%i5) unorder();
                                    2    2
      (%o5)                        a  - a
      (%i6) %th(2);
      (%o6)                          [a]

  Categories: Expressions

Function: verbify (f)

  Returns the verb form of the function name f. See also `verb`, `noun`, and `nounify`.

  Examples:

      (%i1) verbify ('foo);
      (%o1)                          foo
      (%i2) :lisp $%
      $FOO
      (%i2) nounify (foo);
      (%o2)                          foo
      (%i3) :lisp $%
      %FOO

  Categories: Nouns and verbs

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# 7. Operators

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## 7.1 Introduction to operators

It is possible to define new operators with specified precedence, to undefine existing operators, or to redefine the precedence of existing operators. An operator may be unary prefix or unary postfix, binary infix, n-ary infix, matchfix, or nofix. "Matchfix" means a pair of symbols which enclose their argument or arguments, and "nofix" means an operator which takes no arguments. As examples of the different types of operators, there are the following.

unary prefix

  negation `- a`

unary postfix

  factorial `a!`

binary infix

  exponentiation `a^b`

n-ary infix

  addition `a + b`

matchfix

  list construction `[a, b]`

(There are no built-in nofix operators; for an example of such an operator, see `nofix`.)

The mechanism to define a new operator is straightforward. It is only necessary to declare a function as an operator; the operator function might or might not be defined.

An example of user-defined operators is the following. Note that the explicit function call `"dd" (a)` is equivalent to `dd a`, likewise `"<-" (a, b)` is equivalent to `a <- b`. Note also that the functions `"dd"` and `"<-"` are undefined in this example.

    (%i1) prefix ("dd");
    (%o1)                          dd
    (%i2) dd a;
    (%o2)                         dd a
    (%i3) "dd" (a);
    (%o3)                         dd a
    (%i4) infix ("<-");
    (%o4)                          <-
    (%i5) a <- dd b;
    (%o5)                      a <- dd b
    (%i6) "<-" (a, "dd" (b));
    (%o6)                      a <- dd b

The Maxima functions which define new operators are summarized in this table, stating the default left and right binding powers (lbp and rbp, respectively). (Binding power determines operator precedence. However, since left and right binding powers can differ, binding power is somewhat more complicated than precedence.) Some of the operation definition functions take additional arguments; see the function descriptions for details.

`prefix`

  rbp=180

`postfix`

  lbp=180

`infix`

  lbp=180, rbp=180

`nary`

  lbp=180, rbp=180

`matchfix`

  (binding power not applicable)

`nofix`

  (binding power not applicable)

For comparison, here are some built-in operators and their left and right binding powers.

    Operator   lbp     rbp

      :        180     20
      ::       180     20
      :=       180     20
      ::=      180     20
      !        160
      !!       160
      ^        140     139
      .        130     129
      *        120
      /        120     120
      +        100     100
      -        100     134
      =        80      80
      #        80      80
      >        80      80
      >=       80      80
      <        80      80
      <=       80      80
      not              70
      and      65
      or       60
      ,        10
      $        -1
      ;        -1

`remove` and `kill` remove operator properties from an atom. `remove ("a", op)` removes only the operator properties of a. `kill ("a")` removes all properties of a, including the operator properties. Note that the name of the operator must be enclosed in quotation marks.

    (%i1) infix ("##");
    (%o1)                          ##
    (%i2) "##" (a, b) := a^b;
                                         b
    (%o2)                     a ## b := a
    (%i3) 5 ## 3;
    (%o3)                          125
    (%i4) remove ("##", op);
    (%o4)                         done
    (%i5) 5 ## 3;
    Incorrect syntax: # is not a prefix operator
    5 ##
      ^
    (%i5) "##" (5, 3);
    (%o5)                          125
    (%i6) infix ("##");
    (%o6)                          ##
    (%i7) 5 ## 3;
    (%o7)                          125
    (%i8) kill ("##");
    (%o8)                         done
    (%i9) 5 ## 3;
    Incorrect syntax: # is not a prefix operator
    5 ##
      ^
    (%i9) "##" (5, 3);
    (%o9)                       ##(5, 3)

Categories: Operators · Syntax

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## 7.2 Arithmetic operators

Operator: +
Operator: -
Operator: *
Operator: /
Operator: ^

  The symbols `+` `*` `/` and `^` represent addition, multiplication, division, and exponentiation, respectively. The names of these operators are `"+"` `"*"` `"/"` and `"^"`, which may appear where the name of a function or operator is required.

  The symbols `+` and `-` represent unary addition and negation, respectively, and the names of these operators are `"+"` and `"-"`, respectively.

  Subtraction `a - b` is represented within Maxima as addition, `a + (- b)`. Expressions such as `a + (- b)` are displayed as subtraction. Maxima recognizes `"-"` only as the name of the unary negation operator, and not as the name of the binary subtraction operator.

  Division `a / b` is represented within Maxima as multiplication, `a * b^(- 1)`. Expressions such as `a * b^(- 1)` are displayed as division. Maxima recognizes `"/"` as the name of the division operator.

  Addition and multiplication are n-ary, commutative operators. Division and exponentiation are binary, noncommutative operators.

  Maxima sorts the operands of commutative operators to construct a canonical representation. For internal storage, the ordering is determined by `orderlessp`. For display, the ordering for addition is determined by `ordergreatp`, and for multiplication, it is the same as the internal ordering.

  Arithmetic computations are carried out on literal numbers (integers, rationals, ordinary floats, and bigfloats). Except for exponentiation, all arithmetic operations on numbers are simplified to numbers. Exponentiation is simplified to a number if either operand is an ordinary float or bigfloat or if the result is an exact integer or rational; otherwise an exponentiation may be simplified to `sqrt` or another exponentiation or left unchanged.

  Floating-point contagion applies to arithmetic computations: if any operand is a bigfloat, the result is a bigfloat; otherwise, if any operand is an ordinary float, the result is an ordinary float; otherwise, the operands are rationals or integers and the result is a rational or integer.

  Arithmetic computations are a simplification, not an evaluation. Thus arithmetic is carried out in quoted (but simplified) expressions.

  Arithmetic operations are applied element-by-element to lists when the global flag `listarith` is `true`, and always applied element-by-element to matrices. When one operand is a list or matrix and another is an operand of some other type, the other operand is combined with each of the elements of the list or matrix.

  Examples:

  Addition and multiplication are n-ary, commutative operators. Maxima sorts the operands to construct a canonical representation. The names of these operators are `"+"` and `"*"`.

      (%i1) c + g + d + a + b + e + f;
      (%o1)               g + f + e + d + c + b + a
      (%i2) [op (%), args (%)];
      (%o2)              [+, [g, f, e, d, c, b, a]]
      (%i3) c * g * d * a * b * e * f;
      (%o3)                     a b c d e f g
      (%i4) [op (%), args (%)];
      (%o4)              [*, [a, b, c, d, e, f, g]]
      (%i5) apply ("+", [a, 8, x, 2, 9, x, x, a]);
      (%o5)                    3 x + 2 a + 19
      (%i6) apply ("*", [a, 8, x, 2, 9, x, x, a]);
                                       2  3
      (%o6)                       144 a  x

  Division and exponentiation are binary, noncommutative operators. The names of these operators are `"/"` and `"^"`.

      (%i1) [a / b, a ^ b];
                                    a   b
      (%o1)                        [-, a ]
                                    b
      (%i2) [map (op, %), map (args, %)];
      (%o2)              [[/, ^], [[a, b], [a, b]]]
      (%i3) [apply ("/", [a, b]), apply ("^", [a, b])];
                                    a   b
      (%o3)                        [-, a ]
                                    b

  Subtraction and division are represented internally in terms of addition and multiplication, respectively.

      (%i1) [inpart (a - b, 0), inpart (a - b, 1), inpart (a - b, 2)];
      (%o1)                      [+, a, - b]
      (%i2) [inpart (a / b, 0), inpart (a / b, 1), inpart (a / b, 2)];
                                         1
      (%o2)                       [*, a, -]
                                         b

  Computations are carried out on literal numbers. Floating-point contagion applies.

      (%i1) 17 + b - (1/2)*29 + 11^(2/4);
                                             5
      (%o1)                   b + sqrt(11) + -
                                             2
      (%i2) [17 + 29, 17 + 29.0, 17 + 29b0];
      (%o2)                   [46, 46.0, 4.6b1]

  Arithmetic computations are a simplification, not an evaluation.

      (%i1) simp : false;
      (%o1)                         false
      (%i2) '(17 + 29*11/7 - 5^3);
                                    29 11    3
      (%o2)                    17 + ----- - 5
                                      7
      (%i3) simp : true;
      (%o3)                         true
      (%i4) '(17 + 29*11/7 - 5^3);
                                      437
      (%o4)                         - ---
                                       7

  Arithmetic is carried out element-by-element for lists (depending on `listarith`) and matrices.

      (%i1) matrix ([a, x], [h, u]) - matrix ([1, 2], [3, 4]);
                              [ a - 1  x - 2 ]
      (%o1)                   [              ]
                              [ h - 3  u - 4 ]
      (%i2) 5 * matrix ([a, x], [h, u]);
                                [ 5 a  5 x ]
      (%o2)                     [          ]
                                [ 5 h  5 u ]
      (%i3) listarith : false;
      (%o3)                         false
      (%i4) [a, c, m, t] / [1, 7, 2, 9];
                                [a, c, m, t]
      (%o4)                     ------------
                                [1, 7, 2, 9]
      (%i5) [a, c, m, t] ^ x;
                                            x
      (%o5)                     [a, c, m, t]
      (%i6) listarith : true;
      (%o6)                         true
      (%i7) [a, c, m, t] / [1, 7, 2, 9];
                                    c  m  t
      (%o7)                     [a, -, -, -]
                                    7  2  9
      (%i8) [a, c, m, t] ^ x;
                                x   x   x   x
      (%o8)                   [a , c , m , t ]

  Categories: Operators

Operator: **

  Exponentiation operator. Maxima recognizes `**` as the same operator as `^` in input, and it is displayed as `^` in 1-dimensional output, or by placing the exponent as a superscript in 2-dimensional output.

  The `fortran` function displays the exponentiation operator as `**`, whether it was input as `**` or `^`.

  Examples:

      (%i1) is (a**b = a^b);
      (%o1)                         true
      (%i2) x**y + x^z;
                                    z    y
      (%o2)                        x  + x
      (%i3) string (x**y + x^z);
      (%o3)                        x^z+x^y
      (%i4) fortran (x**y + x^z);
            x**z+x**y
      (%o4)                         done

  Categories: Operators

Operator: ^^

  Noncommutative exponentiation operator. `^^` is the exponentiation operator corresponding to noncommutative multiplication `.`, just as the ordinary exponentiation operator `^` corresponds to commutative multiplication `*`.

  Noncommutative exponentiation is displayed by `^^` in 1-dimensional output, and by placing the exponent as a superscript within angle brackets `< >` in 2-dimensional output.

  Examples:

      (%i1) a . a . b . b . b + a * a * a * b * b;
                              3  2    <2>    <3>
      (%o1)                  a  b  + a    . b
      (%i2) string (a . a . b . b . b + a * a * a * b * b);
      (%o2)                  a^3*b^2+a^^2 . b^^3

  Categories: Operators

Operator: .

  The dot operator, for matrix (non-commutative) multiplication. When `"."` is used in this way, spaces should be left on both sides of it, e.g. `A . B` This distinguishes it plainly from a decimal point in a floating point number.

  See also `Dot`, `dot0nscsimp`, `dot0simp`, `dot1simp`, `dotassoc`, `dotconstrules`, `dotdistrib`, `dotexptsimp`, `dotident`, and `dotscrules`.

  Categories: Operators

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## 7.3 Relational operators

Operator: <
Operator: <=
Operator: >=
Operator: >

  The symbols `<` `<=` `>=` and `>` represent less than, less than or equal, greater than or equal, and greater than, respectively. The names of these operators are `"<"` `"<="` `">="` and `">"`, which may appear where the name of a function or operator is required.

  These relational operators are all binary operators; constructs such as `a < b < c` are not recognized by Maxima.

  Relational expressions are evaluated to Boolean values by the functions `is` and `maybe`, and the programming constructs `if`, `while`, and `unless`. Relational expressions are not otherwise evaluated or simplified to Boolean values, although the arguments of relational expressions are evaluated (when evaluation is not otherwise prevented by quotation).

  When a relational expression cannot be evaluated to `true` or `false`, the behavior of `is` and `if` are governed by the global flag `prederror`. When `prederror` is `true`, `is` and `if` trigger an error. When `prederror` is `false`, `is` returns `unknown`, and `if` returns a partially-evaluated conditional expression.

  `maybe` always behaves as if `prederror` were `false`, and `while` and `unless` always behave as if `prederror` were `true`.

  Relational operators do not distribute over lists or other aggregates.

  See also `=`, `#`, `equal`, and `notequal`.

  Examples:

  Relational expressions are evaluated to Boolean values by some functions and programming constructs.

      (%i1) [x, y, z] : [123, 456, 789];
      (%o1)                    [123, 456, 789]
      (%i2) is (x < y);
      (%o2)                         true
      (%i3) maybe (y > z);
      (%o3)                         false
      (%i4) if x >= z then 1 else 0;
      (%o4)                           0
      (%i5) block ([S], S : 0, for i:1 while i <= 100 do S : S + i,
                   return (S));
      (%o5)                         5050

  Relational expressions are not otherwise evaluated or simplified to Boolean values, although the arguments of relational expressions are evaluated.

      (%o1)                    [123, 456, 789]
      (%i2) [x < y, y <= z, z >= y, y > z];
      (%o2)    [123 < 456, 456 <= 789, 789 >= 456, 456 > 789]
      (%i3) map (is, %);
      (%o3)               [true, true, true, false]

  Categories: Operators

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## 7.4 Logical operators

Operator: and

  The logical conjunction operator. `and` is an n-ary infix operator; its operands are Boolean expressions, and its result is a Boolean value.

  `and` forces evaluation (like `is`) of one or more operands, and may force evaluation of all operands.

  Operands are evaluated in the order in which they appear. `and` evaluates only as many of its operands as necessary to determine the result. If any operand is `false`, the result is `false` and no further operands are evaluated.

  The global flag `prederror` governs the behavior of `and` when an evaluated operand cannot be determined to be `true` or `false`. `and` prints an error message when `prederror` is `true`. Otherwise, operands which do not evaluate to `true` or `false` are accepted, and the result is a Boolean expression.

  `and` is not commutative: `a and b` might not be equal to `b and a` due to the treatment of indeterminate operands.

  Categories: Operators

Operator: not

  The logical negation operator. `not` is a prefix operator; its operand is a Boolean expression, and its result is a Boolean value.

  `not` forces evaluation (like `is`) of its operand.

  The global flag `prederror` governs the behavior of `not` when its operand cannot be determined to be `true` or `false`. `not` prints an error message when `prederror` is `true`. Otherwise, operands which do not evaluate to `true` or `false` are accepted, and the result is a Boolean expression.

  Categories: Operators

Operator: or

  The logical disjunction operator. `or` is an n-ary infix operator; its operands are Boolean expressions, and its result is a Boolean value.

  `or` forces evaluation (like `is`) of one or more operands, and may force evaluation of all operands.

  Operands are evaluated in the order in which they appear. `or` evaluates only as many of its operands as necessary to determine the result. If any operand is `true`, the result is `true` and no further operands are evaluated.

  The global flag `prederror` governs the behavior of `or` when an evaluated operand cannot be determined to be `true` or `false`. `or` prints an error message when `prederror` is `true`. Otherwise, operands which do not evaluate to `true` or `false` are accepted, and the result is a Boolean expression.

  `or` is not commutative: `a or b` might not be equal to `b or a` due to the treatment of indeterminate operands.

  Categories: Operators

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## 7.5 Operators for Equations

Operator: #

  Represents the negation of syntactic equality `=`.

  Note that because of the rules for evaluation of predicate expressions (in particular because `not expr` causes evaluation of expr), `not a = b` is equivalent to `is(a # b)`, instead of `a # b`.

  Examples:

      (%i1) a = b;
      (%o1)                         a = b
      (%i2) is (a = b);
      (%o2)                         false
      (%i3) a # b;
      (%o3)                         a # b
      (%i4) not a = b;
      (%o4)                         true
      (%i5) is (a # b);
      (%o5)                         true
      (%i6) is (not a = b);
      (%o6)                         true

  Categories: Operators

Operator: =

  The equation operator.

  An expression `a = b`, by itself, represents an unevaluated equation, which might or might not hold. Unevaluated equations may appear as arguments to `solve` and `algsys` or some other functions.

  The function `is` evaluates `=` to a Boolean value. `is(a = b)` evaluates `a = b` to `true` when a and b are identical. That is, a and b are atoms which are identical, or they are not atoms and their operators are identical and their arguments are identical. Otherwise, `is(a = b)` evaluates to `false`; it never evaluates to `unknown`. When `is(a = b)` is `true`, a and b are said to be syntactically equal, in contrast to equivalent expressions, for which `is(equal(a, b))` is `true`. Expressions can be equivalent and not syntactically equal.

  The negation of `=` is represented by `#`. As with `=`, an expression `a # b`, by itself, is not evaluated. `is(a # b)` evaluates `a # b` to `true` or `false`.

  In addition to `is`, some other operators evaluate `=` and `#` to `true` or `false`, namely `if`, `and`, `or`, and `not`.

  Note that because of the rules for evaluation of predicate expressions (in particular because `not expr` causes evaluation of expr), `not a = b` is equivalent to `is(a # b)`, instead of `a # b`.

  `rhs` and `lhs` return the right-hand and left-hand sides, respectively, of an equation or inequation.

  See also `equal` and `notequal`.

  Examples:

  An expression `a = b`, by itself, represents an unevaluated equation, which might or might not hold.

      (%i1) eq_1 : a * x - 5 * y = 17;
      (%o1)                    a x - 5 y = 17
      (%i2) eq_2 : b * x + 3 * y = 29;
      (%o2)                    3 y + b x = 29
      (%i3) solve ([eq_1, eq_2], [x, y]);
                              196         29 a - 17 b
      (%o3)          [[x = ---------, y = -----------]]
                           5 b + 3 a       5 b + 3 a
      (%i4) subst (%, [eq_1, eq_2]);
               196 a     5 (29 a - 17 b)
      (%o4) [--------- - --------------- = 17,
             5 b + 3 a      5 b + 3 a
                                        196 b     3 (29 a - 17 b)
                                      --------- + --------------- = 29]
                                      5 b + 3 a      5 b + 3 a
      (%i5) ratsimp (%);
      (%o5)                  [17 = 17, 29 = 29]

  `is(a = b)` evaluates `a = b` to `true` when a and b are syntactically equal (that is, identical). Expressions can be equivalent and not syntactically equal.

      (%i1) a : (x + 1) * (x - 1);
      (%o1)                    (x - 1) (x + 1)
      (%i2) b : x^2 - 1;
                                    2
      (%o2)                        x  - 1
      (%i3) [is (a = b), is (a # b)];
      (%o3)                     [false, true]
      (%i4) [is (equal (a, b)), is (notequal (a, b))];
      (%o4)                     [true, false]

  Some operators evaluate `=` and `#` to `true` or `false`.

      (%i1) if expand ((x + y)^2) = x^2 + 2 * x * y + y^2 then FOO else
            BAR;
      (%o1)                          FOO
      (%i2) eq_3 : 2 * x = 3 * x;
      (%o2)                       2 x = 3 x
      (%i3) eq_4 : exp (2) = %e^2;
                                    2     2
      (%o3)                       %e  = %e
      (%i4) [eq_3 and eq_4, eq_3 or eq_4, not eq_3];
      (%o4)                  [false, true, true]

  Because `not expr` causes evaluation of expr, `not a = b` is equivalent to `is(a # b)`.

      (%i1) [2 * x # 3 * x, not (2 * x = 3 * x)];
      (%o1)                   [2 x # 3 x, true]
      (%i2) is (2 * x # 3 * x);
      (%o2)                         true

  Categories: Operators

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## 7.6 Assignment operators

Operator: :

  Assignment operator.

  When the left-hand side is a simple variable (not subscripted), `:` evaluates its right-hand side and associates that value with the left-hand side.

  When the left-hand side is a subscripted element of a list, matrix, declared Maxima array, or Lisp array, the right-hand side is assigned to that element. The subscript must name an existing element; such objects cannot be extended by naming nonexistent elements.

  When the left-hand side is a subscripted element of an undeclared Maxima array, the right-hand side is assigned to that element, if it already exists, or a new element is allocated, if it does not already exist.

  When the left-hand side is a list of simple and/or subscripted variables, the right-hand side must evaluate to a list, and the elements of the right-hand side are assigned to the elements of the left-hand side, in parallel.

  See also `kill` and `remvalue`, which undo the association between the left-hand side and its value.

  Examples:

  Assignment to a simple variable.

      (%i1) a;
      (%o1)                           a
      (%i2) a : 123;
      (%o2)                          123
      (%i3) a;
      (%o3)                          123

  Assignment to an element of a list.

      (%i1) b : [1, 2, 3];
      (%o1)                       [1, 2, 3]
      (%i2) b[3] : 456;
      (%o2)                          456
      (%i3) b;
      (%o3)                      [1, 2, 456]

  Assignment creates an undeclared array.

      (%i1) c[99] : 789;
      (%o1)                          789
      (%i2) c[99];
      (%o2)                          789
      (%i3) c;
      (%o3)                           c
      (%i4) arrayinfo (c);
      (%o4)                   [hashed, 1, [99]]
      (%i5) listarray (c);
      (%o5)                         [789]

  Multiple assignment.

      (%i1) [a, b, c] : [45, 67, 89];
      (%o1)                     [45, 67, 89]
      (%i2) a;
      (%o2)                          45
      (%i3) b;
      (%o3)                          67
      (%i4) c;
      (%o4)                          89

  Multiple assignment is carried out in parallel. The values of `a` and `b` are exchanged in this example.

      (%i1) [a, b] : [33, 55];
      (%o1)                       [33, 55]
      (%i2) [a, b] : [b, a];
      (%o2)                       [55, 33]
      (%i3) a;
      (%o3)                          55
      (%i4) b;
      (%o4)                          33

  Categories: Evaluation · Operators

Operator: ::

  Assignment operator.

  `::` is the same as `:` (which see) except that `::` evaluates its left-hand side as well as its right-hand side.

  Examples:

      (%i1) x : 'foo;
      (%o1)                          foo
      (%i2) x :: 123;
      (%o2)                          123
      (%i3) foo;
      (%o3)                          123
      (%i4) x : '[a, b, c];
      (%o4)                       [a, b, c]
      (%i5) x :: [11, 22, 33];
      (%o5)                     [11, 22, 33]
      (%i6) a;
      (%o6)                          11
      (%i7) b;
      (%o7)                          22
      (%i8) c;
      (%o8)                          33

  Categories: Evaluation · Operators

Operator: ::=

  Macro function definition operator. `::=` defines a function (called a "macro" for historical reasons) which quotes its arguments, and the expression which it returns (called the "macro expansion") is evaluated in the context from which the macro was called. A macro function is otherwise the same as an ordinary function.

  `macroexpand` returns a macro expansion (without evaluating it). `macroexpand (foo (x))` followed by `''%` is equivalent to `foo (x)` when `foo` is a macro function.

  `::=` puts the name of the new macro function onto the global list `macros`. `kill`, `remove`, and `remfunction` unbind macro function definitions and remove names from `macros`.

  `fundef` or `dispfun` return a macro function definition or assign it to a label, respectively.

  Macro functions commonly contain `buildq` and `splice` expressions to construct an expression, which is then evaluated.

  Examples

  A macro function quotes its arguments, so message (1) shows `y - z`, not the value of `y - z`. The macro expansion (the quoted expression `'(print ("(2) x is equal to", x))`) is evaluated in the context from which the macro was called, printing message (2).

      (%i1) x: %pi$
      (%i2) y: 1234$
      (%i3) z: 1729 * w$
      (%i4) printq1 (x) ::= block (print ("(1) x is equal to", x),
            '(print ("(2) x is equal to", x)))$
      (%i5) printq1 (y - z);
      (1) x is equal to y - z
      (2) x is equal to %pi
      (%o5)                                 %pi

  An ordinary function evaluates its arguments, so message (1) shows the value of `y - z`. The return value is not evaluated, so message (2) is not printed until the explicit evaluation `''%`.

      (%i1) x: %pi$
      (%i2) y: 1234$
      (%i3) z: 1729 * w$
      (%i4) printe1 (x) := block (print ("(1) x is equal to", x),
            '(print ("(2) x is equal to", x)))$
      (%i5) printe1 (y - z);
      (1) x is equal to 1234 - 1729 w
      (%o5)                     print((2) x is equal to, x)
      (%i6) ''%;
      (2) x is equal to %pi
      (%o6)                                 %pi

  `macroexpand` returns a macro expansion. `macroexpand (foo (x))` followed by `''%` is equivalent to `foo (x)` when `foo` is a macro function.

      (%i1) x: %pi$
      (%i2) y: 1234$
      (%i3) z: 1729 * w$
      (%i4) g (x) ::= buildq ([x], print ("x is equal to", x))$
      (%i5) macroexpand (g (y - z));
      (%o5)                     print(x is equal to, y - z)
      (%i6) ''%;
      x is equal to 1234 - 1729 w
      (%o6)                            1234 - 1729 w
      (%i7) g (y - z);
      x is equal to 1234 - 1729 w
      (%o7)                            1234 - 1729 w

  Categories: Function definition · Operators

Operator: :=

  The function definition operator.

  `f(x_1, ..., x_n) := expr` defines a function named f with arguments x_1, …, x_n and function body expr. `:=` never evaluates the function body (unless explicitly evaluated by quote-quote `''`). The function body is evaluated every time the function is called.

  `f[x_1, ..., x_n] := expr` defines a so-called array function. Its function body is evaluated just once for each distinct value of its arguments, and that value is returned, without evaluating the function body, whenever the arguments have those values again. (A function of this kind is commonly known as a "memoizing function".)

  `f[x_1, ..., x_n](y_1, ..., y_m) := expr` is a special case of an array function. `f[x_1, ..., x_n]` is an array function which returns a lambda expression with arguments `y_1, ..., y_m`. The function body is evaluated once for each distinct value of `x_1, ..., x_n`, and the body of the lambda expression is that value.

  When the last or only function argument x_n is a list of one element, the function defined by `:=` accepts a variable number of arguments. Actual arguments are assigned one-to-one to formal arguments x_1, …, x_(n - 1), and any further actual arguments, if present, are assigned to x_n as a list.

  All function definitions appear in the same namespace; defining a function `f` within another function `g` does not automatically limit the scope of `f` to `g`. However, `local(f)` makes the definition of function `f` effective only within the block or other compound expression in which `local` appears.

  If some formal argument x_k is a quoted symbol, the function defined by `:=` does not evaluate the corresponding actual argument. Otherwise all actual arguments are evaluated.

  See also `define` and `::=`.

  Examples:

  `:=` never evaluates the function body (unless explicitly evaluated by quote-quote).

      (%i1) expr : cos(y) - sin(x);
      (%o1)                    cos(y) - sin(x)
      (%i2) F1 (x, y) := expr;
      (%o2)                   F1(x, y) := expr
      (%i3) F1 (a, b);
      (%o3)                    cos(y) - sin(x)
      (%i4) F2 (x, y) := ''expr;
      (%o4)              F2(x, y) := cos(y) - sin(x)
      (%i5) F2 (a, b);
      (%o5)                    cos(b) - sin(a)

  `f(x_1, ..., x_n) := ...` defines an ordinary function.

      (%i1) G1(x, y) := (print ("Evaluating G1 for x=", x, "and y=", y), x.y - y.x);
      (%o1) G1(x, y) := (print("Evaluating G1 for x=", x, "and y=",
                                                     y), x . y - y . x)
      (%i2) G1([1, a], [2, b]);
      Evaluating G1 for x= [1, a] and y= [2, b]
      (%o2)                           0
      (%i3) G1([1, a], [2, b]);
      Evaluating G1 for x= [1, a] and y= [2, b]
      (%o3)                           0

  `f[x_1, ..., x_n] := ...` defines an array function.

      (%i1) G2[a] := (print ("Evaluating G2 for a=", a), a^2);
                                                           2
      (%o1)     G2  := (print("Evaluating G2 for a=", a), a )
                  a
      (%i2) G2[1234];
      Evaluating G2 for a= 1234
      (%o2)                        1522756
      (%i3) G2[1234];
      (%o3)                        1522756
      (%i4) G2[2345];
      Evaluating G2 for a= 2345
      (%o4)                        5499025
      (%i5) arrayinfo (G2);
      (%o5)              [hashed, 1, [1234], [2345]]
      (%i6) listarray (G2);
      (%o6)                  [1522756, 5499025]

  `f[x_1, ..., x_n](y_1, ..., y_m) := expr` is a special case of an array function.

      (%i1) G3[n](x) := (print ("Evaluating G3 for n=", n), diff (sin(x)^2, x, n));
      (%o1) G3 (x) := (print("Evaluating G3 for n=", n),
              n
                                                           2
                                                   diff(sin (x), x, n))
      (%i2) G3[2];
      Evaluating G3 for n= 2
                                      2           2
      (%o2)          lambda([x], 2 cos (x) - 2 sin (x))
      (%i3) G3[2];
                                      2           2
      (%o3)          lambda([x], 2 cos (x) - 2 sin (x))
      (%i4) G3[2](1);
                                 2           2
      (%o4)                 2 cos (1) - 2 sin (1)
      (%i5) arrayinfo (G3);
      (%o5)                   [hashed, 1, [2]]
      (%i6) listarray (G3);
                                      2           2
      (%o6)         [lambda([x], 2 cos (x) - 2 sin (x))]

  When the last or only function argument x_n is a list of one element, the function defined by `:=` accepts a variable number of arguments.

      (%i1) H ([L]) := apply ("+", L);
      (%o1)                H([L]) := apply("+", L)
      (%i2) H (a, b, c);
      (%o2)                       c + b + a

  `local` makes a local function definition.

      (%i1) foo (x) := 1 - x;
      (%o1)                    foo(x) := 1 - x
      (%i2) foo (100);
      (%o2)                         - 99
      (%i3) block (local (foo), foo (x) := 2 * x, foo (100));
      (%o3)                          200
      (%i4) foo (100);
      (%o4)                         - 99

  Categories: Function definition · Operators

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## 7.7 User defined operators

Function: infix
`infix` (op)
`infix` (op, lbp, rbp)
`infix` (op, lbp, rbp, lpos, rpos, pos)

  Declares op to be an infix operator. An infix operator is a function of two arguments, with the name of the function written between the arguments. For example, the subtraction operator `-` is an infix operator.

  `infix (op)` declares op to be an infix operator with default binding powers (left and right both equal to 180) and parts of speech (left and right both equal to `any`).

  `infix (op, lbp, rbp)` declares op to be an infix operator with stated left and right binding powers and default parts of speech (left and right both equal to `any`).

  `infix (op, lbp, rbp, lpos, rpos, pos)` declares op to be an infix operator with stated left and right binding powers and parts of speech lpos, rpos, and pos for the left operand, the right operand, and the operator result, respectively.

  "Part of speech", in reference to operator declarations, means expression type. Three types are recognized: `expr`, `clause`, and `any`, indicating an algebraic expression, a Boolean expression, or any kind of expression, respectively. Maxima can detect some syntax errors by comparing the declared part of speech to an actual expression.

  The precedence of op with respect to other operators derives from the left and right binding powers of the operators in question. If the left and right binding powers of op are both greater the left and right binding powers of some other operator, then op takes precedence over the other operator. If the binding powers are not both greater or less, some more complicated relation holds.

  The associativity of op depends on its binding powers. Greater left binding power (lbp) implies an instance of op is evaluated before other operators to its left in an expression, while greater right binding power (rbp) implies an instance of op is evaluated before other operators to its right in an expression. Thus greater lbp makes op right-associative, while greater rbp makes op left-associative. If lbp is equal to rbp, op is left-associative.

  See also Introduction to operators.

  Examples:

  If the left and right binding powers of op are both greater the left and right binding powers of some other operator, then op takes precedence over the other operator.

      (%i1) :lisp (get '$+ 'lbp)
      100
      (%i1) :lisp (get '$+ 'rbp)
      100
      (%i1) infix ("##", 101, 101);
      (%o1)                          ##
      (%i2) "##"(a, b) := sconcat("(", a, ",", b, ")");
      (%o2)       (a ## b) := sconcat("(", a, ",", b, ")")
      (%i3) 1 + a ## b + 2;
      (%o3)                       (a,b) + 3
      (%i4) infix ("##", 99, 99);
      (%o4)                          ##
      (%i5) 1 + a ## b + 2;
      (%o5)                       (a+1,b+2)

  Greater lbp makes op right-associative, while greater rbp makes op left-associative.

      (%i1) infix ("##", 100, 99);
      (%o1)                          ##
      (%i2) "##"(a, b) := sconcat("(", a, ",", b, ")")$
      (%i3) foo ## bar ## baz;
      (%o3)                    (foo,(bar,baz))
      (%i4) infix ("##", 100, 101);
      (%o4)                          ##
      (%i5) foo ## bar ## baz;
      (%o5)                    ((foo,bar),baz)

  Maxima can detect some syntax errors by comparing the declared part of speech to an actual expression.

      (%i1) infix ("##", 100, 99, expr, expr, expr);
      (%o1)                          ##
      (%i2) if x ## y then 1 else 0;
      Incorrect syntax: Found algebraic expression where logical
      expression expected
      if x ## y then
                   ^
      (%i2) infix ("##", 100, 99, expr, expr, clause);
      (%o2)                          ##
      (%i3) if x ## y then 1 else 0;
      (%o3)                if x ## y then 1 else 0

  Categories: Operators · Declarations and inferences · Syntax

Function: matchfix
`matchfix` (ldelimiter, rdelimiter)
`matchfix` (ldelimiter, rdelimiter, arg_pos, pos)

  Declares a matchfix operator with left and right delimiters ldelimiter and rdelimiter. The delimiters are specified as strings.

  A "matchfix" operator is a function of any number of arguments, such that the arguments occur between matching left and right delimiters. The delimiters may be any strings, so long as the parser can distinguish the delimiters from the operands and other expressions and operators. In practice this rules out unparseable delimiters such as `%`, `,`, `$` and `;`, and may require isolating the delimiters with white space. The right delimiter can be the same or different from the left delimiter.

  A left delimiter can be associated with only one right delimiter; two different matchfix operators cannot have the same left delimiter.

  An existing operator may be redeclared as a matchfix operator without changing its other properties. In particular, built-in operators such as addition `+` can be declared matchfix, but operator functions cannot be defined for built-in operators.

  The command `matchfix (ldelimiter, rdelimiter, arg_pos, pos)` declares the argument part-of-speech arg_pos and result part-of-speech pos, and the delimiters ldelimiter and rdelimiter.

  "Part of speech", in reference to operator declarations, means expression type. Three types are recognized: `expr`, `clause`, and `any`, indicating an algebraic expression, a Boolean expression, or any kind of expression, respectively. Maxima can detect some syntax errors by comparing the declared part of speech to an actual expression.

  The function to carry out a matchfix operation is an ordinary user-defined function. The operator function is defined in the usual way with the function definition operator `:=` or `define`. The arguments may be written between the delimiters, or with the left delimiter as a quoted string and the arguments following in parentheses. `dispfun (ldelimiter)` displays the function definition.

  The only built-in matchfix operator is the list constructor `[ ]`. Parentheses `( )` and double-quotes `" "` act like matchfix operators, but are not treated as such by the Maxima parser.

  `matchfix` evaluates its arguments. `matchfix` returns its first argument, ldelimiter.

  Examples:

  Delimiters may be almost any strings.

      (%i1) matchfix ("@@", "~");
      (%o1)                          @@
      (%i2) @@ a, b, c ~;
      (%o2)                      @@a, b, c~
      (%i3) matchfix (">>", "<<");
      (%o3)                          >>
      (%i4) >> a, b, c <<;
      (%o4)                      >>a, b, c<<
      (%i5) matchfix ("foo", "oof");
      (%o5)                          foo
      (%i6) foo a, b, c oof;
      (%o6)                     fooa, b, coof
      (%i7) >> w + foo x, y oof + z << / @@ p, q ~;
                           >>z + foox, yoof + w<<
      (%o7)                ----------------------
                                  @@p, q~

  Matchfix operators are ordinary user-defined functions.

      (%i1) matchfix ("!-", "-!");
      (%o1)                         "!-"
      (%i2) !- x, y -! := x/y - y/x;
                                          x   y
      (%o2)                   !-x, y-! := - - -
                                          y   x
      (%i3) define (!-x, y-!, x/y - y/x);
                                          x   y
      (%o3)                   !-x, y-! := - - -
                                          y   x
      (%i4) define ("!-" (x, y), x/y - y/x);
                                          x   y
      (%o4)                   !-x, y-! := - - -
                                          y   x
      (%i5) dispfun ("!-");
                                          x   y
      (%t5)                   !-x, y-! := - - -
                                          y   x

      (%o5)                         done
      (%i6) !-3, 5-!;
                                      16
      (%o6)                         - --
                                      15
      (%i7) "!-" (3, 5);
                                      16
      (%o7)                         - --
                                      15

  Categories: Syntax · Operators

Function: nary
`nary` (op)
`nary` (op, bp, arg_pos, pos)

  An `nary` operator is used to denote a function of any number of arguments, each of which is separated by an occurrence of the operator, e.g. A+B or A+B+C. The `nary("x")` function is a syntax extension function to declare `x` to be an `nary` operator. Functions may be declared to be `nary`. If `declare(j,nary);` is done, this tells the simplifier to simplify, e.g. `j(j(a,b),j(c,d))` to `j(a, b, c, d)`.

  See also Introduction to operators.

  Categories: Operators · Syntax

Function: nofix
`nofix` (op)
`nofix` (op, pos)

  `nofix` operators are used to denote functions of no arguments. The mere presence of such an operator in a command will cause the corresponding function to be evaluated. For example, when one types "exit;" to exit from a Maxima break, "exit" is behaving similar to a `nofix` operator. The function `nofix("x")` is a syntax extension function which declares `x` to be a `nofix` operator.

  See also Introduction to operators.

  Categories: Operators · Syntax

Function: postfix
`postfix` (op)
`postfix` (op, lbp, lpos, pos)

  `postfix` operators like the `prefix` variety denote functions of a single argument, but in this case the argument immediately precedes an occurrence of the operator in the input string, e.g. 3!. The `postfix("x")` function is a syntax extension function to declare `x` to be a `postfix` operator.

  See also Introduction to operators.

  Categories: Operators · Syntax

Function: prefix
`prefix` (op)
`prefix` (op, rbp, rpos, pos)

  A `prefix` operator is one which signifies a function of one argument, which argument immediately follows an occurrence of the operator. `prefix("x")` is a syntax extension function to declare `x` to be a `prefix` operator.

  See also Introduction to operators.

  Categories: Operators · Syntax

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# 8. Evaluation

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## 8.1 Functions and Variables for Evaluation

Operator: '

  The single quote operator `'` prevents evaluation.

  Applied to a symbol, the single quote prevents evaluation of the symbol.

  Applied to a function call, the single quote prevents evaluation of the function call, although the arguments of the function are still evaluated (if evaluation is not otherwise prevented). The result is the noun form of the function call.

  Applied to a parenthesized expression, the single quote prevents evaluation of all symbols and function calls in the expression. E.g., `'(f(x))` means do not evaluate the expression `f(x)`. `'f(x)` (with the single quote applied to `f` instead of `f(x)`) means return the noun form of `f` applied to `[x]`.

  The single quote does not prevent simplification.

  When the global flag `noundisp` is `true`, nouns display with a single quote. This switch is always `true` when displaying function definitions.

  See also the quote-quote operator `''` and `nouns`.

  Examples:

  Applied to a symbol, the single quote prevents evaluation of the symbol.

      (%i1) aa: 1024;
      (%o1)                         1024
      (%i2) aa^2;
      (%o2)                        1048576
      (%i3) 'aa^2;
                                       2
      (%o3)                          aa
      (%i4) ''%;
      (%o4)                        1048576

  Applied to a function call, the single quote prevents evaluation of the function call. The result is the noun form of the function call.

      (%i1) x0: 5;
      (%o1)                           5
      (%i2) x1: 7;
      (%o2)                           7
      (%i3) integrate (x^2, x, x0, x1);
                                     218
      (%o3)                          ---
                                      3
      (%i4) 'integrate (x^2, x, x0, x1);
                                   7
                                  /
                                  [   2
      (%o4)                       I  x  dx
                                  ]
                                  /
                                   5
      (%i5) %, nouns;
                                     218
      (%o5)                          ---
                                      3

  Applied to a parenthesized expression, the single quote prevents evaluation of all symbols and function calls in the expression.

      (%i1) aa: 1024;
      (%o1)                         1024
      (%i2) bb: 19;
      (%o2)                          19
      (%i3) sqrt(aa) + bb;
      (%o3)                          51
      (%i4) '(sqrt(aa) + bb);
      (%o4)                     bb + sqrt(aa)
      (%i5) ''%;
      (%o5)                          51

  The single quote does not prevent simplification.

      (%i1) sin (17 * %pi) + cos (17 * %pi);
      (%o1)                          - 1
      (%i2) '(sin (17 * %pi) + cos (17 * %pi));
      (%o2)                          - 1

  Maxima considers floating point operations by its in-built mathematical functions to be a simplification.

      (%i1) sin(1.0);
      (%o1)                          .8414709848078965
      (%i2) '(sin(1.0));
      (%o2)                          .8414709848078965

  When the global flag `noundisp` is `true`, nouns display with a single quote.

      (%i1) x:%pi;
      (%o1)                                 %pi
      (%i2) bfloat(x);
      (%o2)                         3.141592653589793b0
      (%i3) sin(x);
      (%o3)                                  0
      (%i4) noundisp;
      (%o4)                                false
      (%i5) 'bfloat(x);
      (%o5)                             bfloat(%pi)
      (%i6) bfloat('x);
      (%o6)                                  x
      (%i7) 'sin(x);
      (%o7)                                  0
      (%i8) sin('x);
      (%o8)                               sin(x)
      (%i9) noundisp : not noundisp;
      (%o9)                                true
      (%i10) 'bfloat(x);
      (%o10)                           'bfloat(%pi)
      (%i11) bfloat('x);
      (%o11)                                 x
      (%i12) 'sin(x);
      (%o12)                                 0
      (%i13) sin('x);
      (%o13)                              sin(x)
      (%i14)

  Categories: Evaluation · Operators

Operator: "

  The quote-quote operator `''` (two single quote marks) modifies evaluation in input expressions.

  Applied to a general expression expr, quote-quote causes the value of expr to be substituted for expr in the input expression.

  Applied to the operator of an expression, quote-quote changes the operator from a noun to a verb (if it is not already a verb).

  The quote-quote operator is applied by the input parser; it is not stored as part of a parsed input expression. The quote-quote operator is always applied as soon as it is parsed, and cannot be quoted. Thus quote-quote causes evaluation when evaluation is otherwise suppressed, such as in function definitions, lambda expressions, and expressions quoted by single quote `'`.

  Quote-quote is recognized by `batch` and `load`.

  See also `ev`, the single-quote operator `'` and `nouns`.

  Examples:

  Applied to a general expression expr, quote-quote causes the value of expr to be substituted for expr in the input expression.

      (%i1) expand ((a + b)^3);
                           3        2      2      3
      (%o1)               b  + 3 a b  + 3 a  b + a
      (%i2) [_, ''_];
                               3    3        2      2      3
      (%o2)     [expand((b + a) ), b  + 3 a b  + 3 a  b + a ]
      (%i3) [%i1, ''%i1];
                               3    3        2      2      3
      (%o3)     [expand((b + a) ), b  + 3 a b  + 3 a  b + a ]
      (%i4) [aa : cc, bb : dd, cc : 17, dd : 29];
      (%o4)                   [cc, dd, 17, 29]
      (%i5) foo_1 (x) := aa - bb * x;
      (%o5)                 foo_1(x) := aa - bb x
      (%i6) foo_1 (10);
      (%o6)                      cc - 10 dd
      (%i7) ''%;
      (%o7)                         - 273
      (%i8) ''(foo_1 (10));
      (%o8)                         - 273
      (%i9) foo_2 (x) := ''aa - ''bb * x;
      (%o9)                 foo_2(x) := cc - dd x
      (%i10) foo_2 (10);
      (%o10)                        - 273
      (%i11) [x0 : x1, x1 : x2, x2 : x3];
      (%o11)                    [x1, x2, x3]
      (%i12) x0;
      (%o12)                         x1
      (%i13) ''x0;
      (%o13)                         x2
      (%i14) '' ''x0;
      (%o14)                         x3

  Applied to the operator of an expression, quote-quote changes the operator from a noun to a verb (if it is not already a verb).

      (%i1) declare (foo, noun);
      (%o1)                         done
      (%i2) foo (x) := x - 1729;
      (%o2)                 ''foo(x) := x - 1729
      (%i3) foo (100);
      (%o3)                       foo(100)
      (%i4) ''foo (100);
      (%o4)                        - 1629

  The quote-quote operator is applied by the input parser; it is not stored as part of a parsed input expression.

      (%i1) [aa : bb, cc : dd, bb : 1234, dd : 5678];
      (%o1)                 [bb, dd, 1234, 5678]
      (%i2) aa + cc;
      (%o2)                        dd + bb
      (%i3) display (_, op (_), args (_));
                                 _ = cc + aa

                               op(cc + aa) = +

                          args(cc + aa) = [cc, aa]

      (%o3)                         done
      (%i4) ''(aa + cc);
      (%o4)                         6912
      (%i5) display (_, op (_), args (_));
                                 _ = dd + bb

                               op(dd + bb) = +

                          args(dd + bb) = [dd, bb]

      (%o5)                         done

  Quote-quote causes evaluation when evaluation is otherwise suppressed, such as in function definitions, lambda expressions, and expressions quoted by single quote `'`.

      (%i1) foo_1a (x) := ''(integrate (log (x), x));
      (%o1)               foo_1a(x) := x log(x) - x
      (%i2) foo_1b (x) := integrate (log (x), x);
      (%o2)           foo_1b(x) := integrate(log(x), x)
      (%i3) dispfun (foo_1a, foo_1b);
      (%t3)               foo_1a(x) := x log(x) - x

      (%t4)           foo_1b(x) := integrate(log(x), x)

      (%o4)                      [%t3, %t4]
      (%i5) integrate (log (x), x);
      (%o5)                     x log(x) - x
      (%i6) foo_2a (x) := ''%;
      (%o6)               foo_2a(x) := x log(x) - x
      (%i7) foo_2b (x) := %;
      (%o7)                    foo_2b(x) := %
      (%i8) dispfun (foo_2a, foo_2b);
      (%t8)               foo_2a(x) := x log(x) - x

      (%t9)                    foo_2b(x) := %

      (%o9)                      [%t7, %t8]
      (%i10) F : lambda ([u], diff (sin (u), u));
      (%o10)             lambda([u], diff(sin(u), u))
      (%i11) G : lambda ([u], ''(diff (sin (u), u)));
      (%o11)                  lambda([u], cos(u))
      (%i12) '(sum (a[k], k, 1, 3) + sum (b[k], k, 1, 3));
      (%o12)         sum(b , k, 1, 3) + sum(a , k, 1, 3)
                          k                  k
      (%i13) '(''(sum (a[k], k, 1, 3)) + ''(sum (b[k], k, 1, 3)));
      (%o13)             b  + a  + b  + a  + b  + a
                          3    3    2    2    1    1

  Categories: Evaluation · Operators

Function: ev (expr, arg_1, …, arg_n)

  Evaluates the expression expr in the environment specified by the arguments arg_1, …, arg_n. The arguments are switches (Boolean flags), assignments, equations, and functions. `ev` returns the result (another expression) of the evaluation.

  The evaluation is carried out in steps, as follows.

  1. First the environment is set up by scanning the arguments which may be any or all of the following.
     - `simp` causes expr to be simplified regardless of the setting of the switch `simp` which inhibits simplification if `false`.
     - `noeval` suppresses the evaluation phase of `ev` (see step (4) below). This is useful in conjunction with the other switches and in causing expr to be resimplified without being reevaluated.
     - `nouns` causes the evaluation of noun forms (typically unevaluated functions such as `'integrate` or `'diff`) in expr.
     - `expand` causes expansion.
     - `expand (m, n)` causes expansion, setting the values of `maxposex` and `maxnegex` to m and n respectively.
     - `detout` causes any matrix inverses computed in expr to have their determinant kept outside of the inverse rather than dividing through each element.
     - `diff` causes all differentiations indicated in expr to be performed.
     - `derivlist (x, y, z, ...)` causes only differentiations with respect to the indicated variables. See also `derivlist`.
     - `risch` causes integrals in expr to be evaluated using the Risch algorithm. See `risch`. The standard integration routine is invoked when using the special symbol `nouns`.
     - `float` causes non-integral rational numbers to be converted to floating point.
     - `numer` causes some mathematical functions (including exponentiation) with numerical arguments to be evaluated in floating point. It causes variables in expr which have been given numervals to be replaced by their values. It also sets the `float` switch on.
     - `pred` causes predicates (expressions which evaluate to `true` or `false`) to be evaluated.
     - `eval` causes an extra post-evaluation of expr to occur. (See step (5) below.) `eval` may occur multiple times. For each instance of `eval`, the expression is evaluated again.
     - `A` where `A` is an atom declared to be an evaluation flag `evflag` causes `A` to be bound to `true` during the evaluation of expr.
     - `V: expression` (or alternately `V=expression`) causes `V` to be bound to the value of `expression` during the evaluation of expr. Note that if `V` is a Maxima option, then `expression` is used for its value during the evaluation of expr. If more than one argument to `ev` is of this type then the binding is done in parallel. If `V` is a non-atomic expression then a substitution rather than a binding is performed.
     - `F` where `F`, a function name, has been declared to be an evaluation function `evfun` causes `F` to be applied to expr.
     - Any other function names, e.g. `sum`, cause evaluation of occurrences of those names in expr as though they were verbs.
     - In addition a function occurring in expr (say `F(x)`) may be defined locally for the purpose of this evaluation of expr by giving `F(x) := expression` as an argument to `ev`.
     - If an atom not mentioned above or a subscripted variable or subscripted expression was given as an argument, it is evaluated and if the result is an equation or assignment then the indicated binding or substitution is performed. If the result is a list then the members of the list are treated as if they were additional arguments given to `ev`. This permits a list of equations to be given (e.g. `[X=1, Y=A**2]`) or a list of names of equations (e.g., `[%t1, %t2]` where `%t1` and `%t2` are equations) such as that returned by `solve`.

     The arguments of `ev` may be given in any order with the exception of substitution equations which are handled in sequence, left to right, and evaluation functions which are composed, e.g., `ev (expr, ratsimp, realpart)` is handled as `realpart (ratsimp (expr))`.

     The `simp`, `numer`, and `float` switches may also be set locally in a block, or globally in Maxima so that they will remain in effect until being reset.

     If expr is a canonical rational expression (CRE), then the expression returned by `ev` is also a CRE, provided the `numer` and `float` switches are not both `true`.

  2. During step (1), a list is made of the non-subscripted variables appearing on the left side of equations in the arguments or in the value of some arguments if the value is an equation. The variables (subscripted variables which do not have associated array functions as well as non-subscripted variables) in the expression expr are replaced by their global values, except for those appearing in this list. Usually, expr is just a label or `%` (as in `%i2` in the example below), so this step simply retrieves the expression named by the label, so that `ev` may work on it.
  3. If any substitutions are indicated by the arguments, they are carried out now.
  4. The resulting expression is then re-evaluated (unless one of the arguments was `noeval`) and simplified according to the arguments. Note that any function calls in expr will be carried out after the variables in it are evaluated and that `ev(F(x))` thus may behave like `F(ev(x))`.
  5. For each instance of `eval` in the arguments, steps (3) and (4) are repeated.

  See also `''`, `at` and `subst`.

  Examples:

      (%i1) sin(x) + cos(y) + (w+1)^2 + 'diff (sin(w), w);
                                           d                    2
      (%o1)              cos(y) + sin(x) + -- (sin(w)) + (w + 1)
                                           dw
      (%i2) ev (%, numer, expand, diff, x=2, y=1);
                                     2
      (%o2)                cos(w) + w  + 2 w + 2.449599732693821

  An alternate top level syntax has been provided for `ev`, whereby one may just type in its arguments, without the `ev()`. That is, one may write simply

      expr, arg_1, ..., arg_n

  This is not permitted as part of another expression, e.g., in functions, blocks, etc.

  Notice the parallel binding process in the following example.

      (%i3) programmode: false;
      (%o3)                                false
      (%i4) x+y, x: a+y, y: 2;
      (%o4)                              y + a + 2
      (%i5) 2*x - 3*y = 3$
      (%i6) -3*x + 2*y = -4$
      (%i7) solve ([%o5, %o6]);
      Solution

                                                1
      (%t7)                               y = - -
                                                5

                                               6
      (%t8)                                x = -
                                               5
      (%o8)                            [[%t7, %t8]]
      (%i8) %o6, %o8;
      (%o8)                              - 4 = - 4
      (%i9) x + 1/x > gamma (1/2);
                                         1
      (%o9)                          x + - > sqrt(%pi)
                                         x
      (%i10) %, numer, x=1/2;
      (%o10)                      2.5 > 1.772453850905516
      (%i11) %, pred;
      (%o11)                               true

  Categories: Evaluation

Special symbol: eval

  As an argument in a call to `ev (expr)`, `eval` causes an extra evaluation of expr. See `ev`.

  Example:

      (%i1) [a:b,b:c,c:d,d:e];
      (%o1)                            [b, c, d, e]
      (%i2) a;
      (%o2)                                  b
      (%i3) ev(a);
      (%o3)                                  c
      (%i4) ev(a),eval;
      (%o4)                                  e
      (%i5) a,eval,eval;
      (%o5)                                  e

  Categories: Evaluation flags

Property: evflag

  When a symbol x has the `evflag` property, the expressions `ev(expr, x)` and `expr, x` (at the interactive prompt) are equivalent to `ev(expr, x = true)`. That is, x is bound to `true` while expr is evaluated.

  The expression `declare(x, evflag)` gives the `evflag` property to the variable x.

  The flags which have the `evflag` property by default are the following:

         algebraic          cauchysum       demoivre
         dotscrules         %emode          %enumer
         exponentialize     exptisolate     factorflag
         float              halfangles      infeval
         isolate_wrt_times  keepfloat       letrat
         listarith          logabs          logarc
         logexpand          lognegint
         m1pbranch          numer_pbranch   programmode
         radexpand          ratalgdenom     ratfac
         ratmx              ratsimpexpons   simp
         simpproduct        simpsum         sumexpand
         trigexpand

  Examples:

      (%i1) sin (1/2);
                                       1
      (%o1)                        sin(-)
                                       2
      (%i2) sin (1/2), float;
      (%o2)                   0.479425538604203
      (%i3) sin (1/2), float=true;
      (%o3)                   0.479425538604203
      (%i4) simp : false;
      (%o4)                         false
      (%i5) 1 + 1;
      (%o5)                         1 + 1
      (%i6) 1 + 1, simp;
      (%o6)                           2
      (%i7) simp : true;
      (%o7)                         true
      (%i8) sum (1/k^2, k, 1, inf);
                                  inf
                                  ====
                                  \     1
      (%o8)                        >    --
                                  /      2
                                  ====  k
                                  k = 1
      (%i9) sum (1/k^2, k, 1, inf), simpsum;
                                       2
                                    %pi
      (%o9)                         ----
                                     6
      (%i10) declare (aa, evflag);
      (%o10)                        done
      (%i11) if aa = true then YES else NO;
      (%o11)                         NO
      (%i12) if aa = true then YES else NO, aa;
      (%o12)                         YES

  Categories: Evaluation flags · Simplification flags and variables

Property: evfun

  When a function F has the `evfun` property, the expressions `ev(expr, F)` and `expr, F` (at the interactive prompt) are equivalent to `F(ev(expr))`.

  If two or more `evfun` functions F, G, etc., are specified, the functions are applied in the order that they are specified.

  The expression `declare(F, evfun)` gives the `evfun` property to the function F. The functions which have the `evfun` property by default are the following:

         bfloat          factor       fullratsimp
         logcontract     polarform    radcan
         ratexpand       ratsimp      rectform
         rootscontract   trigexpand   trigreduce

  Examples:

      (%i1) x^3 - 1;
                                    3
      (%o1)                        x  - 1
      (%i2) x^3 - 1, factor;
                                      2
      (%o2)                 (x - 1) (x  + x + 1)
      (%i3) factor (x^3 - 1);
                                      2
      (%o3)                 (x - 1) (x  + x + 1)
      (%i4) cos(4 * x) / sin(x)^4;
                                  cos(4 x)
      (%o4)                       --------
                                     4
                                  sin (x)
      (%i5) cos(4 * x) / sin(x)^4, trigexpand;
                       4           2       2         4
                    sin (x) - 6 cos (x) sin (x) + cos (x)
      (%o5)         -------------------------------------
                                      4
                                   sin (x)
      (%i6) cos(4 * x) / sin(x)^4, trigexpand, ratexpand;
                                 2         4
                            6 cos (x)   cos (x)
      (%o6)               - --------- + ------- + 1
                                2          4
                             sin (x)    sin (x)
      (%i7) ratexpand (trigexpand (cos(4 * x) / sin(x)^4));
                                 2         4
                            6 cos (x)   cos (x)
      (%o7)               - --------- + ------- + 1
                                2          4
                             sin (x)    sin (x)
      (%i8) declare ([F, G], evfun);
      (%o8)                         done
      (%i9) (aa : bb, bb : cc, cc : dd);
      (%o9)                          dd
      (%i10) aa;
      (%o10)                         bb
      (%i11) aa, F;
      (%o11)                        F(cc)
      (%i12) F (aa);
      (%o12)                        F(bb)
      (%i13) F (ev (aa));
      (%o13)                        F(cc)
      (%i14) aa, F, G;
      (%o14)                      G(F(cc))
      (%i15) G (F (ev (aa)));
      (%o15)                      G(F(cc))

  Categories: Evaluation flags

Option variable: infeval

  Enables "infinite evaluation" mode. `ev` repeatedly evaluates an expression until it stops changing. To prevent a variable, say `X`, from being evaluated away in this mode, simply include `X='X` as an argument to `ev`. Of course expressions such as `ev (X, X=X+1, infeval)` will generate an infinite loop.

  Categories: Evaluation flags

Special symbol: noeval

  `noeval` suppresses the evaluation phase of `ev`. This is useful in conjunction with other switches and in causing expressions to be resimplified without being reevaluated.

  Categories: Evaluation flags

Special symbol: nouns

  `nouns` is an `evflag`. When used as an option to the `ev` command, `nouns` converts all "noun" forms occurring in the expression being `ev`'d to "verbs", i.e., evaluates them. See also `noun`, `nounify`, `verb`, and `verbify`.

  Categories: Evaluation flags · Nouns and verbs

Special symbol: pred

  As an argument in a call to `ev (expr)`, `pred` causes predicates (expressions which evaluate to `true` or `false`) to be evaluated. See `ev`.

  Example:

      (%i1) 1<2;
      (%o1)                                1 < 2
      (%i2) 1<2,pred;
      (%o2)                                true

  Categories: Evaluation flags

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# 9. Simplification

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## 9.1 Introduction to Simplification

Maxima performs a cycle of actions in response to each new user-typed command. This consists of four steps: reading or "parsing" the input, evaluation, simplification and output. Parsing converts a syntactically valid sequence of typed characters into a data structure to be used for the rest of the operations. Evaluation includes replacing names with their assigned values. Simplification means rewriting an expression to be easier for the user or other programs to understand. Output includes displaying computational results in a variety of different formats and notations.

Evaluation and simplification sometimes appear to have similar functionality as they both have the goal of removing "complexity" and system designers have sometimes divided a task so that it is performed partly in each. For example, `integrate(x,x)` evaluates the answer as `x*x/2`, which is then simplified to `x^2/2`.

Evaluation is always present: it is the consequence of having a programming system with functions, subroutines, variables, values, loops, assignments and so on. In the evaluation step, built-in or user-defined function names are replaced by their definitions, variables are replaced by their values. This is largely the same as activities of a conventional programming language, but extended to work with symbolic mathematical data. Because of the generality of the mathematics at hand, there are different possible models of evaluation and so the systems has optional "flags" that can steer the process of evaluation. See section Functions and Variables for Evaluation.

By contrast, the intent of simplification is to maintain the value of an expression while re-formulating its representation to be smaller, simpler to understand, or to conform to particular specifications (like factored, expanded). For example, `sin(0)` to `0` or `x+x to 2*x`. There are several powerful tools to alter the results of simplification, since it is largely in this part of the system that a user can incorporate knowledge of newly introduced functions or symbolic notation into Maxima.

Simplification is generally done at four different levels:

- The internal, built-in automated simplifier,
- Built-in simplification routines that can be explicitly called by the user at selected places in a program or command sequence,
- User-written simplification routines, linked to the simplifier by using "tellsimp" or "tellsimpafter" and called automatically,
- User-written routines that can be explicitly called by the user at selected places in a program or command sequence.

The internal simplifier belongs to the heart of Maxima. It is a large and complicated collection of programs, and it has been refined over many years and by thousands of users. Nevertheless, especially if you are trying out novel ideas or unconventional notation, you may find it helpful to make small (or large) changes to the program yourself. For details see for example the paper at the end of https://people.eecs.berkeley.edu/~fateman/papers/intro5.txt.

Maxima internally represents expressions as "trees" with operators or "roots" like `+`, `*` , `=` and operands ("leaves") which are variables like x, y, z, functions or sub-trees, like `x*y`. Each operator has a simplification program associated with it. `+` (which also covers binary `-` since `a-b = a+(-1)*b)` and `*` (which also covers `/` since `a/b = a*b^(-1)`) have rather elaborate simplification programs. These simplification programs (simplus, simptimes, simpexpt, etc.) are called whenever the simplifier encounters the respective arithmetic operators in an expression tree to be analyzed.

The structure of the simplifier dates back to 1965, and many hands have worked on it through the years. The structure turns out to be, in modern jargon, data- directed, or object-oriented. The program dispatches to the appropriate routine depending on the root of some sub-tree of the expression, recursively. This general notion means you can make modifications to the simplification process by very local changes to the program. In many cases it is conceptually straightforward to add an operator and add its simplification routine without disturbing existing code.

We note that in addition to this general simplifier operating on algebraic expression trees, there are several other representations of expressions in Maxima which have separate methods and simplifiers. For example, the `rat()` function converts polynomials to vectors of coefficients to assist in rapid manipulation of such forms. Other representations include Taylor series and the (rarely used) Poisson series.

All operators introduced by the user initially have no simplification programs associated with them. Maxima does not know anything about function "f" and so typing `f(a,b)` will result in simplifying a,b, but not `f`. Even some built-in operators have no simplifications. For example, `=` does not "simplify" - it is a place-holder with no simplification semantics other than to simplify its two arguments, in this case referred to as the left and right sides. Other parts of Maxima such as the solve program take special note of equations, that is, trees with `=` as the root. (Note - in Maxima, the assignment operation is `:` . That is, `q: 4` sets the value of the symbol q to `4`. Function definition is done with `:=`. )

The general simplifier returns results with an internal flag indicating the expression and each sub-expression has been simplified. This does not guarantee that it is unique over all possible equivalent expressions. That's too hard (theoretically, not possible given the generality of what can be expressed in Maxima). However, some aspects of the expression, such as the ordering of terms in a sum or product, are made uniform. This is important for the other programs to work properly.

You can set a number of option variables which direct Maxima's processing to favor particular kinds of patterns as being goals. You can even use the most extreme option which is to turn the simplifier off by simp:false. We do not recommend this since many internal routines expect their arguments to be simplified. (About the only time it seems plausible to turn off the simplifier is in the rare case that you want to over-ride a built-in simplification. In that case you might temporarily disable the simplifier, put in the new transformation via `tellsimp`, and then re-enable the simplifier by `simp:true`.)

It is more plausible for you to associate user-defined symbolic function names or operators with properties (`additive`, `lassociative`, `oddfun`, `antisymmetric`, `linear`, `outative`, `commutative`, `multiplicative`, `rassociative`, `evenfun`, `nary` and `symmetric`). These options steer the simplifier processing in systematic directions.

For example, `declare(f,oddfun)` specifies that `f` is an odd function. Maxima will simplify `f(-x)` to `-f(x)`. In the case of an even function, that is `declare(g,evenfun)`, Maxima will simplify `g(-x)` to `g(x)`. You can also associate a programming function with a name such as `h(x):=x^2+1`. In that case the evaluator will immediately replace `h(3)` by `10`, and `h(a+1)` by `(a+1)^2+1`, so any properties of `h` will be ignored.

In addition to these directly related properties set up by the user, facts and properties from the actual context may have an impact on the simplifier's behavior, too. See section Introduction to Maximas Database.

Example: `sin(n*%pi)` is simplified to zero, if n is an integer.

    (%i1) sin(n*%pi);
    (%o1)                      sin(%pi n)
    (%i2) declare(n, integer);
    (%o2)                         done
    (%i3) sin(n*%pi);
    (%o3)                           0

If automated simplification is not sufficient, you can consider a variety of built-in, but explicitly called simplfication functions (`ratsimp`, `expand`, `factor`, `radcan` and others). There are also flags that will push simplification into one or another direction. Given `demoivre:true` the simplifier rewrites complex exponentials as trigonometric forms. Given `exponentialize:true` the simplifier tries to do the reverse: rewrite trigonometric forms as complex exponentials.

As everywhere in Maxima, by writing your own functions (be it in the Maxima user language or in the implementation language Lisp) and explicitly calling them at selected places in the program, you can respond to your individual simplification needs. Lisp gives you a handle on all the internal mechanisms, but you rarely need this full generality. "Tellsimp" is designed to generate much of the Lisp internal interface into the simplifier automatically. See See section Rules and Patterns.

Over the years (Maxima/Macsyma's origins date back to about 1966!) users have contributed numerous application packages and tools to extend or alter its functional behavior. Various non-standard and "share" packages exist to modify or extend simplification as well. You are invited to look into this more experimental material where work is still in progress. See section simplification

The following appended material is optional on a first reading, and reading it is not necessary for productive use of Maxima. It is for the curious user who wants to understand what is going on, or the ambitious programmer who might wish to change the (open-source) code. Experimentation with redefining Maxima Lisp code is easily possible: to change the definition of a Lisp program (say the one that simplifies `cos()`, named `simp%cos`), you simply load into Maxima a text file that will overwrite the `simp%cos` function from the maxima package.

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## 9.2 Functions and Variables for Simplification

Property: additive

  If `declare(f,additive)` has been executed, then:

  (1) If `f` is univariate, whenever the simplifier encounters `f` applied to a sum, `f` will be distributed over that sum. I.e. `f(y+x)` will simplify to `f(y)+f(x)`.

  (2) If `f` is a function of 2 or more arguments, additivity is defined as additivity in the first argument to `f`, as in the case of `sum` or `integrate`, i.e. `f(h(x)+g(x),x)` will simplify to `f(h(x),x)+f(g(x),x)`. This simplification does not occur when `f` is applied to expressions of the form `sum(x[i],i,lower-limit,upper-limit)`.

  Example:

      (%i1) F3 (a + b + c);
      (%o1)                     F3(c + b + a)
      (%i2) declare (F3, additive);
      (%o2)                         done
      (%i3) F3 (a + b + c);
      (%o3)                 F3(c) + F3(b) + F3(a)

  Categories: Operators · Declarations and inferences

Property: antisymmetric

  If `declare(h,antisymmetric)` is done, this tells the simplifier that `h` is antisymmetric. E.g. `h(x,z,y)` will simplify to `- h(x, y, z)`. That is, it will give (-1)^n times the result given by `symmetric` or `commutative`, where n is the number of interchanges of two arguments necessary to convert it to that form.

  Examples:

      (%i1) S (b, a);
      (%o1)                        S(b, a)
      (%i2) declare (S, symmetric);
      (%o2)                         done
      (%i3) S (b, a);
      (%o3)                        S(a, b)
      (%i4) S (a, c, e, d, b);
      (%o4)                   S(a, b, c, d, e)
      (%i5) T (b, a);
      (%o5)                        T(b, a)
      (%i6) declare (T, antisymmetric);
      (%o6)                         done
      (%i7) T (b, a);
      (%o7)                       - T(a, b)
      (%i8) T (a, c, e, d, b);
      (%o8)                   T(a, b, c, d, e)

  Categories: Operators · Declarations and inferences

Function: combine (expr)

  Simplifies the sum expr by combining terms with the same denominator into a single term.

  Example:

      (%i1) 1*f/2*b + 2*c/3*a + 3*f/4*b +c/5*b*a;
                            5 b f   a b c   2 a c
      (%o1)                 ----- + ----- + -----
                              4       5       3
      (%i2) combine (%);
                        75 b f + 4 (3 a b c + 10 a c)
      (%o2)             -----------------------------
                                     60

  Categories: Expressions

Property: commutative

  If `declare(h, commutative)` is done, this tells the simplifier that `h` is a commutative function. E.g. `h(x, z, y)` will simplify to `h(x, y, z)`. This is the same as `symmetric`.

  Exemplo:

      (%i1) S (b, a);
      (%o1)                        S(b, a)
      (%i2) S (a, b) + S (b, a);
      (%o2)                   S(b, a) + S(a, b)
      (%i3) declare (S, commutative);
      (%o3)                         done
      (%i4) S (b, a);
      (%o4)                        S(a, b)
      (%i5) S (a, b) + S (b, a);
      (%o5)                       2 S(a, b)
      (%i6) S (a, c, e, d, b);
      (%o6)                   S(a, b, c, d, e)

  Categories: Operators · Declarations and inferences

Function: demoivre (expr)
Option variable: demoivre

  The function `demoivre (expr)` converts one expression without setting the global variable `demoivre`.

  When the variable `demoivre` is `true`, complex exponentials are converted into equivalent expressions in terms of circular functions: `exp (a + b*%i)` simplifies to `%e^a * (cos(b) + %i*sin(b))` if `b` is free of `%i`. `a` and `b` are not expanded.

  The default value of `demoivre` is `false`.

  `exponentialize` converts circular and hyperbolic functions to exponential form. `demoivre` and `exponentialize` cannot both be true at the same time.

  Categories: Complex variables · Trigonometric functions · Hyperbolic functions

Function: distrib (expr)

  Distributes sums over products. It differs from `expand` in that it works at only the top level of an expression, i.e., it doesn't recurse and it is faster than `expand`. It differs from `multthru` in that it expands all sums at that level.

  Examples:

      (%i1) distrib ((a+b) * (c+d));
      (%o1)                 b d + a d + b c + a c
      (%i2) multthru ((a+b) * (c+d));
      (%o2)                 (b + a) d + (b + a) c
      (%i3) distrib (1/((a+b) * (c+d)));
                                      1
      (%o3)                    ---------------
                               (b + a) (d + c)
      (%i4) expand (1/((a+b) * (c+d)), 1, 0);
                                      1
      (%o4)                 ---------------------
                            b d + a d + b c + a c

  Categories: Expressions

Option variable: distribute_over

  Default value: `true`

  `distribute_over` controls the mapping of functions over bags like lists, matrices, and equations. At this time not all Maxima functions have this property. It is possible to look up this property with the command `properties`.

  The mapping of functions is switched off, when setting `distribute_over` to the value `false`.

  Examples:

  The `sin` function maps over a list:

      (%i1) sin([x,1,1.0]);
      (%o1)         [sin(x), sin(1), 0.8414709848078965]

  `mod` is a function with two arguments which maps over lists. Mapping over nested lists is possible too:

      (%i1) mod([x,11,2*a],10);
      (%o1)             [mod(x, 10), 1, 2 mod(a, 5)]
      (%i2) mod([[x,y,z],11,2*a],10);
      (%o2) [[mod(x, 10), mod(y, 10), mod(z, 10)], 1, 2 mod(a, 5)]

  Mapping of the `floor` function over a matrix and an equation:

      (%i1) floor(matrix([a,b],[c,d]));
                           [ floor(a)  floor(b) ]
      (%o1)                [                    ]
                           [ floor(c)  floor(d) ]
      (%i2) floor(a=b);
      (%o2)                  floor(a) = floor(b)

  Functions with more than one argument map over any of the arguments or all arguments:

      (%i1) expintegral_e([1,2],[x,y]);
      (%o1) [[expintegral_e(1, x), expintegral_e(1, y)],
                            [expintegral_e(2, x), expintegral_e(2, y)]]

  Check if a function has the property distribute_over:

      (%i1) properties(abs);
      (%o1) [integral, rule, distributes over bags, noun, gradef,
                                                       system function]

  The mapping of functions is switched off, when setting `distribute_over` to the value `false`.

      (%i1) distribute_over;
      (%o1)                         true
      (%i2) sin([x,1,1.0]);
      (%o2)         [sin(x), sin(1), 0.8414709848078965]
      (%i3) distribute_over : not distribute_over;
      (%o3)                         false
      (%i4) sin([x,1,1.0]);
      (%o4)                   sin([x, 1, 1.0])

  Categories: Simplification flags and variables

Option variable: domain

  Default value: `real`

  When `domain` is set to `complex`, `sqrt (x^2)` will remain `sqrt (x^2)` instead of returning `abs(x)`.

  Categories: Simplification flags and variables

Property: evenfun
Property: oddfun

  `declare(f, evenfun)` or `declare(f, oddfun)` tells Maxima to recognize the function `f` as an even or odd function.

  Examples:

      (%i1) o (- x) + o (x);
      (%o1)                     o(x) + o(- x)
      (%i2) declare (o, oddfun);
      (%o2)                         done
      (%i3) o (- x) + o (x);
      (%o3)                           0
      (%i4) e (- x) - e (x);
      (%o4)                     e(- x) - e(x)
      (%i5) declare (e, evenfun);
      (%o5)                         done
      (%i6) e (- x) - e (x);
      (%o6)                           0

Function: expand
`expand` (expr)
`expand` (expr, p, n)

  Expand expression expr. Products of sums and exponentiated sums are multiplied out, numerators of rational expressions which are sums are split into their respective terms, and multiplication (commutative and non-commutative) are distributed over addition at all levels of expr.

  For polynomials one should usually use `ratexpand` which uses a more efficient algorithm.

  `maxnegex` and `maxposex` control the maximum negative and positive exponents, respectively, which will expand.

  `expand (expr, p, n)` expands expr, using p for `maxposex` and n for `maxnegex`. This is useful in order to expand part but not all of an expression.

  `expon` - the exponent of the largest negative power which is automatically expanded (independent of calls to `expand`). For example if `expon` is 4 then `(x+1)^(-5)` will not be automatically expanded.

  `expop` - the highest positive exponent which is automatically expanded. Thus `(x+1)^3`, when typed, will be automatically expanded only if `expop` is greater than or equal to 3. If it is desired to have `(x+1)^n` expanded where `n` is greater than `expop` then executing `expand ((x+1)^n)` will work only if `maxposex` is not less than `n`.

  `expand(expr, 0, 0)` causes a resimplification of `expr`. `expr` is not reevaluated. In distinction from `ev(expr, noeval)` a special representation (e. g. a CRE form) is removed. See also `ev`.

  The `expand` flag used with `ev` causes expansion.

  The file ``share/simplification/facexp.mac'` contains several related functions (in particular `facsum`, `factorfacsum` and `collectterms`, which are autoloaded) and variables (`nextlayerfactor` and `facsum_combine`) that provide the user with the ability to structure expressions by controlled expansion. Brief function descriptions are available in ``simplification/facexp.usg'`. A demo is available by doing `demo("facexp")`.

  Examples:

      (%i1) expr:(x+1)^2*(y+1)^3;
                                     2        3
      (%o1)                   (x + 1)  (y + 1)
      (%i2) expand(expr);
             2  3        3    3      2  2        2      2      2
      (%o2) x  y  + 2 x y  + y  + 3 x  y  + 6 x y  + 3 y  + 3 x  y
                                                            2
                                           + 6 x y + 3 y + x  + 2 x + 1
      (%i3) expand(expr,2);
                     2        3              3          3
      (%o3)         x  (y + 1)  + 2 x (y + 1)  + (y + 1)
      (%i4) expr:(x+1)^-2*(y+1)^3;
                                         3
                                  (y + 1)
      (%o4)                       --------
                                         2
                                  (x + 1)
      (%i5) expand(expr);
                  3               2
                 y             3 y            3 y             1
      (%o5) ------------ + ------------ + ------------ + ------------
             2              2              2              2
            x  + 2 x + 1   x  + 2 x + 1   x  + 2 x + 1   x  + 2 x + 1
      (%i6) expand(expr,2,2);
                                         3
                                  (y + 1)
      (%o6)                     ------------
                                 2
                                x  + 2 x + 1

  Resimplify an expression without expansion:

      (%i1) expr:(1+x)^2*sin(x);
                                      2
      (%o1)                    (x + 1)  sin(x)
      (%i2) exponentialize:true;
      (%o2)                         true
      (%i3) expand(expr,0,0);
                                  2    %i x     - %i x
                        %i (x + 1)  (%e     - %e      )
      (%o3)           - -------------------------------
                                       2

  Categories: Expressions

Function: expandwrt (expr, x_1, …, x_n)

  Expands expression `expr` with respect to the variables x_1, …, x_n. All products involving the variables appear explicitly. The form returned will be free of products of sums of expressions that are not free of the variables. x_1, …, x_n may be variables, operators, or expressions.

  By default, denominators are not expanded, but this can be controlled by means of the switch `expandwrt_denom`.

  This function is autoloaded from ``simplification/stopex.mac'`.

  Categories: Expressions

Option variable: expandwrt_denom

  Default value: `false`

  `expandwrt_denom` controls the treatment of rational expressions by `expandwrt`. If `true`, then both the numerator and denominator of the expression will be expanded according to the arguments of `expandwrt`, but if `expandwrt_denom` is `false`, then only the numerator will be expanded in that way.

  Categories: Expressions

Function: expandwrt_factored (expr, x_1, …, x_n)

  is similar to `expandwrt`, but treats expressions that are products somewhat differently. `expandwrt_factored` expands only on those factors of `expr` that contain the variables x_1, …, x_n.

  This function is autoloaded from ``simplification/stopex.mac'`.

  Categories: Expressions

Option variable: expon

  Default value: 0

  `expon` is the exponent of the largest negative power which is automatically expanded (independent of calls to `expand`). For example, if `expon` is 4 then `(x+1)^(-5)` will not be automatically expanded.

  Categories: Expressions

Function: exponentialize (expr)
Option variable: exponentialize

  The function `exponentialize (expr)` converts circular and hyperbolic functions in expr to exponentials, without setting the global variable `exponentialize`.

  When the variable `exponentialize` is `true`, all circular and hyperbolic functions are converted to exponential form. The default value is `false`.

  `demoivre` converts complex exponentials into circular functions. `exponentialize` and `demoivre` cannot both be true at the same time.

  Categories: Complex variables · Trigonometric functions · Hyperbolic functions

Option variable: expop

  Default value: 0

  `expop` is the highest positive exponent which is automatically expanded. Thus `(x + 1)^3`, when typed, will be automatically expanded only if `expop` is greater than or equal to 3. If it is desired to have `(x + 1)^n` expanded where `n` is greater than `expop` then executing `expand ((x + 1)^n)` will work only if `maxposex` is not less than n.

  Categories: Expressions

Property: lassociative

  `declare (g, lassociative)` tells the Maxima simplifier that `g` is left-associative. E.g., `g (g (a, b), g (c, d))` will simplify to `g (g (g (a, b), c), d)`.

  Categories: Declarations and inferences · Operators · Simplification

Property: linear

  One of Maxima's operator properties. For univariate `f` so declared, "expansion" `f(x + y)` yields `f(x) + f(y)`, `f(a*x)` yields `a*f(x)` takes place where `a` is a "constant". For functions of two or more arguments, "linearity" is defined to be as in the case of `sum` or `integrate`, i.e., `f (a*x + b, x)` yields `a*f(x,x) + b*f(1,x)` for `a` and `b` free of `x`.

  Example:

      (%i1) declare (f, linear);
      (%o1)                         done
      (%i2) f(x+y);
      (%o2)                      f(y) + f(x)
      (%i3) declare (a, constant);
      (%o3)                         done
      (%i4) f(a*x);
      (%o4)                        a f(x)

  `linear` is equivalent to `additive` and `outative`. See also `opproperties`.

  Example:

      (%i1) 'sum (F(k) + G(k), k, 1, inf);
                             inf
                             ====
                             \
      (%o1)                   >    (G(k) + F(k))
                             /
                             ====
                             k = 1
      (%i2) declare (nounify (sum), linear);
      (%o2)                         done
      (%i3) 'sum (F(k) + G(k), k, 1, inf);
                           inf          inf
                           ====         ====
                           \            \
      (%o3)                 >    G(k) +  >    F(k)
                           /            /
                           ====         ====
                           k = 1        k = 1

  Categories: Declarations and inferences · Operators · Simplification

Option variable: maxnegex

  Default value: 1000

  `maxnegex` is the largest negative exponent which will be expanded by the `expand` command, see also `maxposex`.

  Categories: Expressions

Option variable: maxposex

  Default value: 1000

  `maxposex` is the largest exponent which will be expanded with the `expand` command, see also `maxnegex`.

  Categories: Expressions

Property: multiplicative

  `declare(f, multiplicative)` tells the Maxima simplifier that `f` is multiplicative.

  1. If `f` is univariate, whenever the simplifier encounters `f` applied to a product, `f` distributes over that product. E.g., `f(x*y)` simplifies to `f(x)*f(y)`. This simplification is not applied to expressions of the form `f('product(...))`.
  2. If `f` is a function of 2 or more arguments, multiplicativity is defined as multiplicativity in the first argument to `f`, e.g., `f (g(x) * h(x), x)` simplifies to `f (g(x) ,x) * f (h(x), x)`.

  `declare(nounify(product), multiplicative)` tells Maxima to simplify symbolic products.

  Example:

      (%i1) F2 (a * b * c);
      (%o1)                       F2(a b c)
      (%i2) declare (F2, multiplicative);
      (%o2)                         done
      (%i3) F2 (a * b * c);
      (%o3)                   F2(a) F2(b) F2(c)

  `declare(nounify(product), multiplicative)` tells Maxima to simplify symbolic products.

      (%i1) product (a[i] * b[i], i, 1, n);
                                   n
                                 /===\
                                  ! !
      (%o1)                       ! !  a  b
                                  ! !   i  i
                                 i = 1
      (%i2) declare (nounify (product), multiplicative);
      (%o2)                         done
      (%i3) product (a[i] * b[i], i, 1, n);
                                n         n
                              /===\     /===\
                               ! !       ! !
      (%o3)                  ( ! !  a )  ! !  b
                               ! !   i   ! !   i
                              i = 1     i = 1

  Categories: Declarations and inferences · Expressions · Simplification

Function: multthru
`multthru` (expr)
`multthru` (expr_1, expr_2)

  Multiplies a factor (which should be a sum) of expr by the other factors of expr. That is, expr is `f_1 f_2 ... f_n` where at least one factor, say f_i, is a sum of terms. Each term in that sum is multiplied by the other factors in the product. (Namely all the factors except f_i). `multthru` does not expand exponentiated sums. This function is the fastest way to distribute products (commutative or noncommutative) over sums. Since quotients are represented as products `multthru` can be used to divide sums by products as well.

  `multthru (expr_1, expr_2)` multiplies each term in expr_2 (which should be a sum or an equation) by expr_1. If expr_1 is not itself a sum then this form is equivalent to `multthru (expr_1*expr_2)`.

      (%i1) x/(x-y)^2 - 1/(x-y) - f(x)/(x-y)^3;
                            1        x         f(x)
      (%o1)             - ----- + -------- - --------
                          x - y          2          3
                                  (x - y)    (x - y)
      (%i2) multthru ((x-y)^3, %);
                                 2
      (%o2)             - (x - y)  + x (x - y) - f(x)
      (%i3) ratexpand (%);
                                 2
      (%o3)                   - y  + x y - f(x)
      (%i4) ((a+b)^10*s^2 + 2*a*b*s + (a*b)^2)/(a*b*s^2);
                              10  2              2  2
                       (b + a)   s  + 2 a b s + a  b
      (%o4)            ------------------------------
                                        2
                                   a b s
      (%i5) multthru (%);  /* note that this does not expand (b+a)^10 */
                                              10
                             2   a b   (b + a)
      (%o5)                  - + --- + ---------
                             s    2       a b
                                 s
      (%i6) multthru (a.(b+c.(d+e)+f));
      (%o6)            a . f + a . c . (e + d) + a . b
      (%i7) expand (a.(b+c.(d+e)+f));
      (%o7)         a . f + a . c . e + a . c . d + a . b

  Categories: Expressions

Property: nary

  `declare(f, nary)` tells Maxima to recognize the function `f` as an n-ary function.

  The `nary` declaration is not the same as calling the `nary` function. The sole effect of `declare(f, nary)` is to instruct the Maxima simplifier to flatten nested expressions, for example, to simplify `foo(x, foo(y, z))` to `foo(x, y, z)`. See also `declare`.

  Example:

      (%i1) H (H (a, b), H (c, H (d, e)));
      (%o1)               H(H(a, b), H(c, H(d, e)))
      (%i2) declare (H, nary);
      (%o2)                         done
      (%i3) H (H (a, b), H (c, H (d, e)));
      (%o3)                   H(a, b, c, d, e)

Option variable: negdistrib

  Default value: `true`

  When `negdistrib` is `true`, -1 distributes over an expression. E.g., `-(x + y)` becomes `- y - x`. Setting it to `false` will allow `- (x + y)` to be displayed like that. This is sometimes useful but be very careful: like the `simp` flag, this is one flag you do not want to set to `false` as a matter of course or necessarily for other than local use in your Maxima.

  Example:

      (%i1) negdistrib;
      (%o1)                         true
      (%i2) -(x+y);
      (%o2)                       (- y) - x
      (%i3) negdistrib : not negdistrib ;
      (%o3)                         false
      (%i4) -(x+y);
      (%o4)                       - (y + x)

  Categories: Simplification flags and variables

System variable: opproperties

  `opproperties` is the list of the special operator properties recognized by the Maxima simplifier:

  Example:

      (%i1) opproperties;
      (%o1) [linear, additive, multiplicative, outative, evenfun,
      oddfun, commutative, symmetric, antisymmetric, nary,
      lassociative, rassociative]

  Categories: Global variables · Operators

Property: outative

  `declare(f, outative)` tells the Maxima simplifier that constant factors in the argument of `f` can be pulled out.

  1. If `f` is univariate, whenever the simplifier encounters `f` applied to a product, that product will be partitioned into factors that are constant and factors that are not and the constant factors will be pulled out. E.g., `f(a*x)` will simplify to `a*f(x)` where `a` is a constant. Non-atomic constant factors will not be pulled out.
  2. If `f` is a function of 2 or more arguments, outativity is defined as in the case of `sum` or `integrate`, i.e., `f (a*g(x), x)` will simplify to `a * f(g(x), x)` for `a` free of `x`.

  `sum`, `integrate`, and `limit` are all `outative`.

  Example:

      (%i1) F1 (100 * x);
      (%o1)                       F1(100 x)
      (%i2) declare (F1, outative);
      (%o2)                         done
      (%i3) F1 (100 * x);
      (%o3)                       100 F1(x)
      (%i4) declare (zz, constant);
      (%o4)                         done
      (%i5) F1 (zz * y);
      (%o5)                       zz F1(y)

  Categories: Declarations and inferences · Operators

Function: radcan (expr)

  Simplifies expr, which can contain logs, exponentials, and radicals, by converting it into a form which is canonical over a large class of expressions and a given ordering of variables; that is, all functionally equivalent forms are mapped into a unique form. For a somewhat larger class of expressions, `radcan` produces a regular form. Two equivalent expressions in this class do not necessarily have the same appearance, but their difference can be simplified by `radcan` to zero.

  For some expressions `radcan` is quite time consuming. This is the cost of exploring certain relationships among the components of the expression for simplifications based on factoring and partial-fraction expansions of exponents.

  Examples:

      (%i1) radcan((log(x+x^2)-log(x))^a/log(1+x)^(a/2));
                                          a/2
      (%o1)                     log(x + 1)
      (%i2) radcan((log(1+2*a^x+a^(2*x))/log(1+a^x)));
      (%o2)                           2
      (%i3) radcan((%e^x-1)/(1+%e^(x/2)));
                                    x/2
      (%o3)                       %e    - 1

  Categories: Simplification functions

Option variable: radexpand

  Default value: `true`

  `radexpand` controls some simplifications of radicals.

  When `radexpand` is `all`, causes nth roots of factors of a product which are powers of n to be pulled outside of the radical. E.g. if `radexpand` is `all`, `sqrt (16*x^2)` simplifies to `4*x`.

  More particularly, consider `sqrt (x^2)`.

  - If `radexpand` is `all` or `assume (x > 0)` has been executed, `sqrt(x^2)` simplifies to `x`.
  - If `radexpand` is `true` and `domain` is `real` (its default), `sqrt(x^2)` simplifies to `abs(x)`.
  - If `radexpand` is `false`, or `radexpand` is `true` and `domain` is `complex`, `sqrt(x^2)` is not simplified.

  Note that `domain` only matters when `radexpand` is `true`.

  Categories: Simplification flags and variables

Property: rassociative

  `declare (g, rassociative)` tells the Maxima simplifier that `g` is right-associative. E.g., `g(g(a, b), g(c, d))` simplifies to `g(a, g(b, g(c, d)))`.

  Categories: Declarations and inferences · Operators

Function: scsimp (expr, rule_1, …, rule_n)

  Sequential Comparative Simplification (method due to Stoute). `scsimp` attempts to simplify expr according to the rules rule_1, …, rule_n. If a smaller expression is obtained, the process repeats. Otherwise after all simplifications are tried, it returns the original answer.

  `example (scsimp)` displays some examples.

  Categories: Simplification functions

Option variable: simp

  Default value: `true`

  `simp` enables simplification. This is the default. `simp` is also an `evflag`, which is recognized by the function `ev`. See `ev`.

  When `simp` is used as an `evflag` with a value `false`, the simplification is suppressed only during the evaluation phase of an expression. The flag does not suppress the simplification which follows the evaluation phase.

  Many Maxima functions and operations require simplification to be enabled to work normally. When simplification is disabled, many results will be incomplete, and in addition there may be incorrect results or program errors.

  Examples:

  The simplification is switched off globally. The expression `sin(1.0)` is not simplified to its numerical value. The `simp`-flag switches the simplification on.

      (%i1) simp:false;
      (%o1)                         false
      (%i2) sin(1.0);
      (%o2)                       sin(1.0)
      (%i3) sin(1.0),simp;
      (%o3)                  0.8414709848078965

  The simplification is switched on again. The `simp`-flag cannot suppress the simplification completely. The output shows a simplified expression, but the variable `x` has an unsimplified expression as a value, because the assignment has occurred during the evaluation phase of the expression.

      (%i1) simp:true;
      (%o1)                         true
      (%i2) x:sin(1.0),simp:false;
      (%o2)                  0.8414709848078965
      (%i3) :lisp $x
      ((%SIN) 1.0)

  Categories: Evaluation flags

Property: symmetric

  `declare (h, symmetric)` tells the Maxima simplifier that `h` is a symmetric function. E.g., `h (x, z, y)` simplifies to `h (x, y, z)`.

  `commutative` is synonymous with `symmetric`.

  Categories: Declarations and inferences · Operators

Function: xthru (expr)

  Combines all terms of expr (which should be a sum) over a common denominator without expanding products and exponentiated sums as `ratsimp` does. `xthru` cancels common factors in the numerator and denominator of rational expressions but only if the factors are explicit.

  Sometimes it is better to use `xthru` before `ratsimp`ing an expression in order to cause explicit factors of the gcd of the numerator and denominator to be canceled thus simplifying the expression to be `ratsimp`ed.

  Examples:

      (%i1) ((x+2)^20 - 2*y)/(x+y)^20 + (x+y)^(-19) - x/(x+y)^20;
                                      20
                       1       (x + 2)   - 2 y       x
      (%o1)        --------- + --------------- - ---------
                          19             20             20
                   (y + x)        (y + x)        (y + x)
      (%i2) xthru (%);
                                       20
                                (x + 2)   - y
      (%o2)                     -------------
                                         20
                                  (y + x)

  Categories: Expressions

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# 10. Mathematical Functions

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## 10.1 Functions for Numbers

Function: abs (z)

  The `abs` function represents the mathematical absolute value function and works for both numerical and symbolic values. If the argument, z, is a real or complex number, `abs` returns the absolute value of z. If possible, symbolic expressions using the absolute value function are also simplified.

  Maxima can differentiate, integrate and calculate limits for expressions containing `abs`. The `abs_integrate` package further extends Maxima's ability to calculate integrals involving the abs function. See (%i12) in the examples below.

  When applied to a list or matrix, `abs` automatically distributes over the terms. Similarly, it distributes over both sides of an equation. To alter this behaviour, see the variable `distribute_over`.

  See also `cabs`.

  Examples:

  Calculation of `abs` for real and complex numbers, including numerical constants and various infinities. The first example shows how `abs` distributes over the elements of a list.

      (%i1) abs([-4, 0, 1, 1+%i]);
      (%o1)                  [4, 0, 1, sqrt(2)]

      (%i2) abs((1+%i)*(1-%i));
      (%o2)                           2
      (%i3) abs(%e+%i);
                                      2
      (%o3)                    sqrt(%e  + 1)
      (%i4) abs([inf, infinity, minf]);
      (%o4)                   [inf, inf, inf]

  Simplification of expressions containing `abs`:

      (%i5) abs(x^2);
                                      2
      (%o5)                          x
      (%i6) abs(x^3);
                                   2
      (%o6)                       x  abs(x)

      (%i7) abs(abs(x));
      (%o7)                       abs(x)
      (%i8) abs(conjugate(x));
      (%o8)                       abs(x)

  Integrating and differentiating with the `abs` function. Note that more integrals involving the `abs` function can be performed, if the `abs_integrate` package is loaded. The last example shows the Laplace transform of `abs`: see `laplace`.

      (%i9) diff(x*abs(x),x),expand;
      (%o9)                       2 abs(x)

      (%i10) integrate(abs(x),x);
                                   x abs(x)
      (%o10)                       --------
                                      2

      (%i11) integrate(x*abs(x),x);
                                 /
                                 [
      (%o11)                     I x abs(x) dx
                                 ]
                                 /

      (%i12) load(abs_integrate)$
      (%i13) integrate(x*abs(x),x);
                            2           3
                           x  abs(x)   x  signum(x)
      (%o13)               --------- - ------------
                               2            6

      (%i14) integrate(abs(x),x,-2,%pi);
                                     2
                                  %pi
      (%o14)                      ---- + 2
                                   2

      (%i15) laplace(abs(x),x,s);
                                     1
      (%o15)                         --
                                      2
                                     s

  Categories: Mathematical functions

Function: ceiling (x)

  When x is a real number, return the least integer that is greater than or equal to x.

  If x is a constant expression (`10 * %pi`, for example), `ceiling` evaluates x using big floating point numbers, and applies `ceiling` to the resulting big float. Because `ceiling` uses floating point evaluation, it's possible, although unlikely, that `ceiling` could return an erroneous value for constant inputs. To guard against errors, the floating point evaluation is done using three values for `fpprec`.

  For non-constant inputs, `ceiling` tries to return a simplified value. Here are examples of the simplifications that `ceiling` knows about:

      (%i1) ceiling (ceiling (x));
      (%o1)                      ceiling(x)
      (%i2) ceiling (floor (x));
      (%o2)                       floor(x)
      (%i3) declare (n, integer)$
      (%i4) [ceiling (n), ceiling (abs (n)), ceiling (max (n, 6))];
      (%o4)                [n, abs(n), max(6, n)]
      (%i5) assume (x > 0, x < 1)$
      (%i6) ceiling (x);
      (%o6)                           1
      (%i7) tex (ceiling (a));
      $$\left \lceil a \right \rceil$$
      (%o7)                         false

  The `ceiling` function distributes over lists, matrices and equations. See `distribute_over`.

  Finally, for all inputs that are manifestly complex, `ceiling` returns a noun form.

  If the range of a function is a subset of the integers, it can be declared to be `integervalued`. Both the `ceiling` and `floor` functions can use this information; for example:

      (%i1) declare (f, integervalued)$
      (%i2) floor (f(x));
      (%o2)                         f(x)
      (%i3) ceiling (f(x) - 1);
      (%o3)                       f(x) - 1

  Example use:

      (%i1) unitfrac(r) := block([uf : [], q],
          if not(ratnump(r)) then
             error("unitfrac: argument must be a rational number"),
          while r # 0 do (
              uf : cons(q : 1/ceiling(1/r), uf),
              r : r - q),
          reverse(uf));
      (%o1) unitfrac(r) := block([uf : [], q],
      if not ratnump(r) then error("unitfrac: argument must be a rational number"
                                           1
      ), while r # 0 do (uf : cons(q : ----------, uf), r : r - q),
                                               1
                                       ceiling(-)
                                               r
      reverse(uf))
      (%i2) unitfrac (9/10);
                                  1  1  1
      (%o2)                      [-, -, --]
                                  2  3  15
      (%i3) apply ("+", %);
                                     9
      (%o3)                          --
                                     10
      (%i4) unitfrac (-9/10);
                                        1
      (%o4)                       [- 1, --]
                                        10
      (%i5) apply ("+", %);
                                      9
      (%o5)                         - --
                                      10
      (%i6) unitfrac (36/37);
                              1  1  1  1    1
      (%o6)                  [-, -, -, --, ----]
                              2  3  8  69  6808
      (%i7) apply ("+", %);
                                     36
      (%o7)                          --
                                     37

  Categories: Mathematical functions

Function: entier (x)

  Returns the largest integer less than or equal to x where x is numeric. `fix` (as in `fixnum`) is a synonym for this, so `fix(x)` is precisely the same.

  Categories: Mathematical functions

Function: floor (x)

  When x is a real number, return the largest integer that is less than or equal to x.

  If x is a constant expression (`10 * %pi`, for example), `floor` evaluates x using big floating point numbers, and applies `floor` to the resulting big float. Because `floor` uses floating point evaluation, it's possible, although unlikely, that `floor` could return an erroneous value for constant inputs. To guard against errors, the floating point evaluation is done using three values for `fpprec`.

  For non-constant inputs, `floor` tries to return a simplified value. Here are examples of the simplifications that `floor` knows about:

      (%i1) floor (ceiling (x));
      (%o1)                      ceiling(x)
      (%i2) floor (floor (x));
      (%o2)                       floor(x)
      (%i3) declare (n, integer)$
      (%i4) [floor (n), floor (abs (n)), floor (min (n, 6))];
      (%o4)                [n, abs(n), min(6, n)]
      (%i5) assume (x > 0, x < 1)$
      (%i6) floor (x);
      (%o6)                           0
      (%i7) tex (floor (a));
      $$\left \lfloor a \right \rfloor$$
      (%o7)                         false

  The `floor` function distributes over lists, matrices and equations. See `distribute_over`.

  Finally, for all inputs that are manifestly complex, `floor` returns a noun form.

  If the range of a function is a subset of the integers, it can be declared to be `integervalued`. Both the `ceiling` and `floor` functions can use this information; for example:

      (%i1) declare (f, integervalued)$
      (%i2) floor (f(x));
      (%o2)                         f(x)
      (%i3) ceiling (f(x) - 1);
      (%o3)                       f(x) - 1

  Categories: Mathematical functions

Function: fix (x)

  A synonym for `entier (x)`.

  Categories: Mathematical functions

Function: lmax (L)

  When L is a list or a set, return `apply ('max, args (L))`. When L is not a list or a set, signal an error. See also `lmin` and `max`.

  Categories: Mathematical functions · Lists · Sets

Function: lmin (L)

  When L is a list or a set, return `apply ('min, args (L))`. When L is not a list or a set, signal an error. See also `lmax` and `min`.

  Categories: Mathematical functions · Lists · Sets

Function: max (x_1, …, x_n)

  Return a simplified value for the maximum of the expressions x_1 through x_n. When `get (trylevel, maxmin)`, is 2 or greater, `max` uses the simplification `max (e, -e) --> |e|`. When `get (trylevel, maxmin)` is 3 or greater, max tries to eliminate expressions that are between two other arguments; for example, `max (x, 2*x, 3*x) --> max (x, 3*x)`. To set the value of `trylevel` to 2, use `put (trylevel, 2, maxmin)`.

  See also `min` and `lmax`.

  Categories: Mathematical functions

Function: min (x_1, …, x_n)

  Return a simplified value for the minimum of the expressions `x_1` through `x_n`. When `get (trylevel, maxmin)`, is 2 or greater, `min` uses the simplification `min (e, -e) --> -|e|`. When `get (trylevel, maxmin)` is 3 or greater, `min` tries to eliminate expressions that are between two other arguments; for example, `min (x, 2*x, 3*x) --> min (x, 3*x)`. To set the value of `trylevel` to 2, use `put (trylevel, 2, maxmin)`.

  See also `max` and `lmin`.

  Categories: Mathematical functions

Function: round (x)

  When x is a real number, returns the closest integer to x. Multiples of 1/2 are rounded to the nearest even integer. Evaluation of x is similar to `floor` and `ceiling`.

  The `round` function distributes over lists, matrices and equations. See `distribute_over`.

  Categories: Mathematical functions

Function: signum (x)

  For either real or complex numbers x, the signum function returns 0 if x is zero; for a nonzero numeric input x, the signum function returns `x/abs(x)`.

  For non-numeric inputs, Maxima attempts to determine the sign of the input. When the sign is negative, zero, or positive, `signum` returns -1,0, 1, respectively. For all other values for the sign, `signum` a simplified but equivalent form. The simplifications include reflection (`signum(-x)` gives `-signum(x)`) and multiplicative identity (`signum(x*y)` gives `signum(x) * signum(y)`).

  The `signum` function distributes over a list, a matrix, or an equation. See `distribute_over`.

  Categories: Mathematical functions

Function: truncate (x)

  When x is a real number, return the closest integer to x not greater in absolute value than x. Evaluation of x is similar to `floor` and `ceiling`.

  The `truncate` function distributes over lists, matrices and equations. See `distribute_over`.

  Categories: Mathematical functions

[ < ] | [ > ] |  | [ << ] | [ Up ] | [ >> ] |  |  |  |  | [Top] | [Contents] | [Index] | [ ? ]

## 10.2 Functions for Complex Numbers

Function: cabs (expr)

  Calculates the absolute value of an expression representing a complex number. Unlike the function `abs`, the `cabs` function always decomposes its argument into a real and an imaginary part. If `x` and `y` represent real variables or expressions, the `cabs` function calculates the absolute value of `x + %i*y` as

      (%i1) cabs (1);
      (%o1)                           1
      (%i2) cabs (1 + %i);
      (%o2)                        sqrt(2)
      (%i3) cabs (exp (%i));
      (%o3)                           1
      (%i4) cabs (exp (%pi * %i));
      (%o4)                           1
      (%i5) cabs (exp (3/2 * %pi * %i));
      (%o5)                           1
      (%i6) cabs (17 * exp (2 * %i));
      (%o6)                          17

  If `cabs` returns a noun form this most commonly is caused by some properties of the variables involved not being known:

      (%i1) cabs (a+%i*b);
                                      2    2
      (%o1)                     sqrt(b  + a )
      (%i2) declare(a,real,b,real);
      (%o2)                         done
      (%i3) cabs (a+%i*b);
                                      2    2
      (%o3)                     sqrt(b  + a )
      (%i4) assume(a>0,b>0);
      (%o4)                    [a > 0, b > 0]
      (%i5) cabs (a+%i*b);
                                      2    2
      (%o5)                     sqrt(b  + a )

  The `cabs` function can use known properties like symmetry properties of complex functions to help it calculate the absolute value of an expression. If such identities exist, they can be advertised to `cabs` using function properties. The symmetries that `cabs` understands are: mirror symmetry, conjugate function and complex characteristic.

  `cabs` is a verb function and is not suitable for symbolic calculations. For such calculations (including integration, differentiation and taking limits of expressions containing absolute values), use `abs`.

  The result of `cabs` can include the absolute value function, `abs`, and the arc tangent, `atan2`.

  When applied to a list or matrix, `cabs` automatically distributes over the terms. Similarly, it distributes over both sides of an equation.

  For further ways to compute with complex numbers, see the functions `rectform`, `realpart`, `imagpart`, `carg`, `conjugate` and `polarform`.

  Examples:

  Examples with `sqrt` and `sin`.

      (%i1) cabs(sqrt(1+%i*x));
                                   2     1/4
      (%o1)                      (x  + 1)
      (%i2) cabs(sin(x+%i*y));
                          2        2         2        2
      (%o2)       sqrt(cos (x) sinh (y) + sin (x) cosh (y))

  The error function, `erf`, has mirror symmetry, which is used here in the calculation of the absolute value with a complex argument:

      (%i3) cabs(erf(x+%i*y));
                                                2
                 (erf(%i y + x) - erf(%i y - x))
      (%o3) sqrt(--------------------------------
                                4
                                                                     2
                                      (erf(%i y + x) + erf(%i y - x))
                                    - --------------------------------)
                                                     4

  Maxima knows complex identities for the Bessel functions, which allow it to compute the absolute value for complex arguments. Here is an example for `bessel_j`.

      (%i4) cabs(bessel_j(1,%i));
      (%o4)                 abs(bessel_j(1, %i))

  Categories: Complex variables

Function: carg (z)

  Returns the complex argument of z. The complex argument is an angle `theta` in `(-%pi, %pi]` such that `r exp (theta %i) = z` where `r` is the magnitude of z.

  `carg` is a computational function, not a simplifying function.

  See also `abs` (complex magnitude), `polarform`, `rectform`, `realpart`, and `imagpart`.

  Examples:

      (%i1) carg (1);
      (%o1)                           0
      (%i2) carg (1 + %i);
                                     %pi
      (%o2)                          ---
                                      4
      (%i3) carg (exp (%i));
                                     sin(1)
      (%o3)                     atan(------)
                                     cos(1)
      (%i4) carg (exp (%pi * %i));
      (%o4)                          %pi
      (%i5) carg (exp (3/2 * %pi * %i));
                                      %pi
      (%o5)                         - ---
                                       2
      (%i6) carg (17 * exp (2 * %i));
                                  sin(2)
      (%o6)                  atan(------) + %pi
                                  cos(2)

  If `carg` returns a noun form this most communly is caused by some properties of the variables involved not being known:

      (%i1) carg (a+%i*b);
      (%o1)                      atan2(b, a)
      (%i2) declare(a,real,b,real);
      (%o2)                         done
      (%i3) carg (a+%i*b);
      (%o3)                      atan2(b, a)
      (%i4) assume(a>0,b>0);
      (%o4)                    [a > 0, b > 0]
      (%i5) carg (a+%i*b);
                                        b
      (%o5)                        atan(-)
                                        a

  Categories: Complex variables

Function: conjugate (x)

  Returns the complex conjugate of x.

      (%i1) declare ([aa, bb], real, cc, complex, ii, imaginary);
      (%o1)                         done
      (%i2) conjugate (aa + bb*%i);
      (%o2)                      aa - %i bb
      (%i3) conjugate (cc);
      (%o3)                     conjugate(cc)
      (%i4) conjugate (ii);
      (%o4)                         - ii
      (%i5) conjugate (xx + yy);
      (%o5)                        yy + xx

  Categories: Complex variables

Function: imagpart (expr)

  Returns the imaginary part of the expression expr.

  `imagpart` is a computational function, not a simplifying function.

  See also `abs`, `carg`, `polarform`, `rectform`, and `realpart`.

  Example:

      (%i1) imagpart (a+b*%i);
      (%o1)                           b
      (%i2) imagpart (1+sqrt(2)*%i);
      (%o2)                        sqrt(2)
      (%i3) imagpart (1);
      (%o3)                           0
      (%i4) imagpart (sqrt(2)*%i);
      (%o4)                        sqrt(2)

  Categories: Complex variables

Function: polarform (expr)

  Returns an expression `r %e^(%i theta)` equivalent to expr, such that `r` and `theta` are purely real.

  Example:

      (%i1) polarform(a+b*%i);
                             2    2    %i atan2(b, a)
      (%o1)            sqrt(b  + a ) %e
      (%i2) polarform(1+%i);
                                        %i %pi
                                        ------
                                          4
      (%o2)                   sqrt(2) %e
      (%i3) polarform(1+2*%i);
                                      %i atan(2)
      (%o3)                 sqrt(5) %e

  Categories: Complex variables · Exponential and logarithm functions

Function: realpart (expr)

  Returns the real part of expr. `realpart` and `imagpart` will work on expressions involving trigonometric and hyperbolic functions, as well as square root, logarithm, and exponentiation.

  Example:

      (%i1) realpart (a+b*%i);
      (%o1)                           a
      (%i2) realpart (1+sqrt(2)*%i);
      (%o2)                           1
      (%i3) realpart (sqrt(2)*%i);
      (%o3)                           0
      (%i4) realpart (1);
      (%o4)                           1

  Categories: Complex variables

Function: rectform (expr)

  Returns an expression `a + b %i` equivalent to expr, such that a and b are purely real.

  Example:

      (%i1) rectform(sqrt(2)*%e^(%i*%pi/4));
      (%o1)                        %i + 1
      (%i2) rectform(sqrt(b^2+a^2)*%e^(%i*atan2(b, a)));
      (%o2)                       %i b + a
      (%i3) rectform(sqrt(5)*%e^(%i*atan(2)));
      (%o3)                       2 %i + 1

  Categories: Complex variables

[ < ] | [ > ] |  | [ << ] | [ Up ] | [ >> ] |  |  |  |  | [Top] | [Contents] | [Index] | [ ? ]

## 10.3 Combinatorial Functions

Operator: !!

  The double factorial operator.

  For an integer, float, or rational number `n`, `n!!` evaluates to the product `n (n-2) (n-4) (n-6) ... (n - 2 (k-1))` where `k` is equal to `entier (n/2)`, that is, the largest integer less than or equal to `n/2`. Note that this definition does not coincide with other published definitions for arguments which are not integers.

  For an even (or odd) integer `n`, `n!!` evaluates to the product of all the consecutive even (or odd) integers from 2 (or 1) through `n` inclusive.

  For an argument `n` which is not an integer, float, or rational, `n!!` yields a noun form `genfact (n, n/2, 2)`.

  Categories: Gamma and factorial functions · Operators

Function: binomial (x, y)

  The binomial coefficient `x!/(y! (x - y)!)`. If x and y are integers, then the numerical value of the binomial coefficient is computed. If y, or x - y, is an integer, the binomial coefficient is expressed as a polynomial.

  Examples:

      (%i1) binomial (11, 7);
      (%o1)                          330
      (%i2) 11! / 7! / (11 - 7)!;
      (%o2)                          330
      (%i3) binomial (x, 7);
              (x - 6) (x - 5) (x - 4) (x - 3) (x - 2) (x - 1) x
      (%o3)   -------------------------------------------------
                                    5040
      (%i4) binomial (x + 7, x);
            (x + 1) (x + 2) (x + 3) (x + 4) (x + 5) (x + 6) (x + 7)
      (%o4) -------------------------------------------------------
                                     5040
      (%i5) binomial (11, y);
      (%o5)                    binomial(11, y)

  Categories: Number theory

Function: factcomb (expr)

  Tries to combine the coefficients of factorials in expr with the factorials themselves by converting, for example, `(n + 1)*n!` into `(n + 1)!`.

  `sumsplitfact` if set to `false` will cause `minfactorial` to be applied after a `factcomb`.

  Example:

      (%i1) sumsplitfact;
      (%o1)                         true
      (%i2) (n + 1)*(n + 1)*n!;
                                        2
      (%o2)                      (n + 1)  n!
      (%i3) factcomb (%);
      (%o3)                  (n + 2)! - (n + 1)!
      (%i4) sumsplitfact: not sumsplitfact;
      (%o4)                         false
      (%i5) (n + 1)*(n + 1)*n!;
                                        2
      (%o5)                      (n + 1)  n!
      (%i6) factcomb (%);
      (%o6)                 n (n + 1)! + (n + 1)!

  Categories: Gamma and factorial functions

Function: factorial
Operator: !

  Represents the factorial function. Maxima treats `factorial (x)` the same as `x!`.

  For any complex number `x`, except for negative integers, `x!` is defined as `gamma(x+1)`.

  For an integer `x`, `x!` simplifies to the product of the integers from 1 to `x` inclusive. `0!` simplifies to 1. For a real or complex number in float or bigfloat precision `x`, `x!` simplifies to the value of `gamma (x+1)`. For `x` equal to `n/2` where `n` is an odd integer, `x!` simplifies to a rational factor times `sqrt (%pi)` (since `gamma (1/2)` is equal to `sqrt (%pi)`).

  The option variables `factlim` and `gammalim` control the numerical evaluation of factorials for integer and rational arguments. The functions `minfactorial` and `factcomb` simplifies expressions containing factorials.

  The functions `gamma`, `bffac`, and `cbffac` are varieties of the gamma function. `bffac` and `cbffac` are called internally by `gamma` to evaluate the gamma function for real and complex numbers in bigfloat precision.

  `makegamma` substitutes `gamma` for factorials and related functions.

  Maxima knows the derivative of the factorial function and the limits for specific values like negative integers.

  The option variable `factorial_expand` controls the simplification of expressions like `(n+x)!`, where `n` is an integer.

  See also `binomial`.

  The factorial of an integer is simplified to an exact number unless the operand is greater than `factlim`. The factorial for real and complex numbers is evaluated in float or bigfloat precision.

      (%i1) factlim : 10;
      (%o1)                          10
      (%i2) [0!, (7/2)!, 8!, 20!];
                           105 sqrt(%pi)
      (%o2)            [1, -------------, 40320, 20!]
                                16
      (%i3) [4,77!, (1.0+%i)!];
      (%o3) [4, 77!, 0.3430658398165453 %i + 0.6529654964201667]
      (%i4) [2.86b0!, 1.0b0+%i)!];
      incorrect syntax: Missing ]
      [2.86b0!, 1.0b0+%i)
                       ^

  The factorial of a known constant, or general expression is not simplified. Even so it may be possible to simplify the factorial after evaluating the operand.

      (%i1) [(%i + 1)!, %pi!, %e!, (cos(1) + sin(1))!];
      (%o1)      [(%i + 1)!, %pi!, %e!, (sin(1) + cos(1))!]
      (%i2) ev (%, numer, %enumer);
      (%o2) [0.3430658398165453 %i + 0.6529654964201667,
               7.188082728976031, 4.260820476357003, 1.227580202486819]

  Factorials are simplified, not evaluated. Thus `x!` may be replaced even in a quoted expression.

      (%i1) '([0!, (7/2)!, 4.77!, 8!, 20!]);
                105 sqrt(%pi)
      (%o1) [1, -------------, 81.44668037931197, 40320,
                     16
                                                   2432902008176640000]

  Maxima knows the derivative of the factorial function.

      (%i1) diff(x!,x);
      (%o1)                    x! psi (x + 1)
                                     0

  The option variable `factorial_expand` controls expansion and simplification of expressions with the factorial function.

      (%i1) (n+1)!/n!,factorial_expand:true;
      (%o1)                         n + 1

  Categories: Gamma and factorial functions · Operators

Option variable: factlim

  Default value: 100000

  `factlim` specifies the highest factorial which is automatically expanded. If it is -1 then all integers are expanded.

  Categories: Gamma and factorial functions

Option variable: factorial_expand

  Default value: false

  The option variable `factorial_expand` controls the simplification of expressions like `(n+1)!`, where `n` is an integer. See `factorial` for an example.

  Categories: Gamma and factorial functions

Function: genfact (x, y, z)

  Returns the generalized factorial, defined as `x (x-z) (x - 2 z) ... (x - (y - 1) z)`. Thus, when x is an integer, `genfact (x, x, 1) = x!` and `genfact (x, x/2, 2) = x!!`.

  Categories: Gamma and factorial functions

Function: minfactorial (expr)

  Examines expr for occurrences of two factorials which differ by an integer. `minfactorial` then turns one into a polynomial times the other.

      (%i1) n!/(n+2)!;
                                     n!
      (%o1)                       --------
                                  (n + 2)!
      (%i2) minfactorial (%);
                                      1
      (%o2)                    ---------------
                               (n + 1) (n + 2)

  Categories: Number theory

Option variable: sumsplitfact

  Default value: `true`

  When `sumsplitfact` is `false`, `minfactorial` is applied after a `factcomb`.

      (%i1) sumsplitfact;
      (%o1)                         true
      (%i2) n!/(n+2)!;
                                     n!
      (%o2)                       --------
                                  (n + 2)!
      (%i3) factcomb(%);
                                     n!
      (%o3)                       --------
                                  (n + 2)!
      (%i4) sumsplitfact: not sumsplitfact ;
      (%o4)                         false
      (%i5) n!/(n+2)!;
                                     n!
      (%o5)                       --------
                                  (n + 2)!
      (%i6) factcomb(%);
                                      1
      (%o6)                    ---------------
                               (n + 1) (n + 2)

  Categories: Gamma and factorial functions · Simplification flags and variables

[ < ] | [ > ] |  | [ << ] | [ Up ] | [ >> ] |  |  |  |  | [Top] | [Contents] | [Index] | [ ? ]

## 10.4 Root, Exponential and Logarithmic Functions

Option variable: %e_to_numlog

  Default value: `false`

  When `true`, `r` some rational number, and `x` some expression, `%e^(r*log(x))` will be simplified into `x^r` . It should be noted that the `radcan` command also does this transformation, and more complicated transformations of this ilk as well. The `logcontract` command "contracts" expressions containing `log`.

  Categories: Exponential and logarithm functions · Simplification flags and variables

Option variable: %emode

  Default value: `true`

  When `%emode` is `true`, `%e^(%pi %i x)` is simplified as follows.

  `%e^(%pi %i x)` simplifies to `cos (%pi x) + %i sin (%pi x)` if `x` is a floating point number, an integer, or a multiple of 1/2, 1/3, 1/4, or 1/6, and then further simplified.

  For other numerical `x`, `%e^(%pi %i x)` simplifies to `%e^(%pi %i y)` where `y` is `x - 2 k` for some integer `k` such that `abs(y) < 1`.

  When `%emode` is `false`, no special simplification of `%e^(%pi %i x)` is carried out.

      (%i1) %emode;
      (%o1)                         true
      (%i2) %e^(%pi*%i*1);
      (%o2)                          - 1
      (%i3) %e^(%pi*%i*216/144);
      (%o3)                         - %i
      (%i4) %e^(%pi*%i*192/144);
                                sqrt(3) %i    1
      (%o4)                  (- ----------) - -
                                    2         2
      (%i5) %e^(%pi*%i*180/144);
                                 %i          1
      (%o5)                 (- -------) - -------
                               sqrt(2)    sqrt(2)
      (%i6) %e^(%pi*%i*120/144);
                                %i   sqrt(3)
      (%o6)                     -- - -------
                                2       2
      (%i7) %e^(%pi*%i*121/144);
                                  121 %i %pi
                                  ----------
                                     144
      (%o7)                     %e

  Categories: Exponential and logarithm functions · Simplification flags and variables

Option variable: %enumer

  Default value: `false`

  When `%enumer` is `true`, `%e` is replaced by its numeric value 2.718… whenever `numer` is `true`.

  When `%enumer` is `false`, this substitution is carried out only if the exponent in `%e^x` evaluates to a number.

  See also `ev` and `numer`.

      (%i1) %enumer;
      (%o1)                         false
      (%i2) numer;
      (%o2)                         false
      (%i3) 2*%e;
      (%o3)                         2 %e
      (%i4) %enumer: not %enumer;
      (%o4)                         true
      (%i5) 2*%e;
      (%o5)                         2 %e
      (%i6) numer: not numer;
      (%o6)                         true
      (%i7) 2*%e;
      (%o7)                   5.43656365691809
      (%i8) 2*%e^1;
      (%o8)                   5.43656365691809
      (%i9) 2*%e^x;
                                               x
      (%o9)                 2 2.718281828459045

  Categories: Exponential and logarithm functions · Evaluation flags

Function: exp (x)

  Represents the exponential function. Instances of `exp (x)` in input are simplified to `%e^x`; `exp` does not appear in simplified expressions.

  `demoivre` if `true` causes `%e^(a + b %i)` to simplify to `%e^(a (cos(b) + %i sin(b)))` if `b` is free of `%i`. See `demoivre`.

  `%emode`, when `true`, causes `%e^(%pi %i x)` to be simplified. See `%emode`.

  `%enumer`, when `true` causes `%e` to be replaced by 2.718… whenever `numer` is `true`. See `%enumer`.

      (%i1) demoivre;
      (%o1)                         false
      (%i2) %e^(a + b*%i);
                                   %i b + a
      (%o2)                      %e
      (%i3) demoivre: not demoivre;
      (%o3)                         true
      (%i4) %e^(a + b*%i);
                            a
      (%o4)               %e  (%i sin(b) + cos(b))

  Categories: Exponential and logarithm functions

Function: li [s] (z)

  Represents the polylogarithm function of order s and argument z, defined by the infinite series

                                       inf
                                       ====   k
                                       \     z
                              Li (z) =  >    --
                                s      /      s
                                       ====  k
                                       k = 1

  `li [1]` is `- log (1 - z)`. `li [2]` and `li [3]` are the dilogarithm and trilogarithm functions, respectively.

  When the order is 1, the polylogarithm simplifies to `- log (1 - z)`, which in turn simplifies to a numerical value if z is a real or complex floating point number or the `numer` evaluation flag is present.

  When the order is 2 or 3, the polylogarithm simplifies to a numerical value if z is a real floating point number or the `numer` evaluation flag is present.

  Examples:

      (%i1) assume (x > 0);
      (%o1)                        [x > 0]
      (%i2) integrate ((log (1 - t)) / t, t, 0, x);
      (%o2)                       - li (x)
                                      2
      (%i3) li [2] (7);
      (%o3)                        li (7)
                                     2
      (%i4) li [2] (7), numer;
      (%o4)       1.248273182099423 - 6.113257028817991 %i
      (%i5) li [3] (7);
      (%o5)                        li (7)
                                     3
      (%i6) li [2] (7), numer;
      (%o6)       1.248273182099423 - 6.113257028817991 %i
      (%i7) L : makelist (i / 4.0, i, 0, 8);
      (%o7)   [0.0, 0.25, 0.5, 0.75, 1.0, 1.25, 1.5, 1.75, 2.0]
      (%i8) map (lambda ([x], li [2] (x)), L);
      (%o8) [0.0, 0.2676526390827326, 0.5822405264650125,
      0.978469392930306, 1.644934066848226,
      2.190177011441645 - 0.7010261415046585 %i,
      2.37439527027248 - 1.2738062049196 %i,
      2.448686765338205 - 1.758084848210787 %i,
      2.467401100272339 - 2.177586090303601 %i]
      (%i9) map (lambda ([x], li [3] (x)), L);
      (%o9) [0.0, 0.2584613953442624, 0.537213192678042,
      0.8444258046482203, 1.2020569, 1.642866878950322
       - 0.07821473130035025 %i, 2.060877505514697
       - 0.2582419849982037 %i, 2.433418896388322
       - 0.4919260182322965 %i, 2.762071904015935
       - 0.7546938285978846 %i]

  Categories: Exponential and logarithm functions

Function: log (x)

  Represents the natural (base e) logarithm of x.

  Maxima does not have a built-in function for the base 10 logarithm or other bases. `log10(x) := log(x) / log(10)` is a useful definition.

  Simplification and evaluation of logarithms is governed by several global flags:

  ``logexpand``

    causes `log(a^b)` to become `b*log(a)`. If it is set to `all`, `log(a*b)` will also simplify to `log(a)+log(b)`. If it is set to `super`, then `log(a/b)` will also simplify to `log(a)-log(b)` for rational numbers `a/b`, `a#1`. (`log(1/b)`, for `b` integer, always simplifies.) If it is set to `false`, all of these simplifications will be turned off.

  ``logsimp``

    if `false` then no simplification of `%e` to a power containing `log`'s is done.

  ``lognegint``

    if `true` implements the rule `log(-n)` -> `log(n)+%i*%pi` for `n` a positive integer.

  ``%e_to_numlog``

    when `true`, `r` some rational number, and `x` some expression, the expression `%e^(r*log(x))` will be simplified into `x^r`. It should be noted that the `radcan` command also does this transformation, and more complicated transformations of this as well. The `logcontract` command "contracts" expressions containing `log`.

  Categories: Exponential and logarithm functions

Option variable: logabs

  Default value: `false`

  When doing indefinite integration where logs are generated, e.g. `integrate(1/x,x)`, the answer is given in terms of `log(abs(...))` if `logabs` is `true`, but in terms of `log(...)` if `logabs` is `false`. For definite integration, the `logabs:true` setting is used, because here "evaluation" of the indefinite integral at the endpoints is often needed.

  Categories: Exponential and logarithm functions · Integral calculus · Global flags

Option variable: logarc
Function: logarc (expr)

  When the global variable `logarc` is `true`, inverse circular and hyperbolic functions are replaced by equivalent logarithmic functions. The default value of `logarc` is `false`.

  The function `logarc(expr)` carries out that replacement for an expression expr without setting the global variable `logarc`.

  Categories: Exponential and logarithm functions · Simplification flags and variables · Simplification functions

Option variable: logconcoeffp

  Default value: `false`

  Controls which coefficients are contracted when using `logcontract`. It may be set to the name of a predicate function of one argument. E.g. if you like to generate SQRTs, you can do `logconcoeffp:'logconfun$ logconfun(m):=featurep(m,integer) or ratnump(m)$` . Then `logcontract(1/2*log(x));` will give `log(sqrt(x))`.

  Categories: Exponential and logarithm functions · Simplification flags and variables

Function: logcontract (expr)

  Recursively scans the expression expr, transforming subexpressions of the form `a1*log(b1) + a2*log(b2) + c` into `log(ratsimp(b1^a1 * b2^a2)) + c`

      (%i1) 2*(a*log(x) + 2*a*log(y))$
      (%i2) logcontract(%);
                                       2  4
      (%o2)                     a log(x  y )

  The declaration `declare(n,integer)` causes `logcontract(2*a*n*log(x))` to simplify to `a*log(x^(2*n))`. The coefficients that "contract" in this manner are those such as the 2 and the `n` here which satisfy `featurep(coeff,integer)`. The user can control which coefficients are contracted by setting the option `logconcoeffp` to the name of a predicate function of one argument. E.g. if you like to generate SQRTs, you can do `logconcoeffp:'logconfun$ logconfun(m):=featurep(m,integer) or ratnump(m)$` . Then `logcontract(1/2*log(x));` will give `log(sqrt(x))`.

  Categories: Exponential and logarithm functions

Option variable: logexpand

  Default value: `true`

  If `true`, that is the default value, causes `log(a^b)` to become `b*log(a)`. If it is set to `all`, `log(a*b)` will also simplify to `log(a)+log(b)`. If it is set to `super`, then `log(a/b)` will also simplify to `log(a)-log(b)` for rational numbers `a/b`, `a#1`. (`log(1/b)`, for integer `b`, always simplifies.) If it is set to `false`, all of these simplifications will be turned off.

  When `logexpand` is set to `all` or `super`, the logarithm of a product expression simplifies to a summation of logarithms.

  Examples:

  When `logexpand` is `true`, `log(a^b)` simplifies to `b*log(a)`.

      (%i1) log(n^2), logexpand=true;
      (%o1)                       2 log(n)

  When `logexpand` is `all`, `log(a*b)` simplifies to `log(a)+log(b)`.

      (%i1) log(10*x), logexpand=all;
      (%o1)                   log(x) + log(10)

  When `logexpand` is `super`, `log(a/b)` simplifies to `log(a)-log(b)` for rational numbers `a/b` with `a#1`.

      (%i1) log(a/(n + 1)), logexpand=super;
      (%o1)                  log(a) - log(n + 1)

  When `logexpand` is set to `all` or `super`, the logarithm of a product expression simplifies to a summation of logarithms.

      (%i1) my_product : product (X(i), i, 1, n);
                                   n
                                 /===\
                                  ! !
      (%o1)                       ! !  X(i)
                                  ! !
                                 i = 1
      (%i2) log(my_product), logexpand=all;
                                n
                               ====
                               \
      (%o2)                     >    log(X(i))
                               /
                               ====
                               i = 1
      (%i3) log(my_product), logexpand=super;
                                n
                               ====
                               \
      (%o3)                     >    log(X(i))
                               /
                               ====
                               i = 1

  When `logexpand` is `false`, these simplifications are disabled.

      (%i1) logexpand : false $
      (%i2) log(n^2);
                                        2
      (%o2)                        log(n )
      (%i3) log(10*x);
      (%o3)                       log(10 x)
      (%i4) log(a/(n + 1));
                                       a
      (%o4)                      log(-----)
                                     n + 1
      (%i5) log ('product (X(i), i, 1, n));
                                     n
                                   /===\
                                    ! !
      (%o5)                    log( ! !  X(i))
                                    ! !
                                   i = 1

  Categories: Exponential and logarithm functions · Simplification flags and variables

Option variable: lognegint

  Default value: `false`

  If `true` implements the rule `log(-n)` -> `log(n)+%i*%pi` for `n` a positive integer.

  Categories: Exponential and logarithm functions · Simplification flags and variables

Option variable: logsimp

  Default value: `true`

  If `false` then no simplification of `%e` to a power containing `log`'s is done.

  Categories: Exponential and logarithm functions · Simplification flags and variables

Function: plog (x)

  Represents the principal branch of the complex-valued natural logarithm with `-%pi` < `carg(x)` <= `+%pi` .

  Categories: Exponential and logarithm functions · Complex variables

Function: sqrt (x)

  The square root of x. It is represented internally by `x^(1/2)`. See also `rootscontract` and `radexpand`.

  Categories: Mathematical functions

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## 10.5 Trigonometric Functions

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### 10.5.1 Introduction to Trigonometric

Maxima has many trigonometric functions defined. Not all trigonometric identities are programmed, but it is possible for the user to add many of them using the pattern matching capabilities of the system. The trigonometric functions defined in Maxima are: `acos`, `acosh`, `acot`, `acoth`, `acsc`, `acsch`, `asec`, `asech`, `asin`, `asinh`, `atan`, `atanh`, `cos`, `cosh`, `cot`, `coth`, `csc`, `csch`, `sec`, `sech`, `sin`, `sinh`, `tan`, and `tanh`. There are a number of commands especially for handling trigonometric functions, see `trigexpand`, `trigreduce`, and the switch `trigsign`. Two share packages extend the simplification rules built into Maxima, `ntrig` and `atrig1`. Do `describe(command)` for details.

Categories: Trigonometric functions

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### 10.5.2 Functions and Variables for Trigonometric

Option variable: %piargs

  Default value: `true`

  When `%piargs` is `true`, trigonometric functions are simplified to algebraic constants when the argument is an integer multiple of %pi, %pi/2, %pi/3, %pi/4, or %pi/6.

  Maxima knows some identities which can be applied when %pi, etc., are multiplied by an integer variable (that is, a symbol declared to be integer).

  Examples:

      (%i1) %piargs : false$
      (%i2) [sin (%pi), sin (%pi/2), sin (%pi/3)];
                                      %pi       %pi
      (%o2)            [sin(%pi), sin(---), sin(---)]
                                       2         3
      (%i3) [sin (%pi/4), sin (%pi/5), sin (%pi/6)];
                            %pi       %pi       %pi
      (%o3)            [sin(---), sin(---), sin(---)]
                             4         5         6
      (%i4) %piargs : true$
      (%i5) [sin (%pi), sin (%pi/2), sin (%pi/3)];
                                      sqrt(3)
      (%o5)                    [0, 1, -------]
                                         2
      (%i6) [sin (%pi/4), sin (%pi/5), sin (%pi/6)];
                               1         %pi   1
      (%o6)                [-------, sin(---), -]
                            sqrt(2)       5    2
      (%i7) [cos (%pi/3), cos (10*%pi/3), tan (10*%pi/3),
             cos (sqrt(2)*%pi/3)];
                      1    1               sqrt(2) %pi
      (%o7)          [-, - -, sqrt(3), cos(-----------)]
                      2    2                    3

  Some identities are applied when %pi and %pi/2 are multiplied by an integer variable.

      (%i1) declare (n, integer, m, even)$
      (%i2) [sin (%pi * n), cos (%pi * m), sin (%pi/2 * m),
             cos (%pi/2 * m)];
                                            m/2
      (%o2)                  [0, 1, 0, (- 1)   ]

  Categories: Trigonometric functions · Simplification flags and variables

Option variable: %iargs

  Default value: `true`

  When `%iargs` is `true`, trigonometric functions are simplified to hyperbolic functions when the argument is apparently a multiple of the imaginary unit %i.

  Even when the argument is demonstrably real, the simplification is applied; Maxima considers only whether the argument is a literal multiple of %i.

  Examples:

      (%i1) %iargs : false$
      (%i2) [sin (%i * x), cos (%i * x), tan (%i * x)];
      (%o2)           [sin(%i x), cos(%i x), tan(%i x)]
      (%i3) %iargs : true$
      (%i4) [sin (%i * x), cos (%i * x), tan (%i * x)];
      (%o4)           [%i sinh(x), cosh(x), %i tanh(x)]

  Even when the argument is demonstrably real, the simplification is applied.

      (%i1) declare (x, imaginary)$
      (%i2) [featurep (x, imaginary), featurep (x, real)];
      (%o2)                     [true, false]
      (%i3) sin (%i * x);
      (%o3)                      %i sinh(x)

  Categories: Trigonometric functions · Hyperbolic functions · Simplification flags and variables

Function: acos (x)

  - Arc Cosine.

  Categories: Trigonometric functions

Function: acosh (x)

  - Hyperbolic Arc Cosine.

  Categories: Hyperbolic functions

Function: acot (x)

  - Arc Cotangent.

  Categories: Trigonometric functions

Function: acoth (x)

  - Hyperbolic Arc Cotangent.

  Categories: Hyperbolic functions

Function: acsc (x)

  - Arc Cosecant.

  Categories: Trigonometric functions

Function: acsch (x)

  - Hyperbolic Arc Cosecant.

  Categories: Hyperbolic functions

Function: asec (x)

  - Arc Secant.

  Categories: Trigonometric functions

Function: asech (x)

  - Hyperbolic Arc Secant.

  Categories: Hyperbolic functions

Function: asin (x)

  - Arc Sine.

  Categories: Trigonometric functions

Function: asinh (x)

  - Hyperbolic Arc Sine.

  Categories: Hyperbolic functions

Function: atan (x)

  - Arc Tangent.

  Categories: Trigonometric functions

Function: atan2 (y, x)

  - yields the value of `atan(y/x)` in the interval `-%pi` to `%pi`.

  Categories: Trigonometric functions

Function: atanh (x)

  - Hyperbolic Arc Tangent.

  Categories: Hyperbolic functions

Package: atrig1

  The `atrig1` package contains several additional simplification rules for inverse trigonometric functions. Together with rules already known to Maxima, the following angles are fully implemented: `0`, `%pi/6`, `%pi/4`, `%pi/3`, and `%pi/2`. Corresponding angles in the other three quadrants are also available. Do `load(atrig1);` to use them.

  Categories: Trigonometric functions · Package atrig1

Function: cos (x)

  - Cosine.

  Categories: Trigonometric functions

Function: cosh (x)

  - Hyperbolic Cosine.

  Categories: Hyperbolic functions

Function: cot (x)

  - Cotangent.

  Categories: Trigonometric functions

Function: coth (x)

  - Hyperbolic Cotangent.

  Categories: Hyperbolic functions

Function: csc (x)

  - Cosecant.

  Categories: Trigonometric functions

Function: csch (x)

  - Hyperbolic Cosecant.

  Categories: Hyperbolic functions

Option variable: halfangles

  Default value: `false`

  When `halfangles` is `true`, trigonometric functions of arguments `expr/2` are simplified to functions of expr.

  For a real argument x in the interval `0 < x < 2*%pi` the sine of the half-angle simplifies to a simple formula:

                               sqrt(1 - cos(x))
                               ----------------
                                   sqrt(2)

  A complicated factor is needed to make this formula correct for all complex arguments z:

                 realpart(z)
           floor(-----------)
                    2 %pi
      (- 1)                   (1 - unit_step(- imagpart(z))

                                  realpart(z)            realpart(z)
                            floor(-----------) - ceiling(-----------)
                                     2 %pi                  2 %pi
                      ((- 1)                                          + 1))

  Maxima knows this factor and similar factors for the functions `sin`, `cos`, `sinh`, and `cosh`. For special values of the argument z these factors simplify accordingly.

  Examples:

      (%i1) halfangles : false$
      (%i2) sin (x / 2);
                                       x
      (%o2)                        sin(-)
                                       2
      (%i3) halfangles : true$
      (%i4) sin (x / 2);
                                  x
                          floor(-----)
                                2 %pi
                     (- 1)             sqrt(1 - cos(x))
      (%o4)          ----------------------------------
                                  sqrt(2)
      (%i5) assume(x>0, x<2*%pi)$
      (%i6) sin(x / 2);
                              sqrt(1 - cos(x))
      (%o6)                   ----------------
                                  sqrt(2)

  Categories: Trigonometric functions · Simplification flags and variables

Package: ntrig

  The `ntrig` package contains a set of simplification rules that are used to simplify trigonometric function whose arguments are of the form `f(n %pi/10)` where f is any of the functions `sin`, `cos`, `tan`, `csc`, `sec` and `cot`.

  Categories: Trigonometric functions · Package ntrig

Function: sec (x)

  - Secant.

  Categories: Trigonometric functions

Function: sech (x)

  - Hyperbolic Secant.

  Categories: Hyperbolic functions

Function: sin (x)

  - Sine.

  Categories: Trigonometric functions

Function: sinh (x)

  - Hyperbolic Sine.

  Categories: Hyperbolic functions

Function: tan (x)

  - Tangent.

  Categories: Trigonometric functions

Function: tanh (x)

  - Hyperbolic Tangent.

  Categories: Hyperbolic functions

Function: trigexpand (expr)

  Expands trigonometric and hyperbolic functions of sums of angles and of multiple angles occurring in expr. For best results, expr should be expanded. To enhance user control of simplification, this function expands only one level at a time, expanding sums of angles or multiple angles. To obtain full expansion into sines and cosines immediately, set the switch `trigexpand: true`.

  `trigexpand` is governed by the following global flags:

  `trigexpand`

    If `true` causes expansion of all expressions containing sin's and cos's occurring subsequently.

  `halfangles`

    If `true` causes half-angles to be simplified away.

  `trigexpandplus`

    Controls the "sum" rule for `trigexpand`, expansion of sums (e.g. `sin(x + y)`) will take place only if `trigexpandplus` is `true`.

  `trigexpandtimes`

    Controls the "product" rule for `trigexpand`, expansion of products (e.g. `sin(2 x)`) will take place only if `trigexpandtimes` is `true`.

  Examples:

      (%i1) x+sin(3*x)/sin(x),trigexpand=true,expand;
                               2            2
      (%o1)              (- sin (x)) + 3 cos (x) + x
      (%i2) trigexpand(sin(10*x+y));
      (%o2)          cos(10 x) sin(y) + sin(10 x) cos(y)

  Categories: Trigonometric functions · Simplification functions

Option variable: trigexpandplus

  Default value: `true`

  `trigexpandplus` controls the "sum" rule for `trigexpand`. Thus, when the `trigexpand` command is used or the `trigexpand` switch set to `true`, expansion of sums (e.g. `sin(x+y))` will take place only if `trigexpandplus` is `true`.

  Categories: Trigonometric functions · Simplification flags and variables

Option variable: trigexpandtimes

  Default value: `true`

  `trigexpandtimes` controls the "product" rule for `trigexpand`. Thus, when the `trigexpand` command is used or the `trigexpand` switch set to `true`, expansion of products (e.g. `sin(2*x)`) will take place only if `trigexpandtimes` is `true`.

  Categories: Trigonometric functions · Simplification flags and variables

Option variable: triginverses

  Default value: `true`

  `triginverses` controls the simplification of the composition of trigonometric and hyperbolic functions with their inverse functions.

  If `all`, both e.g. `atan(tan(x))` and `tan(atan(x))` simplify to x.

  If `true`, the `arcfun(fun(x))` simplification is turned off.

  If `false`, both the `arcfun(fun(x))` and `fun(arcfun(x))` simplifications are turned off.

  Categories: Trigonometric functions · Simplification flags and variables

Function: trigreduce
`trigreduce` (expr, x)
`trigreduce` (expr)

  Combines products and powers of trigonometric and hyperbolic sin's and cos's of x into those of multiples of x. It also tries to eliminate these functions when they occur in denominators. If x is omitted then all variables in expr are used.

  See also `poissimp`.

      (%i1) trigreduce(-sin(x)^2+3*cos(x)^2+x);
                     cos(2 x)      cos(2 x)   1        1
      (%o1)          -------- + 3 (-------- + -) + x - -
                        2             2       2        2

  Categories: Trigonometric functions · Simplification functions

Option variable: trigsign

  Default value: `true`

  When `trigsign` is `true`, it permits simplification of negative arguments to trigonometric functions. E.g., `sin(-x)` will become `-sin(x)` only if `trigsign` is `true`.

  Categories: Trigonometric functions · Simplification flags and variables

Function: trigsimp (expr)

  Employs the identities sin(x)^2 + cos(x)^2 = 1 and cosh(x)^2 - sinh(x)^2 = 1 to simplify expressions containing `tan`, `sec`, etc., to `sin`, `cos`, `sinh`, `cosh`.

  `trigreduce`, `ratsimp`, and `radcan` may be able to further simplify the result.

  `demo ("trgsmp.dem")` displays some examples of `trigsimp`.

  Categories: Trigonometric functions · Simplification functions

Function: trigrat (expr)

  Gives a canonical simplified quasilinear form of a trigonometrical expression; expr is a rational fraction of several `sin`, `cos` or `tan`, the arguments of them are linear forms in some variables (or kernels) and `%pi/n` (n integer) with integer coefficients. The result is a simplified fraction with numerator and denominator linear in `sin` and `cos`. Thus `trigrat` linearize always when it is possible.

      (%i1) trigrat(sin(3*a)/sin(a+%pi/3));
      (%o1)            sqrt(3) sin(2 a) + cos(2 a) - 1

  The following example is taken from Davenport, Siret, and Tournier, Calcul Formel, Masson (or in English, Addison-Wesley), section 1.5.5, Morley theorem.

      (%i1) c : %pi/3 - a - b$
      (%i2) bc : sin(a)*sin(3*c)/sin(a+b);
                                                 %pi
                       sin(a) sin(3 ((- b) - a + ---))
                                                  3
      (%o2)            -------------------------------
                                 sin(b + a)
      (%i3) ba : bc, c=a, a=c;
                                               %pi
                          sin(3 a) sin(b + a - ---)
                                                3
      (%o3)               -------------------------
                                        %pi
                                sin(a - ---)
                                         3
      (%i4) ac2 : ba^2 + bc^2 - 2*bc*ba*cos(b);
               2         2         %pi
            sin (3 a) sin (b + a - ---)
                                    3
      (%o4) ---------------------------
                      2     %pi
                   sin (a - ---)
                             3
                                               %pi
       - (2 sin(a) sin(3 a) sin(3 ((- b) - a + ---)) cos(b)
                                                3
                   %pi            %pi
       sin(b + a - ---))/(sin(a - ---) sin(b + a))
                    3              3
            2       2                %pi
         sin (a) sin (3 ((- b) - a + ---))
                                      3
       + ---------------------------------
                       2
                    sin (b + a)
      (%i5) trigrat (ac2);
      (%o5) - (sqrt(3) sin(4 b + 4 a) - cos(4 b + 4 a)
       - 2 sqrt(3) sin(4 b + 2 a) + 2 cos(4 b + 2 a)
       - 2 sqrt(3) sin(2 b + 4 a) + 2 cos(2 b + 4 a)
       + 4 sqrt(3) sin(2 b + 2 a) - 8 cos(2 b + 2 a) - 4 cos(2 b - 2 a)
       + sqrt(3) sin(4 b) - cos(4 b) - 2 sqrt(3) sin(2 b) + 10 cos(2 b)
       + sqrt(3) sin(4 a) - cos(4 a) - 2 sqrt(3) sin(2 a) + 10 cos(2 a)
       - 9)/4

  Categories: Trigonometric functions · Simplification functions

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## 10.6 Random Numbers

Function: make_random_state
`make_random_state` (n)
`make_random_state` (s)
`make_random_state` (true)
`make_random_state` (false)

  A random state object represents the state of the random number generator. The state comprises 627 32-bit words.

  `make_random_state (n)` returns a new random state object created from an integer seed value equal to n modulo 2^32. n may be negative.

  `make_random_state (s)` returns a copy of the random state s.

  `make_random_state (true)` returns a new random state object, using the current computer clock time as the seed.

  `make_random_state (false)` returns a copy of the current state of the random number generator.

  Categories: Random numbers

Function: set_random_state (s)

  Copies s to the random number generator state.

  `set_random_state` always returns `done`.

  Categories: Random numbers

Function: random (x)

  Returns a pseudorandom number. If x is an integer, `random (x)` returns an integer from 0 through `x - 1` inclusive. If x is a floating point number, `random (x)` returns a nonnegative floating point number less than x. `random` complains with an error if x is neither an integer nor a float, or if x is not positive.

  The functions `make_random_state` and `set_random_state` maintain the state of the random number generator.

  The Maxima random number generator is an implementation of the Mersenne twister MT 19937.

  Examples:

      (%i1) s1: make_random_state (654321)$
      (%i2) set_random_state (s1);
      (%o2)                         done
      (%i3) random (1000);
      (%o3)                          768
      (%i4) random (9573684);
      (%o4)                        7657880
      (%i5) random (2^75);
      (%o5)                11804491615036831636390
      (%i6) s2: make_random_state (false)$
      (%i7) random (1.0);
      (%o7)                  0.2310127244107132
      (%i8) random (10.0);
      (%o8)                   4.394553645870825
      (%i9) random (100.0);
      (%o9)                   32.28666704056853
      (%i10) set_random_state (s2);
      (%o10)                        done
      (%i11) random (1.0);
      (%o11)                 0.2310127244107132
      (%i12) random (10.0);
      (%o12)                  4.394553645870825
      (%i13) random (100.0);
      (%o13)                  32.28666704056853

  Categories: Random numbers · Numerical methods

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# 11. Maximas Database

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## 11.1 Introduction to Maximas Database

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## 11.2 Functions and Variables for Properties

Property: alphabetic

  `alphabetic` is a property type recognized by `declare`. The expression `declare(s, alphabetic)` tells Maxima to recognize as alphabetic all of the characters in s, which must be a string.

  See also Identifiers.

  Example:

      (%i1) xx\~yy\`\@ : 1729;
      (%o1)                         1729
      (%i2) declare ("~`@", alphabetic);
      (%o2)                         done
      (%i3) xx~yy`@ + @yy`xx + `xx@@yy~;
      (%o3)               `xx@@yy~ + @yy`xx + 1729
      (%i4) listofvars (%);
      (%o4)                  [@yy`xx, `xx@@yy~]

  Categories: Declarations and inferences

Property: bindtest

  The command `declare(x, bindtest)` tells Maxima to trigger an error when the symbol x is evaluated unbound.

      (%i1) aa + bb;
      (%o1)                        bb + aa
      (%i2) declare (aa, bindtest);
      (%o2)                         done
      (%i3) aa + bb;
      aa unbound variable
       -- an error.  Quitting.  To debug this try debugmode(true);
      (%i4) aa : 1234;
      (%o4)                         1234
      (%i5) aa + bb;
      (%o5)                       bb + 1234

Property: constant

  `declare(a, constant)` declares a to be a constant. The declaration of a symbol to be constant does not prevent the assignment of a nonconstant value to the symbol.

  See `constantp` and `declare`.

  Example:

      (%i1) declare(c, constant);
      (%o1)                         done
      (%i2) constantp(c);
      (%o2)                         true
      (%i3) c : x;
      (%o3)                           x
      (%i4) constantp(c);
      (%o4)                         false

  Categories: Declarations and inferences · Constants

Function: constantp (expr)

  Returns `true` if expr is a constant expression, otherwise returns `false`.

  An expression is considered a constant expression if its arguments are numbers (including rational numbers, as displayed with `/R/`), symbolic constants such as `%pi`, `%e`, and `%i`, variables bound to a constant or declared constant by `declare`, or functions whose arguments are constant.

  `constantp` evaluates its arguments.

  See the property `constant` which declares a symbol to be constant.

  Examples:

      (%i1) constantp (7 * sin(2));
      (%o1)                                true
      (%i2) constantp (rat (17/29));
      (%o2)                                true
      (%i3) constantp (%pi * sin(%e));
      (%o3)                                true
      (%i4) constantp (exp (x));
      (%o4)                                false
      (%i5) declare (x, constant);
      (%o5)                                done
      (%i6) constantp (exp (x));
      (%o6)                                true
      (%i7) constantp (foo (x) + bar (%e) + baz (2));
      (%o7)                                false
      (%i8)

  Categories: Predicate functions · Constants

Function: declare (a_1, p_1, a_2, p_2, …)

  Assigns the atom or list of atoms a_i the property or list of properties p_i. When a_i and/or p_i are lists, each of the atoms gets all of the properties.

  `declare` quotes its arguments. `declare` always returns `done`.

  As noted in the description for each declaration flag, for some flags `featurep(object, feature)` returns `true` if object has been declared to have feature.

  For more information about the features system, see `features`. To remove a property from an atom, use `remove`.

  `declare` recognizes the following properties:

  `additive`

    Tells Maxima to simplify a_i expressions by the substitution `a_i(x + y + z + ...)` `-->` `a_i(x) + a_i(y) + a_i(z) + ...`. The substitution is carried out on the first argument only.

  `alphabetic`

    Tells Maxima to recognize all characters in a_i (which must be a string) as alphabetic characters.

  `antisymmetric, commutative, symmetric`

    Tells Maxima to recognize a_i as a symmetric or antisymmetric function. `commutative` is the same as `symmetric`.

  `bindtest`

    Tells Maxima to trigger an error when a_i is evaluated unbound.

  `constant`

    Tells Maxima to consider a_i a symbolic constant.

  `even, odd`

    Tells Maxima to recognize a_i as an even or odd integer variable.

  `evenfun, oddfun`

    Tells Maxima to recognize a_i as an odd or even function.

  `evflag`

    Makes a_i known to the `ev` function so that a_i is bound to `true` during the execution of `ev` when a_i appears as a flag argument of `ev`. See `evflag`.

  `evfun`

    Makes a_i known to `ev` so that the function named by a_i is applied when a_i appears as a flag argument of `ev`. See `evfun`.

  `feature`

    Tells Maxima to recognize a_i as the name of a feature. Other atoms may then be declared to have the a_i property.

  `increasing, decreasing`

    Tells Maxima to recognize a_i as an increasing or decreasing function.

  `integer, noninteger`

    Tells Maxima to recognize a_i as an integer or noninteger variable.

  `integervalued`

    Tells Maxima to recognize a_i as an integer-valued function.

  `lassociative, rassociative`

    Tells Maxima to recognize a_i as a right-associative or left-associative function.

  `linear`

    Equivalent to declaring a_i both `outative` and `additive`.

  `mainvar`

    Tells Maxima to consider a_i a "main variable". A main variable succeeds all other constants and variables in the canonical ordering of Maxima expressions, as determined by `ordergreatp`.

  `multiplicative`

    Tells Maxima to simplify a_i expressions by the substitution `a_i(x * y * z * ...)` `-->` `a_i(x) * a_i(y) * a_i(z) * ...`. The substitution is carried out on the first argument only.

  `nary`

    Tells Maxima to recognize a_i as an n-ary function.

    The `nary` declaration is not the same as calling the `nary` function. The sole effect of `declare(foo, nary)` is to instruct the Maxima simplifier to flatten nested expressions, for example, to simplify `foo(x, foo(y, z))` to `foo(x, y, z)`.

  `nonarray`

    Tells Maxima to consider a_i not an array. This declaration prevents multiple evaluation of a subscripted variable name.

  `nonscalar`

    Tells Maxima to consider a_i a nonscalar variable. The usual application is to declare a variable as a symbolic vector or matrix.

  `noun`

    Tells Maxima to parse a_i as a noun. The effect of this is to replace instances of a_i with `'a_i` or `nounify(a_i)`, depending on the context.

  `outative`

    Tells Maxima to simplify a_i expressions by pulling constant factors out of the first argument.

    When a_i has one argument, a factor is considered constant if it is a literal or declared constant.

    When a_i has two or more arguments, a factor is considered constant if the second argument is a symbol and the factor is free of the second argument.

  `posfun`

    Tells Maxima to recognize a_i as a positive function.

  `rational, irrational`

    Tells Maxima to recognize a_i as a rational or irrational real variable.

  `real, imaginary, complex`

    Tells Maxima to recognize a_i as a real, pure imaginary, or complex variable.

  `scalar`

    Tells Maxima to consider a_i a scalar variable.

  Examples of the usage of the properties are available in the documentation for each separate description of a property.

  Categories: Declarations and inferences

Property: decreasing
Property: increasing

  The commands `declare(f, decreasing)` or `declare(f, increasing)` tell Maxima to recognize the function f as an decreasing or increasing function.

  See also `declare` for more properties.

  Example:

      (%i1) assume(a > b);
      (%o1)                        [a > b]
      (%i2) is(f(a) > f(b));
      (%o2)                        unknown
      (%i3) declare(f, increasing);
      (%o3)                         done
      (%i4) is(f(a) > f(b));
      (%o4)                         true

  Categories: Declarations and inferences

Property: even
Property: odd

  `declare(a, even)` or `declare(a, odd)` tells Maxima to recognize the symbol a as an even or odd integer variable. The properties `even` and `odd` are not recognized by the functions `evenp`, `oddp`, and `integerp`.

  See also `declare` and `askinteger`.

  Example:

      (%i1) declare(n, even);
      (%o1)                         done
      (%i2) askinteger(n, even);
      (%o2)                          yes
      (%i3) askinteger(n);
      (%o3)                          yes
      (%i4) evenp(n);
      (%o4)                         false

  Categories: Declarations and inferences

Property: feature

  Maxima understands two distinct types of features, system features and features which apply to mathematical expressions. See also `status` for information about system features. See also `features` and `featurep` for information about mathematical features.

  `feature` itself is not the name of a function or variable.

Function: featurep (a, f)

  Attempts to determine whether the object a has the feature f on the basis of the facts in the current database. If so, it returns `true`, else `false`.

  Note that `featurep` returns `false` when neither f nor the negation of f can be established.

  `featurep` evaluates its argument.

  See also `declare` and `features`.

      (%i1) declare (j, even)$
      (%i2) featurep (j, integer);
      (%o2)                           true

  Categories: Predicate functions · Declarations and inferences

Declaration: features

  Maxima recognizes certain mathematical properties of functions and variables. These are called "features".

  `declare (x, foo)` gives the property foo to the function or variable x.

  `declare (foo, feature)` declares a new feature foo. For example, `declare ([red, green, blue], feature)` declares three new features, `red`, `green`, and `blue`.

  The predicate `featurep (x, foo)` returns `true` if x has the foo property, and `false` otherwise.

  The infolist `features` is a list of known features. These are

         integer        noninteger      even
         odd            rational        irrational
         real           imaginary       complex
         analytic       increasing      decreasing
         oddfun         evenfun         posfun
         constant       commutative     lassociative
         rassociative   symmetric       antisymmetric
         integervalued

  plus any user-defined features.

  `features` is a list of mathematical features. There is also a list of non-mathematical, system-dependent features. See `status`.

  Example:

      (%i1) declare (FOO, feature);
      (%o1)                         done
      (%i2) declare (x, FOO);
      (%o2)                         done
      (%i3) featurep (x, FOO);
      (%o3)                         true

  Categories: Declarations and inferences

Function: get (a, i)

  Retrieves the user property indicated by i associated with atom a or returns `false` if a doesn't have property i.

  `get` evaluates its arguments.

  See also `put` and `qput`.

      (%i1) put (%e, 'transcendental, 'type);
      (%o1)                    transcendental
      (%i2) put (%pi, 'transcendental, 'type)$
      (%i3) put (%i, 'algebraic, 'type)$
      (%i4) typeof (expr) := block ([q],
              if numberp (expr)
              then return ('algebraic),
              if not atom (expr)
              then return (maplist ('typeof, expr)),
              q: get (expr, 'type),
              if q=false
              then errcatch (error(expr,"is not numeric.")) else q)$
      (%i5) typeof (2*%e + x*%pi);
      x is not numeric.
      (%o5)  [[transcendental, []], [algebraic, transcendental]]
      (%i6) typeof (2*%e + %pi);
      (%o6)     [transcendental, [algebraic, transcendental]]

  Categories: Declarations and inferences

Property: integer
Property: noninteger

  `declare(a, integer)` or `declare(a, noninteger)` tells Maxima to recognize a as an integer or noninteger variable.

  See also `declare`.

  Example:

      (%i1) declare(n, integer, x, noninteger);
      (%o1)                         done
      (%i2) askinteger(n);
      (%o2)                          yes
      (%i3) askinteger(x);
      (%o3)                          no

  Categories: Declarations and inferences

Property: integervalued

  `declare(f, integervalued)` tells Maxima to recognize f as an integer-valued function.

  See also `declare`.

  Example:

      (%i1) exp(%i)^f(x);
                                    %i f(x)
      (%o1)                      (%e  )
      (%i2) declare(f, integervalued);
      (%o2)                         done
      (%i3) exp(%i)^f(x);
                                    %i f(x)
      (%o3)                       %e

  Categories: Declarations and inferences

Property: nonarray

  The command `declare(a, nonarray)` tells Maxima to consider a not an array. This declaration prevents multiple evaluation, if a is a subscripted variable.

  See also `declare`.

  Example:

      (%i1) a:'b$ b:'c$ c:'d$

      (%i4) a[x];
      (%o4)                          d
                                      x
      (%i5) declare(a, nonarray);
      (%o5)                         done
      (%i6) a[x];
      (%o6)                          a
                                      x

  Categories: Expressions

Property: nonscalar

  Makes atoms behave as does a list or matrix with respect to the dot operator.

  See also `declare`.

  Categories: Declarations and inferences · Vectors · Matrices

Function: nonscalarp (expr)

  Returns `true` if expr is a non-scalar, i.e., it contains atoms declared as non-scalars, lists, or matrices.

  See also the predicate function `scalarp` and `declare`.

  Categories: Predicate functions · Vectors · Matrices

Property: posfun

  `declare (f, posfun)` declares `f` to be a positive function. `is (f(x) > 0)` yields `true`.

  See also `declare`.

  Categories: Declarations and inferences · Operators

Function: printprops
`printprops` (a, i)
`printprops` ([a_1, …, a_n], i)
`printprops` (all, i)

  Displays the property with the indicator i associated with the atom a. a may also be a list of atoms or the atom `all` in which case all of the atoms with the given property will be used. For example, `printprops ([f, g], atvalue)`. `printprops` is for properties that cannot otherwise be displayed, i.e. for `atvalue`, `atomgrad`, `gradef`, and `matchdeclare`.

  Categories: Declarations and inferences · Display functions

Function: properties (a)

  Returns a list of the names of all the properties associated with the atom a.

  Categories: Declarations and inferences

System variable: props

  Default value: `[]`

  `props` are atoms which have any property other than those explicitly mentioned in `infolists`, such as specified by `atvalue`, `matchdeclare`, etc., as well as properties specified in the `declare` function.

  Categories: Declarations and inferences · Global variables

Function: propvars (prop)

  Returns a list of those atoms on the `props` list which have the property indicated by prop. Thus `propvars (atvalue)` returns a list of atoms which have atvalues.

  Categories: Declarations and inferences

Function: put (atom, value, indicator)

  Assigns value to the property (specified by indicator) of atom. indicator may be the name of any property, not just a system-defined property.

  `rem` reverses the effect of `put`.

  `put` evaluates its arguments. `put` returns value.

  See also `qput` and `get`.

  Examples:

      (%i1) put (foo, (a+b)^5, expr);
                                         5
      (%o1)                       (b + a)
      (%i2) put (foo, "Hello", str);
      (%o2)                         Hello
      (%i3) properties (foo);
      (%o3)            [[user properties, str, expr]]
      (%i4) get (foo, expr);
                                         5
      (%o4)                       (b + a)
      (%i5) get (foo, str);
      (%o5)                         Hello

  Categories: Declarations and inferences

Function: qput (atom, value, indicator)

  Assigns value to the property (specified by indicator) of atom. This is the same as `put`, except that the arguments are quoted.

  See also `get`.

  Example:

      (%i1) foo: aa$
      (%i2) bar: bb$
      (%i3) baz: cc$
      (%i4) put (foo, bar, baz);
      (%o4)                          bb
      (%i5) properties (aa);
      (%o5)                [[user properties, cc]]
      (%i6) get (aa, cc);
      (%o6)                          bb
      (%i7) qput (foo, bar, baz);
      (%o7)                          bar
      (%i8) properties (foo);
      (%o8)            [value, [user properties, baz]]
      (%i9) get ('foo, 'baz);
      (%o9)                          bar

  Categories: Declarations and inferences

Property: rational
Property: irrational

  `declare(a, rational)` or `declare(a, irrational)` tells Maxima to recognize a as a rational or irrational real variable.

  See also `declare`.

  Categories: Declarations and inferences

Property: real
Property: imaginary
Property: complex

  `declare(a, real)`, `declare(a, imaginary)`, or `declare(a, complex)` tells Maxima to recognize a as a real, pure imaginary, or complex variable.

  See also `declare`.

  Categories: Declarations and inferences

Function: rem (atom, indicator)

  Removes the property indicated by indicator from atom. `rem` reverses the effect of `put`.

  `rem` returns `done` if atom had an indicator property when `rem` was called, or `false` if it had no such property.

  Categories: Declarations and inferences

Function: remove
`remove` (a_1, p_1, …, a_n, p_n)
`remove` ([a_1, …, a_m], [p_1, …, p_n], …)
`remove` ("a", operator)
`remove` (a, transfun)
`remove` (all, p)

  Removes properties associated with atoms.

  `remove (a_1, p_1, ..., a_n, p_n)` removes property `p_k` from atom `a_k`.

  `remove ([a_1, ..., a_m], [p_1, ..., p_n], ...)` removes properties `p_1, ..., p_n` from atoms a_1, …, a_m. There may be more than one pair of lists.

  `remove (all, p)` removes the property p from all atoms which have it.

  The removed properties may be system-defined properties such as `function`, `macro`, or `mode_declare`. `remove` does not remove properties defined by `put`.

  A property may be `transfun` to remove the translated Lisp version of a function. After executing this, the Maxima version of the function is executed rather than the translated version.

  `remove ("a", operator)` or, equivalently, `remove ("a", op)` removes from a the operator properties declared by `prefix`, `infix`, `nary`, `postfix`, `matchfix`, or `nofix`. Note that the name of the operator must be written as a quoted string.

  `remove` always returns `done` whether or not an atom has a specified property. This behavior is unlike the more specific remove functions `remvalue`, `remarray`, `remfunction`, and `remrule`.

  `remove` quotes its arguments.

  Categories: Declarations and inferences

Property: scalar

  `declare(a, scalar)` tells Maxima to consider a a scalar variable.

  See also `declare`.

  Categories: Declarations and inferences

Function: scalarp (expr)

  Returns `true` if expr is a number, constant, or variable declared `scalar` with `declare`, or composed entirely of numbers, constants, and such variables, but not containing matrices or lists.

  See also the predicate function `nonscalarp`.

  Categories: Predicate functions · Vectors · Matrices

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## 11.3 Functions and Variables for Facts

Function: activate (context_1, …, context_n)

  Activates the contexts context_1, …, context_n. The facts in these contexts are then available to make deductions and retrieve information. The facts in these contexts are not listed by `facts ()`.

  The variable `activecontexts` is the list of contexts which are active by way of the `activate` function.

  Categories: Declarations and inferences

System variable: activecontexts

  Default value: `[]`

  `activecontexts` is a list of the contexts which are active by way of the `activate` function, as opposed to being active because they are subcontexts of the current context.

  Categories: Declarations and inferences

Function: askinteger
`askinteger` (expr, integer)
`askinteger` (expr)
`askinteger` (expr, even)
`askinteger` (expr, odd)

  `askinteger (expr, integer)` attempts to determine from the `assume` database whether expr is an integer. `askinteger` prompts the user if it cannot tell otherwise, and attempt to install the information in the database if possible. `askinteger (expr)` is equivalent to `askinteger (expr, integer)`.

  `askinteger (expr, even)` and `askinteger (expr, odd)` likewise attempt to determine if expr is an even integer or odd integer, respectively.

  Categories: Declarations and inferences

Function: asksign (expr)

  First attempts to determine whether the specified expression is positive, negative, or zero. If it cannot, it asks the user the necessary questions to complete its deduction. The user's answer is recorded in the data base for the duration of the current computation. The return value of `asksign` is one of `pos`, `neg`, or `zero`.

  Categories: Declarations and inferences

Function: assume (pred_1, …, pred_n)

  Adds predicates pred_1, …, pred_n to the current context. If a predicate is inconsistent or redundant with the predicates in the current context, it is not added to the context. The context accumulates predicates from each call to `assume`.

  `assume` returns a list whose elements are the predicates added to the context or the atoms `redundant` or `inconsistent` where applicable.

  The predicates pred_1, …, pred_n can only be expressions with the relational operators `< <= equal notequal >=` and `>`. Predicates cannot be literal equality `=` or literal inequality `#` expressions, nor can they be predicate functions such as `integerp`.

  Compound predicates of the form `pred_1 and ... and pred_n` are recognized, but not `pred_1 or ... or pred_n`. `not pred_k` is recognized if pred_k is a relational predicate. Expressions of the form `not (pred_1 and pred_2)` and `not (pred_1 or pred_2)` are not recognized.

  Maxima's deduction mechanism is not very strong; there are many obvious consequences which cannot be determined by `is`. This is a known weakness.

  `assume` does not handle predicates with complex numbers. If a predicate contains a complex number `assume` returns `inconsistent` or `redunant`.

  `assume` evaluates its arguments.

  See also `is`, `facts`, `forget`, `context`, and `declare`.

  Examples:

      (%i1) assume (xx > 0, yy < -1, zz >= 0);
      (%o1)              [xx > 0, yy < - 1, zz >= 0]
      (%i2) assume (aa < bb and bb < cc);
      (%o2)                  [bb > aa, cc > bb]
      (%i3) facts ();
      (%o3)     [xx > 0, - 1 > yy, zz >= 0, bb > aa, cc > bb]
      (%i4) is (xx > yy);
      (%o4)                         true
      (%i5) is (yy < -yy);
      (%o5)                         true
      (%i6) is (sinh (bb - aa) > 0);
      (%o6)                         true
      (%i7) forget (bb > aa);
      (%o7)                       [bb > aa]
      (%i8) prederror : false;
      (%o8)                         false
      (%i9) is (sinh (bb - aa) > 0);
      (%o9)                        unknown
      (%i10) is (bb^2 < cc^2);
      (%o10)                       unknown

  Categories: Declarations and inferences

Option variable: assumescalar

  Default value: `true`

  `assumescalar` helps govern whether expressions `expr` for which `nonscalarp (expr)` is `false` are assumed to behave like scalars for certain transformations.

  Let `expr` represent any expression other than a list or a matrix, and let `[1, 2, 3]` represent any list or matrix. Then `expr . [1, 2, 3]` yields `[expr, 2 expr, 3 expr]` if `assumescalar` is `true`, or `scalarp (expr)` is `true`, or `constantp (expr)` is `true`.

  If `assumescalar` is `true`, such expressions will behave like scalars only for commutative operators, but not for noncommutative multiplication `.`.

  When `assumescalar` is `false`, such expressions will behave like non-scalars.

  When `assumescalar` is `all`, such expressions will behave like scalars for all the operators listed above.

  Categories: Declarations and inferences

Option variable: assume_pos

  Default value: `false`

  When `assume_pos` is `true` and the sign of a parameter x cannot be determined from the current context or other considerations, `sign` and `asksign (x)` return `true`. This may forestall some automatically-generated `asksign` queries, such as may arise from `integrate` or other computations.

  By default, a parameter is x such that `symbolp (x)` or `subvarp (x)`. The class of expressions considered parameters can be modified to some extent via the variable `assume_pos_pred`.

  `sign` and `asksign` attempt to deduce the sign of expressions from the sign of operands within the expression. For example, if `a` and `b` are both positive, then `a + b` is also positive.

  However, there is no way to bypass all `asksign` queries. In particular, when the `asksign` argument is a difference `x - y` or a logarithm `log(x)`, `asksign` always requests an input from the user, even when `assume_pos` is `true` and `assume_pos_pred` is a function which returns `true` for all arguments.

  Categories: Declarations and inferences

Option variable: assume_pos_pred

  Default value: `false`

  When `assume_pos_pred` is assigned the name of a function or a lambda expression of one argument x, that function is called to determine whether x is considered a parameter for the purpose of `assume_pos`. `assume_pos_pred` is ignored when `assume_pos` is `false`.

  The `assume_pos_pred` function is called by `sign` and `asksign` with an argument x which is either an atom, a subscripted variable, or a function call expression. If the `assume_pos_pred` function returns `true`, x is considered a parameter for the purpose of `assume_pos`.

  By default, a parameter is x such that `symbolp (x)` or `subvarp (x)`.

  See also `assume` and `assume_pos`.

  Examples:

      (%i1) assume_pos: true$
      (%i2) assume_pos_pred: symbolp$
      (%i3) sign (a);
      (%o3)                          pos
      (%i4) sign (a[1]);
      (%o4)                          pnz
      (%i5) assume_pos_pred: lambda ([x], display (x), true)$
      (%i6) asksign (a);
                                    x = a

      (%o6)                          pos
      (%i7) asksign (a[1]);
                                   x = a
                                        1

      (%o7)                          pos
      (%i8) asksign (foo (a));
                                 x = foo(a)

      (%o8)                          pos
      (%i9) asksign (foo (a) + bar (b));
                                 x = foo(a)

                                 x = bar(b)

      (%o9)                          pos
      (%i10) asksign (log (a));
                                    x = a

      Is  a - 1  positive, negative, or zero?

      p;
      (%o10)                         pos
      (%i11) asksign (a - b);
                                    x = a

                                    x = b

                                    x = a

                                    x = b

      Is  b - a  positive, negative, or zero?

      p;
      (%o11)                         neg

  Categories: Declarations and inferences

Option variable: context

  Default value: `initial`

  `context` names the collection of facts maintained by `assume` and `forget`. `assume` adds facts to the collection named by `context`, while `forget` removes facts.

  Binding `context` to a name foo changes the current context to foo. If the specified context foo does not yet exist, it is created automatically by a call to `newcontext`. The specified context is activated automatically.

  See `contexts` for a general description of the context mechanism.

  Categories: Declarations and inferences

Option variable: contexts

  Default value: `[initial, global]`

  `contexts` is a list of the contexts which currently exist, including the currently active context.

  The context mechanism makes it possible for a user to bind together and name a collection of facts, called a context. Once this is done, the user can have Maxima assume or forget large numbers of facts merely by activating or deactivating their context.

  Any symbolic atom can be a context, and the facts contained in that context will be retained in storage until destroyed one by one by calling `forget` or destroyed as a whole by calling `kill` to destroy the context to which they belong.

  Contexts exist in a hierarchy, with the root always being the context `global`, which contains information about Maxima that some functions need. When in a given context, all the facts in that context are "active" (meaning that they are used in deductions and retrievals) as are all the facts in any context which is a subcontext of the active context.

  When a fresh Maxima is started up, the user is in a context called `initial`, which has `global` as a subcontext.

  See also `facts`, `newcontext`, `supcontext`, `killcontext`, `activate`, `deactivate`, `assume`, and `forget`.

  Categories: Declarations and inferences

Function: deactivate (context_1, …, context_n)

  Deactivates the specified contexts context_1, …, context_n.

  Categories: Declarations and inferences

Function: facts
`facts` (item)
`facts` ()

  If item is the name of a context, `facts (item)` returns a list of the facts in the specified context.

  If item is not the name of a context, `facts (item)` returns a list of the facts known about item in the current context. Facts that are active, but in a different context, are not listed.

  `facts ()` (i.e., without an argument) lists the current context.

  Categories: Declarations and inferences

Function: forget
`forget` (pred_1, …, pred_n)
`forget` (L)

  Removes predicates established by `assume`. The predicates may be expressions equivalent to (but not necessarily identical to) those previously assumed.

  `forget (L)`, where L is a list of predicates, forgets each item on the list.

  Categories: Declarations and inferences

Function: is (expr)

  Attempts to determine whether the predicate expr is provable from the facts in the `assume` database.

  If the predicate is provably `true` or `false`, `is` returns `true` or `false`, respectively. Otherwise, the return value is governed by the global flag `prederror`. When `prederror` is `true`, `is` complains with an error message. Otherwise, `is` returns `unknown`.

  `ev(expr, pred)` (which can be written `expr, pred` at the interactive prompt) is equivalent to `is(expr)`.

  See also `assume`, `facts`, and `maybe`.

  Examples:

  `is` causes evaluation of predicates.

      (%i1) %pi > %e;
      (%o1)                       %pi > %e
      (%i2) is (%pi > %e);
      (%o2)                         true

  `is` attempts to derive predicates from the `assume` database.

      (%i1) assume (a > b);
      (%o1)                        [a > b]
      (%i2) assume (b > c);
      (%o2)                        [b > c]
      (%i3) is (a < b);
      (%o3)                         false
      (%i4) is (a > c);
      (%o4)                         true
      (%i5) is (equal (a, c));
      (%o5)                         false

  If `is` can neither prove nor disprove a predicate from the `assume` database, the global flag `prederror` governs the behavior of `is`.

      (%i1) assume (a > b);
      (%o1)                        [a > b]
      (%i2) prederror: true$
      (%i3) is (a > 0);
      Maxima was unable to evaluate the predicate:
      a > 0
       -- an error.  Quitting.  To debug this try debugmode(true);
      (%i4) prederror: false$
      (%i5) is (a > 0);
      (%o5)                        unknown

  Categories: Predicate functions · Declarations and inferences

Function: killcontext (context_1, …, context_n)

  Kills the contexts context_1, …, context_n.

  If one of the contexts is the current context, the new current context will become the first available subcontext of the current context which has not been killed. If the first available unkilled context is `global` then `initial` is used instead. If the `initial` context is killed, a new, empty `initial` context is created.

  `killcontext` refuses to kill a context which is currently active, either because it is a subcontext of the current context, or by use of the function `activate`.

  `killcontext` evaluates its arguments. `killcontext` returns `done`.

  Categories: Declarations and inferences

Function: maybe (expr)

  Attempts to determine whether the predicate expr is provable from the facts in the `assume` database.

  If the predicate is provably `true` or `false`, `maybe` returns `true` or `false`, respectively. Otherwise, `maybe` returns `unknown`.

  `maybe` is functionally equivalent to `is` with `prederror: false`, but the result is computed without actually assigning a value to `prederror`.

  See also `assume`, `facts`, and `is`.

  Examples:

      (%i1) maybe (x > 0);
      (%o1)                        unknown
      (%i2) assume (x > 1);
      (%o2)                        [x > 1]
      (%i3) maybe (x > 0);
      (%o3)                         true

  Categories: Predicate functions · Declarations and inferences

Function: newcontext
`newcontext` (name)
`newcontext` ()

  Creates a new, empty context, called name, which has `global` as its only subcontext. The newly-created context becomes the currently active context.

  If name is not specified, a new name is created (via `gensym`) and returned.

  `newcontext` evaluates its argument. `newcontext` returns name (if specified) or the new context name.

  Categories: Declarations and inferences

Function: sign (expr)

  Attempts to determine the sign of expr on the basis of the facts in the current data base. It returns one of the following answers: `pos` (positive), `neg` (negative), `zero`, `pz` (positive or zero), `nz` (negative or zero), `pn` (positive or negative), or `pnz` (positive, negative, or zero, i.e. nothing known).

  Categories: Declarations and inferences

Function: supcontext
`supcontext` (name, context)
`supcontext` (name)
`supcontext` ()

  Creates a new context, called name, which has context as a subcontext. context must exist.

  If context is not specified, the current context is assumed.

  If name is not specified, a new name is created (via `gensym`) and returned.

  `supcontext` evaluates its argument. `supcontext` returns name (if specified) or the new context name.

  Categories: Declarations and inferences

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## 11.4 Functions and Variables for Predicates

Function: charfun (p)

  Return 0 when the predicate p evaluates to `false`; return 1 when the predicate evaluates to `true`. When the predicate evaluates to something other than `true` or `false` (unknown), return a noun form.

  Examples:

      (%i1) charfun (x < 1);
      (%o1)                    charfun(x < 1)
      (%i2) subst (x = -1, %);
      (%o2)                           1
      (%i3) e : charfun ('"and" (-1 < x, x < 1))$
      (%i4) [subst (x = -1, e), subst (x = 0, e), subst (x = 1, e)];
      (%o4)                       [0, 1, 0]

  Categories: Mathematical functions

Function: compare (x, y)

  Return a comparison operator op (`<`, `<=`, `>`, `>=`, `=`, or `#`) such that `is (x op y)` evaluates to `true`; when either x or y depends on `%i` and `x # y`, return `notcomparable`; when there is no such operator or Maxima isn't able to determine the operator, return `unknown`.

  Examples:

      (%i1) compare (1, 2);
      (%o1)                           <
      (%i2) compare (1, x);
      (%o2)                        unknown
      (%i3) compare (%i, %i);
      (%o3)                           =
      (%i4) compare (%i, %i + 1);
      (%o4)                     notcomparable
      (%i5) compare (1/x, 0);
      (%o5)                           #
      (%i6) compare (x, abs(x));
      (%o6)                          <=

  The function `compare` doesn't try to determine whether the real domains of its arguments are nonempty; thus

      (%i1) compare (acos (x^2 + 1), acos (x^2 + 1) + 1);
      (%o1)                           <

  The real domain of `acos (x^2 + 1)` is empty.

  Categories: Declarations and inferences

Function: equal (a, b)

  Represents equivalence, that is, equal value.

  By itself, `equal` does not evaluate or simplify. The function `is` attempts to evaluate `equal` to a Boolean value. `is(equal(a, b))` returns `true` (or `false`) if and only if a and b are equal (or not equal) for all possible values of their variables, as determined by evaluating `ratsimp(a - b)`; if `ratsimp` returns 0, the two expressions are considered equivalent. Two expressions may be equivalent even if they are not syntactically equal (i.e., identical).

  When `is` fails to reduce `equal` to `true` or `false`, the result is governed by the global flag `prederror`. When `prederror` is `true`, `is` complains with an error message. Otherwise, `is` returns `unknown`.

  In addition to `is`, some other operators evaluate `equal` and `notequal` to `true` or `false`, namely `if`, `and`, `or`, and `not`.

  The negation of `equal` is `notequal`.

  Examples:

  By itself, `equal` does not evaluate or simplify.

      (%i1) equal (x^2 - 1, (x + 1) * (x - 1));
                              2
      (%o1)            equal(x  - 1, (x - 1) (x + 1))
      (%i2) equal (x, x + 1);
      (%o2)                    equal(x, x + 1)
      (%i3) equal (x, y);
      (%o3)                      equal(x, y)

  The function `is` attempts to evaluate `equal` to a Boolean value. `is(equal(a, b))` returns `true` when `ratsimp(a - b)` returns 0. Two expressions may be equivalent even if they are not syntactically equal (i.e., identical).

      (%i1) ratsimp (x^2 - 1 - (x + 1) * (x - 1));
      (%o1)                           0
      (%i2) is (equal (x^2 - 1, (x + 1) * (x - 1)));
      (%o2)                         true
      (%i3) is (x^2 - 1 = (x + 1) * (x - 1));
      (%o3)                         false
      (%i4) ratsimp (x - (x + 1));
      (%o4)                          - 1
      (%i5) is (equal (x, x + 1));
      (%o5)                         false
      (%i6) is (x = x + 1);
      (%o6)                         false
      (%i7) ratsimp (x - y);
      (%o7)                         x - y
      (%i8) is (equal (x, y));
      (%o8)                        unknown
      (%i9) is (x = y);
      (%o9)                         false

  When `is` fails to reduce `equal` to `true` or `false`, the result is governed by the global flag `prederror`.

      (%i1) [aa : x^2 + 2*x + 1, bb : x^2 - 2*x - 1];
                          2             2
      (%o1)             [x  + 2 x + 1, x  - 2 x - 1]
      (%i2) ratsimp (aa - bb);
      (%o2)                        4 x + 2
      (%i3) prederror : true;
      (%o3)                         true
      (%i4) is (equal (aa, bb));
      Maxima was unable to evaluate the predicate:
             2             2
      equal(x  + 2 x + 1, x  - 2 x - 1)
       -- an error.  Quitting.  To debug this try debugmode(true);
      (%i5) prederror : false;
      (%o5)                         false
      (%i6) is (equal (aa, bb));
      (%o6)                        unknown

  Some operators evaluate `equal` and `notequal` to `true` or `false`.

      (%i1) if equal (y, y - 1) then FOO else BAR;
      (%o1)                          BAR
      (%i2) eq_1 : equal (x, x + 1);
      (%o2)                    equal(x, x + 1)
      (%i3) eq_2 : equal (y^2 + 2*y + 1, (y + 1)^2);
                               2                   2
      (%o3)             equal(y  + 2 y + 1, (y + 1) )
      (%i4) [eq_1 and eq_2, eq_1 or eq_2, not eq_1];
      (%o4)                  [false, true, true]

  Because `not expr` causes evaluation of expr, `not equal(a, b)` is equivalent to `is(notequal(a, b))`.

      (%i1) [notequal (2*z, 2*z - 1), not equal (2*z, 2*z - 1)];
      (%o1)            [notequal(2 z, 2 z - 1), true]
      (%i2) is (notequal (2*z, 2*z - 1));
      (%o2)                         true

  Categories: Operators

Function: notequal (a, b)

  Represents the negation of `equal(a, b)`.

  Examples:

      (%i1) equal (a, b);
      (%o1)                      equal(a, b)
      (%i2) maybe (equal (a, b));
      (%o2)                        unknown
      (%i3) notequal (a, b);
      (%o3)                    notequal(a, b)
      (%i4) not equal (a, b);
      (%o4)                    notequal(a, b)
      (%i5) maybe (notequal (a, b));
      (%o5)                        unknown
      (%i6) assume (a > b);
      (%o6)                        [a > b]
      (%i7) equal (a, b);
      (%o7)                      equal(a, b)
      (%i8) maybe (equal (a, b));
      (%o8)                         false
      (%i9) notequal (a, b);
      (%o9)                    notequal(a, b)
      (%i10) maybe (notequal (a, b));
      (%o10)                        true

  Categories: Operators

Function: unknown (expr)

  Returns `true` if and only if expr contains an operator or function not recognized by the Maxima simplifier.

  Categories: Predicate functions · Simplification functions

Function: zeroequiv (expr, v)

  Tests whether the expression expr in the variable v is equivalent to zero, returning `true`, `false`, or `dontknow`.

  `zeroequiv` has these restrictions:

  1. Do not use functions that Maxima does not know how to differentiate and evaluate.
  2. If the expression has poles on the real line, there may be errors in the result (but this is unlikely to occur).
  3. If the expression contains functions which are not solutions to first order differential equations (e.g. Bessel functions) there may be incorrect results.
  4. The algorithm uses evaluation at randomly chosen points for carefully selected subexpressions. This is always a somewhat hazardous business, although the algorithm tries to minimize the potential for error.

  For example `zeroequiv (sin(2 * x) - 2 * sin(x) * cos(x), x)` returns `true` and `zeroequiv (%e^x + x, x)` returns `false`. On the other hand `zeroequiv (log(a * b) - log(a) - log(b), a)` returns `dontknow` because of the presence of an extra parameter `b`.

  Categories: Predicate functions

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# 12. Plotting

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## 12.1 Introduction to Plotting

Maxima uses an external plotting package to make the plots (see the section on `Plotting Formats`). The plotting functions calculate a set of points and pass them to the plotting package together with a set of commands. That information can be passed to the external program either through a pipe or by calling the program with the name of a file where the data has been saved. The data file is given the name `maxout_xxx.format`, where `xxx` is a number that is unique to every concurrently-running instance of Maxima and `format` is the name of the plotting format being used (`gnuplot`, `xmaxima`, `mgnuplot`, `gnuplot_pipes` or `geomview`).

There are to save the plot in a graphic format file. In those cases, the file `maxout_xxx.format` created by Maxima includes commands that will make the external plotting program save the result in a graphic file. The default name for that graphic file is `maxplot.extension`, where `extension` is the extension normally used for the kind of graphic file selected.

The `maxout_xxx.format` and `maxplot.extension` files are created in the directory specified by the system variable `maxima_tempdir`. That location can be changed by assigning to that variable (or to the environment variable MAXIMA_TEMPDIR) a string that represents a valid directory where Maxima can create new files. The output of the Maxima plotting command will be a list with the names of the file(s) created, including their complete path.

If the format used is either `gnuplot` or `xmaxima`, the external programs `gnuplot` or `xmaxima` can be run, giving it the file `maxout_xxx.format` as argument, in order to view again a plot previously created in Maxima. Thus, when a Maxima plotting command fails, the format can be set to `gnuplot` or `xmaxima` and the plain-text file `maxout_xxx.gnuplot` (or `maxout_xxx.xmaxima`) can be inspected to look for the source of the problem.

The additional package draw provides functions similar to the ones described in this section with some extra features. Note that some plotting options have the same name in both plotting packages, but their syntax and behavior is different. To view the documentation for a graphic option `opt`, type `?? opt` in order to choose the information for either of those two packages.

Categories: Plotting

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## 12.2 Plotting Formats

Maxima can use either Gnuplot, Xmaxima or Geomview as graphics program. Gnuplot and Geomview are external programs which must be installed separately, while Xmaxima is distributed with Maxima. There are various different formats for those programs, which can be selected with the option `plot_format` (see also the `Plotting Options` section).

The plotting formats are the following:

- gnuplot (default on Windows)

  Used to launch the external program gnuplot, which must be installed in your system. All plotting commands and data are saved into the file `maxout_xxx.gnuplot`.

- gnuplot_pipes (default on non-Windows platforms)

  This format is not available in Windows platforms. It is similar to the `gnuplot` format except that the commands are sent to gnuplot through a pipe, while the data are saved into the file `maxout_xxx.gnuplot_pipes`. A single gnuplot process is kept open and subsequent plot commands will be sent to the same process, replacing previous plots, unless the gnuplot pipe is closed with the function `gnuplot_close`. When this format is used, the function `gnuplot_replot` can be used to modify a plot that has already displayed on the screen.

  This format is only used to plot to the screen; whenever graphic files are created, the format is silently switched to `gnuplot` and the commands needed to create the graphic file are saved with the data in file `maxout_xxx.gnuplot`.

- mgnuplot

  Mgnuplot is a Tk-based wrapper around gnuplot. It is included in the Maxima distribution. Mgnuplot offers a rudimentary GUI for gnuplot, but has fewer overall features than the plain gnuplot interface. Mgnuplot requires an external gnuplot installation and, in Unix systems, the Tcl/Tk system.

- xmaxima

  Xmaxima is a Tcl/Tk graphical interface for Maxima that can also be used to display plots created when Maxima is run from the console or from other graphical interfaces. To use this format, the xmaxima program, which is distributed together with Maxima, must be installed. If Maxima is being run from the Xmaxima console, the data and commands are passed to xmaxima through the same socket used for the communication between Maxima and the Xmaxima console. When used from a terminal or from graphical interfaces different from Xmaxima, the commands and data are saved in the file `maxout_xxx.xmaxima` and xmaxima is run with the name of that file as argument.

  In previous versions this format used to be called `openmath`; that old name still works as a synonym for `xmaxima`.

- geomview

  Geomview, a Motif based interactive 3D viewing program for Unix, can also be used to display plots created by Maxima. To use this format, the geomview program must be installed.

Categories: Plotting

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## 12.3 Functions and Variables for Plotting

Function: contour_plot (expr, x_range, y_range, options, …)

  It plots the contours (curves of equal value) of expr over the region x_range by y_range. Any additional arguments are treated the same as in `plot3d`.

  This function only works when the plot format is either `gnuplot` or `gnuplot_pipes`. The additional package `implicit_plot`, which works in any graphic format, can also be used to plot contours but a separate expression must be given for each contour.

  Examples:

      (%i1) contour_plot (x^2 + y^2, [x, -4, 4], [y, -4, 4])$

  You can add any options accepted by `plot3d`; for instance, the option `legend` with a value of false, to remove the legend. By default, Gnuplot chooses and displays 3 contours. To increase the number of contours, it is necessary to use a custom `gnuplot_preamble`, as in the next example:

      (%i1) contour_plot (u^3 + v^2, [u, -4, 4], [v, -4, 4],
                    [legend,false],
                    [gnuplot_preamble, "set cntrparam levels 12"])$

  Categories: Plotting

System variable: geomview_command

  This variable stores the name of the command used to run the geomview program when the plot format is `geomview`. Its default value is "geomview". If the geomview program is not found unless you give its complete path or if you want to try a different version of it, you may change the value of this variable. For instance,

      (%i1) geomview_command: "/usr/local/bin/my_geomview"$

  Categories: Plotting

Function: get_plot_option (keyword, index)

  Returns the current default value of the option named keyword, which is a list. The optional argument index must be a positive integer which can be used to extract only one element from the list (element 1 is the name of the option).

  See also `set_plot_option`, `remove_plot_option` and the section on Plotting Options.

System variable: gnuplot_command

  This variable stores the name of the command used to run the gnuplot program when the plot format is `gnuplot`. Its default value is "gnuplot". If the gnuplot program is not found unless you give its complete path or if you want to try a different version of it, you may change the value of this variable. For instance,

      (%i1) gnuplot_command: "/usr/local/bin/my_gnuplot"$

  Categories: Plotting

System variable: gnuplot_file_args

  When a graphic file is going to be created using `gnuplot`, this variable is used to specify the way the file name should be passed to gnuplot. Its default value is "~s", which means that the name of the file will be passed directly. The contents of this variable can be changed in order to add options for the gnuplot program, adding those options before the format directive "~s".

  Categories: Plotting

System variable: gnuplot_view_args

  This variable is used to parse the argument that will be passed to the gnuplot program when the plot format is `gnuplot`. Its default value is "-persist ~s", where "~s" will be replaced with the name of the file where the gnuplot commands have been written (usually "maxout_xxx.gnuplot"). The option `-persist` tells gnuplot to exit after the commands in the file have been executed, without closing the window that displays the plot.

  Those familiar with gnuplot, might want to change the value of this variable. For example, by changing it to:

      (%i1) gnuplot_view_args: "~s -"$

  gnuplot will not be closed after the commands in the file have been executed; thus, the window with the plot will remain, as well as the gnuplot interactive shell where other commands can be issued in order to modify the plot.

  In Windows versions of Gnuplot older than 4.6.3 the behavior of "~s -" and "-persist ~s" were the opposite; namely, "-persist ~s" made the plot window and the gnuplot interactive shell remain, while "~s -" closed the gnuplot shell keeping the plot window. Therefore, when older gnuplot versions are used in Windows, it might be necessary to adjust the value of `gnuplot_view_args`.

  Categories: Plotting

Function: implicit_plot
`implicit_plot` (expr, x_range, y_range)
`implicit_plot` ([expr_1, …, expr_n], x_range, y_range)

  Displays a plot of a function on the real plane, defined implicitly by the expression expr. The domain in the plane is defined by x_range and y_range. Several functions can be represented on the same plot, giving a list [expr_1, …, expr_n] of expressions that define them. This function uses the global format options set up with the `set_plot_option`. Additional options can also be given as extra arguments for the `implicit_plot` command.

  The method used by `implicit_plot` consists of tracking sign changes on the domain given and it can fail for complicated expressions.

  `load(implicit_plot)` loads this function.

  Example:

      (%i1) load(implicit_plot)$
      (%i2) implicit_plot (x^2 = y^3 - 3*y + 1, [x, -4, 4], [y, -4, 4])$

  Categories: Plotting · Share packages · Package implicit_plot

Function: julia (x, y, ...options...)

  Creates a graphic representation of the Julia set for the complex number (x + i y). The two mandatory parameters x and y must be real. This program is part of the additional package `dynamics`, but that package does not have to be loaded; the first time julia is used, it will be loaded automatically.

  Each pixel in the grid is given a color corresponding to the number of iterations it takes the sequence that starts at that point to move out of the convergence circle of radius 2 centered at the origin. The number of pixels in the grid is controlled by the `grid` plot option (default 30 by 30). The maximum number of iterations is set with the option `iterations`. The program uses its own default palette: magenta,violet, blue, cyan, green, yellow, orange, red, brown and black, but it can be changed by adding an explicit `palette` option in the command.

  The default domain used goes from -2 to 2 in both axes and can be changed with the `x` and `y` options. By default, the two axes are shown with the same scale, unless the option `yx_ratio` is used or the option `same_xy` is disabled. Other general plot options are also accepted.

  The following example shows a region of the Julia set for the number -0.55 + i0.6. The option `color_bar_tics` is used to prevent Gnuplot from adjusting the color box up to 40, in which case the points corresponding the maximum 36 iterations would not be black.

      (%i1) julia (-0.55, 0.6, [iterations, 36], [x, -0.3, 0.2],
            [y, 0.3, 0.9], [grid, 400, 400], [color_bar_tics, 0, 6, 36])$

  Categories: Package dynamics · Plotting

Function: make_transform ([var1, var2, var3], fx, fy, fz)

  Returns a function suitable to be used in the option `transform_xy` of plot3d. The three variables var1, var2, var3 are three dummy variable names, which represent the 3 variables given by the plot3d command (first the two independent variables and then the function that depends on those two variables). The three functions fx, fy, fz must depend only on those 3 variables, and will give the corresponding x, y and z coordinates that should be plotted. There are two transformations defined by default: `polar_to_xy` and `spherical_to_xyz`. See the documentation for those two transformations.

  Categories: Plotting

Function: mandelbrot (options)

  Creates a graphic representation of the Mandelbrot set. This program is part of the additional package `dynamics`, but that package does not have to be loaded; the first time mandelbrot is used, the package will be loaded automatically.

  This program can be called without any arguments, in which case it will use a default value of 9 iterations per point, a grid with dimensions set by the `grid` plot option (default 30 by 30) and a region that extends from -2 to 2 in both axes. The options are all the same that plot2d accepts, plus an option `iterations` to change the number of iterations.

  Each pixel in the grid is given a color corresponding to the number of iterations it takes the sequence starting at zero to move out of the convergence circle of radius 2, centered at the origin. The maximum number of iterations is set by the option `iterations`. The program uses its own default palette: magenta,violet, blue, cyan, green, yellow, orange, red, brown and black, but it can be changed by adding an explicit `palette` option in the command. By default, the two axes are shown with the same scale, unless the option `yx_ratio` is used or the option `same_xy` is disabled.

  Example:

  [grid,400,400])$

      (%i1) mandelbrot ([iterations, 30], [x, -2, 1], [y, -1.2, 1.2],
                  [grid,400,400])$

  Categories: Package dynamics · Plotting

System function: polar_to_xy

  It can be given as value for the `transform_xy` option of plot3d. Its effect will be to interpret the two independent variables in plot3d as the distance from the z axis and the azimuthal angle (polar coordinates), and transform them into x and y coordinates.

  Categories: Plotting

Function: plot2d
`plot2d` (plot, x_range, …, options, …)
`plot2d` ([plot_1, …, plot_n], …, options, …)
`plot2d` ([plot_1, …, plot_n], x_range, …, options, …)

  Where plot, plot_1, …, plot_n can be either expressions, function names or a list with the any of the forms: `[discrete, [x1, ..., xn], [y1, ..., yn]]`, `[discrete, [[x1, y1], ..., [xn, ..., yn]]]` or `[parametric, x_expr, y_expr, t_range]`.

  Displays a plot of one or more expressions as a function of one variable or parameter.

  `plot2d` displays one or several plots in two dimensions. When expressions or function name are used to define the plots, they should all depend on only one variable var and the use of x_range will be mandatory, to provide the name of the variable and its minimum and maximum values; the syntax for x_range is: `[variable, min, max]`.

  A plot can also be defined in the discrete or parametric forms. The discrete form is used to plot a set of points with given coordinates. A discrete plot is defined by a list starting with the keyword discrete, followed by one or two lists of values. If two lists are given, they must have the same length; the first list will be interpreted as the x coordinates of the points to be plotted and the second list as the y coordinates. If only one list is given after the discrete keyword, each element on the list could also be a list with two values that correspond to the x and y coordinates of a point, or it could be a sequence of numerical values which will be plotted at consecutive integer values (1,2,3,...) on the x axis.

  A parametric plot is defined by a list starting with the keyword parametric, followed by two expressions or function names and a range for the parameter. The range for the parameter must be a list with the name of the parameter followed by its minimum and maximum values: `[param, min, max]`. The plot will show the path traced out by the point with coordinates given by the two expressions or functions, as param increases from min to max.

  A range for the vertical axis is an optional argument with the form: `[y, min, max]` (the keyword y is always used for the vertical axis). If that option is used, the plot will show that exact vertical range, independently of the values reached by the plot. If the vertical range is not specified, it will be set up according to the minimum and maximum values of the second coordinate of the plot points.

  All other options should also be lists, starting with a keyword and followed by one or more values. See `plot_options`.

  If there are several plots to be plotted, a legend will be written to identity each of the expressions. The labels that should be used in that legend can be given with the option `legend`. If that option is not used, Maxima will create labels from the expressions or function names.

  Examples:

  Plot of a common function:

      (%i1) plot2d (sin(x), [x, -%pi, %pi])$

  If the function grows too fast, it might be necessary to limit the values in the vertical axis using the `y` option:

      (%i1) plot2d (sec(x), [x, -2, 2], [y, -20, 20])$

  When the plot box is disabled, no labels are created for the axes. In that case, instead of using `xlabel` and `ylabel` to set the names of the axes, it is better to use option `label`, which allows more flexibility. Option `yx_ratio` is used to change the default rectangular shape of the plot; in this example the plot will fill a square.

      (%i1) plot2d ( x^2 - 1, [x, -3, 3], [box, false], grid2d,
            [yx_ratio, 1], [axes, solid], [xtics, -2, 4, 2],
            [ytics, 2, 2, 6], [label, ["x", 2.9, -0.3],
            ["x^2-1", 0.1, 8]], [title, "A parabola"])$

  A plot with a logarithmic scale in the vertical axis:

      (%i1) plot2d (exp(3*s), [s, -2, 2], logy)$

  Plotting functions by name:

      (%i1) F(x) := x^2 $
      (%i2) :lisp (defun |$g| (x) (m* x x x))
      $g
      (%i2) H(x) := if x < 0 then x^4 - 1 else 1 - x^5 $
      (%i3) plot2d ([F, G, H], [u, -1, 1], [y, -1.5, 1.5])$

  A plot of the butterfly curve, defined parametrically:

      (%i1) r: (exp(cos(t))-2*cos(4*t)-sin(t/12)^5)$
      (%i2) plot2d([parametric, r*sin(t), r*cos(t), [t, -8*%pi, 8*%pi]])$

  Plot of a circle, using its parametric representation, together with the function -|x|. The circle will only look like a circle if the scale in the two axes is the same, which is done with the option `same_xy`.

      (%i1) plot2d([[parametric, cos(t), sin(t), [t,0,2*%pi]], -abs(x)],
               [x, -sqrt(2), sqrt(2)], same_xy)$

  A plot of 200 random numbers between 0 and 9:

      (%i1) plot2d ([discrete, makelist ( random(10), 200)])$

  A plot of a discrete set of points, defining x and y coordinates separately:

      (%i1) plot2d ([discrete, makelist(i*%pi, i, 1, 5),
                                  [0.6, 0.9, 0.2, 1.3, 1]])$

  In the next example a table with three columns is saved in a file "data.txt" which is then read and the second and third column are plotted on the two axes:

      (%i1) with_stdout ("data.txt", for x:0 thru 10 do
                                   print (x, x^2, x^3))$
      (%i2) data: read_matrix ("data.txt")$
      (%i3) plot2d ([discrete, transpose(data)[2], transpose(data)[3]],
        [style,points], [point_type,diamond], [color,red])$

  A plot of discrete data points together with a continuous function:

      (%i1) xy: [[10, .6], [20, .9], [30, 1.1], [40, 1.3], [50, 1.4]]$
      (%i2) plot2d([[discrete, xy], 2*%pi*sqrt(l/980)], [l,0,50],
              [style, points, lines], [color, red, blue],
              [point_type, asterisk],
              [legend, "experiment", "theory"],
              [xlabel, "pendulum's length (cm)"],
              [ylabel, "period (s)"])$

  See also the section about Plotting Options.

  Categories: Plotting

Function: plot3d
`plot3d` (expr, x_range, y_range, …, options, …)
`plot3d` ([expr_1, …, expr_n], x_range, y_range, …, options, …)

  Displays a plot of one or more surfaces defined as functions of two variables or in parametric form.

  The functions to be plotted may be specified as expressions or function names. The mouse can be used to rotate the plot looking at the surface from different sides.

  Examples:

  Plot of a function of two variables:

      (%i1) plot3d (u^2 - v^2, [u, -2, 2], [v, -3, 3], [grid, 100, 100],
              [mesh_lines_color,false])$

  Use of the `z` option to limit a function that goes to infinity (in this case the function is minus infinity on the x and y axes); this also shows how to plot with only lines and no shading:

      (%i1) plot3d ( log ( x^2*y^2 ), [x, -2, 2], [y, -2, 2], [z, -8, 4],
               [palette, false], [color, magenta])$

  The infinite values of z can also be avoided by choosing a grid that does not fall on any points where the function is undefined, as in the next example, which also shows how to change the palette and how to include a color bar that relates colors to values of the z variable:

      (%i1) plot3d (log (x^2*y^2), [x, -2, 2], [y, -2, 2],[grid, 29, 29],
             [palette, [gradient, red, orange, yellow, green]],
             color_bar, [xtics, 1], [ytics, 1], [ztics, 4],
             [color_bar_tics, 4])$

  Two surfaces in the same plot. Ranges specific to one of the surfaces can be given by placing each expression and its ranges in a separate list; global ranges for the complete plot are also given after the functions definitions.

      (%i1) plot3d ([[-3*x - y, [x, -2, 2], [y, -2, 2]],
         4*sin(3*(x^2 + y^2))/(x^2 + y^2), [x, -3, 3], [y, -3, 3]],
         [x, -4, 4], [y, -4, 4])$

  Plot of a Klein bottle, defined parametrically:

      (%i1) expr_1: 5*cos(x)*(cos(x/2)*cos(y)+sin(x/2)*sin(2*y)+3)-10$
      (%i2) expr_2: -5*sin(x)*(cos(x/2)*cos(y)+sin(x/2)*sin(2*y)+3)$
      (%i3) expr_3: 5*(-sin(x/2)*cos(y)+cos(x/2)*sin(2*y))$
      (%i4) plot3d ([expr_1, expr_2, expr_3], [x, -%pi, %pi],
              [y, -%pi, %pi], [grid, 50, 50])$

  Plot of a "spherical harmonic" function, using the predefined transformation, `spherical_to_xyz` to transform from spherical coordinates to rectangular coordinates. See the documentation for `spherical_to_xyz`.

      (%i1) plot3d (sin(2*theta)*cos(phi), [theta, 0, %pi],
              [phi, 0, 2*%pi],
              [transform_xy, spherical_to_xyz], [grid,30,60],
         [legend,false])$

  Use of the pre-defined function `polar_to_xy` to transform from cylindrical to rectangular coordinates. See the documentation for `polar_to_xy`.

      (%i1) plot3d (r^.33*cos(th/3), [r,0,1], [th,0,6*%pi], [box, false],
         [grid, 12, 80], [transform_xy, polar_to_xy], [legend, false])$

  Plot of a sphere using the transformation from spherical to rectangular coordinates. Option `same_xyz` is used to get the three axes scaled in the same proportion. When transformations are used, it is not convenient to eliminate the mesh lines, because Gnuplot will not show the surface correctly.

      (%i1) plot3d ( 5, [theta, 0, %pi], [phi, 0, 2*%pi], same_xyz,
        [transform_xy, spherical_to_xyz], [mesh_lines_color,blue],
        [palette,[gradient,"#1b1b4e", "#8c8cf8"]], [legend, false])$

  Definition of a function of two-variables using a matrix. Notice the single quote in the definition of the function, to prevent `plot3d` from failing when it realizes that the matrix will require integer indices.

      (%i1) M: matrix([1,2,3,4], [1,2,3,2], [1,2,3,4], [1,2,3,3])$
      (%i2) f(x, y) := float('M [round(x), round(y)])$
      (%i3) plot3d (f(x,y), [x,1,4],[y,1,4],[grid,3,3],[legend,false])$

  By setting the elevation equal to zero, a surface can be seen as a map in which each color represents a different level.

      (%i1) plot3d (cos (-x^2 + y^3/4), [x,-4,4], [y,-4,4], [zlabel,""],
             [mesh_lines_color,false], [elevation,0], [azimuth,0],
             color_bar, [grid,80,80], [ztics,false], [color_bar_tics,1])$

  See also the section about Plotting Options.

  Categories: Plotting

System variable: plot_options

  This option is being kept for compatibility with older versions, but its use is deprecated. To set global plotting options, see their current values or remove options, use `set_plot_option`, `get_plot_option` and `remove_plot_option`.

  Categories: Plotting

Function: remove_plot_option (name)

  Removes the default value of an option. The name of the option must be given.

  See also `set_plot_option`, `get_plot_option` and the section on Plotting Options.

  Categories: Plotting

Function: set_plot_option (option)

  Accepts any of the options listed in the section Plotting Options, and saves them for use in plotting commands. The values of the options set in each plotting command will have precedence, but if those options are not given, the default values set with this function will be used.

  `set_plot_option` evaluates its argument and returns the complete list of options (after modifying the option given). If called without any arguments, it will simply show the list of current default options.

  See also `remove_plot_option`, `get_plot_option` and the section on Plotting Options.

  Example:

  Modification of the `grid` values.

      (%i1) set_plot_option ([grid, 30, 40]);
      (%o1) [[plot_format, gnuplot_pipes], [grid, 30, 40],
      [run_viewer, true], [axes, true], [nticks, 29], [adapt_depth, 5],
      [color, blue, red, green, magenta, black, cyan],
      [point_type, bullet, box, triangle, plus, times, asterisk],
      [palette, [gradient, green, cyan, blue, violet],
      [gradient, magenta, violet, blue, cyan, green, yellow, orange,
      red, brown, black]], [gnuplot_preamble, ], [gnuplot_term, default]]

  Categories: Plotting

System function: spherical_to_xyz

  It can be given as value for the `transform_xy` option of `plot3d`. Its effect will be to interpret the two independent variables and the function in `plot3d` as the spherical coordinates of a point (first, the angle with the z axis, then the angle of the xy projection with the x axis and finally the distance from the origin) and transform them into x, y and z coordinates.

  Categories: Plotting

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## 12.4 Plotting Options

All options consist of a list starting with one of the keywords in this section, followed by one or more values. Some of the options may have different effects in different plotting commands as it will be pointed out in the following list. The options that accept among their possible values true or false, can also be set to true by simply writing their names. For instance, typing logx as an option is equivalent to writing [logx, true].

Plot option: adapt_depth [adapt_depth, integer]

  Default value: `5`

  The maximum number of splittings used by the adaptive plotting routine.

  Categories: Plotting

Plot option: axes [axes, symbol]

  Default value: `true`

  Where symbol can be either `true`, `false`, `x`, `y` or `solid`. If `false`, no axes are shown; if equal to `x` or `y` only the x or y axis will be shown; if it is equal to `true`, both axes will be shown and `solid` will show the two axes with a solid line, rather than the default broken line. This option does not have any effect in the 3 dimensional plots.

  Categories: Plotting

Plot option: azimuth [azimuth, number]

  Default value: `30`

  A plot3d plot can be thought of as starting with the x and y axis in the horizontal and vertical axis, as in plot2d, and the z axis coming out of the screen. The z axis is then rotated around the x axis through an angle equal to `elevation` and then the new xy plane is rotated around the new z axis through an angle `azimuth`. This option sets the value for the azimuth, in degrees.

  See also `elevation`.

  Categories: Plotting

Plot option: box [box, symbol]

  Default value: `true`

  If set to `true`, a bounding box will be drawn for the plot; if set to `false`, no box will be drawn.

  Categories: Plotting

Plot option: color [color, color_1, …, color_n]

  In 2d plots it defines the color (or colors) for the various curves. In `plot3d`, it defines the colors used for the mesh lines of the surfaces, when no palette is being used.

  If there are more curves or surfaces than colors, the colors will be repeated in sequence. The valid colors are `red`, `green`, `blue`, `magenta`, `cyan`, `yellow`, `orange`, `violet`, `brown`, `gray`, `black`, `white`, or a string starting with the character # and followed by six hexadecimal digits: two for the red component, two for green component and two for the blue component. If the name of a given color is unknown color, black will be used instead.

  Categories: Plotting

Plot option: color_bar [color_bar, symbol]

  Default value: `false` in plot3d, `true` in mandelbrot and julia

  Where symbol can be either `true` or `false`. If `true`, whenever `plot3d`, `mandelbrot` or `julia` use a palette to represent different values, a box will be shown on the right, showing the corresponding between colors and values.

  Categories: Plotting

Plot option: color_bar_tics [color_bar_tics, x1, x2, x3]

  Defines the values at which a mark and a number will be placed in the color bar. The first number is the initial value, the second the increments and the third is the last value where a mark is placed. The second and third numbers can be omitted. When only one number is given, it will be used as the increment from an initial value that will be chosen automatically.

  Categories: Plotting

Plot option: elevation [elevation, number]

  Default value: `60`

  A plot3d plot can be thought of as starting with the x and y axis in the horizontal and vertical axis, as in plot2d, and the z axis coming out of the screen. The z axis is then rotated around the x axis through an angle equal to `elevation` and then the new xy plane is rotated around the new z axis through an angle `azimuth`. This option sets the value for the azimuth, in degrees.

  See also `azimuth`.

  Categories: Plotting

Plot option: grid [grid, integer, integer]

  Default value: `30`, `30`

  Sets the number of grid points to use in the x- and y-directions for three-dimensional plotting or for the `julia` and `mandelbrot` programs.

  Categories: Plotting

Plot option: grid2d [grid, value]

  Default value: `false`

  Shows a grid of lines on the xy plane. The points where the grid lines are placed are the same points where tics are marked in the x and y axes, which can be controlled with the `xtics` and `ytics` options.

  Categories: Plotting

Plot option: iterations [grid, value]

  Default value: `9`

  Number of iterations made by the programs mandelbrot and julia.

  Categories: Plotting

Plot option: label [label, [string, x, y], …]

  Writes one or several labels in the points with x, y coordinates indicated after each label.

  Categories: Plotting

Plot option: legend
`legend` [legend, string_1, …, string_n]
`legend` [legend, false]

  It specifies the labels for the plots when various plots are shown. If there are more plots than the number of labels given, they will be repeated. If given the value `false`, no legends will be shown. By default, the names of the expressions or functions will be used, or the words discrete1, discrete2, …, for discrete sets of points.

  Categories: Plotting

Plot option: logx [logx, value]

  Makes the horizontal axes to be scaled logarithmically. It can be either true or false.

  Categories: Plotting

Plot option: logy [logy, value]

  Makes the vertical axes to be scaled logarithmically. It can be either true or false.

  Categories: Plotting

Plot option: mesh_lines_color [mesh_lines_color, color]

  Default value: `black`

  It sets the color used by plot3d to draw the mesh lines, when a palette is being used. It accepts the same colors as for the option `color` (see the list of allowed colors in `color`). It can also be given a value `false` to eliminate completely the mesh lines.

  Categories: Plotting

Plot option: nticks [nticks, integer]

  Default value: `29`

  When plotting functions with `plot2d`, it is gives the initial number of points used by the adaptive plotting routine for plotting functions. When plotting parametric functions with `plot3d`, it sets the number of points that will be shown for the plot.

  Categories: Plotting

Plot option: palette
`palette` [palette, [palette_1], …, [palette_n]]
`palette` [palette, false]

  It can consist of one palette or a list of several palettes. Each palette is a list with a keyword followed by values. If the keyword is gradient, it should be followed by a list of valid colors.

  If the keyword is hue, saturation or value, it must be followed by 4 numbers. The first three numbers, which must be between 0 and 1, define the hue, saturation and value of a basic color to be assigned to the minimum value of z. The keyword specifies which of the three attributes (hue, saturation or value) will be increased according to the values of z. The last number indicates the increase corresponding to the maximum value of z. That last number can be bigger than 1 or negative; the corresponding values of the modified attribute will be rounded modulo 1.

  Gnuplot only uses the first palette in the list; xmaxima will use the palettes in the list sequentially, when several surfaces are plotted together; if the number of palettes is exhausted, they will be repeated sequentially.

  The color of the mesh lines will be given by the option `mesh_lines_color`. If `palette` is given the value `false`, the surfaces will not be shaded but represented with a mesh of curves only. In that case, the colors of the lines will be determined by the option `color`.

  Categories: Plotting

Plot option: plot_format [plot_format, format]

  Default value: `gnuplot`, in Windows systems, or `gnuplot_pipes` in other systems.

  Where format is one of the following: gnuplot, xmaxima, mgnuplot, gnuplot_pipes or geomview.

  It sets the format to be used for plotting.

  Categories: Plotting

Plot option: plot_realpart [plot_realpart, symbol]

  Default value: `false`

  If set to `true`, the functions to be plotted will be considered as complex functions whose real value should be plotted; this is equivalent to plotting `realpart(function)`. If set to `false`, nothing will be plotted when the function does not give a real value. For instance, when `x` is negative, `log(x)` gives a complex value, with real value equal to `log(abs(x))`; if `plot_realpart` were `true`, `log(-5)` would be plotted as `log(5)`, while nothing would be plotted if `plot_realpart` were `false`.

  Categories: Plotting

Plot option: point_type [point_type, type_1, …, type_n]

  In gnuplot, each set of points to be plotted with the style "points" or "linespoints" will be represented with objects taken from this list, in sequential order. If there are more sets of points than objects in this list, they will be repeated sequentially. The possible objects that can be used are: `bullet`, `circle`, `plus`, `times`, `asterisk`, `box`, `square`, `triangle`, `delta`, `wedge`, `nabla`, `diamond`, `lozenge`.

  Categories: Plotting

Plot option: pdf_file [pdf_file, file_name]

  Saves the plot into a PDF file with name equal to file_name, rather than showing it in the screen. By default, the file will be created in the directory defined by the variable `maxima_tempdir`, unless file_name contains the character "/", in which case it will be assumed to contain the complete path where the file should be created. The value of `maxima_tempdir` can be changed to save the file in a different directory. When the option `gnuplot_pdf_term_command` is also given, it will be used to set up Gnuplot's PDF terminal; otherwise, Gnuplot's pdfcairo terminal will be used with solid colored lines of width 3, plot size of 17.2 cm by 12.9 cm and font of 18 points.

  Categories: Plotting

Plot option: png_file [png_file, file_name]

  Saves the plot into a PNG graphics file with name equal to file_name, rather than showing it in the screen. By default, the file will be created in the directory defined by the variable `maxima_tempdir`, unless file_name contains the character "/", in which case it will be assumed to contain the complete path where the file should be created. The value of `maxima_tempdir` can be changed to save the file in a different directory. When the option `gnuplot_png_term_command` is also given, it will be used to set up Gnuplot's PNG terminal; otherwise, Gnuplot's pngcairo terminal will be used, with a font of size 12.

  Categories: Plotting

Plot option: ps_file [ps_file, file_name]

  Saves the plot into a Postscript file with name equal to file_name, rather than showing it in the screen. By default, the file will be created in the directory defined by the variable `maxima_tempdir`, unless file_name contains the character "/", in which case it will be assumed to contain the complete path where the file should be created. The value of `maxima_tempdir` can be changed to save the file in a different directory. When the option `gnuplot_ps_term_command` is also given, it will be used to set up Gnuplot's Postscript terminal; otherwise, Gnuplot's postscript terminal will be used with the EPS option, solid colored lines of width 2, plot size of 16.4 cm by 12.3 cm and font of 24 points.

  Categories: Plotting

Plot option: run_viewer [run_viewer, symbol]

  This option is only used when the plot format is `gnuplot` and the terminal is `default` or when the Gnuplot terminal is set to `dumb` (see `gnuplot_term`) and can have a true or false value.

  If the terminal is `default`, a file `maxout_xxx.gnuplot` (or other name specified with `gnuplot_out_file`) is created with the gnuplot commands necessary to generate the plot. Option `run_viewer` controls whether or not Gnuplot will be launched to execute those commands and show the plot.

  If the terminal is `default`, gnuplot is run to execute the commands in `maxout_xxx.gnuplot`, producing another file `maxplot.txt` (or other name specified with `gnuplot_out_file`). Option `run_viewer` controls whether or not that file, with an ASCII representation of the plot, will be shown in the Maxima or Xmaxima console.

  The default value for this option is true, making the plots to be shown in either the console or a separate graphics window.

  Categories: Plotting

Plot option: same_xy [same_xy , value]

  It can be either true or false. If true, the scales used in the x and y axes will be the same, in either 2d or 3d plots. See also `yx_ratio`.

  Categories: Plotting

Plot option: same_xyz [same_xyz , value]

  It can be either true or false. If true, the scales used in the 3 axes of a 3d plot will be the same.

  Categories: Plotting

Plot option: style
`style` [style, type_1, …, type_n]
`style` [style, [style_1], …, [style_n]]

  The styles that will be used for the various functions or sets of data in a 2d plot. The word style must be followed by one or more styles. If there are more functions and data sets than the styles given, the styles will be repeated. Each style can be either lines for line segments, points for isolated points, linespoints for segments and points, or dots for small isolated dots. Gnuplot accepts also an impulses style.

  Each of the styles can be enclosed inside a list with some additional parameters. lines accepts one or two numbers: the width of the line and an integer that identifies a color. The default color codes are: 1: blue, 2: red, 3: magenta, 4: orange, 5: brown, 6: lime and 7: aqua. If you use Gnuplot with a terminal different than X11, those colors might be different; for example, if you use the option [gnuplot_term, ps], color index 4 will correspond to black, instead of orange.

  points accepts one two or three parameters; the first parameter is the radius of the points, the second parameter is an integer that selects the color, using the same code used for lines and the third parameter is currently used only by Gnuplot and it corresponds to several objects instead of points. The default types of objects are: 1: filled circles, 2: open circles, 3: plus signs, 4: x, 5: *, 6: filled squares, 7: open squares, 8: filled triangles, 9: open triangles, 10: filled inverted triangles, 11: open inverted triangles, 12: filled lozenges and 13: open lozenges.

  linespoints accepts up to four parameters: line width, points radius, color and type of object to replace the points.

  See also `color` and `point_type`.

  Categories: Plotting

Plot option: svg_file [svg_file, file_name]

  Saves the plot into an SVG file with name equal to file_name, rather than showing it in the screen. By default, the file will be created in the directory defined by the variable `maxima_tempdir`, unless file_name contains the character "/", in which case it will be assumed to contain the complete path where the file should be created. The value of `maxima_tempdir` can be changed to save the file in a different directory. When the option `gnuplot_svg_term_command` is also given, it will be used to set up Gnuplot's SVG terminal; otherwise, Gnuplot's svg terminal will be used with font of 14 points.

  Categories: Plotting

Plot option: t [t, min, max]

  Default range for parametric plots.

  Categories: Plotting

Plot option: title [title, text]

  Defines a title that will be written at the top of the plot.

  Categories: Plotting

Plot option: transform_xy [transform_xy, symbol]

  Where symbol is either `false` or the result obtained by using the function `transform_xy`. If different from `false`, it will be used to transform the 3 coordinates in plot3d.

  See `make_transform`, `polar_to_xy` and `spherical_to_xyz`.

  Categories: Plotting

Plot option: x [x, min, max]

  When used as the first option in a `plot2d` command (or any of the first two in `plot3d`), it indicates that the first independent variable is x and it sets its range. It can also be used again after the first option (or after the second option in plot3d) to define the effective horizontal domain that will be shown in the plot.

  Categories: Plotting

Plot option: xlabel [xlabel, string]

  Specifies the string that will label the first axis; if this option is not used, that label will be the name of the independent variable, when plotting functions with `plot2d` or `implicit_plot`, or the name of the first variable, when plotting surfaces with `plot3d` or contours with `contour_plot`, or the first expression in the case of a parametric plot. It can not be used with `set_plot_option`.

  Categories: Plotting

Plot option: xtics [xtics, x1, x2, x3]

  Defines the values at which a mark and a number will be placed in the x axis. The first number is the initial value, the second the increments and the third is the last value where a mark is placed. The second and third numbers can be omitted. When only one number is given, it will be used as the increment from an initial value that will be chosen automatically.

  Categories: Plotting

Plot option: xy_scale [xy_scale, sx, sy]

  In a 2d plot, it defines the ratio of the total size of the Window to the size that will be used for the plot. The two numbers given as arguments are the scale factors for the x and y axes.

  Categories: Plotting

Plot option: y [y, min, max]

  When used as one of the first two options in `plot3d`, it indicates that one of the independent variables is y and it sets its range. Otherwise, it defines the effective domain of the second variable that will be shown in the plot.

  Categories: Plotting

Plot option: ylabel [ylabel, string]

  Specifies the string that will label the second axis; if this option is not used, that label will be "y", when plotting functions with `plot2d` or `implicit_plot`, or the name of the second variable, when plotting surfaces with `plot3d` or contours with `contour_plot`, or the second expression in the case of a parametric plot. It can not be used with `set_plot_option`.

  Categories: Plotting

Plot option: ytics [ytics, y1, y2, y3]

  Defines the values at which a mark and a number will be placed in the y axis. The first number is the initial value, the second the increments and the third is the last value where a mark is placed. The second and third numbers can be omitted. When only one number is given, it will be used as the increment from an initial value that will be chosen automatically

  Categories: Plotting

Plot option: yx_ratio [yx_ratio, r]

  In a 2d plot, the ratio between the vertical and the horizontal sides of the rectangle used to make the plot. See also `same_xy`.

  Categories: Plotting

Plot option: z [z, min, max]

  Used in `plot3d` to set the effective range of values of z that will be shown in the plot.

  Categories: Plotting

Plot option: zlabel [zlabel, string]

  Specifies the string that will label the third axis, when using `plot3d`. If this option is not used, that label will be "z", when plotting surfaces, or the third expression in the case of a parametric plot. It can not be used with `set_plot_option` and it will be ignored by `plot2d` and `implicit_plot`.

  Categories: Plotting

Plot option: zmin [zmin, z]

  In 3d plots, the value of z that will be at the bottom of the plot box.

  Categories: Plotting

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## 12.5 Gnuplot Options

There are several plot options specific to gnuplot. All of them consist of a keyword (the name of the option), followed by a string that should be a valid gnuplot command, to be passed directly to gnuplot. In most cases, there exist a corresponding plotting option that will produce a similar result and whose use is more recommended than the gnuplot specific option.

Plot option: gnuplot_term [gnuplot_term, terminal_name]

  Sets the output terminal type for gnuplot. The argument terminal_name can be a string or one of the following 3 special symbols

  - default (default value)

    Gnuplot output is displayed in a separate graphical window and the gnuplot terminal used will be specified by the value of the option `gnuplot_default_term_command`.

  - dumb

    Gnuplot output is saved to a file `maxout_xxx.gnuplot` using "ASCII art" approximation to graphics. If the option `gnuplot_out_file` is set to filename, the plot will be saved there, instead of the default `maxout_xxx.gnuplot`. The settings for the "dumb" terminal of Gnuplot are given by the value of option `gnuplot_dumb_term_command`. If option `run_viewer` is set to true and the plot_format is `gnuplot` that ASCII representation will also be shown in the Maxima or Xmaxima console.

  - ps

    Gnuplot generates commands in the PostScript page description language. If the option `gnuplot_out_file` is set to filename, gnuplot writes the PostScript commands to filename. Otherwise, it is saved as `maxplot.ps` file. The settings for this terminal are given by the value of the option `gnuplot_dumb_term_command`.

  - A string representing any valid gnuplot term specification

    Gnuplot can generate output in many other graphical formats such as png, jpeg, svg etc. To use those formats, option `gnuplot_term` can be set to any supported gnuplot term name (which must be a symbol) or even a full gnuplot term specification with any valid options (which must be a string). For example `[gnuplot_term, png]` creates output in PNG (Portable Network Graphics) format while `[gnuplot_term, "png size 1000,1000"]` creates PNG of 1000 x 1000 pixels size. If the option `gnuplot_out_file` is set to filename, gnuplot writes the output to filename. Otherwise, it is saved as `maxplot.term` file, where term is gnuplot terminal name.

  Categories: Plotting

Plot option: gnuplot_out_file [gnuplot_out_file, file_name]

  It can be used to replace the default name for the file that contains the commands that will interpreted by gnuplot, when the terminal is set to `default`, or to replace the default name of the graphic file that gnuplot creates, when the terminal is different from `default`. If it contains one or more slashes, "/", the name of the file will be left as it is; otherwise, it will be appended to the path of the temporary directory. The complete name of the files created by the plotting commands is always sent as output of those commands so they can be seen if the command is ended by semi-colon.

  When used in conjunction with the `gnuplot_term` option, it can be used to save the plot in a file, in one of the graphic formats supported by Gnuplot. To create PNG, PDF, Postscript or SVG, it is easier to use options `png_file`, `pdf_file`, `ps_file`, or `svg_file`.

  Categories: Plotting

Plot option: gnuplot_pm3d [gnuplot_pm3d, value]

  With a value of `false`, it can be used to disable the use of PM3D mode, which is enabled by default.

  Categories: Plotting

Plot option: gnuplot_preamble [gnuplot_preamble, string]

  This option inserts gnuplot commands before any other commands sent to Gnuplot. Any valid gnuplot commands may be used. Multiple commands should be separated with a semi-colon. See also `gnuplot_postamble`.

  Categories: Plotting

Plot option: gnuplot_postamble [gnuplot_postamble, string]

  This option inserts gnuplot commands after other commands sent to Gnuplot and right before the plot command is sent. Any valid gnuplot commands may be used. Multiple commands should be separated with a semi-colon. See also `gnuplot_preamble`.

  Categories: Plotting

Plot option: gnuplot_default_term_command

  [gnuplot_default_term_command, command]

  The gnuplot command to set the terminal type for the default terminal. It this option is not set, the command used will be: `"set term wxt size 640,480 font \",12\"; set term pop"`.

  Categories: Plotting

Plot option: gnuplot_dumb_term_command

  [gnuplot_dumb_term_command, command]

  The gnuplot command to set the terminal type for the dumb terminal. It this option is not set, the command used will be: `"set term dumb 79 22"`, which makes the text output 79 characters by 22 characters.

  Categories: Plotting

Plot option: gnuplot_pdf_term_command [gnuplot_pdf_term_command, command]

  The gnuplot command to set the terminal type for the PDF terminal. If this option is not set, the command used will be: `"set term pdfcairo color solid lw 3 size 17.2 cm, 12.9 cm font \",18\""`. See the gnuplot documentation for more information.

  Categories: Plotting

Plot option: gnuplot_png_term_command [gnuplot_png_term_command, command]

  The gnuplot command to set the terminal type for the PNG terminal. If this option is not set, the command used will be: `"set term pngcairo font \",12\""`. See the gnuplot documentation for more information.

  Categories: Plotting

Plot option: gnuplot_ps_term_command [gnuplot_ps_term_command, command]

  The gnuplot command to set the terminal type for the PostScript terminal. If this option is not set, the command used will be: `"set term postscript eps color solid lw 2 size 16.4 cm, 12.3 cm font \",24\""`. See the gnuplot documentation for `set term postscript` for more information.

  Categories: Plotting

Plot option: gnuplot_svg_term_command [gnuplot_svg_term_command, command]

  The gnuplot command to set the terminal type for the SVG terminal. If this option is not set, the command used will be: `"set term svg font \",14\""`. See the gnuplot documentation for more information.

  Categories: Plotting

Plot option: gnuplot_curve_titles

  This is an obsolete option that has been replaced `legend` described above.

  Categories: Plotting

Plot option: gnuplot_curve_styles

  This is an obsolete option that has been replaced by `style`.

  Categories: Plotting

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## 12.6 Gnuplot_pipes Format Functions

Function: gnuplot_start ()

  Opens the pipe to gnuplot used for plotting with the `gnuplot_pipes` format. Is not necessary to manually open the pipe before plotting.

  Categories: Plotting

Function: gnuplot_close ()

  Closes the pipe to gnuplot which is used with the `gnuplot_pipes` format.

  Categories: Plotting

Function: gnuplot_restart ()

  Closes the pipe to gnuplot which is used with the `gnuplot_pipes` format and opens a new pipe.

  Categories: Plotting

Function: gnuplot_replot
`gnuplot_replot` ()
`gnuplot_replot` (s)

  Updates the gnuplot window. If `gnuplot_replot` is called with a gnuplot command in a string s, then `s` is sent to gnuplot before reploting the window.

  Categories: Plotting

Function: gnuplot_reset ()

  Resets the state of gnuplot used with the `gnuplot_pipes` format. To update the gnuplot window call `gnuplot_replot` after `gnuplot_reset`.

  Categories: Plotting

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# 13. File Input and Output

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## 13.1 Comments

A comment in Maxima input is any text between `/*` and `*/`.

The Maxima parser treats a comment as whitespace for the purpose of finding tokens in the input stream; a token always ends at a comment. An input such as `a/* foo */b` contains two tokens, `a` and `b`, and not a single token `ab`. Comments are otherwise ignored by Maxima; neither the content nor the location of comments is stored in parsed input expressions.

Comments can be nested to arbitrary depth. The `/*` and `*/` delimiters form matching pairs. There must be the same number of `/*` as there are `*/`.

Examples:

    (%i1) /* aa is a variable of interest */  aa : 1234;
    (%o1)                         1234
    (%i2) /* Value of bb depends on aa */  bb : aa^2;
    (%o2)                        1522756
    (%i3) /* User-defined infix operator */  infix ("b");
    (%o3)                           b
    (%i4) /* Parses same as a b c, not abc */  a/* foo */b/* bar */c;
    (%o4)                         a b c
    (%i5) /* Comments /* can be nested /* to any depth */ */ */  1 + xyz;
    (%o5)                        xyz + 1

Categories: Syntax

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## 13.2 Files

A file is simply an area on a particular storage device which contains data or text. Files on the disks are figuratively grouped into "directories". A directory is just a list of files. Commands which deal with files are:

       appendfile           batch                 batchload
       closefile            file_output_append    filename_merge
       file_search          file_search_maxima    file_search_lisp
       file_search_demo     file_search_usage     file_search_tests
       file_type            file_type_lisp        file_type_maxima
       load                 load_pathname         loadfile
       loadprint            pathname_directory    pathname_name
       pathname_type        printfile             save
       stringout            with_stdout           writefile

When a file name is passed to functions like `plot2d`, `save`, or `writefile` and the file name does not include a path, Maxima stores the file in the current working directory. The current working directory depends on the system like Windows or Linux and on the installation.

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## 13.3 Functions and Variables for File Input and Output

Function: appendfile (filename)

  Appends a console transcript to filename. `appendfile` is the same as `writefile`, except that the transcript file, if it exists, is always appended.

  `closefile` closes the transcript file opened by `appendfile` or `writefile`.

  Categories: File output · Console interaction

Function: batch
`batch` (filename)
`batch` (filename, `option`)

  `batch(filename)` reads Maxima expressions from filename and evaluates them. `batch` searches for filename in the list `file_search_maxima`. See also `file_search`.

  `batch(filename, `demo`)` is like `demo(filename)`. In this case `batch` searches for filename in the list `file_search_demo`. See `demo`.

  `batch(filename, `test`)` is like `run_testsuite` with the option `display_all=true`. For this case `batch` searches filename in the list `file_search_maxima` and not in the list `file_search_tests` like `run_testsuite`. Furthermore, `run_testsuite` runs tests which are in the list `testsuite_files`. With `batch` it is possible to run any file in a test mode, which can be found in the list `file_search_maxima`. This is useful, when writing a test file.

  filename comprises a sequence of Maxima expressions, each terminated with `;` or `$`. The special variable `%` and the function `%th` refer to previous results within the file. The file may include `:lisp` constructs. Spaces, tabs, and newlines in the file are ignored. A suitable input file may be created by a text editor or by the `stringout` function.

  `batch` reads each input expression from filename, displays the input to the console, computes the corresponding output expression, and displays the output expression. Input labels are assigned to the input expressions and output labels are assigned to the output expressions. `batch` evaluates every input expression in the file unless there is an error. If user input is requested (by `asksign` or `askinteger`, for example) `batch` pauses to collect the requisite input and then continue.

  It may be possible to halt `batch` by typing `control-C` at the console. The effect of `control-C` depends on the underlying Lisp implementation.

  `batch` has several uses, such as to provide a reservoir for working command lines, to give error-free demonstrations, or to help organize one's thinking in solving complex problems.

  `batch` evaluates its argument. `batch` returns the path of filename as a string, when called with no second argument or with the option `demo`. When called with the option `test`, the return value is a an empty list `[]` or a list with filename and the numbers of the tests which have failed.

  See also `load`, `batchload`, and `demo`.

  Categories: Session management · File input

Function: batchload (filename)

  Reads Maxima expressions from filename and evaluates them, without displaying the input or output expressions and without assigning labels to output expressions. Printed output (such as produced by `print` or `describe`)) is displayed, however.

  The special variable `%` and the function `%th` refer to previous results from the interactive interpreter, not results within the file. The file cannot include `:lisp` constructs.

  `batchload` returns the path of filename, as a string. `batchload` evaluates its argument.

  See also `batch`, and `load`.

  Categories: Session management · File input

Function: closefile ()

  Closes the transcript file opened by `writefile` or `appendfile`.

  Categories: File output · Console interaction

Option variable: file_output_append

  Default value: `false`

  `file_output_append` governs whether file output functions append or truncate their output file. When `file_output_append` is `true`, such functions append to their output file. Otherwise, the output file is truncated.

  `save`, `stringout`, and `with_stdout` respect `file_output_append`. Other functions which write output files do not respect `file_output_append`. In particular, plotting and translation functions always truncate their output file, and `tex` and `appendfile` always append.

  Categories: File output · Global flags

Function: filename_merge (path, filename)

  Constructs a modified path from path and filename. If the final component of path is of the form `###.something`, the component is replaced with `filename.something`. Otherwise, the final component is simply replaced by filename.

  The result is a Lisp pathname object.

  Categories: File input · File output

Function: file_search
`file_search` (filename)
`file_search` (filename, pathlist)

  `file_search` searches for the file filename and returns the path to the file (as a string) if it can be found; otherwise `file_search` returns `false`. `file_search (filename)` searches in the default search directories, which are specified by the `file_search_maxima`, `file_search_lisp`, and `file_search_demo` variables.

  `file_search` first checks if the actual name passed exists, before attempting to match it to "wildcard" file search patterns. See `file_search_maxima` concerning file search patterns.

  The argument filename can be a path and file name, or just a file name, or, if a file search directory includes a file search pattern, just the base of the file name (without an extension). For example,

      file_search ("/home/wfs/special/zeta.mac");
      file_search ("zeta.mac");
      file_search ("zeta");

  all find the same file, assuming the file exists and `/home/wfs/special/###.mac` is in `file_search_maxima`.

  `file_search (filename, pathlist)` searches only in the directories specified by pathlist, which is a list of strings. The argument pathlist supersedes the default search directories, so if the path list is given, `file_search` searches only the ones specified, and not any of the default search directories. Even if there is only one directory in pathlist, it must still be given as a one-element list.

  The user may modify the default search directories. See `file_search_maxima`.

  `file_search` is invoked by `load` with `file_search_maxima` and `file_search_lisp` as the search directories.

  Categories: File input

Option variable: file_search_maxima
Option variable: file_search_lisp
Option variable: file_search_demo
Option variable: file_search_usage
Option variable: file_search_tests

  These variables specify lists of directories to be searched by `load`, `demo`, and some other Maxima functions. The default values of these variables name various directories in the Maxima installation.

  The user can modify these variables, either to replace the default values or to append additional directories. For example,

      file_search_maxima: ["/usr/local/foo/###.mac",
          "/usr/local/bar/###.mac"]$

  replaces the default value of `file_search_maxima`, while

      file_search_maxima: append (file_search_maxima,
          ["/usr/local/foo/###.mac", "/usr/local/bar/###.mac"])$

  appends two additional directories. It may be convenient to put such an expression in the file `maxima-init.mac` so that the file search path is assigned automatically when Maxima starts. See also Introduction for Runtime Environment.

  Multiple filename extensions and multiple paths can be specified by special "wildcard" constructions. The string `###` expands into the sought-after name, while a comma-separated list enclosed in curly braces `{foo,bar,baz}` expands into multiple strings. For example, supposing the sought-after name is `neumann`,

      "/home/{wfs,gcj}/###.{lisp,mac}"

  expands into `/home/wfs/neumann.lisp`, `/home/gcj/neumann.lisp`, `/home/wfs/neumann.mac`, and `/home/gcj/neumann.mac`.

  Categories: File input · Global variables

Function: file_type (filename)

  Returns a guess about the content of filename, based on the filename extension. filename need not refer to an actual file; no attempt is made to open the file and inspect the content.

  The return value is a symbol, either `object`, `lisp`, or `maxima`. If the extension is matches one of the values in `file_type_maxima`, `file_type` returns `maxima`. If the extension matches one of the values in `file_type_lisp`, `file_type` returns `lisp`. If none of the above, `file_type` returns `object`.

  See also `pathname_type`.

  See `file_type_maxima` and `file_type_lisp` for the default values.

  Examples:

      (%i2) map('file_type,
                ["test.lisp", "test.mac", "test.dem", "test.txt"]);
      (%o2)            [lisp, maxima, maxima, object]

  Categories: File input

Option variable: file_type_lisp

  Default value: `[l, lsp, lisp]`

  `file_type_lisp` is a list of file extensions that maxima recognizes as denoting a Lisp source file.

  See also `file_type`.

Option variable: file_type_maxima

  Default value: `[mac, mc, demo, dem, dm1, dm2, dm3, dmt, wxm]`

  `file_type_maxima` is a list of file extensions that maxima recognizes as denoting a Maxima source file.

  See also `file_type`.

Function: load (filename)

  Evaluates expressions in filename, thus bringing variables, functions, and other objects into Maxima. The binding of any existing object is clobbered by the binding recovered from filename. To find the file, `load` calls `file_search` with `file_search_maxima` and `file_search_lisp` as the search directories. If `load` succeeds, it returns the name of the file. Otherwise `load` prints an error message.

  `load` works equally well for Lisp code and Maxima code. Files created by `save`, `translate_file`, and `compile_file`, which create Lisp code, and `stringout`, which creates Maxima code, can all be processed by `load`. `load` calls `loadfile` to load Lisp files and `batchload` to load Maxima files.

  `load` does not recognize `:lisp` constructs in Maxima files, and while processing filename, the global variables `_`, `__`, `%`, and `%th` have whatever bindings they had when `load` was called.

  It is also to note that structures will only be read back as structures if they have been defined by `defstruct` before the `load` command is called.

  See also `loadfile`, `batch`, `batchload`, and `demo`. `loadfile` processes Lisp files; `batch`, `batchload`, and `demo` process Maxima files.

  See `file_search` for more detail about the file search mechanism.

  `load` evaluates its argument.

  Categories: Session management · File input

System variable: load_pathname

  Default value: `false`

  When a file is loaded with the functions `load`, `loadfile` or `batchload` the system variable `load_pathname` is bound to the pathname of the file which is processed.

  The variable `load_pathname` can be accessed from the file during the loading.

  Example:

  Suppose we have a batchfile `test.mac` in the directory

  `"/home/dieter/workspace/mymaxima/temp/"` with the following commands

      print("The value of load_pathname is: ", load_pathname)$
      print("End of batchfile")$

  then we get the following output

      (%i1) load("/home/dieter/workspace/mymaxima/temp/test.mac")$
      The value of load_pathname is:
                         /home/dieter/workspace/mymaxima/temp/test.mac
      End of batchfile

  Categories: File input

Function: loadfile (filename)

  Evaluates Lisp expressions in filename. `loadfile` does not invoke `file_search`, so `filename` must include the file extension and as much of the path as needed to find the file.

  `loadfile` can process files created by `save`, `translate_file`, and `compile_file`. The user may find it more convenient to use `load` instead of `loadfile`.

  Categories: Session management · File input

Option variable: loadprint

  Default value: `true`

  `loadprint` tells whether to print a message when a file is loaded.

  - When `loadprint` is `true`, always print a message.
  - When `loadprint` is `'loadfile`, print a message only if a file is loaded by the function `loadfile`.
  - When `loadprint` is `'autoload`, print a message only if a file is automatically loaded. See `setup_autoload`.
  - When `loadprint` is `false`, never print a message.

  Categories: File input · Global flags

Function: directory (path)

  Returns a list of the files and directories found in path in the file system.

  path may contain wildcard characters (i.e., characters which represent unspecified parts of the path), which include at least the asterisk on most systems, and possibly other characters, depending on the system.

  `directory` relies on the Lisp function DIRECTORY, which may have implementation-specific behavior.

  Categories: File input

Function: pathname_directory (pathname)
Function: pathname_name (pathname)
Function: pathname_type (pathname)

  These functions return the components of pathname.

  Examples:

      (%i1) pathname_directory("/home/dieter/maxima/changelog.txt");
      (%o1)                 /home/dieter/maxima/
      (%i2) pathname_name("/home/dieter/maxima/changelog.txt");
      (%o2)                       changelog
      (%i3) pathname_type("/home/dieter/maxima/changelog.txt");
      (%o3)                          txt

  Categories: File input

Function: printfile (path)

  Prints the file named by path to the console. path may be a string or a symbol; if it is a symbol, it is converted to a string.

  If path names a file which is accessible from the current working directory, that file is printed to the console. Otherwise, `printfile` attempts to locate the file by appending path to each of the elements of `file_search_usage` via `filename_merge`.

  `printfile` returns path if it names an existing file, or otherwise the result of a successful filename merge.

  Categories: File input · Console interaction

Function: save
`save` (filename, name_1, name_2, name_3, …)
`save` (filename, values, functions, labels, …)
`save` (filename, [m, n])
`save` (filename, name_1=expr_1, …)
`save` (filename, all)
`save` (filename, name_1=expr_1, name_2=expr_2, …)

  Stores the current values of name_1, name_2, name_3, …, in filename. The arguments are the names of variables, functions, or other objects. If a name has no value or function associated with it, it is ignored. `save` returns filename.

  `save` stores data in the form of Lisp expressions. If filename ends in `.lisp` the data stored by `save` may be recovered by `load (filename)`. See `load`.

  The global flag `file_output_append` governs whether `save` appends or truncates the output file. When `file_output_append` is `true`, `save` appends to the output file. Otherwise, `save` truncates the output file. In either case, `save` creates the file if it does not yet exist.

  The special form `save (filename, values, functions, labels, ...)` stores the items named by `values`, `functions`, `labels`, etc. The names may be any specified by the variable `infolists`. `values` comprises all user-defined variables.

  The special form `save (filename, [m, n])` stores the values of input and output labels m through n. Note that m and n must be literal integers. Input and output labels may also be stored one by one, e.g., `save ("foo.1", %i42, %o42)`. `save (filename, labels)` stores all input and output labels. When the stored labels are recovered, they clobber existing labels.

  The special form `save (filename, name_1=expr_1, name_2=expr_2, ...)` stores the values of expr_1, expr_2, …, with names name_1, name_2, … It is useful to apply this form to input and output labels, e.g., `save ("foo.1", aa=%o88)`. The right-hand side of the equality in this form may be any expression, which is evaluated. This form does not introduce the new names into the current Maxima environment, but only stores them in filename.

  These special forms and the general form of `save` may be mixed at will. For example, `save (filename, aa, bb, cc=42, functions, [11, 17])`.

  The special form `save (filename, all)` stores the current state of Maxima. This includes all user-defined variables, functions, arrays, etc., as well as some automatically defined items. The saved items include system variables, such as `file_search_maxima` or `showtime`, if they have been assigned new values by the user; see `myoptions`.

  `save` evaluates filename and quotes all other arguments.

  Categories: Session management · File output

Function: stringout
`stringout` (filename, expr_1, expr_2, expr_3, …)
`stringout` (filename, [m, n])
`stringout` (filename, input)
`stringout` (filename, functions)
`stringout` (filename, values)

  `stringout` writes expressions to a file in the same form the expressions would be typed for input. The file can then be used as input for the `batch` or `demo` commands, and it may be edited for any purpose. `stringout` can be executed while `writefile` is in progress.

  The global flag `file_output_append` governs whether `stringout` appends or truncates the output file. When `file_output_append` is `true`, `stringout` appends to the output file. Otherwise, `stringout` truncates the output file. In either case, `stringout` creates the file if it does not yet exist.

  The general form of `stringout` writes the values of one or more expressions to the output file. Note that if an expression is a variable, only the value of the variable is written and not the name of the variable. As a useful special case, the expressions may be input labels (`%i1`, `%i2`, `%i3`, …) or output labels (`%o1`, `%o2`, `%o3`, …).

  If `grind` is `true`, `stringout` formats the output using the `grind` format. Otherwise the `string` format is used. See `grind` and `string`.

  The special form `stringout (filename, [m, n])` writes the values of input labels m through n, inclusive.

  The special form `stringout (filename, input)` writes all input labels to the file.

  The special form `stringout (filename, functions)` writes all user-defined functions (named by the global list `functions`)) to the file.

  The special form `stringout (filename, values)` writes all user-assigned variables (named by the global list `values`)) to the file. Each variable is printed as an assignment statement, with the name of the variable, a colon, and its value. Note that the general form of `stringout` does not print variables as assignment statements.

  Categories: Session management · File output

Function: with_stdout
`with_stdout` (f, expr_1, expr_2, expr_3, …)
`with_stdout` (s, expr_1, expr_2, expr_3, …)

  Evaluates expr_1, expr_2, expr_3, … and writes any output thus generated to a file f or output stream s. The evaluated expressions are not written to the output. Output may be generated by `print`, `display`, `grind`, among other functions.

  The global flag `file_output_append` governs whether `with_stdout` appends or truncates the output file f. When `file_output_append` is `true`, `with_stdout` appends to the output file. Otherwise, `with_stdout` truncates the output file. In either case, `with_stdout` creates the file if it does not yet exist.

  `with_stdout` returns the value of its final argument.

  See also `writefile`.

      (%i1) with_stdout ("tmp.out", for i:5 thru 10 do
            print (i, "! yields", i!))$
      (%i2) printfile ("tmp.out")$
      5 ! yields 120
      6 ! yields 720
      7 ! yields 5040
      8 ! yields 40320
      9 ! yields 362880
      10 ! yields 3628800

  Categories: File output

Function: writefile (filename)

  Begins writing a transcript of the Maxima session to filename. All interaction between the user and Maxima is then recorded in this file, just as it appears on the console.

  As the transcript is printed in the console output format, it cannot be reloaded into Maxima. To make a file containing expressions which can be reloaded, see `save` and `stringout`. `save` stores expressions in Lisp form, while `stringout` stores expressions in Maxima form.

  The effect of executing `writefile` when filename already exists depends on the underlying Lisp implementation; the transcript file may be clobbered, or the file may be appended. `appendfile` always appends to the transcript file.

  It may be convenient to execute `playback` after `writefile` to save the display of previous interactions. As `playback` displays only the input and output variables (`%i1`, `%o1`, etc.), any output generated by a print statement in a function (as opposed to a return value) is not displayed by `playback`.

  `closefile` closes the transcript file opened by `writefile` or `appendfile`.

  Categories: File output · Console interaction

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## 13.4 Functions and Variables for TeX Output

Note that the built-in TeX output functionality of wxMaxima makes no use of the functions described here but uses its own implementation instead.

Function: tex
`tex` (expr)
`tex` (expr, destination)
`tex` (expr, false)
`tex` (label)
`tex` (label, destination)
`tex` (label, false)

  Prints a representation of an expression suitable for the TeX document preparation system. The result is a fragment of a document, which can be copied into a larger document but not processed by itself.

  `tex (expr)` prints a TeX representation of expr on the console.

  `tex (label)` prints a TeX representation of the expression named by label and assigns it an equation label (to be displayed to the left of the expression). The TeX equation label is the same as the Maxima label.

  destination may be an output stream or file name. When destination is a file name, `tex` appends its output to the file. The functions `openw` and `opena` create output streams.

  `tex (expr, false)` and `tex (label, false)` return their TeX output as a string.

  `tex` evaluates its first argument after testing it to see if it is a label. Quote-quote `''` forces evaluation of the argument, thereby defeating the test and preventing the label.

  See also `texput`.

  Examples:

      (%i1) integrate (1/(1+x^3), x);
                                          2 x - 1
                        2            atan(-------)
                   log(x  - x + 1)        sqrt(3)    log(x + 1)
      (%o1)      - --------------- + ------------- + ----------
                          6             sqrt(3)          3
      (%i2) tex (%o1);
      $$-{{\log \left(x^2-x+1\right)}\over{6}}+{{\arctan \left({{2\,x-1
       }\over{\sqrt{3}}}\right)}\over{\sqrt{3}}}+{{\log \left(x+1\right)
       }\over{3}}\leqno{\tt (\%o1)}$$
      (%o2)                          (\%o1)
      (%i3) tex (integrate (sin(x), x));
      $$-\cos x$$
      (%o3)                           false
      (%i4) tex (%o1, "foo.tex");
      (%o4)                          (\%o1)

  `tex (expr, false)` returns its TeX output as a string.

      (%i1) S : tex (x * y * z, false);
      (%o1) $$x\,y\,z$$
      (%i2) S;
      (%o2) $$x\,y\,z$$

  Categories: TeX output · File output

Function: tex1 (e)

  Returns a string which represents the TeX output for the expressions e. The TeX output is not enclosed in delimiters for an equation or any other environment.

  Examples:

      (%i1) tex1 (sin(x) + cos(x));
      (%o1)                     \sin x+\cos x

Function: texput
`texput` (a, s)
`texput` (a, f)
`texput` (a, s, operator_type)
`texput` (a, [s_1, s_2], matchfix)
`texput` (a, [s_1, s_2, s_3], matchfix)

  Assign the TeX output for the atom a, which can be a symbol or the name of an operator.

  `texput (a, s)` causes the `tex` function to interpolate the string s into the TeX output in place of a.

  `texput (a, f)` causes the `tex` function to call the function f to generate TeX output. f must accept one argument, which is an expression which has operator a, and must return a string (the TeX output). f may call `tex1` to generate TeX output for the arguments of the input expression.

  `texput (a, s, operator_type)`, where operator_type is `prefix`, `infix`, `postfix`, `nary`, or `nofix`, causes the `tex` function to interpolate s into the TeX output in place of a, and to place the interpolated text in the appropriate position.

  `texput (a, [s_1, s_2], matchfix)` causes the `tex` function to interpolate s_1 and s_2 into the TeX output on either side of the arguments of a. The arguments (if more than one) are separated by commas.

  `texput (a, [s_1, s_2, s_3], matchfix)` causes the `tex` function to interpolate s_1 and s_2 into the TeX output on either side of the arguments of a, with s_3 separating the arguments.

  Examples:

  Assign TeX output for a variable.

      (%i1) texput (me,"\\mu_e");
      (%o1)                         \mu_e
      (%i2) tex (me);
      $$\mu_e$$
      (%o2)                         false

  Assign TeX output for an ordinary function (not an operator).

      (%i1) texput (lcm, "\\mathrm{lcm}");
      (%o1)                     \mathrm{lcm}
      (%i2) tex (lcm (a, b));
      $$\mathrm{lcm}\left(a , b\right)$$
      (%o2)                         false

  Call a function to generate TeX output.

      (%i1) texfoo (e) := block ([a, b], [a, b] : args (e),
        concat("\\left[\\stackrel{",tex1(b),"}{",tex1(a),"}\\right]"))$
      (%i2) texput (foo, texfoo);
      (%o2)                        texfoo
      (%i3) tex (foo (2^x, %pi));
      $$\left[\stackrel{\pi}{2^{x}}\right]$$
      (%o3)                         false

  Assign TeX output for a prefix operator.

      (%i1) prefix ("grad");
      (%o1)                         grad
      (%i2) texput ("grad", " \\nabla ", prefix);
      (%o2)                        \nabla
      (%i3) tex (grad f);
      $$ \nabla f$$
      (%o3)                         false

  Assign TeX output for an infix operator.

      (%i1) infix ("~");
      (%o1)                           ~
      (%i2) texput ("~", " \\times ", infix);
      (%o2)                        \times
      (%i3) tex (a ~ b);
      $$a \times b$$
      (%o3)                         false

  Assign TeX output for a postfix operator.

      (%i1) postfix ("##");
      (%o1)                          ##
      (%i2) texput ("##", "!!", postfix);
      (%o2)                          !!
      (%i3) tex (x ##);
      $$x!!$$
      (%o3)                         false

  Assign TeX output for a nary operator.

      (%i1) nary ("@@");
      (%o1)                          @@
      (%i2) texput ("@@", " \\circ ", nary);
      (%o2)                         \circ
      (%i3) tex (a @@ b @@ c @@ d);
      $$a \circ b \circ c \circ d$$
      (%o3)                         false

  Assign TeX output for a nofix operator.

      (%i1) nofix ("foo");
      (%o1)                          foo
      (%i2) texput ("foo", "\\mathsc{foo}", nofix);
      (%o2)                     \mathsc{foo}
      (%i3) tex (foo);
      $$\mathsc{foo}$$
      (%o3)                         false

  Assign TeX output for a matchfix operator.

      (%i1) matchfix ("<<", ">>");
      (%o1)                          <<
      (%i2) texput ("<<", [" \\langle ", " \\rangle "], matchfix);
      (%o2)                [ \langle ,  \rangle ]
      (%i3) tex (<<a>>);
      $$ \langle a \rangle $$
      (%o3)                         false
      (%i4) tex (<<a, b>>);
      $$ \langle a , b \rangle $$
      (%o4)                         false
      (%i5) texput ("<<", [" \\langle ", " \\rangle ", " \\, | \\,"],
            matchfix);
      (%o5)           [ \langle ,  \rangle ,  \, | \,]
      (%i6) tex (<<a>>);
      $$ \langle a \rangle $$
      (%o6)                         false
      (%i7) tex (<<a, b>>);
      $$ \langle a \, | \,b \rangle $$
      (%o7)                         false

  Categories: TeX output

Function: get_tex_environment (op)
Function: set_tex_environment (op, before, after)

  Customize the TeX environment output by `tex`. As maintained by these functions, the TeX environment comprises two strings: one is printed before any other TeX output, and the other is printed after.

  Only the TeX environment of the top-level operator in an expression is output; TeX environments associated with other operators are ignored.

  `get_tex_environment` returns the TeX environment which is applied to the operator op; returns the default if no other environment has been assigned.

  `set_tex_environment` assigns the TeX environment for the operator op.

  Examples:

      (%i1) get_tex_environment (":=");
      (%o1) [
      \begin{verbatim}
      , ;
      \end{verbatim}
      ]
      (%i2) tex (f (x) := 1 - x);

      \begin{verbatim}
      f(x):=1-x;
      \end{verbatim}

      (%o2)                         false
      (%i3) set_tex_environment (":=", "$$", "$$");
      (%o3)                       [$$, $$]
      (%i4) tex (f (x) := 1 - x);
      $$f(x):=1-x$$
      (%o4)                         false

  Categories: TeX output

Function: get_tex_environment_default ()
Function: set_tex_environment_default (before, after)

  Customize the TeX environment output by `tex`. As maintained by these functions, the TeX environment comprises two strings: one is printed before any other TeX output, and the other is printed after.

  `get_tex_environment_default` returns the TeX environment which is applied to expressions for which the top-level operator has no specific TeX environment (as assigned by `set_tex_environment`).

  `set_tex_environment_default` assigns the default TeX environment.

  Examples:

      (%i1) get_tex_environment_default ();
      (%o1)                       [$$, $$]
      (%i2) tex (f(x) + g(x));
      $$g\left(x\right)+f\left(x\right)$$
      (%o2)                         false
      (%i3) set_tex_environment_default ("\\begin{equation}
      ", "
      \\end{equation}");
      (%o3) [\begin{equation}
      ,
      \end{equation}]
      (%i4) tex (f(x) + g(x));
      \begin{equation}
      g\left(x\right)+f\left(x\right)
      \end{equation}
      (%o4)                         false

  Categories: TeX output

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## 13.5 Functions and Variables for Fortran Output

Option variable: fortindent

  Default value: `0`

  `fortindent` controls the left margin indentation of expressions printed out by the `fortran` command. `0` gives normal printout (i.e., 6 spaces), and positive values will causes the expressions to be printed farther to the right.

  Categories: Translation and compilation

Function: fortran (expr)

  Prints expr as a Fortran statement. The output line is indented with spaces. If the line is too long, `fortran` prints continuation lines. `fortran` prints the exponentiation operator `^` as `**`, and prints a complex number `a + b %i` in the form `(a,b)`.

  expr may be an equation. If so, `fortran` prints an assignment statement, assigning the right-hand side of the equation to the left-hand side. In particular, if the right-hand side of expr is the name of a matrix, then `fortran` prints an assignment statement for each element of the matrix.

  If expr is not something recognized by `fortran`, the expression is printed in `grind` format without complaint. `fortran` does not know about lists, arrays, or functions.

  `fortindent` controls the left margin of the printed lines. `0` is the normal margin (i.e., indented 6 spaces). Increasing `fortindent` causes expressions to be printed further to the right.

  When `fortspaces` is `true`, `fortran` fills out each printed line with spaces to 80 columns.

  `fortran` evaluates its arguments; quoting an argument defeats evaluation. `fortran` always returns `done`.

  See also the function `f90` for printing one or more expressions as a Fortran 90 program.

  Examples:

      (%i1) expr: (a + b)^12$
      (%i2) fortran (expr);
            (b+a)**12
      (%o2)                         done
      (%i3) fortran ('x=expr);
            x = (b+a)**12
      (%o3)                         done
      (%i4) fortran ('x=expand (expr));
            x = b**12+12*a*b**11+66*a**2*b**10+220*a**3*b**9+495*a**4*b**8+792
           1   *a**5*b**7+924*a**6*b**6+792*a**7*b**5+495*a**8*b**4+220*a**9*b
           2   **3+66*a**10*b**2+12*a**11*b+a**12
      (%o4)                         done
      (%i5) fortran ('x=7+5*%i);
            x = (7,5)
      (%o5)                         done
      (%i6) fortran ('x=[1,2,3,4]);
            x = [1,2,3,4]
      (%o6)                         done
      (%i7) f(x) := x^2$
      (%i8) fortran (f);
            f
      (%o8)                         done

  Categories: Translation and compilation

Option variable: fortspaces

  Default value: `false`

  When `fortspaces` is `true`, `fortran` fills out each printed line with spaces to 80 columns.

  Categories: Translation and compilation

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# 14. Polynomials

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## 14.1 Introduction to Polynomials

Polynomials are stored in Maxima either in General Form or as Canonical Rational Expressions (CRE) form. The latter is a standard form, and is used internally by operations such as factor, ratsimp, and so on.

Canonical Rational Expressions constitute a kind of representation which is especially suitable for expanded polynomials and rational functions (as well as for partially factored polynomials and rational functions when RATFAC is set to `true`). In this CRE form an ordering of variables (from most to least main) is assumed for each expression. Polynomials are represented recursively by a list consisting of the main variable followed by a series of pairs of expressions, one for each term of the polynomial. The first member of each pair is the exponent of the main variable in that term and the second member is the coefficient of that term which could be a number or a polynomial in another variable again represented in this form. Thus the principal part of the CRE form of 3*X^2-1 is (X 2 3 0 -1) and that of 2*X*Y+X-3 is (Y 1 (X 1 2) 0 (X 1 1 0 -3)) assuming Y is the main variable, and is (X 1 (Y 1 2 0 1) 0 -3) assuming X is the main variable. "Main"-ness is usually determined by reverse alphabetical order. The "variables" of a CRE expression needn't be atomic. In fact any subexpression whose main operator is not + - * / or ^ with integer power will be considered a "variable" of the expression (in CRE form) in which it occurs. For example the CRE variables of the expression X+SIN(X+1)+2*SQRT(X)+1 are X, SQRT(X), and SIN(X+1). If the user does not specify an ordering of variables by using the RATVARS function Maxima will choose an alphabetic one. In general, CRE's represent rational expressions, that is, ratios of polynomials, where the numerator and denominator have no common factors, and the denominator is positive. The internal form is essentially a pair of polynomials (the numerator and denominator) preceded by the variable ordering list. If an expression to be displayed is in CRE form or if it contains any subexpressions in CRE form, the symbol /R/ will follow the line label. See the RAT function for converting an expression to CRE form. An extended CRE form is used for the representation of Taylor series. The notion of a rational expression is extended so that the exponents of the variables can be positive or negative rational numbers rather than just positive integers and the coefficients can themselves be rational expressions as described above rather than just polynomials. These are represented internally by a recursive polynomial form which is similar to and is a generalization of CRE form, but carries additional information such as the degree of truncation. As with CRE form, the symbol /T/ follows the line label of such expressions.

Categories: Polynomials · Rational expressions

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## 14.2 Functions and Variables for Polynomials

Option variable: algebraic

  Default value: `false`

  `algebraic` must be set to `true` in order for the simplification of algebraic integers to take effect.

  Categories: Simplification flags and variables

Option variable: berlefact

  Default value: `true`

  When `berlefact` is `false` then the Kronecker factoring algorithm will be used otherwise the Berlekamp algorithm, which is the default, will be used.

  Categories: Polynomials

Function: bezout (p1, p2, x)

  an alternative to the `resultant` command. It returns a matrix. `determinant` of this matrix is the desired resultant.

  Examples:

      (%i1) bezout(a*x+b, c*x^2+d, x);
                               [ b c  - a d ]
      (%o1)                    [            ]
                               [  a     b   ]
      (%i2) determinant(%);
                                  2      2
      (%o2)                      a  d + b  c
      (%i3) resultant(a*x+b, c*x^2+d, x);
                                  2      2
      (%o3)                      a  d + b  c

  Categories: Polynomials

Function: bothcoef (expr, x)

  Returns a list whose first member is the coefficient of x in expr (as found by `ratcoef` if expr is in CRE form otherwise by `coeff`) and whose second member is the remaining part of expr. That is, `[A, B]` where `expr = A*x + B`.

  Example:

      (%i1) islinear (expr, x) := block ([c],
              c: bothcoef (rat (expr, x), x),
              is (freeof (x, c) and c[1] # 0))$
      (%i2) islinear ((r^2 - (x - r)^2)/x, x);
      (%o2)                         true

  Categories: Polynomials

Function: coeff
`coeff` (expr, x, n)
`coeff` (expr, x)

  Returns the coefficient of `x^n` in expr, where expr is a polynomial or a monomial term in x. Other than `ratcoef` `coeff` is a strictly syntactical operation and will only find literal instances of `x^n` in the internal representation of expr.

  `coeff(expr, x^n)` is equivalent to `coeff(expr, x, n)`. `coeff(expr, x, 0)` returns the remainder of expr which is free of x. If omitted, n is assumed to be 1.

  x may be a simple variable or a subscripted variable, or a subexpression of expr which comprises an operator and all of its arguments.

  It may be possible to compute coefficients of expressions which are equivalent to expr by applying `expand` or `factor`. `coeff` itself does not apply `expand` or `factor` or any other function.

  `coeff` distributes over lists, matrices, and equations.

  See also `ratcoef`.

  Examples:

  `coeff` returns the coefficient `x^n` in expr.

      (%i1) coeff (b^3*a^3 + b^2*a^2 + b*a + 1, a^3);
                                      3
      (%o1)                          b

  `coeff(expr, x^n)` is equivalent to `coeff(expr, x, n)`.

      (%i1) coeff (c[4]*z^4 - c[3]*z^3 - c[2]*z^2 + c[1]*z, z, 3);
      (%o1)                         - c
                                       3
      (%i2) coeff (c[4]*z^4 - c[3]*z^3 - c[2]*z^2 + c[1]*z, z^3);
      (%o2)                         - c
                                       3

  `coeff(expr, x, 0)` returns the remainder of expr which is free of x.

      (%i1) coeff (a*u + b^2*u^2 + c^3*u^3, b, 0);
                                  3  3
      (%o1)                      c  u  + a u

  x may be a simple variable or a subscripted variable, or a subexpression of expr which comprises an operator and all of its arguments.

      (%i1) coeff (h^4 - 2*%pi*h^2 + 1, h, 2);
      (%o1)                        - 2 %pi
      (%i2) coeff (v[1]^4 - 2*%pi*v[1]^2 + 1, v[1], 2);
      (%o2)                        - 2 %pi
      (%i3) coeff (sin(1+x)*sin(x) + sin(1+x)^3*sin(x)^3, sin(1+x)^3);
                                      3
      (%o3)                        sin (x)
      (%i4) coeff ((d - a)^2*(b + c)^3 + (a + b)^4*(c - d), a + b, 4);
      (%o4)                         c - d

  `coeff` itself does not apply `expand` or `factor` or any other function.

      (%i1) coeff (c*(a + b)^3, a);
      (%o1)                           0
      (%i2) expand (c*(a + b)^3);
                       3          2        2        3
      (%o2)           b  c + 3 a b  c + 3 a  b c + a  c
      (%i3) coeff (%, a);
                                      2
      (%o3)                        3 b  c
      (%i4) coeff (b^3*c + 3*a*b^2*c + 3*a^2*b*c + a^3*c, (a + b)^3);
      (%o4)                           0
      (%i5) factor (b^3*c + 3*a*b^2*c + 3*a^2*b*c + a^3*c);
                                        3
      (%o5)                      (b + a)  c
      (%i6) coeff (%, (a + b)^3);
      (%o6)                           c

  `coeff` distributes over lists, matrices, and equations.

      (%i1) coeff ([4*a, -3*a, 2*a], a);
      (%o1)                      [4, - 3, 2]
      (%i2) coeff (matrix ([a*x, b*x], [-c*x, -d*x]), x);
                                [  a    b  ]
      (%o2)                     [          ]
                                [ - c  - d ]
      (%i3) coeff (a*u - b*v = 7*u + 3*v, u);
      (%o3)                         a = 7

  Categories: Polynomials

Function: content (p_1, x_1, …, x_n)

  Returns a list whose first element is the greatest common divisor of the coefficients of the terms of the polynomial p_1 in the variable x_n (this is the content) and whose second element is the polynomial p_1 divided by the content.

  Examples:

      (%i1) content (2*x*y + 4*x^2*y^2, y);
                                         2
      (%o1)                   [2 x, 2 x y  + y]

  Categories: Polynomials

Function: denom (expr)

  Returns the denominator of the rational expression expr.

  See also `num`

      (%i1) g1:(x+2)*(x+1)/((x+3)^2);
                               (x + 1) (x + 2)
      (%o1)                    ---------------
                                         2
                                  (x + 3)
      (%i2) denom(g1);
                                         2
      (%o2)                       (x + 3)
      (%i3) g2:sin(x)/10*cos(x)/y;
                                cos(x) sin(x)
      (%o3)                     -------------
                                    10 y
      (%i4) denom(g2);
      (%o4)                         10 y

  Categories: Expressions

Function: divide (p_1, p_2, x_1, …, x_n)

  computes the quotient and remainder of the polynomial p_1 divided by the polynomial p_2, in a main polynomial variable, x_n. The other variables are as in the `ratvars` function. The result is a list whose first element is the quotient and whose second element is the remainder.

  Examples:

      (%i1) divide (x + y, x - y, x);
      (%o1)                       [1, 2 y]
      (%i2) divide (x + y, x - y);
      (%o2)                      [- 1, 2 x]

  Note that `y` is the main variable in the second example.

  Categories: Polynomials

Function: eliminate ([eqn_1, …, eqn_n], [x_1, …, x_k])

  Eliminates variables from equations (or expressions assumed equal to zero) by taking successive resultants. This returns a list of `n - k` expressions with the k variables x_1, …, x_k eliminated. First x_1 is eliminated yielding `n - 1` expressions, then `x_2` is eliminated, etc. If `k = n` then a single expression in a list is returned free of the variables x_1, …, x_k. In this case `solve` is called to solve the last resultant for the last variable.

  Example:

      (%i1) expr1: 2*x^2 + y*x + z;
                                            2
      (%o1)                    z + x y + 2 x
      (%i2) expr2: 3*x + 5*y - z - 1;
      (%o2)                  - z + 5 y + 3 x - 1
      (%i3) expr3: z^2 + x - y^2 + 5;
                                2    2
      (%o3)                    z  - y  + x + 5
      (%i4) eliminate ([expr3, expr2, expr1], [y, z]);
                   8         7         6          5          4
      (%o4) [7425 x  - 1170 x  + 1299 x  + 12076 x  + 22887 x

                                          3         2
                                  - 5154 x  - 1291 x  + 7688 x + 15376]

  Categories: Polynomials · Algebraic equations

Function: ezgcd (p_1, p_2, p_3, …)

  Returns a list whose first element is the greatest common divisor of the polynomials p_1, p_2, p_3, … and whose remaining elements are the polynomials divided by the greatest common divisor. This always uses the `ezgcd` algorithm.

  See also `gcd`, `gcdex`, `gcdivide`, and `poly_gcd`.

  Examples:

  The three polynomials have the greatest common divisor `2*x-3`. The gcd is first calculated with the function `gcd` and then with the function `ezgcd`.

      (%i1) p1 : 6*x^3-17*x^2+14*x-3;
                              3       2
      (%o1)                6 x  - 17 x  + 14 x - 3
      (%i2) p2 : 4*x^4-14*x^3+12*x^2+2*x-3;
                          4       3       2
      (%o2)            4 x  - 14 x  + 12 x  + 2 x - 3
      (%i3) p3 : -8*x^3+14*x^2-x-3;
                                3       2
      (%o3)                - 8 x  + 14 x  - x - 3

      (%i4) gcd(p1, gcd(p2, p3));
      (%o4)                        2 x - 3

      (%i5) ezgcd(p1, p2, p3);
                         2               3      2           2
      (%o5) [2 x - 3, 3 x  - 4 x + 1, 2 x  - 4 x  + 1, - 4 x  + x + 1]

  Categories: Polynomials

Option variable: facexpand

  Default value: `true`

  `facexpand` controls whether the irreducible factors returned by `factor` are in expanded (the default) or recursive (normal CRE) form.

  Categories: Polynomials

Function: factor
`factor` (expr)
`factor` (expr, p)

  Factors the expression expr, containing any number of variables or functions, into factors irreducible over the integers. `factor (expr, p)` factors expr over the field of rationals with an element adjoined whose minimum polynomial is p.

  `factor` uses `ifactors` function for factoring integers.

  `factorflag` if `false` suppresses the factoring of integer factors of rational expressions.

  `dontfactor` may be set to a list of variables with respect to which factoring is not to occur. (It is initially empty). Factoring also will not take place with respect to any variables which are less important (using the variable ordering assumed for CRE form) than those on the `dontfactor` list.

  `savefactors` if `true` causes the factors of an expression which is a product of factors to be saved by certain functions in order to speed up later factorizations of expressions containing some of the same factors.

  `berlefact` if `false` then the Kronecker factoring algorithm will be used otherwise the Berlekamp algorithm, which is the default, will be used.

  `intfaclim` if `true` maxima will give up factorization of integers if no factor is found after trial divisions and Pollard's rho method. If set to `false` (this is the case when the user calls `factor` explicitly), complete factorization of the integer will be attempted. The user's setting of `intfaclim` is used for internal calls to `factor`. Thus, `intfaclim` may be reset to prevent Maxima from taking an inordinately long time factoring large integers.

  See also `collectterms`.

  Examples:

      (%i1) factor (2^63 - 1);
                          2
      (%o1)              7  73 127 337 92737 649657
      (%i2) factor (-8*y - 4*x + z^2*(2*y + x));
      (%o2)               (2 y + x) (z - 2) (z + 2)
      (%i3) -1 - 2*x - x^2 + y^2 + 2*x*y^2 + x^2*y^2;
                      2  2        2    2    2
      (%o3)          x  y  + 2 x y  + y  - x  - 2 x - 1
      (%i4) block ([dontfactor: [x]], factor (%/36/(1 + 2*y + y^2)));
                             2
                           (x  + 2 x + 1) (y - 1)
      (%o4)                ----------------------
                                 36 (y + 1)
      (%i5) factor (1 + %e^(3*x));
                            x         2 x     x
      (%o5)              (%e  + 1) (%e    - %e  + 1)
      (%i6) factor (1 + x^4, a^2 - 2);
                          2              2
      (%o6)             (x  - a x + 1) (x  + a x + 1)
      (%i7) factor (-y^2*z^2 - x*z^2 + x^2*y^2 + x^3);
                             2
      (%o7)              - (y  + x) (z - x) (z + x)
      (%i8) (2 + x)/(3 + x)/(b + x)/(c + x)^2;
                                   x + 2
      (%o8)               ------------------------
                                                 2
                          (x + 3) (x + b) (x + c)
      (%i9) ratsimp (%);
                      4                  3
      (%o9) (x + 2)/(x  + (2 c + b + 3) x

           2                       2             2                   2
       + (c  + (2 b + 6) c + 3 b) x  + ((b + 3) c  + 6 b c) x + 3 b c )
      (%i10) partfrac (%, x);
                 2                   4                3
      (%o10) - (c  - 4 c - b + 6)/((c  + (- 2 b - 6) c

           2              2         2                2
       + (b  + 12 b + 9) c  + (- 6 b  - 18 b) c + 9 b ) (x + c))

                       c - 2
       - ---------------------------------
           2                             2
         (c  + (- b - 3) c + 3 b) (x + c)

                               b - 2
       + -------------------------------------------------
                   2             2       3      2
         ((b - 3) c  + (6 b - 2 b ) c + b  - 3 b ) (x + b)

                               1
       - ----------------------------------------------
                   2
         ((b - 3) c  + (18 - 6 b) c + 9 b - 27) (x + 3)
      (%i11) map ('factor, %);
                    2
                   c  - 4 c - b + 6                 c - 2
      (%o11) - ------------------------- - ------------------------
                      2        2                                  2
               (c - 3)  (c - b)  (x + c)   (c - 3) (c - b) (x + c)

                             b - 2                        1
                  + ------------------------ - ------------------------
                                   2                          2
                    (b - 3) (c - b)  (x + b)   (b - 3) (c - 3)  (x + 3)
      (%i12) ratsimp ((x^5 - 1)/(x - 1));
                             4    3    2
      (%o12)                x  + x  + x  + x + 1
      (%i13) subst (a, x, %);
                             4    3    2
      (%o13)                a  + a  + a  + a + 1
      (%i14) factor (%th(2), %);
                             2        3        3    2
      (%o14)   (x - a) (x - a ) (x - a ) (x + a  + a  + a + 1)
      (%i15) factor (1 + x^12);
                             4        8    4
      (%o15)               (x  + 1) (x  - x  + 1)
      (%i16) factor (1 + x^99);
                       2            6    3
      (%o16) (x + 1) (x  - x + 1) (x  - x  + 1)

         10    9    8    7    6    5    4    3    2
       (x   - x  + x  - x  + x  - x  + x  - x  + x  - x + 1)

         20    19    17    16    14    13    11    10    9    7    6
       (x   + x   - x   - x   + x   + x   - x   - x   - x  + x  + x

          4    3            60    57    51    48    42    39    33
       - x  - x  + x + 1) (x   + x   - x   - x   + x   + x   - x

          30    27    21    18    12    9    3
       - x   - x   + x   + x   - x   - x  + x  + 1)

  Categories: Polynomials

Option variable: factorflag

  Default value: `false`

  When `factorflag` is `false`, suppresses the factoring of integer factors of rational expressions.

  Categories: Polynomials

Function: factorout (expr, x_1, x_2, …)

  Rearranges the sum expr into a sum of terms of the form `f (x_1, x_2, …)*g` where `g` is a product of expressions not containing any x_i and `f` is factored.

  Note that the option variable `keepfloat` is ignored by `factorout`.

  Example:

      (%i1) expand (a*(x+1)*(x-1)*(u+1)^2);
                   2  2          2      2      2
      (%o1)     a u  x  + 2 a u x  + a x  - a u  - 2 a u - a
      (%i2) factorout(%,x);
               2
      (%o2) a u  (x - 1) (x + 1) + 2 a u (x - 1) (x + 1)
                                                    + a (x - 1) (x + 1)

  Categories: Expressions

Function: factorsum (expr)

  Tries to group terms in factors of expr which are sums into groups of terms such that their sum is factorable. `factorsum` can recover the result of `expand ((x + y)^2 + (z + w)^2)` but it can't recover `expand ((x + 1)^2 + (x + y)^2)` because the terms have variables in common.

  Example:

      (%i1) expand ((x + 1)*((u + v)^2 + a*(w + z)^2));
                 2      2                            2      2
      (%o1) a x z  + a z  + 2 a w x z + 2 a w z + a w  x + v  x

                                           2        2    2            2
                              + 2 u v x + u  x + a w  + v  + 2 u v + u
      (%i2) factorsum (%);
                                         2          2
      (%o2)            (x + 1) (a (z + w)  + (v + u) )

  Categories: Expressions

Function: fasttimes (p_1, p_2)

  Returns the product of the polynomials p_1 and p_2 by using a special algorithm for multiplication of polynomials. `p_1` and `p_2` should be multivariate, dense, and nearly the same size. Classical multiplication is of order `n_1 n_2` where `n_1` is the degree of `p_1` and `n_2` is the degree of `p_2`. `fasttimes` is of order `max (n_1, n_2)^1.585`.

  Categories: Polynomials

Function: fullratsimp (expr)

  `fullratsimp` repeatedly applies `ratsimp` followed by non-rational simplification to an expression until no further change occurs, and returns the result.

  When non-rational expressions are involved, one call to `ratsimp` followed as is usual by non-rational ("general") simplification may not be sufficient to return a simplified result. Sometimes, more than one such call may be necessary. `fullratsimp` makes this process convenient.

  `fullratsimp (expr, x_1, ..., x_n)` takes one or more arguments similar to `ratsimp` and `rat`.

  Example:

      (%i1) expr: (x^(a/2) + 1)^2*(x^(a/2) - 1)^2/(x^a - 1);
                             a/2     2   a/2     2
                           (x    - 1)  (x    + 1)
      (%o1)                -----------------------
                                    a
                                   x  - 1
      (%i2) ratsimp (expr);
                                2 a      a
                               x    - 2 x  + 1
      (%o2)                    ---------------
                                    a
                                   x  - 1
      (%i3) fullratsimp (expr);
                                    a
      (%o3)                        x  - 1
      (%i4) rat (expr);
                             a/2 4       a/2 2
                           (x   )  - 2 (x   )  + 1
      (%o4)/R/             -----------------------
                                    a
                                   x  - 1

  Categories: Simplification functions · Rational expressions

Function: fullratsubst (a, b, c)

  is the same as `ratsubst` except that it calls itself recursively on its result until that result stops changing. This function is useful when the replacement expression and the replaced expression have one or more variables in common.

  `fullratsubst` will also accept its arguments in the format of `lratsubst`. That is, the first argument may be a single substitution equation or a list of such equations, while the second argument is the expression being processed.

  `load ("lrats")` loads `fullratsubst` and `lratsubst`.

  Examples:

      (%i1) load ("lrats")$

  - `subst` can carry out multiple substitutions. `lratsubst` is analogous to `subst`.

      (%i2) subst ([a = b, c = d], a + c);
      (%o2)                         d + b
      (%i3) lratsubst ([a^2 = b, c^2 = d], (a + e)*c*(a + c));
      (%o3)                (d + a c) e + a d + b c

  - If only one substitution is desired, then a single equation may be given as first argument.

      (%i4) lratsubst (a^2 = b, a^3);
      (%o4)                          a b

  - `fullratsubst` is equivalent to `ratsubst` except that it recurses until its result stops changing.

      (%i5) ratsubst (b*a, a^2, a^3);
                                     2
      (%o5)                         a  b
      (%i6) fullratsubst (b*a, a^2, a^3);
                                       2
      (%o6)                         a b

  - `fullratsubst` also accepts a list of equations or a single equation as first argument.

      (%i7) fullratsubst ([a^2 = b, b^2 = c, c^2 = a], a^3*b*c);
      (%o7)                           b
      (%i8) fullratsubst (a^2 = b*a, a^3);
                                       2
      (%o8)                         a b

  - `fullratsubst` may cause an indefinite recursion.

      (%i9) errcatch (fullratsubst (b*a^2, a^2, a^3));

      *** - Lisp stack overflow. RESET

  Categories: Rational expressions

Function: gcd (p_1, p_2, x_1, …)

  Returns the greatest common divisor of p_1 and p_2. The flag `gcd` determines which algorithm is employed. Setting `gcd` to `ez`, `subres`, `red`, or `spmod` selects the `ezgcd`, subresultant `prs`, reduced, or modular algorithm, respectively. If `gcd` `false` then `gcd (p_1, p_2, x)` always returns 1 for all x. Many functions (e.g. `ratsimp`, `factor`, etc.) cause gcd's to be taken implicitly. For homogeneous polynomials it is recommended that `gcd` equal to `subres` be used. To take the gcd when an algebraic is present, e.g., `gcd (x^2 - 2*sqrt(2)* x + 2, x - sqrt(2))`, the option variable `algebraic` must be `true` and `gcd` must not be `ez`.

  The `gcd` flag, default: `spmod`, if `false` will also prevent the greatest common divisor from being taken when expressions are converted to canonical rational expression (CRE) form. This will sometimes speed the calculation if gcds are not required.

  See also `ezgcd`, `gcdex`, `gcdivide`, and `poly_gcd`.

  Example:

      (%i1) p1:6*x^3+19*x^2+19*x+6;
                              3       2
      (%o1)                6 x  + 19 x  + 19 x + 6
      (%i2) p2:6*x^5+13*x^4+12*x^3+13*x^2+6*x;
                        5       4       3       2
      (%o2)          6 x  + 13 x  + 12 x  + 13 x  + 6 x
      (%i3) gcd(p1, p2);
                                  2
      (%o3)                    6 x  + 13 x + 6
      (%i4) p1/gcd(p1, p2), ratsimp;
      (%o4)                         x + 1
      (%i5) p2/gcd(p1, p2), ratsimp;
                                    3
      (%o5)                        x  + x

  `ezgcd` returns a list whose first element is the greatest common divisor of the polynomials p_1 and p_2, and whose remaining elements are the polynomials divided by the greatest common divisor.

      (%i6) ezgcd(p1, p2);
                          2                     3
      (%o6)           [6 x  + 13 x + 6, x + 1, x  + x]

  Categories: Polynomials · Rational expressions

Function: gcdex
`gcdex` (f, g)
`gcdex` (f, g, x)

  Returns a list `[a, b, u]` where u is the greatest common divisor (gcd) of f and g, and u is equal to `a f + b g`. The arguments f and g should be univariate polynomials, or else polynomials in x a supplied main variable since we need to be in a principal ideal domain for this to work. The gcd means the gcd regarding f and g as univariate polynomials with coefficients being rational functions in the other variables.

  `gcdex` implements the Euclidean algorithm, where we have a sequence of `L[i]: [a[i], b[i], r[i]]` which are all perpendicular to `[f, g, -1]` and the next one is built as if `q = quotient(r[i]/r[i+1])` then `L[i+2]: L[i] - q L[i+1]`, and it terminates at `L[i+1]` when the remainder `r[i+2]` is zero.

  The arguments f and g can be integers. For this case the function `igcdex` is called by `gcdex`.

  See also `ezgcd`, `gcd`, `gcdivide`, and `poly_gcd`.

  Examples:

      (%i1) gcdex (x^2 + 1, x^3 + 4);
                             2
                            x  + 4 x - 1  x + 4
      (%o1)/R/           [- ------------, -----, 1]
                                 17        17
      (%i2) % . [x^2 + 1, x^3 + 4, -1];
      (%o2)/R/                        0

  Note that the gcd in the following is `1` since we work in `k(y)[x]`, not the `y+1` we would expect in `k[y, x]`.

      (%i1) gcdex (x*(y + 1), y^2 - 1, x);
                                     1
      (%o1)/R/                 [0, ------, 1]
                                    2
                                   y  - 1

  Categories: Polynomials · Rational expressions

Function: gcfactor (n)

  Factors the Gaussian integer n over the Gaussian integers, i.e., numbers of the form `a + b `%i`` where a and b are rational integers (i.e., ordinary integers). Factors are normalized by making a and b non-negative.

  Categories: Integers

Function: gfactor (expr)

  Factors the polynomial expr over the Gaussian integers (that is, the integers with the imaginary unit `%i` adjoined). This is like `factor (expr, a^2+1)` where a is `%i`.

  Example:

      (%i1) gfactor (x^4 - 1);
      (%o1)           (x - 1) (x + 1) (x - %i) (x + %i)

  Categories: Polynomials

Function: gfactorsum (expr)

  is similar to `factorsum` but applies `gfactor` instead of `factor`.

  Categories: Expressions

Function: hipow (expr, x)

  Returns the highest explicit exponent of x in expr. x may be a variable or a general expression. If x does not appear in expr, `hipow` returns `0`.

  `hipow` does not consider expressions equivalent to `expr`. In particular, `hipow` does not expand `expr`, so `hipow (expr, x)` and `hipow (expand (expr, x))` may yield different results.

  Examples:

      (%i1) hipow (y^3 * x^2 + x * y^4, x);
      (%o1)                           2
      (%i2) hipow ((x + y)^5, x);
      (%o2)                           1
      (%i3) hipow (expand ((x + y)^5), x);
      (%o3)                           5
      (%i4) hipow ((x + y)^5, x + y);
      (%o4)                           5
      (%i5) hipow (expand ((x + y)^5), x + y);
      (%o5)                           0

  Categories: Expressions

Option variable: intfaclim

  Default value: true

  If `true`, maxima will give up factorization of integers if no factor is found after trial divisions and Pollard's rho method and factorization will not be complete.

  When `intfaclim` is `false` (this is the case when the user calls `factor` explicitly), complete factorization will be attempted. `intfaclim` is set to `false` when factors are computed in `divisors`, `divsum` and `totient`.

  Internal calls to `factor` respect the user-specified value of `intfaclim`. Setting `intfaclim` to `true` may reduce the time spent factoring large integers.

  Categories: Integers

Option variable: keepfloat

  Default value: `false`

  When `keepfloat` is `true`, prevents floating point numbers from being rationalized when expressions which contain them are converted to canonical rational expression (CRE) form.

  Note that the function `solve` and those functions calling it (`eigenvalues`, for example) currently ignore this flag, converting floating point numbers anyway.

  Examples:

      (%i1) rat(x/2.0);
      rat: replaced 0.5 by 1/2 = 0.5
                                      x
      (%o1)/R/                        -
                                      2
      (%i2) rat(x/2.0), keepfloat;
      (%o2)/R/                      0.5 x

  `solve` ignores `keepfloat`:

      (%i1) solve(1.0-x,x), keepfloat;
      rat: replaced 1.0 by 1/1 = 1.0
      (%o1)                        [x = 1]

  Categories: Numerical evaluation

Function: lopow (expr, x)

  Returns the lowest exponent of x which explicitly appears in expr. Thus

      (%i1) lopow ((x+y)^2 + (x+y)^a, x+y);
      (%o1)                       min(a, 2)

  Categories: Expressions

Function: lratsubst (L, expr)

  is analogous to `subst (L, expr)` except that it uses `ratsubst` instead of `subst`.

  The first argument of `lratsubst` is an equation or a list of equations identical in format to that accepted by `subst`. The substitutions are made in the order given by the list of equations, that is, from left to right.

  `load ("lrats")` loads `fullratsubst` and `lratsubst`.

  Examples:

      (%i1) load ("lrats")$

  - `subst` can carry out multiple substitutions. `lratsubst` is analogous to `subst`.

      (%i2) subst ([a = b, c = d], a + c);
      (%o2)                         d + b
      (%i3) lratsubst ([a^2 = b, c^2 = d], (a + e)*c*(a + c));
      (%o3)                (d + a c) e + a d + b c

  - If only one substitution is desired, then a single equation may be given as first argument.

      (%i4) lratsubst (a^2 = b, a^3);
      (%o4)                          a b

  Categories: Polynomials · Rational expressions

Option variable: modulus

  Default value: `false`

  When `modulus` is a positive number p, operations on rational numbers (as returned by `rat` and related functions) are carried out modulo p, using the so-called "balanced" modulus system in which `n modulo p` is defined as an integer k in `[-(p-1)/2, ..., 0, ..., (p-1)/2]` when p is odd, or `[-(p/2 - 1), ..., 0, ...., p/2]` when p is even, such that `a p + k` equals n for some integer a.

  If expr is already in canonical rational expression (CRE) form when `modulus` is reset, then you may need to re-rat expr, e.g., `expr: rat (ratdisrep (expr))`, in order to get correct results.

  Typically `modulus` is set to a prime number. If `modulus` is set to a positive non-prime integer, this setting is accepted, but a warning message is displayed. Maxima signals an error, when zero or a negative integer is assigned to `modulus`.

  Examples:

      (%i1) modulus:7;
      (%o1)                           7
      (%i2) polymod([0,1,2,3,4,5,6,7]);
      (%o2)            [0, 1, 2, 3, - 3, - 2, - 1, 0]
      (%i3) modulus:false;
      (%o3)                         false
      (%i4) poly:x^6+x^2+1;
                                  6    2
      (%o4)                      x  + x  + 1
      (%i5) factor(poly);
                                  6    2
      (%o5)                      x  + x  + 1
      (%i6) modulus:13;
      (%o6)                          13
      (%i7) factor(poly);
                            2        4      2
      (%o7)               (x  + 6) (x  - 6 x  - 2)
      (%i8) polymod(%);
                                  6    2
      (%o8)                      x  + x  + 1

  Categories: Integers

Function: num (expr)

  Returns the numerator of expr if it is a ratio. If expr is not a ratio, expr is returned.

  `num` evaluates its argument.

  See also `denom`

      (%i1) g1:(x+2)*(x+1)/((x+3)^2);
                               (x + 1) (x + 2)
      (%o1)                    ---------------
                                         2
                                  (x + 3)
      (%i2) num(g1);
      (%o2)                    (x + 1) (x + 2)
      (%i3) g2:sin(x)/10*cos(x)/y;
                                cos(x) sin(x)
      (%o3)                     -------------
                                    10 y
      (%i4) num(g2);
      (%o4)                     cos(x) sin(x)

  Categories: Expressions

Function: polydecomp (p, x)

  Decomposes the polynomial p in the variable x into the functional composition of polynomials in x. `polydecomp` returns a list `[p_1, ..., p_n]` such that

      lambda ([x], p_1) (lambda ([x], p_2) (... (lambda ([x], p_n) (x))
        ...))

  is equal to p. The degree of p_i is greater than 1 for i less than n.

  Such a decomposition is not unique.

  Examples:

      (%i1) polydecomp (x^210, x);
                                7   5   3   2
      (%o1)                   [x , x , x , x ]
      (%i2) p : expand (subst (x^3 - x - 1, x, x^2 - a));
                      6      4      3    2
      (%o2)          x  - 2 x  - 2 x  + x  + 2 x - a + 1
      (%i3) polydecomp (p, x);
                              2       3
      (%o3)                 [x  - a, x  - x - 1]

  The following function composes `L = [e_1, ..., e_n]` as functions in `x`; it is the inverse of polydecomp:

      (%i1) compose (L, x) :=
        block ([r : x], for e in L do r : subst (e, x, r), r) $

  Re-express above example using `compose`:

      (%i1) polydecomp (compose ([x^2 - a, x^3 - x - 1], x), x);
                                2       3
      (%o1)          [compose([x  - a, x  - x - 1], x)]

  Note that though `compose (polydecomp (p, x), x)` always returns p (unexpanded), `polydecomp (compose ([p_1, ..., p_n], x), x)` does not necessarily return `[p_1, ..., p_n]`:

      (%i1) polydecomp (compose ([x^2 + 2*x + 3, x^2], x), x);
                                 2             2
      (%o1)           [compose([x  + 2 x + 3, x ], x)]
      (%i2) polydecomp (compose ([x^2 + x + 1, x^2 + x + 1], x), x);
                              2           2
      (%o2)        [compose([x  + x + 1, x  + x + 1], x)]

  Categories: Polynomials

Function: polymod
`polymod` (p)
`polymod` (p, m)

  Converts the polynomial p to a modular representation with respect to the current modulus which is the value of the variable `modulus`.

  `polymod (p, m)` specifies a modulus m to be used instead of the current value of `modulus`.

  See `modulus`.

  Categories: Polynomials

Function: quotient
`quotient` (p_1, p_2)
`quotient` (p_1, p_2, x_1, …, x_n)

  Returns the polynomial p_1 divided by the polynomial p_2. The arguments x_1, …, x_n are interpreted as in `ratvars`.

  `quotient` returns the first element of the two-element list returned by `divide`.

  Categories: Polynomials

Function: rat
`rat` (expr)
`rat` (expr, x_1, …, x_n)

  Converts expr to canonical rational expression (CRE) form by expanding and combining all terms over a common denominator and cancelling out the greatest common divisor of the numerator and denominator, as well as converting floating point numbers to rational numbers within a tolerance of `ratepsilon`. The variables are ordered according to the x_1, …, x_n, if specified, as in `ratvars`.

  `rat` does not generally simplify functions other than addition `+`, subtraction `-`, multiplication `*`, division `/`, and exponentiation to an integer power, whereas `ratsimp` does handle those cases. Note that atoms (numbers and variables) in CRE form are not the same as they are in the general form. For example, `rat(x)- x` yields `rat(0)` which has a different internal representation than 0.

  When `ratfac` is `true`, `rat` yields a partially factored form for CRE. During rational operations the expression is maintained as fully factored as possible without an actual call to the factor package. This should always save space and may save some time in some computations. The numerator and denominator are still made relatively prime (e.g., `rat((x^2 - 1)^4/(x + 1)^2)` yields `(x - 1)^4 (x + 1)^2` when `ratfac` is `true`), but the factors within each part may not be relatively prime.

  `ratprint` if `false` suppresses the printout of the message informing the user of the conversion of floating point numbers to rational numbers.

  `keepfloat` if `true` prevents floating point numbers from being converted to rational numbers.

  See also `ratexpand` and `ratsimp`.

  Examples:

      (%i1) ((x - 2*y)^4/(x^2 - 4*y^2)^2 + 1)*(y + a)*(2*y + x) /
            (4*y^2 + x^2);
                                                 4
                                        (x - 2 y)
                    (y + a) (2 y + x) (------------ + 1)
                                         2      2 2
                                       (x  - 4 y )
      (%o1)         ------------------------------------
                                    2    2
                                 4 y  + x
      (%i2) rat (%, y, a, x);
                                  2 a + 2 y
      (%o2)/R/                    ---------
                                   x + 2 y

  Categories: Rational expressions

Option variable: ratalgdenom

  Default value: `true`

  When `ratalgdenom` is `true`, allows rationalization of denominators with respect to radicals to take effect. `ratalgdenom` has an effect only when canonical rational expressions (CRE) are used in algebraic mode.

  Categories: Simplification flags and variables

Function: ratcoef
`ratcoef` (expr, x, n)
`ratcoef` (expr, x)

  Returns the coefficient of the expression `x^n` in the expression expr. If omitted, n is assumed to be 1.

  The return value is free (except possibly in a non-rational sense) of the variables in x. If no coefficient of this type exists, 0 is returned.

  `ratcoef` expands and rationally simplifies its first argument and thus it may produce answers different from those of `coeff` which is purely syntactic. Thus `ratcoef ((x + 1)/y + x, x)` returns `(y + 1)/y` whereas `coeff` returns 1.

  `ratcoef (expr, x, 0)`, viewing expr as a sum, returns a sum of those terms which do not contain x. Therefore if x occurs to any negative powers, `ratcoef` should not be used.

  Since expr is rationally simplified before it is examined, coefficients may not appear quite the way they were envisioned.

  Example:

      (%i1) s: a*x + b*x + 5$
      (%i2) ratcoef (s, a + b);
      (%o2)                           x

  Categories: Polynomials · Rational expressions

Function: ratdenom (expr)

  Returns the denominator of expr, after coercing expr to a canonical rational expression (CRE). The return value is a CRE.

  expr is coerced to a CRE by `rat` if it is not already a CRE. This conversion may change the form of expr by putting all terms over a common denominator.

  `denom` is similar, but returns an ordinary expression instead of a CRE. Also, `denom` does not attempt to place all terms over a common denominator, and thus some expressions which are considered ratios by `ratdenom` are not considered ratios by `denom`.

  Categories: Rational expressions

Option variable: ratdenomdivide

  Default value: `true`

  When `ratdenomdivide` is `true`, `ratexpand` expands a ratio in which the numerator is a sum into a sum of ratios, all having a common denominator. Otherwise, `ratexpand` collapses a sum of ratios into a single ratio, the numerator of which is the sum of the numerators of each ratio.

  Examples:

      (%i1) expr: (x^2 + x + 1)/(y^2 + 7);
                                  2
                                 x  + x + 1
      (%o1)                      ----------
                                    2
                                   y  + 7
      (%i2) ratdenomdivide: true$
      (%i3) ratexpand (expr);
                             2
                            x        x        1
      (%o3)               ------ + ------ + ------
                           2        2        2
                          y  + 7   y  + 7   y  + 7
      (%i4) ratdenomdivide: false$
      (%i5) ratexpand (expr);
                                  2
                                 x  + x + 1
      (%o5)                      ----------
                                    2
                                   y  + 7
      (%i6) expr2: a^2/(b^2 + 3) + b/(b^2 + 3);
                                           2
                                 b        a
      (%o6)                    ------ + ------
                                2        2
                               b  + 3   b  + 3
      (%i7) ratexpand (expr2);
                                        2
                                   b + a
      (%o7)                        ------
                                    2
                                   b  + 3

  Categories: Simplification flags and variables · Rational expressions

Function: ratdiff (expr, x)

  Differentiates the rational expression expr with respect to x. expr must be a ratio of polynomials or a polynomial in x. The argument x may be a variable or a subexpression of expr.

  The result is equivalent to `diff`, although perhaps in a different form. `ratdiff` may be faster than `diff`, for rational expressions.

  `ratdiff` returns a canonical rational expression (CRE) if `expr` is a CRE. Otherwise, `ratdiff` returns a general expression.

  `ratdiff` considers only the dependence of expr on x, and ignores any dependencies established by `depends`.

  Example:

      (%i1) expr: (4*x^3 + 10*x - 11)/(x^5 + 5);
                                 3
                              4 x  + 10 x - 11
      (%o1)                   ----------------
                                    5
                                   x  + 5
      (%i2) ratdiff (expr, x);
                          7       5       4       2
                       8 x  + 40 x  - 55 x  - 60 x  - 50
      (%o2)          - ---------------------------------
                                10       5
                               x   + 10 x  + 25
      (%i3) expr: f(x)^3 - f(x)^2 + 7;
                               3       2
      (%o3)                   f (x) - f (x) + 7
      (%i4) ratdiff (expr, f(x));
                                 2
      (%o4)                   3 f (x) - 2 f(x)
      (%i5) expr: (a + b)^3 + (a + b)^2;
                                    3          2
      (%o5)                  (b + a)  + (b + a)
      (%i6) ratdiff (expr, a + b);
                          2                    2
      (%o6)            3 b  + (6 a + 2) b + 3 a  + 2 a

  Categories: Rational expressions

Function: ratdisrep (expr)

  Returns its argument as a general expression. If expr is a general expression, it is returned unchanged.

  Typically `ratdisrep` is called to convert a canonical rational expression (CRE) into a general expression. This is sometimes convenient if one wishes to stop the "contagion", or use rational functions in non-rational contexts.

  See also `totaldisrep`.

  Categories: Rational expressions

Function: ratexpand (expr)
Option variable: ratexpand

  Expands expr by multiplying out products of sums and exponentiated sums, combining fractions over a common denominator, cancelling the greatest common divisor of the numerator and denominator, then splitting the numerator (if a sum) into its respective terms divided by the denominator.

  The return value of `ratexpand` is a general expression, even if expr is a canonical rational expression (CRE).

  The switch `ratexpand` if `true` will cause CRE expressions to be fully expanded when they are converted back to general form or displayed, while if it is `false` then they will be put into a recursive form. See also `ratsimp`.

  When `ratdenomdivide` is `true`, `ratexpand` expands a ratio in which the numerator is a sum into a sum of ratios, all having a common denominator. Otherwise, `ratexpand` collapses a sum of ratios into a single ratio, the numerator of which is the sum of the numerators of each ratio.

  When `keepfloat` is `true`, prevents floating point numbers from being rationalized when expressions which contain them are converted to canonical rational expression (CRE) form.

  Examples:

      (%i1) ratexpand ((2*x - 3*y)^3);
                           3         2       2        3
      (%o1)          - 27 y  + 54 x y  - 36 x  y + 8 x
      (%i2) expr: (x - 1)/(x + 1)^2 + 1/(x - 1);
                               x - 1       1
      (%o2)                   -------- + -----
                                     2   x - 1
                              (x + 1)
      (%i3) expand (expr);
                          x              1           1
      (%o3)          ------------ - ------------ + -----
                      2              2             x - 1
                     x  + 2 x + 1   x  + 2 x + 1
      (%i4) ratexpand (expr);
                              2
                           2 x                 2
      (%o4)           --------------- + ---------------
                       3    2            3    2
                      x  + x  - x - 1   x  + x  - x - 1

  Categories: Rational expressions

Option variable: ratfac

  Default value: `false`

  When `ratfac` is `true`, canonical rational expressions (CRE) are manipulated in a partially factored form.

  During rational operations the expression is maintained as fully factored as possible without calling `factor`. This should always save space and may save time in some computations. The numerator and denominator are made relatively prime, for example `factor ((x^2 - 1)^4/(x + 1)^2)` yields `(x - 1)^4 (x + 1)^2`, but the factors within each part may not be relatively prime.

  In the `ctensr` (Component Tensor Manipulation) package, Ricci, Einstein, Riemann, and Weyl tensors and the scalar curvature are factored automatically when `ratfac` is `true`. `ratfac` should only be set for cases where the tensorial components are known to consist of few terms.

  The `ratfac` and `ratweight` schemes are incompatible and may not both be used at the same time.

  Categories: Rational expressions

Function: ratnumer (expr)

  Returns the numerator of expr, after coercing expr to a canonical rational expression (CRE). The return value is a CRE.

  expr is coerced to a CRE by `rat` if it is not already a CRE. This conversion may change the form of expr by putting all terms over a common denominator.

  `num` is similar, but returns an ordinary expression instead of a CRE. Also, `num` does not attempt to place all terms over a common denominator, and thus some expressions which are considered ratios by `ratnumer` are not considered ratios by `num`.

  Categories: Rational expressions

Function: ratp (expr)

  Returns `true` if expr is a canonical rational expression (CRE) or extended CRE, otherwise `false`.

  CRE are created by `rat` and related functions. Extended CRE are created by `taylor` and related functions.

  Categories: Predicate functions · Rational expressions

Option variable: ratprint

  Default value: `true`

  When `ratprint` is `true`, a message informing the user of the conversion of floating point numbers to rational numbers is displayed.

  Categories: Rational expressions · Numerical evaluation · Console interaction

Function: ratsimp (expr)
Function: ratsimp (expr, x_1, …, x_n)

  Simplifies the expression expr and all of its subexpressions, including the arguments to non-rational functions. The result is returned as the quotient of two polynomials in a recursive form, that is, the coefficients of the main variable are polynomials in the other variables. Variables may include non-rational functions (e.g., `sin (x^2 + 1)`) and the arguments to any such functions are also rationally simplified.

  `ratsimp (expr, x_1, ..., x_n)` enables rational simplification with the specification of variable ordering as in `ratvars`.

  When `ratsimpexpons` is `true`, `ratsimp` is applied to the exponents of expressions during simplification.

  See also `ratexpand`. Note that `ratsimp` is affected by some of the flags which affect `ratexpand`.

  Examples:

      (%i1) sin (x/(x^2 + x)) = exp ((log(x) + 1)^2 - log(x)^2);
                                               2      2
                         x         (log(x) + 1)  - log (x)
      (%o1)        sin(------) = %e
                        2
                       x  + x
      (%i2) ratsimp (%);
                                   1          2
      (%o2)                  sin(-----) = %e x
                                 x + 1
      (%i3) ((x - 1)^(3/2) - (x + 1)*sqrt(x - 1))/sqrt((x - 1)*(x + 1));
                             3/2
                      (x - 1)    - sqrt(x - 1) (x + 1)
      (%o3)           --------------------------------
                           sqrt((x - 1) (x + 1))
      (%i4) ratsimp (%);
                                 2 sqrt(x - 1)
      (%o4)                    - -------------
                                       2
                                 sqrt(x  - 1)
      (%i5) x^(a + 1/a), ratsimpexpons: true;
                                     2
                                    a  + 1
                                    ------
                                      a
      (%o5)                        x

  Categories: Simplification functions · Rational expressions

Option variable: ratsimpexpons

  Default value: `false`

  When `ratsimpexpons` is `true`, `ratsimp` is applied to the exponents of expressions during simplification.

  Categories: Simplification flags and variables · Rational expressions

Option variable: radsubstflag

  Default value: `false`

  `radsubstflag`, if `true`, permits `ratsubst` to make substitutions such as `u` for `sqrt (x)` in `x`.

  Categories: Simplification flags and variables

Function: ratsubst (a, b, c)

  Substitutes a for b in c and returns the resulting expression. b may be a sum, product, power, etc.

  `ratsubst` knows something of the meaning of expressions whereas `subst` does a purely syntactic substitution. Thus `subst (a, x + y, x + y + z)` returns `x + y + z` whereas `ratsubst` returns `z + a`.

  When `radsubstflag` is `true`, `ratsubst` makes substitutions for radicals in expressions which don't explicitly contain them.

  `ratsubst` ignores the value `true` of the option variable `keepfloat`.

  Examples:

      (%i1) ratsubst (a, x*y^2, x^4*y^3 + x^4*y^8);
                                    3      4
      (%o1)                      a x  y + a
      (%i2) cos(x)^4 + cos(x)^3 + cos(x)^2 + cos(x) + 1;
                     4         3         2
      (%o2)       cos (x) + cos (x) + cos (x) + cos(x) + 1
      (%i3) ratsubst (1 - sin(x)^2, cos(x)^2, %);
                  4           2                     2
      (%o3)    sin (x) - 3 sin (x) + cos(x) (2 - sin (x)) + 3
      (%i4) ratsubst (1 - cos(x)^2, sin(x)^2, sin(x)^4);
                              4           2
      (%o4)                cos (x) - 2 cos (x) + 1
      (%i5) radsubstflag: false$
      (%i6) ratsubst (u, sqrt(x), x);
      (%o6)                           x
      (%i7) radsubstflag: true$
      (%i8) ratsubst (u, sqrt(x), x);
                                      2
      (%o8)                          u

  Categories: Rational expressions

Function: ratvars (x_1, …, x_n)
Function: ratvars ()
System variable: ratvars

  Declares main variables x_1, …, x_n for rational expressions. x_n, if present in a rational expression, is considered the main variable. Otherwise, x_[n-1] is considered the main variable if present, and so on through the preceding variables to x_1, which is considered the main variable only if none of the succeeding variables are present.

  If a variable in a rational expression is not present in the `ratvars` list, it is given a lower priority than x_1.

  The arguments to `ratvars` can be either variables or non-rational functions such as `sin(x)`.

  The variable `ratvars` is a list of the arguments of the function `ratvars` when it was called most recently. Each call to the function `ratvars` resets the list. `ratvars ()` clears the list.

  Categories: Rational expressions

Option variable: ratvarswitch

  Default value: `true`

  Maxima keeps an internal list in the Lisp variable `VARLIST` of the main variables for rational expressions. If `ratvarswitch` is `true`, every evaluation starts with a fresh list `VARLIST`. This is the default behavior. Otherwise, the main variables from previous evaluations are not removed from the internal list `VARLIST`.

  The main variables, which are declared with the function `ratvars` are not affected by the option variable `ratvarswitch`.

  Examples:

  If `ratvarswitch` is `true`, every evaluation starts with a fresh list `VARLIST`.

      (%i1) ratvarswitch:true$

      (%i2) rat(2*x+y^2);
                                   2
      (%o2)/R/                    y  + 2 x
      (%i3) :lisp varlist
      ($X $Y)

      (%i3) rat(2*a+b^2);
                                   2
      (%o3)/R/                    b  + 2 a

      (%i4) :lisp varlist
      ($A $B)

  If `ratvarswitch` is `false`, the main variables from the last evaluation are still present.

      (%i4) ratvarswitch:false$

      (%i5) rat(2*x+y^2);
                                   2
      (%o5)/R/                    y  + 2 x
      (%i6) :lisp varlist
      ($X $Y)

      (%i6) rat(2*a+b^2);
                                   2
      (%o6)/R/                    b  + 2 a

      (%i7) :lisp varlist
      ($A $B $X $Y)

  Categories: Rational expressions · Global flags

Function: ratweight
`ratweight` (x_1, w_1, …, x_n, w_n)
`ratweight` ()

  Assigns a weight w_i to the variable x_i. This causes a term to be replaced by 0 if its weight exceeds the value of the variable `ratwtlvl` (default yields no truncation). The weight of a term is the sum of the products of the weight of a variable in the term times its power. For example, the weight of `3 x_1^2 x_2` is `2 w_1 + w_2`. Truncation according to `ratwtlvl` is carried out only when multiplying or exponentiating canonical rational expressions (CRE).

  `ratweight ()` returns the cumulative list of weight assignments.

  Note: The `ratfac` and `ratweight` schemes are incompatible and may not both be used at the same time.

  Examples:

      (%i1) ratweight (a, 1, b, 1);
      (%o1)                     [a, 1, b, 1]
      (%i2) expr1: rat(a + b + 1)$
      (%i3) expr1^2;
                        2                  2
      (%o3)/R/         b  + (2 a + 2) b + a  + 2 a + 1
      (%i4) ratwtlvl: 1$
      (%i5) expr1^2;
      (%o5)/R/                  2 b + 2 a + 1

  Categories: Rational expressions

System variable: ratweights

  Default value: `[]`

  `ratweights` is the list of weights assigned by `ratweight`. The list is cumulative: each call to `ratweight` places additional items in the list.

  `kill (ratweights)` and `save (ratweights)` both work as expected.

  Categories: Rational expressions

Option variable: ratwtlvl

  Default value: `false`

  `ratwtlvl` is used in combination with the `ratweight` function to control the truncation of canonical rational expressions (CRE). For the default value of `false`, no truncation occurs.

  Categories: Rational expressions

Function: remainder
`remainder` (p_1, p_2)
`remainder` (p_1, p_2, x_1, …, x_n)

  Returns the remainder of the polynomial p_1 divided by the polynomial p_2. The arguments x_1, …, x_n are interpreted as in `ratvars`.

  `remainder` returns the second element of the two-element list returned by `divide`.

  Categories: Polynomials

Function: resultant (p_1, p_2, x)

  The function `resultant` computes the resultant of the two polynomials p_1 and p_2, eliminating the variable x. The resultant is a determinant of the coefficients of x in p_1 and p_2, which equals zero if and only if p_1 and p_2 have a non-constant factor in common.

  If p_1 or p_2 can be factored, it may be desirable to call `factor` before calling `resultant`.

  The option variable `resultant` controls which algorithm will be used to compute the resultant. See the option variable `resultant`.

  The function `bezout` takes the same arguments as `resultant` and returns a matrix. The determinant of the return value is the desired resultant.

  Examples:

      (%i1) resultant(2*x^2+3*x+1, 2*x^2+x+1, x);
      (%o1)                           8
      (%i2) resultant(x+1, x+1, x);
      (%o2)                           0
      (%i3) resultant((x+1)*x, (x+1), x);
      (%o3)                           0
      (%i4) resultant(a*x^2+b*x+1, c*x + 2, x);
                               2
      (%o4)                   c  - 2 b c + 4 a

      (%i5) bezout(a*x^2+b*x+1, c*x+2, x);
                              [ 2 a  2 b - c ]
      (%o5)                   [              ]
                              [  c      2    ]
      (%i6) determinant(%);
      (%o6)                   4 a - (2 b - c) c

  Categories: Polynomials

Option variable: resultant

  Default value: `subres`

  The option variable `resultant` controls which algorithm will be used to compute the resultant with the function `resultant`. The possible values are:

  `subres`

    for the subresultant polynomial remainder sequence (PRS) algorithm,

  `mod`

    (not enabled) for the modular resultant algorithm, and

  `red`

    for the reduced polynomial remainder sequence (PRS) algorithm.

  On most problems the default value `subres` should be best.

  Categories: Polynomials

Option variable: savefactors

  Default value: `false`

  When `savefactors` is `true`, causes the factors of an expression which is a product of factors to be saved by certain functions in order to speed up later factorizations of expressions containing some of the same factors.

  Categories: Polynomials

Function: showratvars (expr)

  Returns a list of the canonical rational expression (CRE) variables in expression `expr`.

  See also `ratvars`.

  Categories: Rational expressions · Display functions

Function: sqfr (expr)

  is similar to `factor` except that the polynomial factors are "square-free." That is, they have factors only of degree one. This algorithm, which is also used by the first stage of `factor`, utilizes the fact that a polynomial has in common with its n'th derivative all its factors of degree greater than n. Thus by taking greatest common divisors with the polynomial of the derivatives with respect to each variable in the polynomial, all factors of degree greater than 1 can be found.

  Example:

      (%i1) sqfr (4*x^4 + 4*x^3 - 3*x^2 - 4*x - 1);
                                      2   2
      (%o1)                  (2 x + 1)  (x  - 1)

  Categories: Polynomials

Function: tellrat
`tellrat` (p_1, …, p_n)
`tellrat` ()

  Adds to the ring of algebraic integers known to Maxima the elements which are the solutions of the polynomials p_1, …, p_n. Each argument p_i is a polynomial with integer coefficients.

  `tellrat (x)` effectively means substitute 0 for x in rational functions.

  `tellrat ()` returns a list of the current substitutions.

  `algebraic` must be set to `true` in order for the simplification of algebraic integers to take effect.

  Maxima initially knows about the imaginary unit `%i` and all roots of integers.

  There is a command `untellrat` which takes kernels and removes `tellrat` properties.

  When `tellrat`'ing a multivariate polynomial, e.g., `tellrat (x^2 - y^2)`, there would be an ambiguity as to whether to substitute `y^2` for `x^2` or vice versa. Maxima picks a particular ordering, but if the user wants to specify which, e.g. `tellrat (y^2 = x^2)` provides a syntax which says replace `y^2` by `x^2`.

  Examples:

      (%i1) 10*(%i + 1)/(%i + 3^(1/3));
                                 10 (%i + 1)
      (%o1)                      -----------
                                        1/3
                                  %i + 3
      (%i2) ev (ratdisrep (rat(%)), algebraic);
                   2/3      1/3              2/3      1/3
      (%o2)    (4 3    - 2 3    - 4) %i + 2 3    + 4 3    - 2
      (%i3) tellrat (1 + a + a^2);
                                  2
      (%o3)                     [a  + a + 1]
      (%i4) 1/(a*sqrt(2) - 1) + a/(sqrt(3) + sqrt(2));
                            1                 a
      (%o4)           ------------- + -----------------
                      sqrt(2) a - 1   sqrt(3) + sqrt(2)
      (%i5) ev (ratdisrep (rat(%)), algebraic);
               (7 sqrt(3) - 10 sqrt(2) + 2) a - 2 sqrt(2) - 1
      (%o5)    ----------------------------------------------
                                     7
      (%i6) tellrat (y^2 = x^2);
                              2    2   2
      (%o6)                 [y  - x , a  + a + 1]

  Categories: Polynomials · Rational expressions

Function: totaldisrep (expr)

  Converts every subexpression of expr from canonical rational expressions (CRE) to general form and returns the result. If expr is itself in CRE form then `totaldisrep` is identical to `ratdisrep`.

  `totaldisrep` may be useful for ratdisrepping expressions such as equations, lists, matrices, etc., which have some subexpressions in CRE form.

  Categories: Rational expressions

Function: untellrat (x_1, …, x_n)

  Removes `tellrat` properties from x_1, …, x_n.

  Categories: Polynomials · Rational expressions

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# 15. Special Functions

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## 15.1 Introduction to Special Functions

Special function notation follows:

    bessel_j (index, expr)         Bessel function, 1st kind
    bessel_y (index, expr)         Bessel function, 2nd kind
    bessel_i (index, expr)         Modified Bessel function, 1st kind
    bessel_k (index, expr)         Modified Bessel function, 2nd kind

    hankel_1 (v,z)                 Hankel function of the 1st kind
    hankel_2 (v,z)                 Hankel function of the 2nd kind
    struve_h (v,z)                 Struve H function
    struve_l (v,z)                 Struve L function

    assoc_legendre_p[v,u] (z)      Legendre function of degree v and order u
    assoc_legendre_q[v,u] (z)      Legendre function, 2nd kind

    %f[p,q] ([], [], expr)         Generalized Hypergeometric function
    gamma (z)                      Gamma function
    gamma_incomplete_lower (a,z)   Lower incomplete gamma function
    gamma_incomplete (a,z)         Tail of incomplete gamma function
    hypergeometric (l1, l2, z)     Hypergeometric function
    slommel
    %m[u,k] (z)                    Whittaker function, 1st kind
    %w[u,k] (z)                    Whittaker function, 2nd kind
    erfc (z)                       Complement of the erf function

    expintegral_e (v,z)            Exponential integral E
    expintegral_e1 (z)             Exponential integral E1
    expintegral_ei (z)             Exponential integral Ei
    expintegral_li (z)             Logarithmic integral Li
    expintegral_si (z)             Exponential integral Si
    expintegral_ci (z)             Exponential integral Ci
    expintegral_shi (z)            Exponential integral Shi
    expintegral_chi (z)            Exponential integral Chi

    kelliptic (z)                  Complete elliptic integral of the first
                                   kind (K)
    parabolic_cylinder_d (v,z)     Parabolic cylinder D function

Categories: Bessel functions · Airy functions · Special functions

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## 15.2 Bessel Functions

Function: bessel_j (v, z)

  The Bessel function of the first kind of order v and argument z.

  `bessel_j` is defined as

                      inf
                      ====       k  - v - 2 k  v + 2 k
                      \     (- 1)  2          z
                       >    --------------------------
                      /        k! gamma(v + k + 1)
                      ====
                      k = 0

  although the infinite series is not used for computations.

  Categories: Bessel functions · Special functions

Function: bessel_y (v, z)

  The Bessel function of the second kind of order v and argument z.

  `bessel_y` is defined as

                    cos(%pi v) bessel_j(v, z) - bessel_j(-v, z)
                    -------------------------------------------
                                   sin(%pi v)

  when v is not an integer. When v is an integer n, the limit as v approaches n is taken.

  Categories: Bessel functions · Special functions

Function: bessel_i (v, z)

  The modified Bessel function of the first kind of order v and argument z.

  `bessel_i` is defined as

                          inf
                          ====   - v - 2 k  v + 2 k
                          \     2          z
                           >    -------------------
                          /     k! gamma(v + k + 1)
                          ====
                          k = 0

  although the infinite series is not used for computations.

  Categories: Bessel functions · Special functions

Function: bessel_k (v, z)

  The modified Bessel function of the second kind of order v and argument z.

  `bessel_k` is defined as

                 %pi csc(%pi v) (bessel_i(-v, z) - bessel_i(v, z))
                 -------------------------------------------------
                                        2

  when v is not an integer. If v is an integer n, then the limit as v approaches n is taken.

  Categories: Bessel functions · Special functions

Function: hankel_1 (v, z)

  The Hankel function of the first kind of order v and argument z (A&S 9.1.3). `hankel_1` is defined as

         bessel_j(v,z) + %i * bessel_y(v,z)

  Maxima evaluates `hankel_1` numerically for a complex order v and complex argument z in float precision. The numerical evaluation in bigfloat precision is not supported.

  When `besselexpand` is `true`, `hankel_1` is expanded in terms of elementary functions when the order v is half of an odd integer. See `besselexpand`.

  Maxima knows the derivative of `hankel_1` wrt the argument z.

  Examples:

  Numerical evaluation:

      (%i1) hankel_1(1,0.5);
      (%o1)        0.24226845767487 - 1.471472392670243 %i
      (%i2) hankel_1(1,0.5+%i);
      (%o2)       - 0.25582879948621 %i - 0.23957560188301

  Expansion of `hankel_1` when `besselexpand` is `true`:

      (%i1) hankel_1(1/2,z),besselexpand:true;
                     sqrt(2) sin(z) - sqrt(2) %i cos(z)
      (%o1)          ----------------------------------
                             sqrt(%pi) sqrt(z)

  Derivative of `hankel_1` wrt the argument z. The derivative wrt the order v is not supported. Maxima returns a noun form:

      (%i1) diff(hankel_1(v,z),z);
                   hankel_1(v - 1, z) - hankel_1(v + 1, z)
      (%o1)        ---------------------------------------
                                      2
      (%i2) diff(hankel_1(v,z),v);
                             d
      (%o2)                  -- (hankel_1(v, z))
                             dv

  Categories: Bessel functions · Special functions

Function: hankel_2 (v, z)

  The Hankel function of the second kind of order v and argument z (A&S 9.1.4). `hankel_2` is defined as

         bessel_j(v,z) - %i * bessel_y(v,z)

  Maxima evaluates `hankel_2` numerically for a complex order v and complex argument z in float precision. The numerical evaluation in bigfloat precision is not supported.

  When `besselexpand` is `true`, `hankel_2` is expanded in terms of elementary functions when the order v is half of an odd integer. See `besselexpand`.

  Maxima knows the derivative of `hankel_2` wrt the argument z.

  For examples see `hankel_1`.

  Categories: Bessel functions · Special functions

Option variable: besselexpand

  Default value: `false`

  Controls expansion of the Bessel functions when the order is half of an odd integer. In this case, the Bessel functions can be expanded in terms of other elementary functions. When `besselexpand` is `true`, the Bessel function is expanded.

      (%i1) besselexpand: false$
      (%i2) bessel_j (3/2, z);
                                          3
      (%o2)                      bessel_j(-, z)
                                          2
      (%i3) besselexpand: true$
      (%i4) bessel_j (3/2, z);
                                              sin(z)   cos(z)
                             sqrt(2) sqrt(z) (------ - ------)
                                                 2       z
                                                z
      (%o4)                  ---------------------------------
                                         sqrt(%pi)

  Categories: Bessel functions · Simplification flags and variables Special functions

Function: scaled_bessel_i (v, z)

  The scaled modified Bessel function of the first kind of order v and argument z. That is, scaled_bessel_i(v,z) = exp(-abs(z))*bessel_i(v, z). This function is particularly useful for calculating bessel_i for large z, which is large. However, maxima does not otherwise know much about this function. For symbolic work, it is probably preferable to work with the expression `exp(-abs(z))*bessel_i(v, z)`.

  Categories: Bessel functions

Function: scaled_bessel_i0 (z)

  Identical to `scaled_bessel_i(0,z)`.

  Categories: Bessel functions · Special functions

Function: scaled_bessel_i1 (z)

  Identical to `scaled_bessel_i(1,z)`.

  Bessel functions · Special functions

Function: %s [u,v] (z)

  Lommel's little s[u,v](z) function. Probably Gradshteyn & Ryzhik 8.570.1.

  Bessel functions · Special functions

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## 15.3 Airy Functions

The Airy functions Ai(x) and Bi(x) are defined in Abramowitz and Stegun, Handbook of Mathematical Functions, Section 10.4.

`y = Ai(x)` and `y = Bi(x)` are two linearly independent solutions of the Airy differential equation `diff (y(x), x, 2) - x y(x) = 0`.

If the argument `x` is a real or complex floating point number, the numerical value of the function is returned.

Function: airy_ai (x)

  The Airy function Ai(x). (A&S 10.4.2)

  The derivative `diff (airy_ai(x), x)` is `airy_dai(x)`.

  See also `airy_bi`, `airy_dai`, `airy_dbi`.

  Categories: Airy functions · Special functions

Function: airy_dai (x)

  The derivative of the Airy function Ai `airy_ai(x)`.

  See `airy_ai`.

  Categories: Airy functions · Special functions

Function: airy_bi (x)

  The Airy function Bi(x). (A&S 10.4.3)

  The derivative `diff (airy_bi(x), x)` is `airy_dbi(x)`.

  See `airy_ai`, `airy_dbi`.

  Categories: Airy functions · Special functions

Function: airy_dbi (x)

  The derivative of the Airy Bi function `airy_bi(x)`.

  See `airy_ai` and `airy_bi`.

  Categories: Airy functions · Special functions

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## 15.4 Gamma and factorial Functions

The gamma function and the related beta, psi and incomplete gamma functions are defined in Abramowitz and Stegun, Handbook of Mathematical Functions, Chapter 6.

Function: bffac (expr, n)

  Bigfloat version of the factorial (shifted gamma) function. The second argument is how many digits to retain and return, it's a good idea to request a couple of extra.

  Categories: Gamma and factorial functions · Numerical evaluation

Function: bfpsi (n, z, fpprec)
Function: bfpsi0 (z, fpprec)

  `bfpsi` is the polygamma function of real argument z and integer order n. `bfpsi0` is the digamma function. `bfpsi0 (z, fpprec)` is equivalent to `bfpsi (0, z, fpprec)`.

  These functions return bigfloat values. fpprec is the bigfloat precision of the return value.

  Categories: Gamma and factorial functions · Numerical evaluation

Function: cbffac (z, fpprec)

  Complex bigfloat factorial.

  `load ("bffac")` loads this function.

  Categories: Gamma and factorial functions · Complex variables · Numerical evaluation

Function: gamma (z)

  The basic definition of the gamma function (A&S 6.1.1) is

                               inf
                              /
                              [     z - 1   - t
                   gamma(z) = I    t      %e    dt
                              ]
                              /
                               0

  Maxima simplifies `gamma` for positive integer and positive and negative rational numbers. For half integral values the result is a rational number times `sqrt(%pi)`. The simplification for integer values is controlled by `factlim`. For integers greater than `factlim` the numerical result of the factorial function, which is used to calculate `gamma`, will overflow. The simplification for rational numbers is controlled by `gammalim` to avoid internal overflow. See `factlim` and `gammalim`.

  For negative integers `gamma` is not defined.

  Maxima can evalute `gamma` numerically for real and complex values in float and bigfloat precision.

  `gamma` has mirror symmetry.

  When `gamma_expand` is `true`, Maxima expands `gamma` for arguments `z+n` and `z-n` where `n` is an integer.

  Maxima knows the derivate of `gamma`.

  Examples:

  Simplification for integer, half integral, and rational numbers:

      (%i1) map('gamma,[1,2,3,4,5,6,7,8,9]);
      (%o1)        [1, 1, 2, 6, 24, 120, 720, 5040, 40320]
      (%i2) map('gamma,[1/2,3/2,5/2,7/2]);
                          sqrt(%pi)  3 sqrt(%pi)  15 sqrt(%pi)
      (%o2)   [sqrt(%pi), ---------, -----------, ------------]
                              2           4            8
      (%i3) map('gamma,[2/3,5/3,7/3]);
                                        2           1
                                2 gamma(-)  4 gamma(-)
                            2           3           3
      (%o3)          [gamma(-), ----------, ----------]
                            3       3           9

  Numerical evaluation for real and complex values:

      (%i4) map('gamma,[2.5,2.5b0]);
      (%o4)     [1.329340388179137, 1.3293403881791370205b0]
      (%i5) map('gamma,[1.0+%i,1.0b0+%i]);
      (%o5) [0.498015668118356 - .1549498283018107 %i,
                4.9801566811835604272b-1 - 1.5494982830181068513b-1 %i]

  `gamma` has mirror symmetry:

      (%i6) declare(z,complex)$
      (%i7) conjugate(gamma(z));
      (%o7)                  gamma(conjugate(z))

  Maxima expands `gamma(z+n)` and `gamma(z-n)`, when `gamma_expand` is `true`:

      (%i8) gamma_expand:true$

      (%i9) [gamma(z+1),gamma(z-1),gamma(z+2)/gamma(z+1)];
                                     gamma(z)
      (%o9)             [z gamma(z), --------, z + 1]
                                      z - 1

  The deriviative of `gamma`:

      (%i10) diff(gamma(z),z);
      (%o10)                  psi (z) gamma(z)
                                 0

  See also `makegamma`.

  The Euler-Mascheroni constant is `%gamma`.

  Categories: Gamma and factorial functions · Special functions

Function: log_gamma (z)

  The natural logarithm of the gamma function.

  Categories: Gamma and factorial functions · Special functions

Function: gamma_incomplete_lower (a, z)

  The lower incomplete gamma function (A&S 6.5.2):

                                          z
                                         /
                                         [  a - 1   - t
          gamma_incomplete_lower(a, z) = I t      %e    dt
                                         ]
                                         /
                                          0

  See also `gamma_incomplete` (upper incomplete gamma function).

  Categories: Gamma and factorial functions · Special functions

Function: gamma_incomplete (a, z)

  The incomplete upper gamma function (A&S 6.5.3):

                                    inf
                                   /
                                   [     a - 1   - t
          gamma_incomplete(a, z) = I    t      %e    dt
                                   ]
                                   /
                                    z

  See also `gamma_expand` for controlling how `gamma_incomplete` is expressed in terms of elementary functions and `erfc`.

  Also see the related functions `gamma_incomplete_regularized` and `gamma_incomplete_generalized`.

  Categories: Gamma and factorial functions · Special functions

Function: gamma_incomplete_regularized (a, z)

  The regularized incomplete upper gamma function (A&S 6.5.1):

      gamma_incomplete_regularized(a, z) =
                                              gamma_incomplete(a, z)
                                              ----------------------
                                                     gamma(a)

  See also `gamma_expand` for controlling how `gamma_incomplete` is expressed in terms of elementary functions and `erfc`.

  Also see `gamma_incomplete`.

  Categories: Gamma and factorial functions · Special functions

Function: gamma_incomplete_generalized (a, z1, z1)

  The generalized incomplete gamma function.

      gamma_incomplete_generalized(a, z1, z2) =
                                                     z2
                                                    /
                                                    [    a - 1   - t
                                                    I   t      %e    dt
                                                    ]
                                                    /
                                                     z1

  Also see `gamma_incomplete` and `gamma_incomplete_regularized`.

  Categories: Gamma and factorial functions · Special functions

Option variable: gamma_expand

  Default value: `false`

  `gamma_expand` controls expansion of `gamma_incomplete`. When `gamma_expand` is `true`, `gamma_incomplete(v,z)` is expanded in terms of `z`, `exp(z)`, and `erfc(z)` when possible.

      (%i1) gamma_incomplete(2,z);
      (%o1)                       gamma_incomplete(2, z)
      (%i2) gamma_expand:true;
      (%o2)                                true
      (%i3) gamma_incomplete(2,z);
                                                 - z
      (%o3)                            (z + 1) %e
      (%i4) gamma_incomplete(3/2,z);
                                    - z   sqrt(%pi) erfc(sqrt(z))
      (%o4)               sqrt(z) %e    + -----------------------
                                                     2

  Categories: Gamma and factorial functions · Simplification flags and variables

Option variable: gammalim

  Default value: 10000

  `gammalim` controls simplification of the gamma function for integral and rational number arguments. If the absolute value of the argument is not greater than `gammalim`, then simplification will occur. Note that the `factlim` switch controls simplification of the result of `gamma` of an integer argument as well.

  Categories: Gamma and factorial functions · Simplification flags and variables

Function: makegamma (expr)

  Transforms instances of binomial, factorial, and beta functions in expr into gamma functions.

  See also `makefact`.

  Categories: Gamma and factorial functions

Function: beta (a, b)

  The beta function is defined as `gamma(a) gamma(b)/gamma(a+b)` (A&S 6.2.1).

  Maxima simplifies the beta function for positive integers and rational numbers, which sum to an integer. When `beta_args_sum_to_integer` is `true`, Maxima simplifies also general expressions which sum to an integer.

  For a or b equal to zero the beta function is not defined.

  In general the beta function is not defined for negative integers as an argument. The exception is for a=-n, n a positive integer and b a positive integer with b<=n, it is possible to define an analytic continuation. Maxima gives for this case a result.

  When `beta_expand` is `true`, expressions like `beta(a+n,b)` and `beta(a-n,b)` or `beta(a,b+n)` and `beta(a,b-n)` with `n` an integer are simplified.

  Maxima can evaluate the beta function for real and complex values in float and bigfloat precision. For numerical evaluation Maxima uses `log_gamma`:

                 - log_gamma(b + a) + log_gamma(b) + log_gamma(a)
               %e

  Maxima knows that the beta function is symmetric and has mirror symmetry.

  Maxima knows the derivatives of the beta function with respect to a or b.

  To express the beta function as a ratio of gamma functions see `makegamma`.

  Examples:

  Simplification, when one of the arguments is an integer:

      (%i1) [beta(2,3),beta(2,1/3),beta(2,a)];
                                     1   9      1
      (%o1)                         [--, -, ---------]
                                     12  4  a (a + 1)

  Simplification for two rational numbers as arguments which sum to an integer:

      (%i2) [beta(1/2,5/2),beta(1/3,2/3),beta(1/4,3/4)];
                                3 %pi   2 %pi
      (%o2)                    [-----, -------, sqrt(2) %pi]
                                  8    sqrt(3)

  When setting `beta_args_sum_to_integer` to `true` more general expression are simplified, when the sum of the arguments is an integer:

      (%i3) beta_args_sum_to_integer:true$
      (%i4) beta(a+1,-a+2);
                                      %pi (a - 1) a
      (%o4)                         ------------------
                                    2 sin(%pi (2 - a))

  The possible results, when one of the arguments is a negative integer:

      (%i5) [beta(-3,1),beta(-3,2),beta(-3,3)];
                                          1  1    1
      (%o5)                            [- -, -, - -]
                                          3  6    3

  `beta(a+n,b)` or `beta(a-n)` with `n` an integer simplifies when `beta_expand` is `true`:

      (%i6) beta_expand:true$
      (%i7) [beta(a+1,b),beta(a-1,b),beta(a+1,b)/beta(a,b+1)];
                          a beta(a, b)  beta(a, b) (b + a - 1)  a
      (%o7)              [------------, ----------------------, -]
                             b + a              a - 1           b

  Beta is not defined, when one of the arguments is zero:

      (%i7) beta(0,b);
      beta: expected nonzero arguments; found 0, b
       -- an error.  To debug this try debugmode(true);

  Numercial evaluation for real and complex arguments in float or bigfloat precision:

      (%i8) beta(2.5,2.3);
      (%o8) .08694748611299981

      (%i9) beta(2.5,1.4+%i);
      (%o9) 0.0640144950796695 - .1502078053286415 %i

      (%i10) beta(2.5b0,2.3b0);
      (%o10) 8.694748611299969b-2

      (%i11) beta(2.5b0,1.4b0+%i);
      (%o11) 6.401449507966944b-2 - 1.502078053286415b-1 %i

  Beta is symmetric and has mirror symmetry:

      (%i14) beta(a,b)-beta(b,a);
      (%o14)                                 0
      (%i15) declare(a,complex,b,complex)$
      (%i16) conjugate(beta(a,b));
      (%o16)                 beta(conjugate(a), conjugate(b))

  The derivative of the beta function wrt `a`:

      (%i17) diff(beta(a,b),a);
      (%o17)               - beta(a, b) (psi (b + a) - psi (a))
                                            0             0

  Categories: Gamma and factorial functions

Function: beta_incomplete (a, b, z)

  The basic definition of the incomplete beta function (A&S 6.6.1) is

              z
             /
             [         b - 1  a - 1
             I  (1 - t)      t      dt
             ]
             /
              0

  This definition is possible for realpart(a)>0 and realpart(b)>0 and abs(z)<1. For other values the incomplete beta function can be defined through a generalized hypergeometric function:

         gamma(a) hypergeometric_generalized([a, 1 - b], [a + 1], z) z

  (See functions.wolfram.com for a complete definition of the incomplete beta function.)

  For negative integers a = -n and positive integers b=m with m<=n the incomplete beta function is defined through

                                  m - 1           k
                                  ====  (1 - m)  z
                            n - 1 \            k
                           z       >    -----------
                                  /     k! (n - k)
                                  ====
                                  k = 0

  Maxima uses this definition to simplify `beta_incomplete` for a a negative integer.

  For a a positive integer, `beta_incomplete` simplifies for any argument b and z and for b a positive integer for any argument a and z, with the exception of a a negative integer.

  For z=0 and realpart(a)>0, `beta_incomplete` has the specific value zero. For z=1 and realpart(b)>0, `beta_incomplete` simplifies to the beta function `beta(a,b)`.

  Maxima evaluates `beta_incomplete` numerically for real and complex values in float or bigfloat precision. For the numerical evaluation an expansion of the incomplete beta function in continued fractions is used.

  When the option variable `beta_expand` is `true`, Maxima expands expressions like `beta_incomplete(a+n,b,z)` and `beta_incomplete(a-n,b,z)` where n is a positive integer.

  Maxima knows the derivatives of `beta_incomplete` with respect to the variables a, b and z and the integral with respect to the variable z.

  Examples:

  Simplification for a a positive integer:

      (%i1) beta_incomplete(2,b,z);
                                             b
                                  1 - (1 - z)  (b z + 1)
      (%o1)                       ----------------------
                                        b (b + 1)

  Simplification for b a positive integer:

      (%i2) beta_incomplete(a,2,z);
                                                     a
                                    (a (1 - z) + 1) z
      (%o2)                         ------------------
                                        a (a + 1)

  Simplification for a and b a positive integer:

      (%i3) beta_incomplete(3,2,z);
                                                     3
                                    (3 (1 - z) + 1) z
      (%o3)                         ------------------
                                            12

  a is a negative integer and b<=(-a), Maxima simplifies:

      (%i4) beta_incomplete(-3,1,z);
                                             1
      (%o4)                              - ----
                                              3
                                           3 z

  For the specific values z=0 and z=1, Maxima simplifies:

      (%i5) assume(a>0,b>0)$
      (%i6) beta_incomplete(a,b,0);
      (%o6)                                 0
      (%i7) beta_incomplete(a,b,1);
      (%o7)                            beta(a, b)

  Numerical evaluation in float or bigfloat precision:

      (%i8) beta_incomplete(0.25,0.50,0.9);
      (%o8)                          4.594959440269333
      (%i9)  fpprec:25$
      (%i10) beta_incomplete(0.25,0.50,0.9b0);
      (%o10)                    4.594959440269324086971203b0

  For abs(z)>1 `beta_incomplete` returns a complex result:

      (%i11) beta_incomplete(0.25,0.50,1.7);
      (%o11)              5.244115108584249 - 1.45518047787844 %i

  Results for more general complex arguments:

      (%i14) beta_incomplete(0.25+%i,1.0+%i,1.7+%i);
      (%o14)             2.726960675662536 - .3831175704269199 %i
      (%i15) beta_incomplete(1/2,5/4*%i,2.8+%i);
      (%o15)             13.04649635168716 %i - 5.802067956270001
      (%i16)

  Expansion, when `beta_expand` is `true`:

      (%i23) beta_incomplete(a+1,b,z),beta_expand:true;
                                                             b  a
                         a beta_incomplete(a, b, z)   (1 - z)  z
      (%o23)             -------------------------- - -----------
                                   b + a                 b + a

      (%i24) beta_incomplete(a-1,b,z),beta_expand:true;
                                                                 b  a - 1
                 beta_incomplete(a, b, z) (- b - a + 1)   (1 - z)  z
      (%o24)     -------------------------------------- - ---------------
                                 1 - a                         1 - a

  Derivative and integral for `beta_incomplete`:

      (%i34) diff(beta_incomplete(a, b, z), z);
                                    b - 1  a - 1
      (%o34)                 (1 - z)      z
      (%i35) integrate(beta_incomplete(a, b, z), z);
                    b  a
             (1 - z)  z
      (%o35) ----------- + beta_incomplete(a, b, z) z
                b + a
                                             a beta_incomplete(a, b, z)
                                           - --------------------------
                                                       b + a
      (%i36) factor(diff(%, z));
      (%o36)              beta_incomplete(a, b, z)

  Categories: Gamma and factorial functions

Function: beta_incomplete_regularized (a, b, z)

  The regularized incomplete beta function (A&S 6.6.2), defined as

      beta_incomplete_regularized(a, b, z) =
                                            beta_incomplete(a, b, z)
                                            ------------------------
                                                   beta(a, b)

  As for `beta_incomplete` this definition is not complete. See functions.wolfram.com for a complete definition of `beta_incomplete_regularized`.

  `beta_incomplete_regularized` simplifies a or b a positive integer.

  For z=0 and realpart(a)>0, `beta_incomplete_regularized` has the specific value 0. For z=1 and realpart(b)>0, `beta_incomplete_regularized` simplifies to 1.

  Maxima can evaluate `beta_incomplete_regularized` for real and complex arguments in float and bigfloat precision.

  When `beta_expand` is `true`, Maxima expands `beta_incomplete_regularized` for arguments a+n or a-n, where n is an integer.

  Maxima knows the derivatives of `beta_incomplete_regularized` with respect to the variables a, b, and z and the integral with respect to the variable z.

  Examples:

  Simplification for a or b a positive integer:

      (%i1) beta_incomplete_regularized(2,b,z);
                                             b
      (%o1)                       1 - (1 - z)  (b z + 1)

      (%i2) beta_incomplete_regularized(a,2,z);
                                                     a
      (%o2)                         (a (1 - z) + 1) z

      (%i3) beta_incomplete_regularized(3,2,z);
                                                     3
      (%o3)                         (3 (1 - z) + 1) z

  For the specific values z=0 and z=1, Maxima simplifies:

      (%i4) assume(a>0,b>0)$
      (%i5) beta_incomplete_regularized(a,b,0);
      (%o5)                                 0
      (%i6) beta_incomplete_regularized(a,b,1);
      (%o6)                                 1

  Numerical evaluation for real and complex arguments in float and bigfloat precision:

      (%i7) beta_incomplete_regularized(0.12,0.43,0.9);
      (%o7)                         .9114011367359802
      (%i8) fpprec:32$
      (%i9) beta_incomplete_regularized(0.12,0.43,0.9b0);
      (%o9)               9.1140113673598075519946998779975b-1
      (%i10) beta_incomplete_regularized(1+%i,3/3,1.5*%i);
      (%o10)             .2865367499935403 %i - 0.122995963334684
      (%i11) fpprec:20$
      (%i12) beta_incomplete_regularized(1+%i,3/3,1.5b0*%i);
      (%o12)      2.8653674999354036142b-1 %i - 1.2299596333468400163b-1

  Expansion, when `beta_expand` is `true`:

      (%i13) beta_incomplete_regularized(a+1,b,z);
                                                           b  a
                                                    (1 - z)  z
      (%o13) beta_incomplete_regularized(a, b, z) - ------------
                                                    a beta(a, b)
      (%i14) beta_incomplete_regularized(a-1,b,z);
      (%o14) beta_incomplete_regularized(a, b, z)
                                                           b  a - 1
                                                    (1 - z)  z
                                               - ----------------------
                                                 beta(a, b) (b + a - 1)

  The derivative and the integral wrt z:

      (%i15) diff(beta_incomplete_regularized(a,b,z),z);
                                    b - 1  a - 1
                             (1 - z)      z
      (%o15)                 -------------------
                                 beta(a, b)
      (%i16) integrate(beta_incomplete_regularized(a,b,z),z);
      (%o16) beta_incomplete_regularized(a, b, z) z
                                                                 b  a
                                                          (1 - z)  z
                a (beta_incomplete_regularized(a, b, z) - ------------)
                                                          a beta(a, b)
              - -------------------------------------------------------
                                         b + a

  Categories: Gamma and factorial functions

Function: beta_incomplete_generalized (a, b, z1, z2)

  The basic definition of the generalized incomplete beta function is

                   z2
                 /
                 [          b - 1  a - 1
                 I   (1 - t)      t      dt
                 ]
                 /
                  z1

  Maxima simplifies `beta_incomplete_regularized` for a and b a positive integer.

  For realpart(a)>0 and z1=0 or z2=0, Maxima simplifies `beta_incomplete_generalized` to `beta_incomplete`. For realpart(b)>0 and z1=1 or z2=1, Maxima simplifies to an expression with `beta` and `beta_incomplete`.

  Maxima evaluates `beta_incomplete_regularized` for real and complex values in float and bigfloat precision.

  When `beta_expand` is `true`, Maxima expands `beta_incomplete_generalized` for a+n and a-n, n a positive integer.

  Maxima knows the derivative of `beta_incomplete_generalized` with respect to the variables a, b, z1, and z2 and the integrals with respect to the variables z1 and z2.

  Examples:

  Maxima simplifies `beta_incomplete_generalized` for a and b a positive integer:

      (%i1) beta_incomplete_generalized(2,b,z1,z2);
                         b                      b
                 (1 - z1)  (b z1 + 1) - (1 - z2)  (b z2 + 1)
      (%o1)      -------------------------------------------
                                  b (b + 1)
      (%i2) beta_incomplete_generalized(a,2,z1,z2);
                                    a                      a
                 (a (1 - z2) + 1) z2  - (a (1 - z1) + 1) z1
      (%o2)      -------------------------------------------
                                  a (a + 1)
      (%i3) beta_incomplete_generalized(3,2,z1,z2);
                    2      2                       2      2
            (1 - z1)  (3 z1  + 2 z1 + 1) - (1 - z2)  (3 z2  + 2 z2 + 1)
      (%o3) -----------------------------------------------------------
                                        12

  Simplification for specific values z1=0, z2=0, z1=1, or z2=1:

      (%i4) assume(a > 0, b > 0)$
      (%i5) beta_incomplete_generalized(a,b,z1,0);
      (%o5)                    - beta_incomplete(a, b, z1)

      (%i6) beta_incomplete_generalized(a,b,0,z2);
      (%o6)                    - beta_incomplete(a, b, z2)

      (%i7) beta_incomplete_generalized(a,b,z1,1);
      (%o7)              beta(a, b) - beta_incomplete(a, b, z1)

      (%i8) beta_incomplete_generalized(a,b,1,z2);
      (%o8)              beta_incomplete(a, b, z2) - beta(a, b)

  Numerical evaluation for real arguments in float or bigfloat precision:

      (%i9) beta_incomplete_generalized(1/2,3/2,0.25,0.31);
      (%o9)                        .09638178086368676

      (%i10) fpprec:32$
      (%i10) beta_incomplete_generalized(1/2,3/2,0.25,0.31b0);
      (%o10)               9.6381780863686935309170054689964b-2

  Numerical evaluation for complex arguments in float or bigfloat precision:

      (%i11) beta_incomplete_generalized(1/2+%i,3/2+%i,0.25,0.31);
      (%o11)           - .09625463003205376 %i - .003323847735353769
      (%i12) fpprec:20$
      (%i13) beta_incomplete_generalized(1/2+%i,3/2+%i,0.25,0.31b0);
      (%o13)     - 9.6254630032054178691b-2 %i - 3.3238477353543591914b-3

  Expansion for a+n or a-n, n a positive integer, when `beta_expand` is `true`:

      (%i14) beta_expand:true$

      (%i15) beta_incomplete_generalized(a+1,b,z1,z2);

                     b   a           b   a
             (1 - z1)  z1  - (1 - z2)  z2
      (%o15) -----------------------------
                         b + a
                            a beta_incomplete_generalized(a, b, z1, z2)
                          + -------------------------------------------
                                               b + a
      (%i16) beta_incomplete_generalized(a-1,b,z1,z2);

             beta_incomplete_generalized(a, b, z1, z2) (- b - a + 1)
      (%o16) -------------------------------------------------------
                                      1 - a
                                          b   a - 1           b   a - 1
                                  (1 - z2)  z2      - (1 - z1)  z1
                                - -------------------------------------
                                                  1 - a

  Derivative wrt the variable z1 and integrals wrt z1 and z2:

      (%i17) diff(beta_incomplete_generalized(a,b,z1,z2),z1);
                                     b - 1   a - 1
      (%o17)               - (1 - z1)      z1
      (%i18) integrate(beta_incomplete_generalized(a,b,z1,z2),z1);
      (%o18) beta_incomplete_generalized(a, b, z1, z2) z1
                                        + beta_incomplete(a + 1, b, z1)
      (%i19) integrate(beta_incomplete_generalized(a,b,z1,z2),z2);
      (%o19) beta_incomplete_generalized(a, b, z1, z2) z2
                                        - beta_incomplete(a + 1, b, z2)

  Categories: Gamma and factorial functions

Option variable: beta_expand

  Default value: false

  When `beta_expand` is `true`, `beta(a,b)` and related functions are expanded for arguments like a+n or a-n, where n is an integer.

  Categories: Gamma and factorial functions · Simplification flags and variables

Option variable: beta_args_sum_to_integer

  Default value: false

  When `beta_args_sum_to_integer` is `true`, Maxima simplifies `beta(a,b)`, when the arguments a and b sum to an integer.

  Categories: Gamma and factorial functions · Simplification flags and variables

Function: psi [n](x)

  The derivative of `log (gamma (x))` of order `n+1`. Thus, `psi[0](x)` is the first derivative, `psi[1](x)` is the second derivative, etc.

  Maxima does not know how, in general, to compute a numerical value of `psi`, but it can compute some exact values for rational args. Several variables control what range of rational args `psi` will return an exact value, if possible. See `maxpsiposint`, `maxpsinegint`, `maxpsifracnum`, and `maxpsifracdenom`. That is, x must lie between `maxpsinegint` and `maxpsiposint`. If the absolute value of the fractional part of x is rational and has a numerator less than `maxpsifracnum` and has a denominator less than `maxpsifracdenom`, `psi` will return an exact value.

  The function `bfpsi` in the `bffac` package can compute numerical values.

  Categories: Gamma and factorial functions

Option variable: maxpsiposint

  Default value: 20

  `maxpsiposint` is the largest positive value for which `psi[n](x)` will try to compute an exact value.

  Categories: Gamma and factorial functions

Option variable: maxpsinegint

  Default value: -10

  `maxpsinegint` is the most negative value for which `psi[n](x)` will try to compute an exact value. That is if x is less than `maxnegint`, `psi[n](x)` will not return simplified answer, even if it could.

  Categories: Gamma and factorial functions

Option variable: maxpsifracnum

  Default value: 6

  Let x be a rational number less than one of the form `p/q`. If `p` is greater than `maxpsifracnum`, then `psi[n](x)` will not try to return a simplified value.

  Categories: Gamma and factorial functions

Option variable: maxpsifracdenom

  Default value: 6

  Let x be a rational number less than one of the form `p/q`. If `q` is greater than `maxpsifracdenom`, then `psi[n](x)` will not try to return a simplified value.

  Categories: Gamma and factorial functions

Function: makefact (expr)

  Transforms instances of binomial, gamma, and beta functions in expr into factorials.

  See also `makegamma`.

  Categories: Gamma and factorial functions

Function: numfactor (expr)

  Returns the numerical factor multiplying the expression expr, which should be a single term.

  `content` returns the greatest common divisor (gcd) of all terms in a sum.

      (%i1) gamma (7/2);
                                15 sqrt(%pi)
      (%o1)                     ------------
                                     8
      (%i2) numfactor (%);
                                     15
      (%o2)                          --
                                     8

  Categories: Expressions

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## 15.5 Exponential Integrals

The Exponential Integral and related funtions are defined in Abramowitz and Stegun, Handbook of Mathematical Functions, Chapter 5

Function: expintegral_e1 (z)

  The Exponential Integral E1(z) (A&S 5.1.1) defined as

  Categories: Exponential Integrals · Special functions

Function: expintegral_ei (z)

  The Exponential Integral Ei(z) (A&S 5.1.2)

  Categories: Exponential Integrals · Special functions

Function: expintegral_li (z)

  The Exponential Integral Li(z) (A&S 5.1.3)

  Categories: Exponential Integrals · Special functions

Function: expintegral_e (n,z)

  The Exponential Integral En(z) (A&S 5.1.4) defined as

  Categories: Exponential Integrals · Special functions

Function: expintegral_si (z)

  The Exponential Integral Si(z) (A&S 5.2.1) defined as

  Categories: Exponential Integrals · Special functions

Function: expintegral_ci (z)

  The Exponential Integral Ci(z) (A&S 5.2.2) defined as

  Categories: Exponential Integrals · Special functions

Function: expintegral_shi (z)

  The Exponential Integral Shi(z) (A&S 5.2.3) defined as

  Categories: Exponential Integrals · Special functions

Function: expintegral_chi (z)

  The Exponential Integral Chi(z) (A&S 5.2.4) defined as

  Categories: Exponential Integrals · Special functions

Option variable: expintrep

  Default value: false

  Change the representation of one of the exponential integrals, expintegral_e(m, z), expintegral_e1, or expintegral_ei to an equivalent form if possible.

  Possible values for expintrep are false, gamma_incomplete, expintegral_e1, expintegral_ei, expintegral_li, expintegral_trig, or expintegral_hyp.

  false means that the representation is not changed. Other values indicate the representation is to be changed to use the function specified where expintegral_trig means expintegral_si, expintegral_ci, and expintegral_hyp means expintegral_shi or expintegral_chi.

  Categories: Exponential Integrals

Option variable: expintexpand

  Default value: false

  Expand the Exponential Integral E[n](z) for half integral values in terms of Erfc or Erf and for positive integers in terms of Ei

  Exponential Integrals

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## 15.6 Error Function

The Error function and related funtions are defined in Abramowitz and Stegun, Handbook of Mathematical Functions, Chapter 7

Function: erf (z)

  The Error Function erf(z) (A&S 7.1.1)

  See also flag `erfflag`.

  Special functions

Function: erfc (z)

  The Complementary Error Function erfc(z) (A&S 7.1.2)

  `erfc(z) = 1-erf(z)`

  Special functions

Function: erfi (z)

  The Imaginary Error Function.

  `erfi(z) = -%i*erf(%i*z)`

  Special functions

Function: erf_generalized (z1,z2)

  Generalized Error function Erf(z1,z2)

  Special functions

Function: fresnel_c (z)

  The Fresnel Integral C(z) = integrate(cos((%pi/2)*t^2),t,0,z). (A&S 7.3.1)

  The simplification fresnel_c(-x) = -fresnel_c(x) is applied when flag `trigsign` is true.

  The simplification fresnel_c(%i*x) = %i*fresnel_c(x) is applied when flag `%iargs` is true.

  See flags `erf_representation` and `hypergeometric_representation`.

  Special functions

Function: fresnel_s (z)

  The Fresnel Integral S(z) = integrate(sin((%pi/2)*t^2),t,0,z). (A&S 7.3.2)

  The simplification fresnel_s(-x) = -fresnel_s(x) is applied when flag `trigsign` is true.

  The simplification fresnel_s(%i*x) = -%i*fresnel_s(x) is applied when flag `%iargs` is true.

  See flags `erf_representation` and `hypergeometric_representation`.

  Special functions

Option variable: erf_representation

  Default value: false

  When T erfc, erfi, erf_generalized, fresnel_s and fresnel_c are transformed to erf.

Option variable: hypergeometric_representation

  Default value: false

  Enables transformation to a Hypergeometric representation for fresnel_s and fresnel_c

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## 15.7 Struve Functions

The Struve functions are defined in Abramowitz and Stegun, Handbook of Mathematical Functions, Chapter 12.

Function: struve_h (v, z)

  The Struve Function H of order v and argument z. (A&S 12.1.1)

  Categories: Special functions

Function: struve_l (v, z)

  The Modified Struve Function L of order v and argument z. (A&S 12.2.1)

  Categories: Special functions

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## 15.8 Hypergeometric Functions

The Hypergeometric Functions are defined in Abramowitz and Stegun, Handbook of Mathematical Functions, Chapters 13 and 15.

Maxima has very limited knowledge of these functions. They can be returned from function `hgfred`.

Function: %m [k,u] (z)

  Whittaker M function `M[k,u](z) = exp(-z/2)*z^(1/2+u)*M(1/2+u-k,1+2*u,z)`. (A&S 13.1.32)

  Special functions

Function: %w [k,u] (z)

  Whittaker W function. (A&S 13.1.33)

  Special functions

Function: %f [p,q] ([a],[b],z)

  The pFq(a1,a2,..ap;b1,b2,..bq;z) hypergeometric function, where `a` a list of length `p` and `b` a list of length `q`.

  Bessel functions · Special functions

Function: hypergeometric ([a1, ..., ap],[b1, ... ,bq], x)

  The hypergeometric function. Unlike Maxima's `%f` hypergeometric function, the function `hypergeometric` is a simplifying function; also, `hypergeometric` supports complex double and big floating point evaluation. For the Gauss hypergeometric function, that is p = 2 and q = 1, floating point evaluation outside the unit circle is supported, but in general, it is not supported.

  When the option variable `expand_hypergeometric` is true (default is false) and one of the arguments `a1` through `ap` is a negative integer (a polynomial case), `hypergeometric` returns an expanded polynomial.

  Examples:

      (%i1)  hypergeometric([],[],x);
      (%o1) %e^x

  Polynomial cases automatically expand when `expand_hypergeometric` is true:

      (%i2) hypergeometric([-3],[7],x);
      (%o2) hypergeometric([-3],[7],x)

      (%i3) hypergeometric([-3],[7],x), expand_hypergeometric : true;
      (%o3) -x^3/504+3*x^2/56-3*x/7+1

  Both double float and big float evaluation is supported:

      (%i4) hypergeometric([5.1],[7.1 + %i],0.42);
      (%o4)       1.346250786375334 - 0.0559061414208204 %i
      (%i5) hypergeometric([5,6],[8], 5.7 - %i);
      (%o5)     .007375824009774946 - .001049813688578674 %i
      (%i6) hypergeometric([5,6],[8], 5.7b0 - %i), fpprec : 30;
      (%o6) 7.37582400977494674506442010824b-3
                                - 1.04981368857867315858055393376b-3 %i

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## 15.9 Parabolic Cylinder Functions

The Parabolic Cylinder Functions are defined in Abramowitz and Stegun, Handbook of Mathematical Functions, Chapter 19.

Maxima has very limited knowledge of these functions. They can be returned from function `hgfred`.

Function: parabolic_cylinder_d (v, z)

  The parabolic cylinder function `parabolic_cylinder_d(v,z)`. (A&S 19.3.1)

  Special functions

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## 15.10 Functions and Variables for Special Functions

Function: specint (exp(- s*t) * expr, t)

  Compute the Laplace transform of expr with respect to the variable t. The integrand expr may contain special functions.

  The following special functions are handled by `specint`: incomplete gamma function, error functions (but not the error function `erfi`, it is easy to transform `erfi` e.g. to the error function `erf`), exponential integrals, bessel functions (including products of bessel functions), hankel functions, hermite and the laguerre polynomials.

  Furthermore, `specint` can handle the hypergeometric function `%f[p,q]([],[],z)`, the whittaker function of the first kind `%m[u,k](z)` and of the second kind `%w[u,k](z)`.

  The result may be in terms of special functions and can include unsimplified hypergeometric functions.

  When `laplace` fails to find a Laplace transform, `specint` is called. Because `laplace` knows more general rules for Laplace transforms, it is preferable to use `laplace` and not `specint`.

  `demo(hypgeo)` displays several examples of Laplace transforms computed by `specint`.

  Examples:

      (%i1) assume (p > 0, a > 0)$
      (%i2) specint (t^(1/2) * exp(-a*t/4) * exp(-p*t), t);
                                 sqrt(%pi)
      (%o2)                     ------------
                                       a 3/2
                                2 (p + -)
                                       4
      (%i3) specint (t^(1/2) * bessel_j(1, 2 * a^(1/2) * t^(1/2))
                    * exp(-p*t), t);
                                         - a/p
                               sqrt(a) %e
      (%o3)                    ---------------
                                      2
                                     p

  Examples for exponential integrals:

      (%i4) assume(s>0,a>0,s-a>0)$
      (%i5) ratsimp(specint(%e^(a*t)
                            *(log(a)+expintegral_e1(a*t))*%e^(-s*t),t));
                                   log(s)
      (%o5)                        ------
                                   s - a
      (%i6) logarc:true$

      (%i7) gamma_expand:true$

      radcan(specint((cos(t)*expintegral_si(t)
                           -sin(t)*expintegral_ci(t))*%e^(-s*t),t));
                                   log(s)
      (%o8)                        ------
                                    2
                                   s  + 1
      ratsimp(specint((2*t*log(a)+2/a*sin(a*t)
                            -2*t*expintegral_ci(a*t))*%e^(-s*t),t));
                                     2    2
                                log(s  + a )
      (%o9)                     ------------
                                      2
                                     s

  Results when using the expansion of `gamma_incomplete` and when changing the representation to `expintegral_e1`:

      (%i10) assume(s>0)$
      (%i11) specint(1/sqrt(%pi*t)*unit_step(t-k)*%e^(-s*t),t);
                                                  1
                                  gamma_incomplete(-, k s)
                                                  2
      (%o11)                      ------------------------
                                     sqrt(%pi) sqrt(s)

      (%i12) gamma_expand:true$
      (%i13) specint(1/sqrt(%pi*t)*unit_step(t-k)*%e^(-s*t),t);
                                    erfc(sqrt(k) sqrt(s))
      (%o13)                        ---------------------
                                           sqrt(s)

      (%i14) expintrep:expintegral_e1$
      (%i15) ratsimp(specint(1/(t+a)^2*%e^(-s*t),t));
                                    a s
                              a s %e    expintegral_e1(a s) - 1
      (%o15)                - ---------------------------------
                                              a

  Categories: Laplace transform

Function: hypergeometric_simp (e)

  `hypergeometric_simp` simplifies hypergeometric functions by applying `hgfred` to the arguments of any hypergeometric functions in the expression e.

  Only instances of `hypergeometric` are affected; any `%f`, `%w`, and `%m` in the expression e are not affected. Any unsimplified hypergeometric functions are returned unchanged (instead of changing to `%f` as `hgfred` would).

  `load(hypergeometric);` loads this function.

  See also `hgfred`.

  Examples:

      (%i1) load ("hypergeometric") $
      (%i2) foo : [hypergeometric([1,1], [2], z), hypergeometric([1/2], [1], z)];
      (%o2) [hypergeometric([1, 1], [2], z),
                                                           1
                                           hypergeometric([-], [1], z)]
                                                           2
      (%i3) hypergeometric_simp (foo);
                       log(1 - z)              z    z/2
      (%o3)         [- ----------, bessel_i(0, -) %e   ]
                           z                   2
      (%i4) bar : hypergeometric([n], [m], z + 1);
      (%o4)            hypergeometric([n], [m], z + 1)
      (%i5) hypergeometric_simp (bar);
      (%o5)            hypergeometric([n], [m], z + 1)

  Categories: Hypergeometric functions · Simplification functions · Special functions

Function: hgfred (a, b, t)

  Simplify the generalized hypergeometric function in terms of other, simpler, forms. a is a list of numerator parameters and b is a list of the denominator parameters.

  If `hgfred` cannot simplify the hypergeometric function, it returns an expression of the form `%f[p,q]([a], [b], x)` where p is the number of elements in a, and q is the number of elements in b. This is the usual `pFq` generalized hypergeometric function.

      (%i1) assume(not(equal(z,0)));
      (%o1)                          [notequal(z, 0)]
      (%i2) hgfred([v+1/2],[2*v+1],2*%i*z);

                           v/2                               %i z
                          4    bessel_j(v, z) gamma(v + 1) %e
      (%o2)               ---------------------------------------
                                             v
                                            z
      (%i3) hgfred([1,1],[2],z);

                                         log(1 - z)
      (%o3)                            - ----------
                                             z
      (%i4) hgfred([a,a+1/2],[3/2],z^2);

                                     1 - 2 a          1 - 2 a
                              (z + 1)        - (1 - z)
      (%o4)                   -------------------------------
                                       2 (1 - 2 a) z

  It can be beneficial to load orthopoly too as the following example shows. Note that L is the generalized Laguerre polynomial.

      (%i5) load(orthopoly)$
      (%i6) hgfred([-2],[a],z);

                                          (a - 1)
                                       2 L       (z)
                                          2
      (%o6)                            -------------
                                         a (a + 1)
      (%i7) ev(%);

                                        2
                                       z        2 z
      (%o7)                         --------- - --- + 1
                                    a (a + 1)    a

Function: lambert_w (z)

  The principal branch of Lambert's W function W(z), the solution of `z = W(z) * exp(W(z))`. (DLMF 4.13)

  Special functions

Function: generalized_lambert_w (k, z)

  The k-th branch of Lambert's W function W(z), the solution of `z = W(z) * exp(W(z))`. (DLMF 4.13)

  The principal branch, denoted Wp(z) in DLMF, is `lambert_w(z) = generalized_lambert_w(0,z)`.

  The other branch with real values, denoted Wm(z) in DLMF, is `generalized_lambert_w(-1,z)`.

  Special functions

Function: nzeta (z)

  The Plasma Dispersion Function `nzeta(z) = %i*sqrt(%pi)*exp(-z^2)*(1-erf(-%i*z))`

  Special functions

Function: nzetar (z)

  Returns `realpart(nzeta(z))`.

  Special functions

Function: nzetai (z)

  Returns `imagpart(nzeta(z))`.

  Special functions

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# 16. Elliptic Functions

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## 16.1 Introduction to Elliptic Functions and Integrals

Maxima includes support for Jacobian elliptic functions and for complete and incomplete elliptic integrals. This includes symbolic manipulation of these functions and numerical evaluation as well. Definitions of these functions and many of their properties can by found in Abramowitz and Stegun, Chapter 16-17. As much as possible, we use the definitions and relationships given there.

In particular, all elliptic functions and integrals use the parameter m instead of the modulus k or the modular angle \alpha. This is one area where we differ from Abramowitz and Stegun who use the modular angle for the elliptic functions. The following relationships are true:

The elliptic functions and integrals are primarily intended to support symbolic computation. Therefore, most of derivatives of the functions and integrals are known. However, if floating-point values are given, a floating-point result is returned.

Support for most of the other properties of elliptic functions and integrals other than derivatives has not yet been written.

Some examples of elliptic functions:

    (%i1) jacobi_sn (u, m);
    (%o1)                    jacobi_sn(u, m)
    (%i2) jacobi_sn (u, 1);
    (%o2)                        tanh(u)
    (%i3) jacobi_sn (u, 0);
    (%o3)                        sin(u)
    (%i4) diff (jacobi_sn (u, m), u);
    (%o4)            jacobi_cn(u, m) jacobi_dn(u, m)
    (%i5) diff (jacobi_sn (u, m), m);
    (%o5) jacobi_cn(u, m) jacobi_dn(u, m)

          elliptic_e(asin(jacobi_sn(u, m)), m)
     (u - ------------------------------------)/(2 m)
                         1 - m

                2
       jacobi_cn (u, m) jacobi_sn(u, m)
     + --------------------------------
                  2 (1 - m)

Some examples of elliptic integrals:

    (%i1) elliptic_f (phi, m);
    (%o1)                  elliptic_f(phi, m)
    (%i2) elliptic_f (phi, 0);
    (%o2)                          phi
    (%i3) elliptic_f (phi, 1);
                                   phi   %pi
    (%o3)                  log(tan(--- + ---))
                                    2     4
    (%i4) elliptic_e (phi, 1);
    (%o4)                       sin(phi)
    (%i5) elliptic_e (phi, 0);
    (%o5)                          phi
    (%i6) elliptic_kc (1/2);
                                         1
    (%o6)                    elliptic_kc(-)
                                         2
    (%i7) makegamma (%);
                                     2 1
                                gamma (-)
                                       4
    (%o7)                      -----------
                               4 sqrt(%pi)
    (%i8) diff (elliptic_f (phi, m), phi);
                                    1
    (%o8)                 ---------------------
                                        2
                          sqrt(1 - m sin (phi))
    (%i9) diff (elliptic_f (phi, m), m);
           elliptic_e(phi, m) - (1 - m) elliptic_f(phi, m)
    (%o9) (-----------------------------------------------
                                  m

                                     cos(phi) sin(phi)
                                 - ---------------------)/(2 (1 - m))
                                                 2
                                   sqrt(1 - m sin (phi))

Support for elliptic functions and integrals was written by Raymond Toy. It is placed under the terms of the General Public License (GPL) that governs the distribution of Maxima.

Categories: Elliptic functions

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## 16.2 Functions and Variables for Elliptic Functions

Function: jacobi_sn (u, m)

  The Jacobian elliptic function sn(u,m).

  Categories: Elliptic functions

Function: jacobi_cn (u, m)

  The Jacobian elliptic function cn(u,m).

  Categories: Elliptic functions

Function: jacobi_dn (u, m)

  The Jacobian elliptic function dn(u,m).

  Categories: Elliptic functions

Function: jacobi_ns (u, m)

  The Jacobian elliptic function ns(u,m) = 1/sn(u,m).

  Categories: Elliptic functions

Function: jacobi_sc (u, m)

  The Jacobian elliptic function sc(u,m) = sn(u,m)/cn(u,m).

  Categories: Elliptic functions

Function: jacobi_sd (u, m)

  The Jacobian elliptic function sd(u,m) = sn(u,m)/dn(u,m).

  Categories: Elliptic functions

Function: jacobi_nc (u, m)

  The Jacobian elliptic function nc(u,m) = 1/cn(u,m).

  Categories: Elliptic functions

Function: jacobi_cs (u, m)

  The Jacobian elliptic function cs(u,m) = cn(u,m)/sn(u,m).

  Categories: Elliptic functions

Function: jacobi_cd (u, m)

  The Jacobian elliptic function cd(u,m) = cn(u,m)/dn(u,m).

  Categories: Elliptic functions

Function: jacobi_nd (u, m)

  The Jacobian elliptic function nd(u,m) = 1/dn(u,m).

  Categories: Elliptic functions

Function: jacobi_ds (u, m)

  The Jacobian elliptic function ds(u,m) = dn(u,m)/sn(u,m).

  Categories: Elliptic functions

Function: jacobi_dc (u, m)

  The Jacobian elliptic function dc(u,m) = dn(u,m)/cn(u,m).

  Categories: Elliptic functions

Function: inverse_jacobi_sn (u, m)

  The inverse of the Jacobian elliptic function sn(u,m).

  Categories: Elliptic functions

Function: inverse_jacobi_cn (u, m)

  The inverse of the Jacobian elliptic function cn(u,m).

  Categories: Elliptic functions

Function: inverse_jacobi_dn (u, m)

  The inverse of the Jacobian elliptic function dn(u,m).

  Categories: Elliptic functions

Function: inverse_jacobi_ns (u, m)

  The inverse of the Jacobian elliptic function ns(u,m).

  Categories: Elliptic functions

Function: inverse_jacobi_sc (u, m)

  The inverse of the Jacobian elliptic function sc(u,m).

  Categories: Elliptic functions

Function: inverse_jacobi_sd (u, m)

  The inverse of the Jacobian elliptic function sd(u,m).

  Categories: Elliptic functions

Function: inverse_jacobi_nc (u, m)

  The inverse of the Jacobian elliptic function nc(u,m).

  Categories: Elliptic functions

Function: inverse_jacobi_cs (u, m)

  The inverse of the Jacobian elliptic function cs(u,m).

  Categories: Elliptic functions

Function: inverse_jacobi_cd (u, m)

  The inverse of the Jacobian elliptic function cd(u,m).

  Categories: Elliptic functions

Function: inverse_jacobi_nd (u, m)

  The inverse of the Jacobian elliptic function nd(u,m).

  Categories: Elliptic functions

Function: inverse_jacobi_ds (u, m)

  The inverse of the Jacobian elliptic function ds(u,m).

  Categories: Elliptic functions

Function: inverse_jacobi_dc (u, m)

  The inverse of the Jacobian elliptic function dc(u,m).

  Categories: Elliptic functions

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## 16.3 Functions and Variables for Elliptic Integrals

Function: elliptic_f (phi, m)

  The incomplete elliptic integral of the first kind, defined as

  integrate(1/sqrt(1 - m*sin(x)^2), x, 0, phi)

  See also elliptic_e and elliptic_kc.

  Categories: Elliptic integrals

Function: elliptic_e (phi, m)

  The incomplete elliptic integral of the second kind, defined as

  elliptic_e(phi, m) = integrate(sqrt(1 - m*sin(x)^2), x, 0, phi)

  See also elliptic_f and elliptic_ec.

  Categories: Elliptic integrals

Function: elliptic_eu (u, m)

  The incomplete elliptic integral of the second kind, defined as

  integrate(dn(v,m)^2,v,0,u) = integrate(sqrt(1-m*t^2)/sqrt(1-t^2), t, 0, tau)

  where tau = sn(u,m).

  This is related to elliptic_e by

  elliptic_eu(u, m) = elliptic_e(asin(sn(u,m)),m)

  See also elliptic_e.

  Elliptic integrals

Function: elliptic_pi (n, phi, m)

  The incomplete elliptic integral of the third kind, defined as

  integrate(1/(1-n*sin(x)^2)/sqrt(1 - m*sin(x)^2), x, 0, phi)

  Categories: Elliptic integrals

Function: elliptic_kc (m)

  The complete elliptic integral of the first kind, defined as

  integrate(1/sqrt(1 - m*sin(x)^2), x, 0, %pi/2)

  For certain values of m, the value of the integral is known in terms of Gamma functions. Use `makegamma` to evaluate them.

  Categories: Elliptic integrals

Function: elliptic_ec (m)

  The complete elliptic integral of the second kind, defined as

  integrate(sqrt(1 - m*sin(x)^2), x, 0, %pi/2)

  For certain values of m, the value of the integral is known in terms of Gamma functions. Use `makegamma` to evaluate them.

  Categories: Elliptic integrals

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# 17. Limits

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## 17.1 Functions and Variables for Limits

Option variable: lhospitallim

  Default value: 4

  `lhospitallim` is the maximum number of times L'Hospital's rule is used in `limit`. This prevents infinite looping in cases like `limit (cot(x)/csc(x), x, 0)`.

  Categories: Limits

Function: limit
`limit` (expr, x, val, dir)
`limit` (expr, x, val)
`limit` (expr)

  Computes the limit of expr as the real variable x approaches the value val from the direction dir. dir may have the value `plus` for a limit from above, `minus` for a limit from below, or may be omitted (implying a two-sided limit is to be computed).

  `limit` uses the following special symbols: `inf` (positive infinity) and `minf` (negative infinity). On output it may also use `und` (undefined), `ind` (indefinite but bounded) and `infinity` (complex infinity).

  `infinity` (complex infinity) is returned when the limit of the absolute value of the expression is positive infinity, but the limit of the expression itself is not positive infinity or negative infinity. This includes cases where the limit of the complex argument is a constant, as in `limit(log(x), x, minf)`, cases where the complex argument oscillates, as in `limit((-2)^x, x, inf)`, and cases where the complex argument is different for either side of a two-sided limit, as in `limit(1/x, x, 0)` and `limit(log(x), x, 0)`.

  `lhospitallim` is the maximum number of times L'Hospital's rule is used in `limit`. This prevents infinite looping in cases like `limit (cot(x)/csc(x), x, 0)`.

  `tlimswitch` when true will allow the `limit` command to use Taylor series expansion when necessary.

  `limsubst` prevents `limit` from attempting substitutions on unknown forms. This is to avoid bugs like `limit (f(n)/f(n+1), n, inf)` giving 1. Setting `limsubst` to `true` will allow such substitutions.

  `limit` with one argument is often called upon to simplify constant expressions, for example, `limit (inf-1)`.

  `example (limit)` displays some examples.

  For the method see Wang, P., "Evaluation of Definite Integrals by Symbolic Manipulation", Ph.D. thesis, MAC TR-92, October 1971.

  Categories: Limits

Option variable: limsubst

  Default value: `false`

  prevents `limit` from attempting substitutions on unknown forms. This is to avoid bugs like `limit (f(n)/f(n+1), n, inf)` giving 1. Setting `limsubst` to `true` will allow such substitutions.

  Categories: Limits

Function: tlimit
`tlimit` (expr, x, val, dir)
`tlimit` (expr, x, val)
`tlimit` (expr)

  Take the limit of the Taylor series expansion of `expr` in `x` at `val` from direction `dir`.

  Categories: Limits

Option variable: tlimswitch

  Default value: `true`

  When `tlimswitch` is `true`, the `limit` command will use a Taylor series expansion if the limit of the input expression cannot be computed directly. This allows evaluation of limits such as `limit(x/(x-1)-1/log(x),x,1,plus)`. When `tlimswitch` is `false` and the limit of input expression cannot be computed directly, `limit` will return an unevaluated limit expression.

  Categories: Limits

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# 18. Differentiation

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## 18.1 Functions and Variables for Differentiation

Function: antid (expr, x, u(x))

  Returns a two-element list, such that an antiderivative of expr with respect to x can be constructed from the list. The expression expr may contain an unknown function u and its derivatives.

  Let L, a list of two elements, be the return value of `antid`. Then `L[1] + 'integrate (L[2], x)` is an antiderivative of expr with respect to x.

  When `antid` succeeds entirely, the second element of the return value is zero. Otherwise, the second element is nonzero, and the first element is nonzero or zero. If `antid` cannot make any progress, the first element is zero and the second nonzero.

  `load ("antid")` loads this function. The `antid` package also defines the functions `nonzeroandfreeof` and `linear`.

  `antid` is related to `antidiff` as follows. Let L, a list of two elements, be the return value of `antid`. Then the return value of `antidiff` is equal to `L[1] + 'integrate (L[2], x)` where x is the variable of integration.

  Examples:

      (%i1) load ("antid")$
      (%i2) expr: exp (z(x)) * diff (z(x), x) * y(x);
                                  z(x)  d
      (%o2)                y(x) %e     (-- (z(x)))
                                        dx
      (%i3) a1: antid (expr, x, z(x));
                             z(x)      z(x)  d
      (%o3)          [y(x) %e    , - %e     (-- (y(x)))]
                                             dx
      (%i4) a2: antidiff (expr, x, z(x));
                                  /
                           z(x)   [   z(x)  d
      (%o4)         y(x) %e     - I %e     (-- (y(x))) dx
                                  ]         dx
                                  /
      (%i5) a2 - (first (a1) + 'integrate (second (a1), x));
      (%o5)                           0
      (%i6) antid (expr, x, y(x));
                                   z(x)  d
      (%o6)             [0, y(x) %e     (-- (z(x)))]
                                         dx
      (%i7) antidiff (expr, x, y(x));
                        /
                        [        z(x)  d
      (%o7)             I y(x) %e     (-- (z(x))) dx
                        ]              dx
                        /

  Categories: Integral calculus

Function: antidiff (expr, x, u(x))

  Returns an antiderivative of expr with respect to x. The expression expr may contain an unknown function u and its derivatives.

  When `antidiff` succeeds entirely, the resulting expression is free of integral signs (that is, free of the `integrate` noun). Otherwise, `antidiff` returns an expression which is partly or entirely within an integral sign. If `antidiff` cannot make any progress, the return value is entirely within an integral sign.

  `load ("antid")` loads this function. The `antid` package also defines the functions `nonzeroandfreeof` and `linear`.

  `antidiff` is related to `antid` as follows. Let L, a list of two elements, be the return value of `antid`. Then the return value of `antidiff` is equal to `L[1] + 'integrate (L[2], x)` where x is the variable of integration.

  Examples:

      (%i1) load ("antid")$
      (%i2) expr: exp (z(x)) * diff (z(x), x) * y(x);
                                  z(x)  d
      (%o2)                y(x) %e     (-- (z(x)))
                                        dx
      (%i3) a1: antid (expr, x, z(x));
                             z(x)      z(x)  d
      (%o3)          [y(x) %e    , - %e     (-- (y(x)))]
                                             dx
      (%i4) a2: antidiff (expr, x, z(x));
                                  /
                           z(x)   [   z(x)  d
      (%o4)         y(x) %e     - I %e     (-- (y(x))) dx
                                  ]         dx
                                  /
      (%i5) a2 - (first (a1) + 'integrate (second (a1), x));
      (%o5)                           0
      (%i6) antid (expr, x, y(x));
                                   z(x)  d
      (%o6)             [0, y(x) %e     (-- (z(x)))]
                                         dx
      (%i7) antidiff (expr, x, y(x));
                        /
                        [        z(x)  d
      (%o7)             I y(x) %e     (-- (z(x))) dx
                        ]              dx
                        /

  Categories: Integral calculus

Function: at
`at` (expr, [eqn_1, …, eqn_n])
`at` (expr, eqn)

  Evaluates the expression expr with the variables assuming the values as specified for them in the list of equations `[eqn_1, ..., eqn_n]` or the single equation eqn.

  If a subexpression depends on any of the variables for which a value is specified but there is no `atvalue` specified and it can't be otherwise evaluated, then a noun form of the `at` is returned which displays in a two-dimensional form.

  `at` carries out multiple substitutions in parallel.

  See also `atvalue`. For other functions which carry out substitutions, see also `subst` and `ev`.

  Examples:

      (%i1) atvalue (f(x,y), [x = 0, y = 1], a^2);
                                      2
      (%o1)                          a
      (%i2) atvalue ('diff (f(x,y), x), x = 0, 1 + y);
      (%o2)                        @2 + 1
      (%i3) printprops (all, atvalue);
                                      !
                        d             !
                       --- (f(@1, @2))!       = @2 + 1
                       d@1            !
                                      !@1 = 0

                                           2
                                f(0, 1) = a

      (%o3)                         done
      (%i4) diff (4*f(x, y)^2 - u(x, y)^2, x);
                        d                          d
      (%o4)  8 f(x, y) (-- (f(x, y))) - 2 u(x, y) (-- (u(x, y)))
                        dx                         dx
      (%i5) at (%, [x = 0, y = 1]);
                                                  !
                       2              d           !
      (%o5)        16 a  - 2 u(0, 1) (-- (u(x, 1))!     )
                                      dx          !
                                                  !x = 0

  Note that in the last line `y` is treated differently to `x` as `y` isn't used as a differentiation variable.

  The difference between `subst`, `at` and `ev` can be seen in the following example:

      (%i1) e1:I(t)=C*diff(U(t),t)$
      (%i2) e2:U(t)=L*diff(I(t),t)$
      (%i3) at(e1,e2);
                                     !
                            d        !
      (%o3)       I(t) = C (-- (U(t))!                    )
                            dt       !          d
                                     !U(t) = L (-- (I(t)))
                                                dt
      (%i4) subst(e2,e1);
                                  d      d
      (%o4)             I(t) = C (-- (L (-- (I(t)))))
                                  dt     dt
      (%i5) ev(e1,e2,diff);
                                        2
                                       d
      (%o5)                I(t) = C L (--- (I(t)))
                                         2
                                       dt

  Categories: Evaluation · Differential equations

Property: atomgrad

  `atomgrad` is the atomic gradient property of an expression. This property is assigned by `gradef`.

  Categories: Differential calculus

Function: atvalue
`atvalue` (expr, [x_1 = a_1, …, x_m = a_m], c)
`atvalue` (expr, x_1 = a_1, c)

  Assigns the value c to expr at the point `x = a`. Typically boundary values are established by this mechanism.

  expr is a function evaluation, `f(x_1, ..., x_m)`, or a derivative, `diff (f(x_1, ..., x_m), x_1, n_1, ..., x_n, n_m)` in which the function arguments explicitly appear. n_i is the order of differentiation with respect to x_i.

  The point at which the atvalue is established is given by the list of equations `[x_1 = a_1, ..., x_m = a_m]`. If there is a single variable x_1, the sole equation may be given without enclosing it in a list.

  `printprops ([f_1, f_2, ...], atvalue)` displays the atvalues of the functions `f_1, f_2, ...` as specified by calls to `atvalue`. `printprops (f, atvalue)` displays the atvalues of one function f. `printprops (all, atvalue)` displays the atvalues of all functions for which atvalues are defined.

  The symbols `@1`, `@2`, … represent the variables x_1, x_2, … when atvalues are displayed.

  `atvalue` evaluates its arguments. `atvalue` returns c, the atvalue.

  See also `at`.

  Examples:

      (%i1) atvalue (f(x,y), [x = 0, y = 1], a^2);
                                      2
      (%o1)                          a
      (%i2) atvalue ('diff (f(x,y), x), x = 0, 1 + y);
      (%o2)                        @2 + 1
      (%i3) printprops (all, atvalue);
                                      !
                        d             !
                       --- (f(@1, @2))!       = @2 + 1
                       d@1            !
                                      !@1 = 0

                                           2
                                f(0, 1) = a

      (%o3)                         done
      (%i4) diff (4*f(x,y)^2 - u(x,y)^2, x);
                        d                          d
      (%o4)  8 f(x, y) (-- (f(x, y))) - 2 u(x, y) (-- (u(x, y)))
                        dx                         dx
      (%i5) at (%, [x = 0, y = 1]);
                                                  !
                       2              d           !
      (%o5)        16 a  - 2 u(0, 1) (-- (u(x, 1))!     )
                                      dx          !
                                                  !x = 0

  Categories: Differential equations · Declarations and inferences

Function: cartan

  The exterior calculus of differential forms is a basic tool of differential geometry developed by Elie Cartan and has important applications in the theory of partial differential equations. The `cartan` package implements the functions `ext_diff` and `lie_diff`, along with the operators `~` (wedge product) and `|` (contraction of a form with a vector.) Type `demo (tensor)` to see a brief description of these commands along with examples.

  `cartan` was implemented by F.B. Estabrook and H.D. Wahlquist.

  Categories: Differential geometry

Function: del (x)

  `del (x)` represents the differential of the variable x.

  `diff` returns an expression containing `del` if an independent variable is not specified. In this case, the return value is the so-called "total differential".

  Examples:

      (%i1) diff (log (x));
                                   del(x)
      (%o1)                        ------
                                     x
      (%i2) diff (exp (x*y));
                           x y              x y
      (%o2)            x %e    del(y) + y %e    del(x)
      (%i3) diff (x*y*z);
      (%o3)         x y del(z) + x z del(y) + y z del(x)

  Categories: Differential calculus

Function: delta (t)

  The Dirac Delta function.

  Currently only `laplace` knows about the `delta` function.

  Example:

      (%i1) laplace (delta (t - a) * sin(b*t), t, s);
      Is  a  positive, negative, or zero?

      p;
                                         - a s
      (%o1)                   sin(a b) %e

  Categories: Mathematical functions · Laplace transform

System variable: dependencies
Function: dependencies (f_1, …, f_n)

  The variable `dependencies` is the list of atoms which have functional dependencies, assigned by `depends`, the function `dependencies`, or `gradef`. The `dependencies` list is cumulative: each call to `depends`, `dependencies`, or `gradef` appends additional items. The default value of `dependencies` is `[]`.

  The function `dependencies(f_1, …, f_n)` appends f_1, …, f_n, to the `dependencies` list, where f_1, …, f_n are expressions of the form `f(x_1, …, x_m)`, and x_1, …, x_m are any number of arguments.

  `dependencies(f(x_1, …, x_m))` is equivalent to `depends(f, [x_1, …, x_m])`.

  See also `depends` and `gradef`.

      (%i1) dependencies;
      (%o1)                          []
      (%i2) depends (foo, [bar, baz]);
      (%o2)                    [foo(bar, baz)]
      (%i3) depends ([g, h], [a, b, c]);
      (%o3)               [g(a, b, c), h(a, b, c)]
      (%i4) dependencies;
      (%o4)        [foo(bar, baz), g(a, b, c), h(a, b, c)]
      (%i5) dependencies (quux (x, y), mumble (u));
      (%o5)                [quux(x, y), mumble(u)]
      (%i6) dependencies;
      (%o6) [foo(bar, baz), g(a, b, c), h(a, b, c), quux(x, y),
                                                             mumble(u)]
      (%i7) remove (quux, dependency);
      (%o7)                         done
      (%i8) dependencies;
      (%o8)  [foo(bar, baz), g(a, b, c), h(a, b, c), mumble(u)]

  Categories: Declarations and inferences · Global variables

Function: depends (f_1, x_1, …, f_n, x_n)

  Declares functional dependencies among variables for the purpose of computing derivatives. In the absence of declared dependence, `diff (f, x)` yields zero. If `depends (f, x)` is declared, `diff (f, x)` yields a symbolic derivative (that is, a `diff` noun).

  Each argument f_1, x_1, etc., can be the name of a variable or array, or a list of names. Every element of f_i (perhaps just a single element) is declared to depend on every element of x_i (perhaps just a single element). If some f_i is the name of an array or contains the name of an array, all elements of the array depend on x_i.

  `diff` recognizes indirect dependencies established by `depends` and applies the chain rule in these cases.

  `remove (f, dependency)` removes all dependencies declared for f.

  `depends` returns a list of the dependencies established. The dependencies are appended to the global variable `dependencies`. `depends` evaluates its arguments.

  `diff` is the only Maxima command which recognizes dependencies established by `depends`. Other functions (`integrate`, `laplace`, etc.) only recognize dependencies explicitly represented by their arguments. For example, `integrate` does not recognize the dependence of `f` on `x` unless explicitly represented as `integrate (f(x), x)`.

  `depends(f, [x_1, …, x_n])` is equivalent to `dependencies(f(x_1, …, x_n))`.

      (%i1) depends ([f, g], x);
      (%o1)                     [f(x), g(x)]
      (%i2) depends ([r, s], [u, v, w]);
      (%o2)               [r(u, v, w), s(u, v, w)]
      (%i3) depends (u, t);
      (%o3)                        [u(t)]
      (%i4) dependencies;
      (%o4)      [f(x), g(x), r(u, v, w), s(u, v, w), u(t)]
      (%i5) diff (r.s, u);
                               dr           ds
      (%o5)                    -- . s + r . --
                               du           du

      (%i6) diff (r.s, t);
                            dr du           ds du
      (%o6)                 -- -- . s + r . -- --
                            du dt           du dt

      (%i7) remove (r, dependency);
      (%o7)                         done
      (%i8) diff (r.s, t);
                                      ds du
      (%o8)                       r . -- --
                                      du dt

  Categories: Differential calculus · Declarations and inferences

Option variable: derivabbrev

  Default value: `false`

  When `derivabbrev` is `true`, symbolic derivatives (that is, `diff` nouns) are displayed as subscripts. Otherwise, derivatives are displayed in the Leibniz notation `dy/dx`.

  Categories: Differential calculus · Global flags

Function: derivdegree (expr, y, x)

  Returns the highest degree of the derivative of the dependent variable y with respect to the independent variable x occurring in expr.

  Example:

      (%i1) 'diff (y, x, 2) + 'diff (y, z, 3) + 'diff (y, x) * x^2;
                               3     2
                              d y   d y    2 dy
      (%o1)                   --- + --- + x  --
                                3     2      dx
                              dz    dx
      (%i2) derivdegree (%, y, x);
      (%o2)                           2

  Categories: Differential calculus · Expressions

Function: derivlist (var_1, …, var_k)

  Causes only differentiations with respect to the indicated variables, within the `ev` command.

  Categories: Differential calculus · Evaluation

Option variable: derivsubst

  Default value: `false`

  When `derivsubst` is `true`, a non-syntactic substitution such as `subst (x, 'diff (y, t), 'diff (y, t, 2))` yields `'diff (x, t)`.

  Categories: Differential calculus · Expressions

Function: diff
`diff` (expr, x_1, n_1, …, x_m, n_m)
`diff` (expr, x, n)
`diff` (expr, x)
`diff` (expr)

  Returns the derivative or differential of expr with respect to some or all variables in expr.

  `diff (expr, x, n)` returns the n'th derivative of expr with respect to x.

  `diff (expr, x_1, n_1, ..., x_m, n_m)` returns the mixed partial derivative of expr with respect to x_1, …, x_m. It is equivalent to `diff (... (diff (expr, x_m, n_m) ...), x_1, n_1)`.

  `diff (expr, x)` returns the first derivative of expr with respect to the variable x.

  `diff (expr)` returns the total differential of expr, that is, the sum of the derivatives of expr with respect to each its variables times the differential `del` of each variable. No further simplification of `del` is offered.

  The noun form of `diff` is required in some contexts, such as stating a differential equation. In these cases, `diff` may be quoted (as `'diff`) to yield the noun form instead of carrying out the differentiation.

  When `derivabbrev` is `true`, derivatives are displayed as subscripts. Otherwise, derivatives are displayed in the Leibniz notation, `dy/dx`.

  Examples:

      (%i1) diff (exp (f(x)), x, 2);
                           2
                    f(x)  d               f(x)  d         2
      (%o1)       %e     (--- (f(x))) + %e     (-- (f(x)))
                            2                   dx
                          dx
      (%i2) derivabbrev: true$
      (%i3) 'integrate (f(x, y), y, g(x), h(x));
                               h(x)
                              /
                              [
      (%o3)                   I     f(x, y) dy
                              ]
                              /
                               g(x)
      (%i4) diff (%, x);
             h(x)
            /
            [
      (%o4) I     f(x, y)  dy + f(x, h(x)) h(x)  - f(x, g(x)) g(x)
            ]            x                     x                  x
            /
             g(x)

  For the tensor package, the following modifications have been incorporated:

  (1) The derivatives of any indexed objects in expr will have the variables x_i appended as additional arguments. Then all the derivative indices will be sorted.

  (2) The x_i may be integers from 1 up to the value of the variable `dimension` [default value: 4]. This will cause the differentiation to be carried out with respect to the x_i'th member of the list `coordinates` which should be set to a list of the names of the coordinates, e.g., `[x, y, z, t]`. If `coordinates` is bound to an atomic variable, then that variable subscripted by x_i will be used for the variable of differentiation. This permits an array of coordinate names or subscripted names like `X[1]`, `X[2]`, … to be used. If `coordinates` has not been assigned a value, then the variables will be treated as in (1) above.

  Categories: Differential calculus

Special symbol: diff

  When `diff` is present as an `evflag` in call to `ev`, all differentiations indicated in `expr` are carried out.

Function: dscalar (f)

  Applies the scalar d'Alembertian to the scalar function f.

  `load ("ctensor")` loads this function.

  Categories: Differential calculus · Package ctensor

Function: express (expr)

  Expands differential operator nouns into expressions in terms of partial derivatives. `express` recognizes the operators `grad`, `div`, `curl`, `laplacian`. `express` also expands the cross product `~`.

  Symbolic derivatives (that is, `diff` nouns) in the return value of express may be evaluated by including `diff` in the `ev` function call or command line. In this context, `diff` acts as an `evfun`.

  `load ("vect")` loads this function.

  Examples:

      (%i1) load ("vect")$
      (%i2) grad (x^2 + y^2 + z^2);
                                    2    2    2
      (%o2)                  grad (z  + y  + x )
      (%i3) express (%);
             d    2    2    2   d    2    2    2   d    2    2    2
      (%o3) [-- (z  + y  + x ), -- (z  + y  + x ), -- (z  + y  + x )]
             dx                 dy                 dz
      (%i4) ev (%, diff);
      (%o4)                    [2 x, 2 y, 2 z]
      (%i5) div ([x^2, y^2, z^2]);
                                    2   2   2
      (%o5)                   div [x , y , z ]
      (%i6) express (%);
                         d    2    d    2    d    2
      (%o6)              -- (z ) + -- (y ) + -- (x )
                         dz        dy        dx
      (%i7) ev (%, diff);
      (%o7)                    2 z + 2 y + 2 x
      (%i8) curl ([x^2, y^2, z^2]);
                                     2   2   2
      (%o8)                   curl [x , y , z ]
      (%i9) express (%);
             d    2    d    2   d    2    d    2   d    2    d    2
      (%o9) [-- (z ) - -- (y ), -- (x ) - -- (z ), -- (y ) - -- (x )]
             dy        dz       dz        dx       dx        dy
      (%i10) ev (%, diff);
      (%o10)                      [0, 0, 0]
      (%i11) laplacian (x^2 * y^2 * z^2);
                                        2  2  2
      (%o11)                laplacian (x  y  z )
      (%i12) express (%);
               2                2                2
              d     2  2  2    d     2  2  2    d     2  2  2
      (%o12)  --- (x  y  z ) + --- (x  y  z ) + --- (x  y  z )
                2                2                2
              dz               dy               dx
      (%i13) ev (%, diff);
                            2  2      2  2      2  2
      (%o13)             2 y  z  + 2 x  z  + 2 x  y
      (%i14) [a, b, c] ~ [x, y, z];
      (%o14)                [a, b, c] ~ [x, y, z]
      (%i15) express (%);
      (%o15)          [b z - c y, c x - a z, a y - b x]

  Categories: Differential calculus · Vectors · Operators

Function: gradef
`gradef` (f(x_1, …, x_n), g_1, …, g_m)
`gradef` (a, x, expr)

  Defines the partial derivatives (i.e., the components of the gradient) of the function f or variable a.

  `gradef (f(x_1, ..., x_n), g_1, ..., g_m)` defines `df/dx_i` as g_i, where g_i is an expression; g_i may be a function call, but not the name of a function. The number of partial derivatives m may be less than the number of arguments n, in which case derivatives are defined with respect to x_1 through x_m only.

  `gradef (a, x, expr)` defines the derivative of variable a with respect to x as expr. This also establishes the dependence of a on x (via `depends (a, x)`).

  The first argument `f(x_1, ..., x_n)` or a is quoted, but the remaining arguments g_1, ..., g_m are evaluated. `gradef` returns the function or variable for which the partial derivatives are defined.

  `gradef` can redefine the derivatives of Maxima's built-in functions. For example, `gradef (sin(x), sqrt (1 - sin(x)^2))` redefines the derivative of `sin`.

  `gradef` cannot define partial derivatives for a subscripted function.

  `printprops ([f_1, ..., f_n], gradef)` displays the partial derivatives of the functions f_1, ..., f_n, as defined by `gradef`.

  `printprops ([a_n, ..., a_n], atomgrad)` displays the partial derivatives of the variables a_n, ..., a_n, as defined by `gradef`.

  `gradefs` is the list of the functions for which partial derivatives have been defined by `gradef`. `gradefs` does not include any variables for which partial derivatives have been defined by `gradef`.

  Gradients are needed when, for example, a function is not known explicitly but its first derivatives are and it is desired to obtain higher order derivatives.

  Categories: Differential calculus · Declarations and inferences

System variable: gradefs

  Default value: `[]`

  `gradefs` is the list of the functions for which partial derivatives have been defined by `gradef`. `gradefs` does not include any variables for which partial derivatives have been defined by `gradef`.

  Categories: Differential calculus · Declarations and inferences

Function: laplace (expr, t, s)

  Attempts to compute the Laplace transform of expr with respect to the variable t and transform parameter s.

  `laplace` recognizes in expr the functions `delta`, `exp`, `log`, `sin`, `cos`, `sinh`, `cosh`, and `erf`, as well as `derivative`, `integrate`, `sum`, and `ilt`. If laplace fails to find a transform the function `specint` is called. `specint` can find the laplace transform for expressions with special functions like the bessel functions `bessel_j`, `bessel_i`, … and can handle the `unit_step` function. See also `specint`.

  If `specint` cannot find a solution too, a noun `laplace` is returned.

  expr may also be a linear, constant coefficient differential equation in which case `atvalue` of the dependent variable is used. The required atvalue may be supplied either before or after the transform is computed. Since the initial conditions must be specified at zero, if one has boundary conditions imposed elsewhere he can impose these on the general solution and eliminate the constants by solving the general solution for them and substituting their values back.

  `laplace` recognizes convolution integrals of the form `integrate (f(x) * g(t - x), x, 0, t)`; other kinds of convolutions are not recognized.

  Functional relations must be explicitly represented in expr; implicit relations, established by `depends`, are not recognized. That is, if f depends on x and y, `f (x, y)` must appear in expr.

  See also `ilt`,, the inverse Laplace transform.

  Examples:

      (%i1) laplace (exp (2*t + a) * sin(t) * t, t, s);
                                  a
                                %e  (2 s - 4)
      (%o1)                    ---------------
                                 2           2
                               (s  - 4 s + 5)
      (%i2) laplace ('diff (f (x), x), x, s);
      (%o2)             s laplace(f(x), x, s) - f(0)
      (%i3) diff (diff (delta (t), t), t);
                                2
                               d
      (%o3)                    --- (delta(t))
                                 2
                               dt
      (%i4) laplace (%, t, s);
                                  !
                     d            !         2
      (%o4)        - -- (delta(t))!      + s  - delta(0) s
                     dt           !
                                  !t = 0
      (%i5) assume(a>0)$
      (%i6) laplace(gamma_incomplete(a,t),t,s),gamma_expand:true;
                                                    - a - 1
                               gamma(a)   gamma(a) s
      (%o6)                    -------- - -----------------
                                  s            1     a
                                              (- + 1)
                                               s
      (%i7) factor(laplace(gamma_incomplete(1/2,t),t,s));
                                                    s + 1
                            sqrt(%pi) (sqrt(s) sqrt(-----) - 1)
                                                      s
      (%o7)                 -----------------------------------
                                      3/2      s + 1
                                     s    sqrt(-----)
                                                 s
      (%i8) assume(exp(%pi*s)>1)$
      (%i9) laplace(sum((-1)^n*unit_step(t-n*%pi)*sin(t),n,0,inf),t,s),
               simpsum;
                               %i                         %i
                    ------------------------ - ------------------------
                                    - %pi s                    - %pi s
                    (s + %i) (1 - %e       )   (s - %i) (1 - %e       )
      (%o9)         ---------------------------------------------------
                                             2
      (%i9) factor(%);
                                            %pi s
                                          %e
      (%o9)                   -------------------------------
                                                   %pi s
                              (s - %i) (s + %i) (%e      - 1)

  Categories: Laplace transform · Differential equations

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# 19. Integration

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## 19.1 Introduction to Integration

Maxima has several routines for handling integration. The `integrate` function makes use of most of them. There is also the `antid` package, which handles an unspecified function (and its derivatives, of course). For numerical uses, there is a set of adaptive integrators from QUADPACK, named `quad_qag`, `quad_qags`, etc., which are described under the heading `QUADPACK`. Hypergeometric functions are being worked on, see `specint` for details. Generally speaking, Maxima only handles integrals which are integrable in terms of the "elementary functions" (rational functions, trigonometrics, logs, exponentials, radicals, etc.) and a few extensions (error function, dilogarithm). It does not handle integrals in terms of unknown functions such as `g(x)` and `h(x)`.

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## 19.2 Functions and Variables for Integration

Function: changevar (expr, f(x,y), y, x)

  Makes the change of variable given by `f(x,y) = 0` in all integrals occurring in expr with integration with respect to x. The new variable is y.

  The change of variable can also be written `f(x) = g(y)`.

      (%i1) assume(a > 0)$
      (%i2) 'integrate (%e**sqrt(a*y), y, 0, 4);
                            4
                           /
                           [    sqrt(a) sqrt(y)
      (%o2)                I  %e                dy
                           ]
                           /
                            0
      (%i3) changevar (%, y-z^2/a, z, y);
                            0
                           /
                           [                abs(z)
                         2 I            z %e       dz
                           ]
                           /
                            - 2 sqrt(a)
      (%o3)            - ----------------------------
                                      a

  An expression containing a noun form, such as the instances of `'integrate` above, may be evaluated by `ev` with the `nouns` flag. For example, the expression returned by `changevar` above may be evaluated by `ev (%o3, nouns)`.

  `changevar` may also be used to changes in the indices of a sum or product. However, it must be realized that when a change is made in a sum or product, this change must be a shift, i.e., `i = j+ ...`, not a higher degree function. E.g.,

      (%i4) sum (a[i]*x^(i-2), i, 0, inf);
                               inf
                               ====
                               \         i - 2
      (%o4)                     >    a  x
                               /      i
                               ====
                               i = 0
      (%i5) changevar (%, i-2-n, n, i);
                              inf
                              ====
                              \               n
      (%o5)                    >      a      x
                              /        n + 2
                              ====
                              n = - 2

  Categories: Integral calculus

Function: dblint (f, r, s, a, b)

  A double-integral routine which was written in top-level Maxima and then translated and compiled to machine code. Use `load ("dblint")` to access this package. It uses the Simpson's rule method in both the x and y directions to calculate

      /b /s(x)
      |  |
      |  |    f(x,y) dy dx
      |  |
      /a /r(x)

  The function f must be a translated or compiled function of two variables, and r and s must each be a translated or compiled function of one variable, while a and b must be floating point numbers. The routine has two global variables which determine the number of divisions of the x and y intervals: `dblint_x` and `dblint_y`, both of which are initially 10, and can be changed independently to other integer values (there are `2*dblint_x+1` points computed in the x direction, and `2*dblint_y+1` in the y direction). The routine subdivides the X axis and then for each value of X it first computes `r(x)` and `s(x)`; then the Y axis between `r(x)` and `s(x)` is subdivided and the integral along the Y axis is performed using Simpson's rule; then the integral along the X axis is done using Simpson's rule with the function values being the Y-integrals. This procedure may be numerically unstable for a great variety of reasons, but is reasonably fast: avoid using it on highly oscillatory functions and functions with singularities (poles or branch points in the region). The Y integrals depend on how far apart `r(x)` and `s(x)` are, so if the distance `s(x) - r(x)` varies rapidly with X, there may be substantial errors arising from truncation with different step-sizes in the various Y integrals. One can increase `dblint_x` and `dblint_y` in an effort to improve the coverage of the region, at the expense of computation time. The function values are not saved, so if the function is very time-consuming, you will have to wait for re-computation if you change anything (sorry). It is required that the functions f, r, and s be either translated or compiled prior to calling `dblint`. This will result in orders of magnitude speed improvement over interpreted code in many cases!

  `demo (dblint)` executes a demonstration of `dblint` applied to an example problem.

  Categories: Integral calculus

Function: defint (expr, x, a, b)

  Attempts to compute a definite integral. `defint` is called by `integrate` when limits of integration are specified, i.e., when `integrate` is called as `integrate (expr, x, a, b)`. Thus from the user's point of view, it is sufficient to call `integrate`.

  `defint` returns a symbolic expression, either the computed integral or the noun form of the integral. See `quad_qag` and related functions for numerical approximation of definite integrals.

  Categories: Integral calculus

Option variable: erfflag

  Default value: `true`

  When `erfflag` is `false`, prevents `risch` from introducing the `erf` function in the answer if there were none in the integrand to begin with.

  Categories: Integral calculus

Function: ilt (expr, s, t)

  Computes the inverse Laplace transform of expr with respect to s and parameter t. expr must be a ratio of polynomials whose denominator has only linear and quadratic factors. By using the functions `laplace` and `ilt` together with the `solve` or `linsolve` functions the user can solve a single differential or convolution integral equation or a set of them.

      (%i1) 'integrate (sinh(a*x)*f(t-x), x, 0, t) + b*f(t) = t**2;
                    t
                   /
                   [                                    2
      (%o1)        I  f(t - x) sinh(a x) dx + b f(t) = t
                   ]
                   /
                    0
      (%i2) laplace (%, t, s);
                                     a laplace(f(t), t, s)   2
      (%o2)  b laplace(f(t), t, s) + --------------------- = --
                                             2    2           3
                                            s  - a           s
      (%i3) linsolve ([%], ['laplace(f(t), t, s)]);
                                              2      2
                                           2 s  - 2 a
      (%o3)     [laplace(f(t), t, s) = --------------------]
                                          5         2     3
                                       b s  + (a - a  b) s
      (%i4) ilt (rhs (first (%)), s, t);
      Is  a b (a b - 1)  positive, negative, or zero?

      pos;
                     sqrt(a b (a b - 1)) t
              2 cosh(---------------------)       2
                               b               a t
      (%o4) - ----------------------------- + -------
                    3  2      2               a b - 1
                   a  b  - 2 a  b + a

                                                             2
                                                   + ------------------
                                                      3  2      2
                                                     a  b  - 2 a  b + a

  Categories: Laplace transform

Option variable: intanalysis

  Default value: `true`

  When `true`, definite integration tries to find poles in the integrand in the interval of integration. If there are, then the integral is evaluated appropriately as a principal value integral. If intanalysis is `false`, this check is not performed and integration is done assuming there are no poles.

  See also `ldefint`.

  Examples:

  Maxima can solve the following integrals, when `intanalysis` is set to `false`:

      (%i1) integrate(1/(sqrt(x)+1),x,0,1);
                                      1
                                     /
                                     [       1
      (%o1)                          I  ----------- dx
                                     ]  sqrt(x) + 1
                                     /
                                      0

      (%i2) integrate(1/(sqrt(x)+1),x,0,1),intanalysis:false;
      (%o2)                            2 - 2 log(2)

      (%i3) integrate(cos(a)/sqrt((tan(a))^2 +1),a,-%pi/2,%pi/2);
      The number 1 isn't in the domain of atanh
       -- an error. To debug this try: debugmode(true);

      (%i4) intanalysis:false$
      (%i5) integrate(cos(a)/sqrt((tan(a))^2+1),a,-%pi/2,%pi/2);
                                            %pi
      (%o5)                                 ---
                                             2

  Categories: Integral calculus

Function: integrate
`integrate` (expr, x)
`integrate` (expr, x, a, b)

  Attempts to symbolically compute the integral of expr with respect to x. `integrate (expr, x)` is an indefinite integral, while `integrate (expr, x, a, b)` is a definite integral, with limits of integration a and b. The limits should not contain x, although `integrate` does not enforce this restriction. a need not be less than b. If b is equal to a, `integrate` returns zero.

  See `quad_qag` and related functions for numerical approximation of definite integrals. See `residue` for computation of residues (complex integration). See `antid` for an alternative means of computing indefinite integrals.

  The integral (an expression free of `integrate`) is returned if `integrate` succeeds. Otherwise the return value is the noun form of the integral (the quoted operator `'integrate`) or an expression containing one or more noun forms. The noun form of `integrate` is displayed with an integral sign.

  In some circumstances it is useful to construct a noun form by hand, by quoting `integrate` with a single quote, e.g., `'integrate (expr, x)`. For example, the integral may depend on some parameters which are not yet computed. The noun may be applied to its arguments by `ev (i, nouns)` where i is the noun form of interest.

  `integrate` handles definite integrals separately from indefinite, and employs a range of heuristics to handle each case. Special cases of definite integrals include limits of integration equal to zero or infinity (`inf` or `minf`), trigonometric functions with limits of integration equal to zero and `%pi` or `2 %pi`, rational functions, integrals related to the definitions of the `beta` and `psi` functions, and some logarithmic and trigonometric integrals. Processing rational functions may include computation of residues. If an applicable special case is not found, an attempt will be made to compute the indefinite integral and evaluate it at the limits of integration. This may include taking a limit as a limit of integration goes to infinity or negative infinity; see also `ldefint`.

  Special cases of indefinite integrals include trigonometric functions, exponential and logarithmic functions, and rational functions. `integrate` may also make use of a short table of elementary integrals.

  `integrate` may carry out a change of variable if the integrand has the form `f(g(x)) * diff(g(x), x)`. `integrate` attempts to find a subexpression `g(x)` such that the derivative of `g(x)` divides the integrand. This search may make use of derivatives defined by the `gradef` function. See also `changevar` and `antid`.

  If none of the preceding heuristics find the indefinite integral, the Risch algorithm is executed. The flag `risch` may be set as an `evflag`, in a call to `ev` or on the command line, e.g., `ev (integrate (expr, x), risch)` or `integrate (expr, x), risch`. If `risch` is present, `integrate` calls the `risch` function without attempting heuristics first. See also `risch`.

  `integrate` works only with functional relations represented explicitly with the `f(x)` notation. `integrate` does not respect implicit dependencies established by the `depends` function.

  `integrate` may need to know some property of a parameter in the integrand. `integrate` will first consult the `assume` database, and, if the variable of interest is not there, `integrate` will ask the user. Depending on the question, suitable responses are `yes;` or `no;`, or `pos;`, `zero;`, or `neg;`.

  `integrate` is not, by default, declared to be linear. See `declare` and `linear`.

  `integrate` attempts integration by parts only in a few special cases.

  Examples:

  - Elementary indefinite and definite integrals.

        (%i1) integrate (sin(x)^3, x);
                                   3
                                cos (x)
        (%o1)                   ------- - cos(x)
                                   3
        (%i2) integrate (x/ sqrt (b^2 - x^2), x);
                                         2    2
        (%o2)                    - sqrt(b  - x )
        (%i3) integrate (cos(x)^2 * exp(x), x, 0, %pi);
                                       %pi
                                   3 %e      3
        (%o3)                      ------- - -
                                      5      5
        (%i4) integrate (x^2 * exp(-x^2), x, minf, inf);
                                    sqrt(%pi)
        (%o4)                       ---------
                                        2

  - Use of `assume` and interactive query.

        (%i1) assume (a > 1)$
        (%i2) integrate (x**a/(x+1)**(5/2), x, 0, inf);
            2 a + 2
        Is  -------  an integer?
               5

        no;
        Is  2 a - 3  positive, negative, or zero?

        neg;
                                           3
        (%o2)                  beta(a + 1, - - a)
                                           2

  - Change of variable. There are two changes of variable in this example: one using a derivative established by `gradef`, and one using the derivation `diff(r(x))` of an unspecified function `r(x)`.

        (%i3) gradef (q(x), sin(x**2));
        (%o3)                         q(x)
        (%i4) diff (log (q (r (x))), x);
                              d               2
                             (-- (r(x))) sin(r (x))
                              dx
        (%o4)                ----------------------
                                    q(r(x))
        (%i5) integrate (%, x);
        (%o5)                     log(q(r(x)))

  - Return value contains the `'integrate` noun form. In this example, Maxima can extract one factor of the denominator of a rational function, but cannot factor the remainder or otherwise find its integral. `grind` shows the noun form `'integrate` in the result. See also `integrate_use_rootsof` for more on integrals of rational functions.

        (%i1) expand ((x-4) * (x^3+2*x+1));
                            4      3      2
        (%o1)              x  - 4 x  + 2 x  - 7 x - 4
        (%i2) integrate (1/%, x);
                                      /  2
                                      [ x  + 4 x + 18
                                      I ------------- dx
                                      ]  3
                         log(x - 4)   / x  + 2 x + 1
        (%o2)            ---------- - ------------------
                             73               73
        (%i3) grind (%);
        log(x-4)/73-('integrate((x^2+4*x+18)/(x^3+2*x+1),x))/73$

  - Defining a function in terms of an integral. The body of a function is not evaluated when the function is defined. Thus the body of `f_1` in this example contains the noun form of `integrate`. The quote-quote operator `''` causes the integral to be evaluated, and the result becomes the body of `f_2`.

        (%i1) f_1 (a) := integrate (x^3, x, 1, a);
                                             3
        (%o1)           f_1(a) := integrate(x , x, 1, a)
        (%i2) ev (f_1 (7), nouns);
        (%o2)                          600
        (%i3) /* Note parentheses around integrate(...) here */
              f_2 (a) := ''(integrate (x^3, x, 1, a));
                                           4
                                          a    1
        (%o3)                   f_2(a) := -- - -
                                          4    4
        (%i4) f_2 (7);
        (%o4)                          600

  Categories: Integral calculus

System variable: integration_constant

  Default value: `%c`

  When a constant of integration is introduced by indefinite integration of an equation, the name of the constant is constructed by concatenating `integration_constant` and `integration_constant_counter`.

  `integration_constant` may be assigned any symbol.

  Examples:

      (%i1) integrate (x^2 = 1, x);
                                 3
                                x
      (%o1)                     -- = x + %c1
                                3
      (%i2) integration_constant : 'k;
      (%o2)                           k
      (%i3) integrate (x^2 = 1, x);
                                  3
                                 x
      (%o3)                      -- = x + k2
                                 3

  Categories: Integral calculus

System variable: integration_constant_counter

  Default value: 0

  When a constant of integration is introduced by indefinite integration of an equation, the name of the constant is constructed by concatenating `integration_constant` and `integration_constant_counter`.

  `integration_constant_counter` is incremented before constructing the next integration constant.

  Examples:

      (%i1) integrate (x^2 = 1, x);
                                 3
                                x
      (%o1)                     -- = x + %c1
                                3
      (%i2) integrate (x^2 = 1, x);
                                 3
                                x
      (%o2)                     -- = x + %c2
                                3
      (%i3) integrate (x^2 = 1, x);
                                 3
                                x
      (%o3)                     -- = x + %c3
                                3
      (%i4) reset (integration_constant_counter);
      (%o4)            [integration_constant_counter]
      (%i5) integrate (x^2 = 1, x);
                                 3
                                x
      (%o5)                     -- = x + %c1
                                3

  Categories: Integral calculus

Option variable: integrate_use_rootsof

  Default value: `false`

  When `integrate_use_rootsof` is `true` and the denominator of a rational function cannot be factored, `integrate` returns the integral in a form which is a sum over the roots (not yet known) of the denominator.

  For example, with `integrate_use_rootsof` set to `false`, `integrate` returns an unsolved integral of a rational function in noun form:

      (%i1) integrate_use_rootsof: false$
      (%i2) integrate (1/(1+x+x^5), x);
              /  2
              [ x  - 4 x + 5
              I ------------ dx                            2 x + 1
              ]  3    2                2            5 atan(-------)
              / x  - x  + 1       log(x  + x + 1)          sqrt(3)
      (%o2)   ----------------- - --------------- + ---------------
                      7                 14             7 sqrt(3)

  Now we set the flag to be true and the unsolved part of the integral will be expressed as a summation over the roots of the denominator of the rational function:

      (%i3) integrate_use_rootsof: true$
      (%i4) integrate (1/(1+x+x^5), x);
            ====        2
            \       (%r4  - 4 %r4 + 5) log(x - %r4)
             >      -------------------------------
            /                    2
            ====            3 %r4  - 2 %r4
                              3      2
            %r4 in rootsof(%r4  - %r4  + 1, %r4)
      (%o4) ----------------------------------------------------------
                     7

                                                            2 x + 1
                                        2            5 atan(-------)
                                   log(x  + x + 1)          sqrt(3)
                                 - --------------- + ---------------
                                         14             7 sqrt(3)

  Alternatively the user may compute the roots of the denominator separately, and then express the integrand in terms of these roots, e.g., `1/((x - a)*(x - b)*(x - c))` or `1/((x^2 - (a+b)*x + a*b)*(x - c))` if the denominator is a cubic polynomial. Sometimes this will help Maxima obtain a more useful result.

  Categories: Integral calculus

Function: ldefint (expr, x, a, b)

  Attempts to compute the definite integral of expr by using `limit` to evaluate the indefinite integral of expr with respect to x at the upper limit b and at the lower limit a. If it fails to compute the definite integral, `ldefint` returns an expression containing limits as noun forms.

  `ldefint` is not called from `integrate`, so executing `ldefint (expr, x, a, b)` may yield a different result than `integrate (expr, x, a, b)`. `ldefint` always uses the same method to evaluate the definite integral, while `integrate` may employ various heuristics and may recognize some special cases.

  Categories: Integral calculus

Function: potential (givengradient)

  The calculation makes use of the global variable `potentialzeroloc[0]` which must be `nonlist` or of the form

      [indeterminatej=expressionj, indeterminatek=expressionk, ...]

  the former being equivalent to the nonlist expression for all right-hand sides in the latter. The indicated right-hand sides are used as the lower limit of integration. The success of the integrations may depend upon their values and order. `potentialzeroloc` is initially set to 0.

Function: residue (expr, z, z_0)

  Computes the residue in the complex plane of the expression expr when the variable z assumes the value z_0. The residue is the coefficient of `(z - z_0)^(-1)` in the Laurent series for expr.

      (%i1) residue (s/(s**2+a**2), s, a*%i);
                                      1
      (%o1)                           -
                                      2
      (%i2) residue (sin(a*x)/x**4, x, 0);
                                       3
                                      a
      (%o2)                         - --
                                      6

  Categories: Integral calculus · Complex variables

Function: risch (expr, x)

  Integrates expr with respect to x using the transcendental case of the Risch algorithm. (The algebraic case of the Risch algorithm has not been implemented.) This currently handles the cases of nested exponentials and logarithms which the main part of `integrate` can't do. `integrate` will automatically apply `risch` if given these cases.

  `erfflag`, if `false`, prevents `risch` from introducing the `erf` function in the answer if there were none in the integrand to begin with.

      (%i1) risch (x^2*erf(x), x);
                                                              2
                   3                      2                - x
              %pi x  erf(x) + (sqrt(%pi) x  + sqrt(%pi)) %e
      (%o1)   -------------------------------------------------
                                    3 %pi
      (%i2) diff(%, x), ratsimp;
                                   2
      (%o2)                       x  erf(x)

  Categories: Integral calculus

Function: tldefint (expr, x, a, b)

  Equivalent to `ldefint` with `tlimswitch` set to `true`.

  Categories: Integral calculus

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## 19.3 Introduction to QUADPACK

QUADPACK is a collection of functions for the numerical computation of one-dimensional definite integrals. It originated from a joint project of R. Piessens (1), E. de Doncker (2), C. Ueberhuber (3), and D. Kahaner (4).

The QUADPACK library included in Maxima is an automatic translation (via the program `f2cl`) of the Fortran source code of QUADPACK as it appears in the SLATEC Common Mathematical Library, Version 4.1 (5). The SLATEC library is dated July 1993, but the QUADPACK functions were written some years before. There is another version of QUADPACK at Netlib (6); it is not clear how that version differs from the SLATEC version.

The QUADPACK functions included in Maxima are all automatic, in the sense that these functions attempt to compute a result to a specified accuracy, requiring an unspecified number of function evaluations. Maxima's Lisp translation of QUADPACK also includes some non-automatic functions, but they are not exposed at the Maxima level.

Further information about QUADPACK can be found in the QUADPACK book (7).

[ < ] | [ > ] |  | [ << ] | [ Up ] | [ >> ] |  |  |  |  | [Top] | [Contents] | [Index] | [ ? ]

### 19.3.1 Overview

`quad_qag`

  Integration of a general function over a finite interval. `quad_qag` implements a simple globally adaptive integrator using the strategy of Aind (Piessens, 1973). The caller may choose among 6 pairs of Gauss-Kronrod quadrature formulae for the rule evaluation component. The high-degree rules are suitable for strongly oscillating integrands.

`quad_qags`

  Integration of a general function over a finite interval. `quad_qags` implements globally adaptive interval subdivision with extrapolation (de Doncker, 1978) by the Epsilon algorithm (Wynn, 1956).

`quad_qagi`

  Integration of a general function over an infinite or semi-infinite interval. The interval is mapped onto a finite interval and then the same strategy as in `quad_qags` is applied.

`quad_qawo`

  Integration of cos(omega x) f(x) or sin(omega x) f(x) over a finite interval, where omega is a constant. The rule evaluation component is based on the modified Clenshaw-Curtis technique. `quad_qawo` applies adaptive subdivision with extrapolation, similar to `quad_qags`.

`quad_qawf`

  Calculates a Fourier cosine or Fourier sine transform on a semi-infinite interval. The same approach as in `quad_qawo` is applied on successive finite intervals, and convergence acceleration by means of the Epsilon algorithm (Wynn, 1956) is applied to the series of the integral contributions.

`quad_qaws`

  Integration of w(x) f(x) over a finite interval [a, b], where w is a function of the form (x - a)^alpha (b - x)^beta v(x) and v(x) is 1 or log(x - a) or log(b - x) or log(x - a) log(b - x), and alpha > -1 and beta > -1.

  A globally adaptive subdivision strategy is applied, with modified Clenshaw-Curtis integration on the subintervals which contain a or b.

`quad_qawc`

  Computes the Cauchy principal value of f(x)/(x - c) over a finite interval (a, b) and specified c. The strategy is globally adaptive, and modified Clenshaw-Curtis integration is used on the subranges which contain the point x = c.

`quad_qagp`

  Basically the same as `quad_qags` but points of singularity or discontinuity of the integrand must be supplied. This makes it easier for the integrator to produce a good solution.

Categories: Integral calculus · Numerical methods · Share packages · Package quadpack

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## 19.4 Functions and Variables for QUADPACK

Function: quad_qag
`quad_qag` (f(x), x, a, b, key, [epsrel, epsabs, limit])
`quad_qag` (f, x, a, b, key, [epsrel, epsabs, limit])

  Integration of a general function over a finite interval. `quad_qag` implements a simple globally adaptive integrator using the strategy of Aind (Piessens, 1973). The caller may choose among 6 pairs of Gauss-Kronrod quadrature formulae for the rule evaluation component. The high-degree rules are suitable for strongly oscillating integrands.

  `quad_qag` computes the integral

  integrate (f(x), x, a, b)

  The function to be integrated is f(x), with dependent variable x, and the function is to be integrated between the limits a and b. key is the integrator to be used and should be an integer between 1 and 6, inclusive. The value of key selects the order of the Gauss-Kronrod integration rule. High-order rules are suitable for strongly oscillating integrands.

  The integrand may be specified as the name of a Maxima or Lisp function or operator, a Maxima lambda expression, or a general Maxima expression.

  The numerical integration is done adaptively by subdividing the integration region into sub-intervals until the desired accuracy is achieved.

  The keyword arguments are optional and may be specified in any order. They all take the form `key=val`. The keyword arguments are:

  `epsrel`

    Desired relative error of approximation. Default is 1d-8.

  `epsabs`

    Desired absolute error of approximation. Default is 0.

  `limit`

    Size of internal work array. limit is the maximum number of subintervals to use. Default is 200.

  `quad_qag` returns a list of four elements:

  The error code (fourth element of the return value) can have the values:

  `0`

    if no problems were encountered;

  `1`

    if too many sub-intervals were done;

  `2`

    if excessive roundoff error is detected;

  `3`

    if extremely bad integrand behavior occurs;

  `6`

    if the input is invalid.

  Examples:

      (%i1) quad_qag (x^(1/2)*log(1/x), x, 0, 1, 3, 'epsrel=5d-8);
      (%o1)    [.4444444444492108, 3.1700968502883E-9, 961, 0]
      (%i2) integrate (x^(1/2)*log(1/x), x, 0, 1);
                                      4
      (%o2)                           -
                                      9

  Categories: Numerical methods · Package quadpack

Function: quad_qags
`quad_qags` (f(x), x, a, b, [epsrel, epsabs, limit])
`quad_qags` (f, x, a, b, [epsrel, epsabs, limit])

  Integration of a general function over a finite interval. `quad_qags` implements globally adaptive interval subdivision with extrapolation (de Doncker, 1978) by the Epsilon algorithm (Wynn, 1956).

  `quad_qags` computes the integral

  integrate (f(x), x, a, b)

  The function to be integrated is f(x), with dependent variable x, and the function is to be integrated between the limits a and b.

  The integrand may be specified as the name of a Maxima or Lisp function or operator, a Maxima lambda expression, or a general Maxima expression.

  The keyword arguments are optional and may be specified in any order. They all take the form `key=val`. The keyword arguments are:

  `epsrel`

    Desired relative error of approximation. Default is 1d-8.

  `epsabs`

    Desired absolute error of approximation. Default is 0.

  `limit`

    Size of internal work array. limit is the maximum number of subintervals to use. Default is 200.

  `quad_qags` returns a list of four elements:

  The error code (fourth element of the return value) can have the values:

  `0`

    no problems were encountered;

  `1`

    too many sub-intervals were done;

  `2`

    excessive roundoff error is detected;

  `3`

    extremely bad integrand behavior occurs;

  `4`

    failed to converge

  `5`

    integral is probably divergent or slowly convergent

  `6`

    if the input is invalid.

  Examples:

      (%i1) quad_qags (x^(1/2)*log(1/x), x, 0, 1, 'epsrel=1d-10);
      (%o1)   [.4444444444444448, 1.11022302462516E-15, 315, 0]

  Note that `quad_qags` is more accurate and efficient than `quad_qag` for this integrand.

  Categories: Numerical methods · Package quadpack

Function: quad_qagi
`quad_qagi` (f(x), x, a, b, [epsrel, epsabs, limit])
`quad_qagi` (f, x, a, b, [epsrel, epsabs, limit])

  Integration of a general function over an infinite or semi-infinite interval. The interval is mapped onto a finite interval and then the same strategy as in `quad_qags` is applied.

  `quad_qagi` evaluates one of the following integrals

  integrate (f(x), x, a, inf)

  integrate (f(x), x, minf, a)

  integrate (f(x), x, minf, inf)

  using the Quadpack QAGI routine. The function to be integrated is f(x), with dependent variable x, and the function is to be integrated over an infinite range.

  The integrand may be specified as the name of a Maxima or Lisp function or operator, a Maxima lambda expression, or a general Maxima expression.

  One of the limits of integration must be infinity. If not, then `quad_qagi` will just return the noun form.

  The keyword arguments are optional and may be specified in any order. They all take the form `key=val`. The keyword arguments are:

  `epsrel`

    Desired relative error of approximation. Default is 1d-8.

  `epsabs`

    Desired absolute error of approximation. Default is 0.

  `limit`

    Size of internal work array. limit is the maximum number of subintervals to use. Default is 200.

  `quad_qagi` returns a list of four elements:

  The error code (fourth element of the return value) can have the values:

  `0`

    no problems were encountered;

  `1`

    too many sub-intervals were done;

  `2`

    excessive roundoff error is detected;

  `3`

    extremely bad integrand behavior occurs;

  `4`

    failed to converge

  `5`

    integral is probably divergent or slowly convergent

  `6`

    if the input is invalid.

  Examples:

      (%i1) quad_qagi (x^2*exp(-4*x), x, 0, inf, 'epsrel=1d-8);
      (%o1)        [0.03125, 2.95916102995002E-11, 105, 0]
      (%i2) integrate (x^2*exp(-4*x), x, 0, inf);
                                     1
      (%o2)                          --
                                     32

  Categories: Numerical methods · Package quadpack

Function: quad_qawc
`quad_qawc` (f(x), x, c, a, b, [epsrel, epsabs, limit])
`quad_qawc` (f, x, c, a, b, [epsrel, epsabs, limit])

  Computes the Cauchy principal value of f(x)/(x - c) over a finite interval. The strategy is globally adaptive, and modified Clenshaw-Curtis integration is used on the subranges which contain the point x = c.

  `quad_qawc` computes the Cauchy principal value of

  integrate (f(x)/(x - c), x, a, b)

  using the Quadpack QAWC routine. The function to be integrated is `f(x)/(x - c)`, with dependent variable x, and the function is to be integrated over the interval a to b.

  The integrand may be specified as the name of a Maxima or Lisp function or operator, a Maxima lambda expression, or a general Maxima expression.

  The keyword arguments are optional and may be specified in any order. They all take the form `key=val`. The keyword arguments are:

  `epsrel`

    Desired relative error of approximation. Default is 1d-8.

  `epsabs`

    Desired absolute error of approximation. Default is 0.

  `limit`

    Size of internal work array. limit is the maximum number of subintervals to use. Default is 200.

  `quad_qawc` returns a list of four elements:

  The error code (fourth element of the return value) can have the values:

  `0`

    no problems were encountered;

  `1`

    too many sub-intervals were done;

  `2`

    excessive roundoff error is detected;

  `3`

    extremely bad integrand behavior occurs;

  `6`

    if the input is invalid.

  Examples:

      (%i1) quad_qawc (2^(-5)*((x-1)^2+4^(-5))^(-1), x, 2, 0, 5,
                       'epsrel=1d-7);
      (%o1)    [- 3.130120337415925, 1.306830140249558E-8, 495, 0]
      (%i2) integrate (2^(-alpha)*(((x-1)^2 + 4^(-alpha))*(x-2))^(-1),
            x, 0, 5);
      Principal Value
                             alpha
              alpha       9 4                 9
             4      log(------------- + -------------)
                            alpha           alpha
                        64 4      + 4   64 4      + 4
      (%o2) (-----------------------------------------
                              alpha
                           2 4      + 2

             3 alpha                       3 alpha
             -------                       -------
                2            alpha/2          2          alpha/2
          2 4        atan(4 4       )   2 4        atan(4       )   alpha
        - --------------------------- - -------------------------)/2
                    alpha                        alpha
                 2 4      + 2                 2 4      + 2
      (%i3) ev (%, alpha=5, numer);
      (%o3)                    - 3.130120337415917

  Categories: Numerical methods · Package quadpack

Function: quad_qawf
`quad_qawf` (f(x), x, a, omega, trig, [epsabs, limit, maxp1, limlst])
`quad_qawf` (f, x, a, omega, trig, [epsabs, limit, maxp1, limlst])

  Calculates a Fourier cosine or Fourier sine transform on a semi-infinite interval using the Quadpack QAWF function. The same approach as in `quad_qawo` is applied on successive finite intervals, and convergence acceleration by means of the Epsilon algorithm (Wynn, 1956) is applied to the series of the integral contributions.

  `quad_qawf` computes the integral

  integrate (f(x)*w(x), x, a, inf)

  The weight function w is selected by trig:

  `cos`

    w(x) = cos (omega x)

  `sin`

    w(x) = sin (omega x)

  The integrand may be specified as the name of a Maxima or Lisp function or operator, a Maxima lambda expression, or a general Maxima expression.

  The keyword arguments are optional and may be specified in any order. They all take the form `key=val`. The keyword arguments are:

  `epsabs`

    Desired absolute error of approximation. Default is 1d-10.

  `limit`

    Size of internal work array. (limit - limlst)/2 is the maximum number of subintervals to use. Default is 200.

  `maxp1`

    Maximum number of Chebyshev moments. Must be greater than 0. Default is 100.

  `limlst`

    Upper bound on the number of cycles. Must be greater than or equal to 3. Default is 10.

  `quad_qawf` returns a list of four elements:

  The error code (fourth element of the return value) can have the values:

  `0`

    no problems were encountered;

  `1`

    too many sub-intervals were done;

  `2`

    excessive roundoff error is detected;

  `3`

    extremely bad integrand behavior occurs;

  `6`

    if the input is invalid.

  Examples:

      (%i1) quad_qawf (exp(-x^2), x, 0, 1, 'cos, 'epsabs=1d-9);
      (%o1)   [.6901942235215714, 2.84846300257552E-11, 215, 0]
      (%i2) integrate (exp(-x^2)*cos(x), x, 0, inf);
                                - 1/4
                              %e      sqrt(%pi)
      (%o2)                   -----------------
                                      2
      (%i3) ev (%, numer);
      (%o3)                   .6901942235215714

  Categories: Numerical methods · Package quadpack

Function: quad_qawo
`quad_qawo` (f(x), x, a, b, omega, trig, [epsrel, epsabs, limit, maxp1, limlst])
`quad_qawo` (f, x, a, b, omega, trig, [epsrel, epsabs, limit, maxp1, limlst])

  Integration of cos (omega x) f(x) or sin (omega x) f(x) over a finite interval, where omega is a constant. The rule evaluation component is based on the modified Clenshaw-Curtis technique. `quad_qawo` applies adaptive subdivision with extrapolation, similar to `quad_qags`.

  `quad_qawo` computes the integral using the Quadpack QAWO routine:

  integrate (f(x)*w(x), x, a, b)

  The weight function w is selected by trig:

  `cos`

    w(x) = cos (omega x)

  `sin`

    w(x) = sin (omega x)

  The integrand may be specified as the name of a Maxima or Lisp function or operator, a Maxima lambda expression, or a general Maxima expression.

  The keyword arguments are optional and may be specified in any order. They all take the form `key=val`. The keyword arguments are:

  `epsrel`

    Desired relative error of approximation. Default is 1d-8.

  `epsabs`

    Desired absolute error of approximation. Default is 0.

  `limit`

    Size of internal work array. limit/2 is the maximum number of subintervals to use. Default is 200.

  `maxp1`

    Maximum number of Chebyshev moments. Must be greater than 0. Default is 100.

  `limlst`

    Upper bound on the number of cycles. Must be greater than or equal to 3. Default is 10.

  `quad_qawo` returns a list of four elements:

  The error code (fourth element of the return value) can have the values:

  `0`

    no problems were encountered;

  `1`

    too many sub-intervals were done;

  `2`

    excessive roundoff error is detected;

  `3`

    extremely bad integrand behavior occurs;

  `6`

    if the input is invalid.

  Examples:

      (%i1) quad_qawo (x^(-1/2)*exp(-2^(-2)*x), x, 1d-8, 20*2^2, 1, cos);
      (%o1)     [1.376043389877692, 4.72710759424899E-11, 765, 0]
      (%i2) rectform (integrate (x^(-1/2)*exp(-2^(-alpha)*x) * cos(x),
            x, 0, inf));
                         alpha/2 - 1/2            2 alpha
              sqrt(%pi) 2              sqrt(sqrt(2        + 1) + 1)
      (%o2)   -----------------------------------------------------
                                     2 alpha
                               sqrt(2        + 1)
      (%i3) ev (%, alpha=2, numer);
      (%o3)                     1.376043390090716

  Categories: Numerical methods · Package quadpack

Function: quad_qaws
`quad_qaws` (f(x), x, a, b, alpha, beta, wfun, [epsrel, epsabs, limit])
`quad_qaws` (f, x, a, b, alpha, beta, wfun, [epsrel, epsabs, limit])

  Integration of w(x) f(x) over a finite interval, where w(x) is a certain algebraic or logarithmic function. A globally adaptive subdivision strategy is applied, with modified Clenshaw-Curtis integration on the subintervals which contain the endpoints of the interval of integration.

  `quad_qaws` computes the integral using the Quadpack QAWS routine:

  integrate (f(x)*w(x), x, a, b)

  The weight function w is selected by wfun:

  `1`

    w(x) = (x - a)^alpha (b - x)^beta

  `2`

    w(x) = (x - a)^alpha (b - x)^beta log(x - a)

  `3`

    w(x) = (x - a)^alpha (b - x)^beta log(b - x)

  `4`

    w(x) = (x - a)^alpha (b - x)^beta log(x - a) log(b - x)

  The integrand may be specified as the name of a Maxima or Lisp function or operator, a Maxima lambda expression, or a general Maxima expression.

  The keyword arguments are optional and may be specified in any order. They all take the form `key=val`. The keyword arguments are:

  `epsrel`

    Desired relative error of approximation. Default is 1d-8.

  `epsabs`

    Desired absolute error of approximation. Default is 0.

  `limit`

    Size of internal work array. limitis the maximum number of subintervals to use. Default is 200.

  `quad_qaws` returns a list of four elements:

  The error code (fourth element of the return value) can have the values:

  `0`

    no problems were encountered;

  `1`

    too many sub-intervals were done;

  `2`

    excessive roundoff error is detected;

  `3`

    extremely bad integrand behavior occurs;

  `6`

    if the input is invalid.

  Examples:

      (%i1) quad_qaws (1/(x+1+2^(-4)), x, -1, 1, -0.5, -0.5, 1,
                       'epsabs=1d-9);
      (%o1)     [8.750097361672832, 1.24321522715422E-10, 170, 0]
      (%i2) integrate ((1-x*x)^(-1/2)/(x+1+2^(-alpha)), x, -1, 1);
             alpha
      Is  4 2      - 1  positive, negative, or zero?

      pos;
                                alpha         alpha
                         2 %pi 2      sqrt(2 2      + 1)
      (%o2)              -------------------------------
                                     alpha
                                  4 2      + 2
      (%i3) ev (%, alpha=4, numer);
      (%o3)                     8.750097361672829

  Categories: Numerical methods · Package quadpack

Function: quad_qagp
`quad_qagp` (f(x), x, a, b, points, [epsrel, epsabs, limit])
`quad_qagp` (f, x, a, b, points, [epsrel, epsabs, limit])

  Integration of a general function over a finite interval. `quad_qagp` implements globally adaptive interval subdivision with extrapolation (de Doncker, 1978) by the Epsilon algorithm (Wynn, 1956).

  `quad_qagp` computes the integral

  integrate (f(x), x, a, b)

  The function to be integrated is f(x), with dependent variable x, and the function is to be integrated between the limits a and b.

  The integrand may be specified as the name of a Maxima or Lisp function or operator, a Maxima lambda expression, or a general Maxima expression.

  To help the integrator, the user must supply a list of points where the integrand is singular or discontinous.

  The keyword arguments are optional and may be specified in any order. They all take the form `key=val`. The keyword arguments are:

  `epsrel`

    Desired relative error of approximation. Default is 1d-8.

  `epsabs`

    Desired absolute error of approximation. Default is 0.

  `limit`

    Size of internal work array. limit is the maximum number of subintervals to use. Default is 200.

  `quad_qagp` returns a list of four elements:

  The error code (fourth element of the return value) can have the values:

  `0`

    no problems were encountered;

  `1`

    too many sub-intervals were done;

  `2`

    excessive roundoff error is detected;

  `3`

    extremely bad integrand behavior occurs;

  `4`

    failed to converge

  `5`

    integral is probably divergent or slowly convergent

  `6`

    if the input is invalid.

  Examples:

      (%i1) quad_qagp(x^3*log(abs((x^2-1)*(x^2-2))),x,0,3,[1,sqrt(2)]);
      (%o1)   [52.74074838347143, 2.6247632689546663e-7, 1029, 0]
      (%i2) quad_qags(x^3*log(abs((x^2-1)*(x^2-2))), x, 0, 3);
      (%o2)   [52.74074847951494, 4.088443219529836e-7, 1869, 0]

  The integrand has singularities at `1` and `sqrt(2)` so we supply these points to `quad_qagp`. We also note that `quad_qagp` is more accurate and more efficient that `quad_qags`.

  Categories: Numerical methods · Package quadpack

Function: quad_control (parameter, [value])

  Control error handling for quadpack. The parameter should be one of the following symbols:

  `current_error`

    The current error number

  `control`

    Controls if messages are printed or not. If it is set to zero or less, messages are suppressed.

  `max_message`

    The maximum number of times any message is to be printed.

  If value is not given, then the current value of the parameter is returned. If value is given, the value of parameter is set to the given value.

  Categories: Numerical methods · Package quadpack

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# 20. Equations

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## 20.1 Functions and Variables for Equations

System variable: %rnum

  Default value: `0`

  `%rnum` is the counter for the `%r` variables introduced in solutions by `solve` and `algsys`.. The next `%r` variable is numbered `%rnum+1`.

  See also `%rnum_list`.

  Categories: Algebraic equations

System variable: %rnum_list

  Default value: `[]`

  `%rnum_list` is the list of variables introduced in solutions by `solve` and `algsys`. `%r` variables are added to `%rnum_list` in the order they are created. This is convenient for doing substitutions into the solution later on.

  See also `%rnum`.

  It's recommended to use this list rather than doing `concat ('%r, j)`.

      (%i1) solve ([x + y = 3], [x,y]);
      (%o1)              [[x = 3 - %r1, y = %r1]]
      (%i2) %rnum_list;
      (%o2)                       [%r1]
      (%i3) sol : solve ([x + 2*y + 3*z = 4], [x,y,z]);
      (%o3)   [[x = - 2 %r3 - 3 %r2 + 4, y = %r3, z = %r2]]
      (%i4) %rnum_list;
      (%o4)                     [%r2, %r3]
      (%i5) for i : 1 thru length (%rnum_list) do
              sol : subst (t[i], %rnum_list[i], sol)$
      (%i6) sol;
      (%o6)     [[x = - 2 t  - 3 t  + 4, y = t , z = t ]]
                           2      1           2       1

  Categories: Algebraic equations

Option variable: algepsilon

  Default value: 10^8

  `algepsilon` is used by `algsys`.

  Categories: Algebraic equations

Option variable: algexact

  Default value: `false`

  `algexact` affects the behavior of `algsys` as follows:

  If `algexact` is `true`, `algsys` always calls `solve` and then uses `realroots` on `solve`'s failures.

  If `algexact` is `false`, `solve` is called only if the eliminant was not univariate, or if it was a quadratic or biquadratic.

  Thus `algexact: true` does not guarantee only exact solutions, just that `algsys` will first try as hard as it can to give exact solutions, and only yield approximations when all else fails.

  Categories: Algebraic equations

Function: algsys
`algsys` ([expr_1, …, expr_m], [x_1, …, x_n])
`algsys` ([eqn_1, …, eqn_m], [x_1, …, x_n])

  Solves the simultaneous polynomials expr_1, …, expr_m or polynomial equations eqn_1, …, eqn_m for the variables x_1, …, x_n. An expression expr is equivalent to an equation `expr = 0`. There may be more equations than variables or vice versa.

  `algsys` returns a list of solutions, with each solution given as a list of equations stating values of the variables x_1, …, x_n which satisfy the system of equations. If `algsys` cannot find a solution, an empty list `[]` is returned.

  The symbols `%r1`, `%r2`, …, are introduced as needed to represent arbitrary parameters in the solution; these variables are also appended to the list `%rnum_list`.

  The method is as follows:

  1. First the equations are factored and split into subsystems.
  2. For each subsystem S_i, an equation E and a variable x are selected. The variable is chosen to have lowest nonzero degree. Then the resultant of E and E_j with respect to x is computed for each of the remaining equations E_j in the subsystem S_i. This yields a new subsystem S_i' in one fewer variables, as x has been eliminated. The process now returns to (1).
  3. Eventually, a subsystem consisting of a single equation is obtained. If the equation is multivariate and no approximations in the form of floating point numbers have been introduced, then `solve` is called to find an exact solution.

     In some cases, `solve` is not be able to find a solution, or if it does the solution may be a very large expression.

     If the equation is univariate and is either linear, quadratic, or biquadratic, then again `solve` is called if no approximations have been introduced. If approximations have been introduced or the equation is not univariate and neither linear, quadratic, or biquadratic, then if the switch `realonly` is `true`, the function `realroots` is called to find the real-valued solutions. If `realonly` is `false`, then `allroots` is called which looks for real and complex-valued solutions.

     If `algsys` produces a solution which has fewer significant digits than required, the user can change the value of `algepsilon` to a higher value.

     If `algexact` is set to `true`, `solve` will always be called.

  4. Finally, the solutions obtained in step (3) are substituted into previous levels and the solution process returns to (1).

  When `algsys` encounters a multivariate equation which contains floating point approximations (usually due to its failing to find exact solutions at an earlier stage), then it does not attempt to apply exact methods to such equations and instead prints the message: "`algsys` cannot solve - system too complicated."

  Interactions with `radcan` can produce large or complicated expressions. In that case, it may be possible to isolate parts of the result with `pickapart` or `reveal`.

  Occasionally, `radcan` may introduce an imaginary unit `%i` into a solution which is actually real-valued.

  Examples:

      (%i1) e1: 2*x*(1 - a1) - 2*(x - 1)*a2;
      (%o1)              2 (1 - a1) x - 2 a2 (x - 1)
      (%i2) e2: a2 - a1;
      (%o2)                        a2 - a1
      (%i3) e3: a1*(-y - x^2 + 1);
                                         2
      (%o3)                   a1 (- y - x  + 1)
      (%i4) e4: a2*(y - (x - 1)^2);
                                             2
      (%o4)                   a2 (y - (x - 1) )
      (%i5) algsys ([e1, e2, e3, e4], [x, y, a1, a2]);
      (%o5) [[x = 0, y = %r1, a1 = 0, a2 = 0],

                                        [x = 1, y = 0, a1 = 1, a2 = 1]]
      (%i6) e1: x^2 - y^2;
                                    2    2
      (%o6)                        x  - y
      (%i7) e2: -1 - y + 2*y^2 - x + x^2;
                               2        2
      (%o7)                 2 y  - y + x  - x - 1
      (%i8) algsys ([e1, e2], [x, y]);
                       1            1
      (%o8) [[x = - -------, y = -------],
                    sqrt(3)      sqrt(3)

              1              1             1        1
      [x = -------, y = - -------], [x = - -, y = - -], [x = 1, y = 1]]
           sqrt(3)        sqrt(3)          3        3

  Categories: Algebraic equations

Function: allroots
`allroots` (expr)
`allroots` (eqn)

  Computes numerical approximations of the real and complex roots of the polynomial expr or polynomial equation eqn of one variable.

  The flag `polyfactor` when `true` causes `allroots` to factor the polynomial over the real numbers if the polynomial is real, or over the complex numbers, if the polynomial is complex.

  `allroots` may give inaccurate results in case of multiple roots. If the polynomial is real, `allroots (%i*p)` may yield more accurate approximations than `allroots (p)`, as `allroots` invokes a different algorithm in that case.

  `allroots` rejects non-polynomials. It requires that the numerator after `rat`'ing should be a polynomial, and it requires that the denominator be at most a complex number. As a result of this `allroots` will always return an equivalent (but factored) expression, if `polyfactor` is `true`.

  For complex polynomials an algorithm by Jenkins and Traub is used (Algorithm 419, Comm. ACM, vol. 15, (1972), p. 97). For real polynomials the algorithm used is due to Jenkins (Algorithm 493, ACM TOMS, vol. 1, (1975), p.178).

  Examples:

      (%i1) eqn: (1 + 2*x)^3 = 13.5*(1 + x^5);
                                  3          5
      (%o1)              (2 x + 1)  = 13.5 (x  + 1)
      (%i2) soln: allroots (eqn);
      (%o2) [x = .8296749902129361, x = - 1.015755543828121,

      x = .9659625152196369 %i - .4069597231924075,

      x = - .9659625152196369 %i - .4069597231924075, x = 1.0]
      (%i3) for e in soln
              do (e2: subst (e, eqn), disp (expand (lhs(e2) - rhs(e2))));
                            - 3.5527136788005E-15

                           - 5.32907051820075E-15

               4.44089209850063E-15 %i - 4.88498130835069E-15

              - 4.44089209850063E-15 %i - 4.88498130835069E-15

                             3.5527136788005E-15

      (%o3)                         done
      (%i4) polyfactor: true$
      (%i5) allroots (eqn);
      (%o5) - 13.5 (x - 1.0) (x - .8296749902129361)

                                 2
       (x + 1.015755543828121) (x  + .8139194463848151 x

       + 1.098699797110288)

  Categories: Polynomials · Numerical methods

Function: bfallroots
`bfallroots` (expr)
`bfallroots` (eqn)

  Computes numerical approximations of the real and complex roots of the polynomial expr or polynomial equation eqn of one variable.

  In all respects, `bfallroots` is identical to `allroots` except that `bfallroots` computes the roots using bigfloats. See `allroots` for more information.

  Categories: Polynomials · Numerical methods

Option variable: backsubst

  Default value: `true`

  When `backsubst` is `false`, prevents back substitution in `linsolve` after the equations have been triangularized. This may be helpful in very big problems where back substitution would cause the generation of extremely large expressions.

      (%i1) eq1 : x + y + z = 6$
      (%i2) eq2 : x - y + z = 2$
      (%i3) eq3 : x + y - z = 0$
      (%i4) backsubst : false$
      (%i5) linsolve ([eq1, eq2, eq3], [x,y,z]);
      (%o5)             [x = z - y, y = 2, z = 3]
      (%i6) backsubst : true$
      (%i7) linsolve ([eq1, eq2, eq3], [x,y,z]);
      (%o7)               [x = 1, y = 2, z = 3]

  Categories: Algebraic equations

Option variable: breakup

  Default value: `true`

  When `breakup` is `true`, `solve` expresses solutions of cubic and quartic equations in terms of common subexpressions, which are assigned to intermediate expression labels (`%t1`, `%t2`, etc.). Otherwise, common subexpressions are not identified.

  `breakup: true` has an effect only when `programmode` is `false`.

  Examples:

      (%i1) programmode: false$
      (%i2) breakup: true$
      (%i3) solve (x^3 + x^2 - 1);

                              sqrt(23)    25 1/3
      (%t3)                  (--------- + --)
                              6 sqrt(3)   54
      Solution:

                                            sqrt(3) %i   1
                                            ---------- - -
                      sqrt(3) %i   1            2        2   1
      (%t4)    x = (- ---------- - -) %t3 + -------------- - -
                          2        2            9 %t3        3

                                            sqrt(3) %i   1
                                          - ---------- - -
                    sqrt(3) %i   1              2        2   1
      (%t5)    x = (---------- - -) %t3 + ---------------- - -
                        2        2             9 %t3         3

                                         1     1
      (%t6)                  x = %t3 + ----- - -
                                       9 %t3   3
      (%o6)                    [%t4, %t5, %t6]
      (%i6) breakup: false$
      (%i7) solve (x^3 + x^2 - 1);
      Solution:

                   sqrt(3) %i   1
                   ---------- - -
                       2        2        sqrt(23)    25 1/3
      (%t7) x = --------------------- + (--------- + --)
                   sqrt(23)    25 1/3    6 sqrt(3)   54
                9 (--------- + --)
                   6 sqrt(3)   54

                                                    sqrt(3) %i   1    1
                                                 (- ---------- - -) - -
                                                        2        2    3
                 sqrt(23)    25 1/3  sqrt(3) %i   1
      (%t8) x = (--------- + --)    (---------- - -)
                 6 sqrt(3)   54          2        2

                                                  sqrt(3) %i   1
                                                - ---------- - -
                                                      2        2      1
                                            + --------------------- - -
                                                 sqrt(23)    25 1/3   3
                                              9 (--------- + --)
                                                 6 sqrt(3)   54
                  sqrt(23)    25 1/3             1             1
      (%t9)  x = (--------- + --)    + --------------------- - -
                  6 sqrt(3)   54          sqrt(23)    25 1/3   3
                                       9 (--------- + --)
                                          6 sqrt(3)   54
      (%o9)                    [%t7, %t8, %t9]

  Categories: Algebraic equations

Function: dimension
`dimension` (eqn)
`dimension` (eqn_1, …, eqn_n)

  `dimen` is a package for dimensional analysis. `load ("dimen")` loads this package. `demo ("dimen")` displays a short demonstration.

  Categories: Share packages

Option variable: dispflag

  Default value: `true`

  If set to `false` within a `block` will inhibit the display of output generated by the solve functions called from within the `block`. Termination of the `block` with a dollar sign, $, sets `dispflag` to `false`.

  Categories: Algebraic equations · Display flags and variables

Function: funcsolve (eqn, g(t))

  Returns `[g(t) = ...]` or `[]`, depending on whether or not there exists a rational function `g(t)` satisfying eqn, which must be a first order, linear polynomial in (for this case) `g(t)` and `g(t+1)`

      (%i1) eqn: (n + 1)*f(n) - (n + 3)*f(n + 1)/(n + 1) =
            (n - 1)/(n + 2);
                                  (n + 3) f(n + 1)   n - 1
      (%o1)        (n + 1) f(n) - ---------------- = -----
                                       n + 1         n + 2
      (%i2) funcsolve (eqn, f(n));

      Dependent equations eliminated:  (4 3)
                                         n
      (%o2)                f(n) = ---------------
                                  (n + 1) (n + 2)

  Warning: this is a very rudimentary implementation - many safety checks and obvious generalizations are missing.

  Categories: Algebraic equations

Option variable: globalsolve

  Default value: `false`

  When `globalsolve` is `true`, solved-for variables are assigned the solution values found by `linsolve`, and by `solve` when solving two or more linear equations.

  When `globalsolve` is `false`, solutions found by `linsolve` and by `solve` when solving two or more linear equations are expressed as equations, and the solved-for variables are not assigned.

  When solving anything other than two or more linear equations, `solve` ignores `globalsolve`. Other functions which solve equations (e.g., `algsys`) always ignore `globalsolve`.

  Examples:

      (%i1) globalsolve: true$
      (%i2) solve ([x + 3*y = 2, 2*x - y = 5], [x, y]);
      Solution

                                       17
      (%t2)                        x : --
                                       7

                                         1
      (%t3)                        y : - -
                                         7
      (%o3)                     [[%t2, %t3]]
      (%i3) x;
                                     17
      (%o3)                          --
                                     7
      (%i4) y;
                                       1
      (%o4)                          - -
                                       7
      (%i5) globalsolve: false$
      (%i6) kill (x, y)$
      (%i7) solve ([x + 3*y = 2, 2*x - y = 5], [x, y]);
      Solution

                                       17
      (%t7)                        x = --
                                       7

                                         1
      (%t8)                        y = - -
                                         7
      (%o8)                     [[%t7, %t8]]
      (%i8) x;
      (%o8)                           x
      (%i9) y;
      (%o9)                           y

  Categories: Linear equations

Function: ieqn (ie, unk, tech, n, guess)

  `inteqn` is a package for solving integral equations. `load ("inteqn")` loads this package.

  ie is the integral equation; unk is the unknown function; tech is the technique to be tried from those given above (tech = `first` means: try the first technique which finds a solution; tech = `all` means: try all applicable techniques); n is the maximum number of terms to take for `taylor`, `neumann`, `firstkindseries`, or `fredseries` (it is also the maximum depth of recursion for the differentiation method); guess is the initial guess for `neumann` or `firstkindseries`.

  Default values for the 2nd thru 5th parameters are:

  unk: `p(x)`, where p is the first function encountered in an integrand which is unknown to Maxima and x is the variable which occurs as an argument to the first occurrence of p found outside of an integral in the case of `secondkind` equations, or is the only other variable besides the variable of integration in `firstkind` equations. If the attempt to search for x fails, the user will be asked to supply the independent variable.

  tech: `first`

  n: 1

  guess: `none` which will cause `neumann` and `firstkindseries` to use `f(x)` as an initial guess.

  Categories: Integral equations

Option variable: ieqnprint

  Default value: `true`

  `ieqnprint` governs the behavior of the result returned by the `ieqn` command. When `ieqnprint` is `false`, the lists returned by the `ieqn` function are of the form

  [solution, technique used, nterms, flag]

  where flag is absent if the solution is exact.

  Otherwise, it is the word `approximate` or `incomplete` corresponding to an inexact or non-closed form solution, respectively. If a series method was used, nterms gives the number of terms taken (which could be less than the n given to `ieqn` if an error prevented generation of further terms).

  Categories: Integral equations

Function: lhs (expr)

  Returns the left-hand side (that is, the first argument) of the expression expr, when the operator of expr is one of the relational operators `< <= = # equal notequal >= >`, one of the assignment operators `:= ::= : ::`, or a user-defined binary infix operator, as declared by `infix`.

  When expr is an atom or its operator is something other than the ones listed above, `lhs` returns expr.

  See also `rhs`.

  Examples:

      (%i1) e: aa + bb = cc;
      (%o1)                     bb + aa = cc
      (%i2) lhs (e);
      (%o2)                        bb + aa
      (%i3) rhs (e);
      (%o3)                          cc
      (%i4) [lhs (aa < bb), lhs (aa <= bb), lhs (aa >= bb),
             lhs (aa > bb)];
      (%o4)                   [aa, aa, aa, aa]
      (%i5) [lhs (aa = bb), lhs (aa # bb), lhs (equal (aa, bb)),
             lhs (notequal (aa, bb))];
      (%o5)                   [aa, aa, aa, aa]
      (%i6) e1: '(foo(x) := 2*x);
      (%o6)                     foo(x) := 2 x
      (%i7) e2: '(bar(y) ::= 3*y);
      (%o7)                    bar(y) ::= 3 y
      (%i8) e3: '(x : y);
      (%o8)                         x : y
      (%i9) e4: '(x :: y);
      (%o9)                        x :: y
      (%i10) [lhs (e1), lhs (e2), lhs (e3), lhs (e4)];
      (%o10)               [foo(x), bar(y), x, x]
      (%i11) infix ("][");
      (%o11)                         ][
      (%i12) lhs (aa ][ bb);
      (%o12)                         aa

  Categories: Expressions

Function: linsolve ([expr_1, …, expr_m], [x_1, …, x_n])

  Solves the list of simultaneous linear equations for the list of variables. The expressions must each be polynomials in the variables and may be equations.

  When `globalsolve` is `true`, each solved-for variable is bound to its value in the solution of the equations.

  When `backsubst` is `false`, `linsolve` does not carry out back substitution after the equations have been triangularized. This may be necessary in very big problems where back substitution would cause the generation of extremely large expressions.

  When `linsolve_params` is `true`, `linsolve` also generates the `%r` symbols used to represent arbitrary parameters described in the manual under `algsys`. Otherwise, `linsolve` solves an under-determined system of equations with some variables expressed in terms of others.

  When `programmode` is `false`, `linsolve` displays the solution with intermediate expression (`%t`) labels, and returns the list of labels.

      (%i1) e1: x + z = y;
      (%o1)                       z + x = y
      (%i2) e2: 2*a*x - y = 2*a^2;
                                             2
      (%o2)                   2 a x - y = 2 a
      (%i3) e3: y - 2*z = 2;
      (%o3)                      y - 2 z = 2
      (%i4) [globalsolve: false, programmode: true];
      (%o4)                     [false, true]
      (%i5) linsolve ([e1, e2, e3], [x, y, z]);
      (%o5)            [x = a + 1, y = 2 a, z = a - 1]
      (%i6) [globalsolve: false, programmode: false];
      (%o6)                    [false, false]
      (%i7) linsolve ([e1, e2, e3], [x, y, z]);
      Solution

      (%t7)                       z = a - 1

      (%t8)                        y = 2 a

      (%t9)                       x = a + 1
      (%o9)                    [%t7, %t8, %t9]
      (%i9) ''%;
      (%o9)            [z = a - 1, y = 2 a, x = a + 1]
      (%i10) [globalsolve: true, programmode: false];
      (%o10)                    [true, false]
      (%i11) linsolve ([e1, e2, e3], [x, y, z]);
      Solution

      (%t11)                      z : a - 1

      (%t12)                       y : 2 a

      (%t13)                      x : a + 1
      (%o13)                 [%t11, %t12, %t13]
      (%i13) ''%;
      (%o13)           [z : a - 1, y : 2 a, x : a + 1]
      (%i14) [x, y, z];
      (%o14)                 [a + 1, 2 a, a - 1]
      (%i15) [globalsolve: true, programmode: true];
      (%o15)                    [true, true]
      (%i16) linsolve ([e1, e2, e3], '[x, y, z]);
      (%o16)           [x : a + 1, y : 2 a, z : a - 1]
      (%i17) [x, y, z];
      (%o17)                 [a + 1, 2 a, a - 1]

  Categories: Linear equations

Option variable: linsolvewarn

  Default value: `true`

  When `linsolvewarn` is `true`, `linsolve` prints a message "Dependent equations eliminated".

  Categories: Linear equations

Option variable: linsolve_params

  Default value: `true`

  When `linsolve_params` is `true`, `linsolve` also generates the `%r` symbols used to represent arbitrary parameters described in the manual under `algsys`. Otherwise, `linsolve` solves an under-determined system of equations with some variables expressed in terms of others.

  Categories: Linear equations

System variable: multiplicities

  Default value: `not_set_yet`

  `multiplicities` is set to a list of the multiplicities of the individual solutions returned by `solve` or `realroots`.

  Categories: Algebraic equations · Polynomials

Function: nroots (p, low, high)

  Returns the number of real roots of the real univariate polynomial p in the half-open interval `(low, high]`. The endpoints of the interval may be `minf` or `inf`.

  `nroots` uses the method of Sturm sequences.

      (%i1) p: x^10 - 2*x^4 + 1/2$
      (%i2) nroots (p, -6, 9.1);
      (%o2)                           4

  Categories: Polynomials · Numerical methods

Function: nthroot (p, n)

  where p is a polynomial with integer coefficients and n is a positive integer returns `q`, a polynomial over the integers, such that `q^n = p` or prints an error message indicating that p is not a perfect nth power. This routine is much faster than `factor` or even `sqfr`.

  Categories: Polynomials

Option variable: polyfactor

  Default value: `false`

  The option variable `polyfactor` when `true` causes `allroots` and `bfallroots` to factor the polynomial over the real numbers if the polynomial is real, or over the complex numbers, if the polynomial is complex.

  See `allroots` for an example.

  Categories: Polynomials · Numerical methods

Option variable: programmode

  Default value: `true`

  When `programmode` is `true`, `solve`, `realroots`, `allroots`, and `linsolve` return solutions as elements in a list. (Except when `backsubst` is set to `false`, in which case `programmode: false` is assumed.)

  When `programmode` is `false`, `solve`, etc. create intermediate expression labels `%t1`, `%t2`, etc., and assign the solutions to them.

  Categories: Algebraic equations · Polynomials

Option variable: realonly

  Default value: `false`

  When `realonly` is `true`, `algsys` returns only those solutions which are free of `%i`.

  Categories: Algebraic equations

Function: realroots
`realroots` (expr, bound)
`realroots` (eqn, bound)
`realroots` (expr)
`realroots` (eqn)

  Computes rational approximations of the real roots of the polynomial expr or polynomial equation eqn of one variable, to within a tolerance of bound. Coefficients of expr or eqn must be literal numbers; symbol constants such as `%pi` are rejected.

  `realroots` assigns the multiplicities of the roots it finds to the global variable `multiplicities`.

  `realroots` constructs a Sturm sequence to bracket each root, and then applies bisection to refine the approximations. All coefficients are converted to rational equivalents before searching for roots, and computations are carried out by exact rational arithmetic. Even if some coefficients are floating-point numbers, the results are rational (unless coerced to floats by the `float` or `numer` flags).

  When bound is less than 1, all integer roots are found exactly. When bound is unspecified, it is assumed equal to the global variable `rootsepsilon`.

  When the global variable `programmode` is `true`, `realroots` returns a list of the form `[x = x_1, x = x_2, ...]`. When `programmode` is `false`, `realroots` creates intermediate expression labels `%t1`, `%t2`, …, assigns the results to them, and returns the list of labels.

  Examples:

      (%i1) realroots (-1 - x + x^5, 5e-6);
                                     612003
      (%o1)                     [x = ------]
                                     524288
      (%i2) ev (%[1], float);
      (%o2)                 x = 1.167303085327148
      (%i3) ev (-1 - x + x^5, %);
      (%o3)                - 7.396496210176905E-6

      (%i1) realroots (expand ((1 - x)^5 * (2 - x)^3 * (3 - x)), 1e-20);
      (%o1)                 [x = 1, x = 2, x = 3]
      (%i2) multiplicities;
      (%o2)                       [5, 3, 1]

  Categories: Polynomials · Numerical methods

Function: rhs (expr)

  Returns the right-hand side (that is, the second argument) of the expression expr, when the operator of expr is one of the relational operators `< <= = # equal notequal >= >`, one of the assignment operators `:= ::= : ::`, or a user-defined binary infix operator, as declared by `infix`.

  When expr is an atom or its operator is something other than the ones listed above, `rhs` returns 0.

  See also `lhs`.

  Examples:

      (%i1) e: aa + bb = cc;
      (%o1)                     bb + aa = cc
      (%i2) lhs (e);
      (%o2)                        bb + aa
      (%i3) rhs (e);
      (%o3)                          cc
      (%i4) [rhs (aa < bb), rhs (aa <= bb), rhs (aa >= bb),
             rhs (aa > bb)];
      (%o4)                   [bb, bb, bb, bb]
      (%i5) [rhs (aa = bb), rhs (aa # bb), rhs (equal (aa, bb)),
             rhs (notequal (aa, bb))];
      (%o5)                   [bb, bb, bb, bb]
      (%i6) e1: '(foo(x) := 2*x);
      (%o6)                     foo(x) := 2 x
      (%i7) e2: '(bar(y) ::= 3*y);
      (%o7)                    bar(y) ::= 3 y
      (%i8) e3: '(x : y);
      (%o8)                         x : y
      (%i9) e4: '(x :: y);
      (%o9)                        x :: y
      (%i10) [rhs (e1), rhs (e2), rhs (e3), rhs (e4)];
      (%o10)                  [2 x, 3 y, y, y]
      (%i11) infix ("][");
      (%o11)                         ][
      (%i12) rhs (aa ][ bb);
      (%o12)                         bb

  Categories: Expressions

Option variable: rootsconmode

  Default value: `true`

  `rootsconmode` governs the behavior of the `rootscontract` command. See `rootscontract` for details.

  Categories: Expressions · Simplification flags and variables

Function: rootscontract (expr)

  Converts products of roots into roots of products. For example, `rootscontract (sqrt(x)*y^(3/2))` yields `sqrt(x*y^3)`.

  When `radexpand` is `true` and `domain` is `real`, `rootscontract` converts `abs` into `sqrt`, e.g., `rootscontract (abs(x)*sqrt(y))` yields `sqrt(x^2*y)`.

  There is an option `rootsconmode` affecting `rootscontract` as follows:

      Problem            Value of        Result of applying
                        rootsconmode        rootscontract

      x^(1/2)*y^(3/2)      false          (x*y^3)^(1/2)
      x^(1/2)*y^(1/4)      false          x^(1/2)*y^(1/4)
      x^(1/2)*y^(1/4)      true           (x*y^(1/2))^(1/2)
      x^(1/2)*y^(1/3)      true           x^(1/2)*y^(1/3)
      x^(1/2)*y^(1/4)      all            (x^2*y)^(1/4)
      x^(1/2)*y^(1/3)      all            (x^3*y^2)^(1/6)

  When `rootsconmode` is `false`, `rootscontract` contracts only with respect to rational number exponents whose denominators are the same. The key to the `rootsconmode: true` examples is simply that 2 divides into 4 but not into 3. `rootsconmode: all` involves taking the least common multiple of the denominators of the exponents.

  `rootscontract` uses `ratsimp` in a manner similar to `logcontract`.

  Examples:

      (%i1) rootsconmode: false$
      (%i2) rootscontract (x^(1/2)*y^(3/2));
                                         3
      (%o2)                      sqrt(x y )
      (%i3) rootscontract (x^(1/2)*y^(1/4));
                                         1/4
      (%o3)                     sqrt(x) y
      (%i4) rootsconmode: true$
      (%i5) rootscontract (x^(1/2)*y^(1/4));
      (%o5)                    sqrt(x sqrt(y))
      (%i6) rootscontract (x^(1/2)*y^(1/3));
                                         1/3
      (%o6)                     sqrt(x) y
      (%i7) rootsconmode: all$
      (%i8) rootscontract (x^(1/2)*y^(1/4));
                                    2   1/4
      (%o8)                       (x  y)
      (%i9) rootscontract (x^(1/2)*y^(1/3));
                                   3  2 1/6
      (%o9)                      (x  y )
      (%i10) rootsconmode: false$
      (%i11) rootscontract (sqrt(sqrt(x) + sqrt(1 + x))
                          *sqrt(sqrt(1 + x) - sqrt(x)));
      (%o11)                          1
      (%i12) rootsconmode: true$
      (%i13) rootscontract (sqrt(5+sqrt(5)) - 5^(1/4)*sqrt(1+sqrt(5)));
      (%o13)                          0

  Categories: Simplification functions

Option variable: rootsepsilon

  Default value: 1.0e-7

  `rootsepsilon` is the tolerance which establishes the confidence interval for the roots found by the `realroots` function.

  Categories: Polynomials · Numerical methods

Function: solve
`solve` (expr, x)
`solve` (expr)
`solve` ([eqn_1, …, eqn_n], [x_1, …, x_n])

  Solves the algebraic equation expr for the variable x and returns a list of solution equations in x. If expr is not an equation, the equation `expr = 0` is assumed in its place. x may be a function (e.g. `f(x)`), or other non-atomic expression except a sum or product. x may be omitted if expr contains only one variable. expr may be a rational expression, and may contain trigonometric functions, exponentials, etc.

  The following method is used:

  Let E be the expression and X be the variable. If E is linear in X then it is trivially solved for X. Otherwise if E is of the form `A*X^N + B` then the result is `(-B/A)^1/N)` times the `N`'th roots of unity.

  If E is not linear in X then the gcd of the exponents of X in E (say N) is divided into the exponents and the multiplicity of the roots is multiplied by N. Then `solve` is called again on the result. If E factors then `solve` is called on each of the factors. Finally `solve` will use the quadratic, cubic, or quartic formulas where necessary.

  In the case where E is a polynomial in some function of the variable to be solved for, say `F(X)`, then it is first solved for `F(X)` (call the result C), then the equation `F(X)=C` can be solved for X provided the inverse of the function F is known.

  `breakup` if `false` will cause `solve` to express the solutions of cubic or quartic equations as single expressions rather than as made up of several common subexpressions which is the default.

  `multiplicities` - will be set to a list of the multiplicities of the individual solutions returned by `solve`, `realroots`, or `allroots`. Try `apropos (solve)` for the switches which affect `solve`. `describe` may then by used on the individual switch names if their purpose is not clear.

  `solve ([eqn_1, ..., eqn_n], [x_1, ..., x_n])` solves a system of simultaneous (linear or non-linear) polynomial equations by calling `linsolve` or `algsys` and returns a list of the solution lists in the variables. In the case of `linsolve` this list would contain a single list of solutions. It takes two lists as arguments. The first list represents the equations to be solved; the second list is a list of the unknowns to be determined. If the total number of variables in the equations is equal to the number of equations, the second argument-list may be omitted.

  When `programmode` is `false`, `solve` displays solutions with intermediate expression (`%t`) labels, and returns the list of labels.

  When `globalsolve` is `true` and the problem is to solve two or more linear equations, each solved-for variable is bound to its value in the solution of the equations.

  Examples:

      (%i1) solve (asin (cos (3*x))*(f(x) - 1), x);

      solve: using arc-trig functions to get a solution.
      Some solutions will be lost.
                                  %pi
      (%o1)                  [x = ---, f(x) = 1]
                                   6
      (%i2) ev (solve (5^f(x) = 125, f(x)), solveradcan);
                                      log(125)
      (%o2)                   [f(x) = --------]
                                       log(5)
      (%i3) [4*x^2 - y^2 = 12, x*y - x = 2];
                            2    2
      (%o3)             [4 x  - y  = 12, x y - x = 2]

      (%i4) solve (%, [x, y]);
      (%o4) [[x = 2, y = 2], [x = .5202594388652008 %i
       - .1331240357358706, y = .07678378523787788
       - 3.608003221870287 %i], [x = - .5202594388652008 %i
       - .1331240357358706, y = 3.608003221870287 %i
       + .07678378523787788], [x = - 1.733751846381093,
      y = - .1535675710019696]]

      (%i5) solve (1 + a*x + x^3, x);

                                             3
                    sqrt(3) %i   1   sqrt(4 a  + 27)   1 1/3
      (%o5) [x = (- ---------- - -) (--------------- - -)
                        2        2      6 sqrt(3)      2

              sqrt(3) %i   1
             (---------- - -) a
                  2        2
       - --------------------------, x =
                    3
            sqrt(4 a  + 27)   1 1/3
         3 (--------------- - -)
               6 sqrt(3)      2

                                3
       sqrt(3) %i   1   sqrt(4 a  + 27)   1 1/3
      (---------- - -) (--------------- - -)
           2        2      6 sqrt(3)      2

               sqrt(3) %i   1
            (- ---------- - -) a
                   2        2
       - --------------------------, x =
                    3
            sqrt(4 a  + 27)   1 1/3
         3 (--------------- - -)
               6 sqrt(3)      2

               3
       sqrt(4 a  + 27)   1 1/3               a
      (--------------- - -)    - --------------------------]
          6 sqrt(3)      2                  3
                                    sqrt(4 a  + 27)   1 1/3
                                 3 (--------------- - -)
                                       6 sqrt(3)      2
      (%i6) solve (x^3 - 1);
                   sqrt(3) %i - 1        sqrt(3) %i + 1
      (%o6)   [x = --------------, x = - --------------, x = 1]
                         2                     2
      (%i7) solve (x^6 - 1);
                 sqrt(3) %i + 1      sqrt(3) %i - 1
      (%o7) [x = --------------, x = --------------, x = - 1,
                       2                   2

                           sqrt(3) %i + 1        sqrt(3) %i - 1
                     x = - --------------, x = - --------------, x = 1]
                                 2                     2
      (%i8) ev (x^6 - 1, %[1]);
                                            6
                            (sqrt(3) %i + 1)
      (%o8)                 ----------------- - 1
                                   64
      (%i9) expand (%);
      (%o9)                           0
      (%i10) x^2 - 1;
                                    2
      (%o10)                       x  - 1
      (%i11) solve (%, x);
      (%o11)                  [x = - 1, x = 1]
      (%i12) ev (%th(2), %[1]);
      (%o12)                          0

  The symbols `%r` are used to denote arbitrary constants in a solution.

      (%i1) solve([x+y=1,2*x+2*y=2],[x,y]);

      solve: dependent equations eliminated: (2)
      (%o1)                      [[x = 1 - %r1, y = %r1]]

  See `algsys` and `%rnum_list` for more information.

  Categories: Algebraic equations

Option variable: solvedecomposes

  Default value: `true`

  When `solvedecomposes` is `true`, `solve` calls `polydecomp` if asked to solve polynomials.

  Categories: Algebraic equations

Option variable: solveexplicit

  Default value: `false`

  When `solveexplicit` is `true`, inhibits `solve` from returning implicit solutions, that is, solutions of the form `F(x) = 0` where `F` is some function.

  Categories: Algebraic equations

Option variable: solvefactors

  Default value: `true`

  When `solvefactors` is `false`, `solve` does not try to factor the expression. The `false` setting may be desired in some cases where factoring is not necessary.

  Categories: Algebraic equations

Option variable: solvenullwarn

  Default value: `true`

  When `solvenullwarn` is `true`, `solve` prints a warning message if called with either a null equation list or a null variable list. For example, `solve ([], [])` would print two warning messages and return `[]`.

  Categories: Algebraic equations

Option variable: solveradcan

  Default value: `false`

  When `solveradcan` is `true`, `solve` calls `radcan` which makes `solve` slower but will allow certain problems containing exponentials and logarithms to be solved.

  Categories: Algebraic equations

Option variable: solvetrigwarn

  Default value: `true`

  When `solvetrigwarn` is `true`, `solve` may print a message saying that it is using inverse trigonometric functions to solve the equation, and thereby losing solutions.

  Categories: Algebraic equations

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# 21. Differential Equations

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## 21.1 Introduction to Differential Equations

This section describes the functions available in Maxima to obtain analytic solutions for some specific types of first and second-order equations. To obtain a numerical solution for a system of differential equations, see the additional package `dynamics`. For graphical representations in phase space, see the additional package `plotdf`.

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## 21.2 Functions and Variables for Differential Equations

Function: bc2 (solution, xval1, yval1, xval2, yval2)

  Solves a boundary value problem for a second order differential equation. Here: solution is a general solution to the equation, as found by `ode2`; xval1 specifies the value of the independent variable in a first point, in the form `x = x1`, and yval1 gives the value of the dependent variable in that point, in the form `y = y1`. The expressions xval2 and yval2 give the values for these variables at a second point, using the same form.

  See `ode2` for an example of its usage.

  Categories: Differential equations

Function: desolve
`desolve` (eqn, x)
`desolve` ([eqn_1, ..., eqn_n], [x_1, ..., x_n])

  The function `desolve` solves systems of linear ordinary differential equations using Laplace transform. Here the eqn's are differential equations in the dependent variables x_1, ..., x_n. The functional dependence of x_1, ..., x_n on an independent variable, for instance x, must be explicitly indicated in the variables and its derivatives. For example, this would not be the correct way to define two equations:

      eqn_1: 'diff(f,x,2) = sin(x) + 'diff(g,x);
      eqn_2: 'diff(f,x) + x^2 - f = 2*'diff(g,x,2);

  The correct way would be:

      eqn_1: 'diff(f(x),x,2) = sin(x) + 'diff(g(x),x);
      eqn_2: 'diff(f(x),x) + x^2 - f(x) = 2*'diff(g(x),x,2);

  The call to the function `desolve` would then be

      desolve([eqn_1, eqn_2], [f(x),g(x)]);

  If initial conditions at `x=0` are known, they can be supplied before calling `desolve` by using `atvalue`.

      (%i1) 'diff(f(x),x)='diff(g(x),x)+sin(x);
                       d           d
      (%o1)            -- (f(x)) = -- (g(x)) + sin(x)
                       dx          dx
      (%i2) 'diff(g(x),x,2)='diff(f(x),x)-cos(x);
                        2
                       d            d
      (%o2)            --- (g(x)) = -- (f(x)) - cos(x)
                         2          dx
                       dx
      (%i3) atvalue('diff(g(x),x),x=0,a);
      (%o3)                           a
      (%i4) atvalue(f(x),x=0,1);
      (%o4)                           1
      (%i5) desolve([%o1,%o2],[f(x),g(x)]);
                        x
      (%o5) [f(x) = a %e  - a + 1, g(x) =

                                                      x
                                         cos(x) + a %e  - a + g(0) - 1]
      (%i6) [%o1,%o2],%o5,diff;
                   x       x      x                x
      (%o6)   [a %e  = a %e , a %e  - cos(x) = a %e  - cos(x)]

  If `desolve` cannot obtain a solution, it returns `false`.

  Categories: Differential equations · Laplace transform

Function: ic1 (solution, xval, yval)

  Solves initial value problems for first order differential equations. Here solution is a general solution to the equation, as found by `ode2`, xval gives an initial value for the independent variable in the form `x = x0`, and yval gives the initial value for the dependent variable in the form `y = y0`.

  See `ode2` for an example of its usage.

  Categories: Differential equations

Function: ic2 (solution, xval, yval, dval)

  Solves initial value problems for second-order differential equations. Here solution is a general solution to the equation, as found by `ode2`, xval gives the initial value for the independent variable in the form `x = x0`, yval gives the initial value of the dependent variable in the form `y = y0`, and dval gives the initial value for the first derivative of the dependent variable with respect to independent variable, in the form `diff(y,x) = dy0` (`diff` does not have to be quoted).

  See `ode2` for an example of its usage.

  Categories: Differential equations

Function: ode2 (eqn, dvar, ivar)

  The function `ode2` solves an ordinary differential equation (ODE) of first or second order. It takes three arguments: an ODE given by eqn, the dependent variable dvar, and the independent variable ivar. When successful, it returns either an explicit or implicit solution for the dependent variable. `%c` is used to represent the integration constant in the case of first-order equations, and `%k1` and `%k2` the constants for second-order equations. The dependence of the dependent variable on the independent variable does not have to be written explicitly, as in the case of `desolve`, but the independent variable must always be given as the third argument.

  If `ode2` cannot obtain a solution for whatever reason, it returns `false`, after perhaps printing out an error message. The methods implemented for first order equations in the order in which they are tested are: linear, separable, exact - perhaps requiring an integrating factor, homogeneous, Bernoulli's equation, and a generalized homogeneous method. The types of second-order equations which can be solved are: constant coefficients, exact, linear homogeneous with non-constant coefficients which can be transformed to constant coefficients, the Euler or equi-dimensional equation, equations solvable by the method of variation of parameters, and equations which are free of either the independent or of the dependent variable so that they can be reduced to two first order linear equations to be solved sequentially.

  In the course of solving ODE's, several variables are set purely for informational purposes: `method` denotes the method of solution used (e.g., `linear`), `intfactor` denotes any integrating factor used, `odeindex` denotes the index for Bernoulli's method or for the generalized homogeneous method, and `yp` denotes the particular solution for the variation of parameters technique.

  In order to solve initial value problems (IVP) functions `ic1` and `ic2` are available for first and second order equations, and to solve second-order boundary value problems (BVP) the function `bc2` can be used.

  Example:

      (%i1) x^2*'diff(y,x) + 3*y*x = sin(x)/x;
                            2 dy           sin(x)
      (%o1)                x  -- + 3 x y = ------
                              dx             x
      (%i2) ode2(%,y,x);
                                   %c - cos(x)
      (%o2)                    y = -----------
                                        3
                                       x
      (%i3) ic1(%o2,x=%pi,y=0);
                                    cos(x) + 1
      (%o3)                   y = - ----------
                                         3
                                        x
      (%i4) 'diff(y,x,2) + y*'diff(y,x)^3 = 0;
                               2
                              d y      dy 3
      (%o4)                   --- + y (--)  = 0
                                2      dx
                              dx
      (%i5) ode2(%,y,x);
                            3
                           y  + 6 %k1 y
      (%o5)                ------------ = x + %k2
                                6
      (%i6) ratsimp(ic2(%o5,x=0,y=0,'diff(y,x)=2));
                                   3
                                2 y  - 3 y
      (%o6)                   - ---------- = x
                                    6
      (%i7) bc2(%o5,x=0,y=1,x=1,y=3);
                               3
                              y  - 10 y       3
      (%o7)                   --------- = x - -
                                  6           2

  Categories: Differential equations

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# 22. Numerical

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## 22.1 Introduction to fast Fourier transform

The `fft` package comprises functions for the numerical (not symbolic) computation of the fast Fourier transform.

Categories: Fourier transform · Numerical methods · Share packages · Package fft

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## 22.2 Functions and Variables for fast Fourier transform

Function: polartorect (r, t)

  Translates complex values of the form `r %e^(%i t)` to the form `a + b %i`, where r is the magnitude and t is the phase. r and t are 1-dimensional arrays of the same size. The array size need not be a power of 2.

  The original values of the input arrays are replaced by the real and imaginary parts, `a` and `b`, on return. The outputs are calculated as

      a = r cos(t)
      b = r sin(t)

  `polartorect` is the inverse function of `recttopolar`.

  `load(fft)` loads this function. See also `fft`.

  Categories: Package fft · Complex variables

Function: recttopolar (a, b)

  Translates complex values of the form `a + b %i` to the form `r %e^(%i t)`, where a is the real part and b is the imaginary part. a and b are 1-dimensional arrays of the same size. The array size need not be a power of 2.

  The original values of the input arrays are replaced by the magnitude and angle, `r` and `t`, on return. The outputs are calculated as

      r = sqrt(a^2 + b^2)
      t = atan2(b, a)

  The computed angle is in the range `-%pi` to `%pi`.

  `recttopolar` is the inverse function of `polartorect`.

  `load(fft)` loads this function. See also `fft`.

  Categories: Package fft · Complex variables

Function: inverse_fft (y)

  Computes the inverse complex fast Fourier transform. y is a list or array (named or unnamed) which contains the data to transform. The number of elements must be a power of 2. The elements must be literal numbers (integers, rationals, floats, or bigfloats) or symbolic constants, or expressions `a + b*%i` where `a` and `b` are literal numbers or symbolic constants.

  `inverse_fft` returns a new object of the same type as y, which is not modified. Results are always computed as floats or expressions `a + b*%i` where `a` and `b` are floats. If bigfloat precision is needed the function `bf_inverse_fft` can be used instead as a drop-in replacement of `inverse_fft` that is slower, but supports bfloats.

  The inverse discrete Fourier transform is defined as follows. Let `x` be the output of the inverse transform. Then for `j` from 0 through `n - 1`,

      x[j] = sum(y[k] exp(-2 %i %pi j k / n), k, 0, n - 1)

  As there are various sign and normalization conventions possible, this definition of the transform may differ from that used by other mathematical software.

  `load(fft)` loads this function.

  See also `fft` (forward transform), `recttopolar`, and `polartorect`.

  Examples:

  Real data.

      (%i1) load ("fft") $
      (%i2) fpprintprec : 4 $
      (%i3) L : [1, 2, 3, 4, -1, -2, -3, -4] $
      (%i4) L1 : inverse_fft (L);
      (%o4) [0.0, 14.49 %i - .8284, 0.0, 2.485 %i + 4.828, 0.0,
                             4.828 - 2.485 %i, 0.0, - 14.49 %i - .8284]
      (%i5) L2 : fft (L1);
      (%o5) [1.0, 2.0 - 2.168L-19 %i, 3.0 - 7.525L-20 %i,
      4.0 - 4.256L-19 %i, - 1.0, 2.168L-19 %i - 2.0,
      7.525L-20 %i - 3.0, 4.256L-19 %i - 4.0]
      (%i6) lmax (abs (L2 - L));
      (%o6)                       3.545L-16

  Complex data.

      (%i1) load ("fft") $
      (%i2) fpprintprec : 4 $
      (%i3) L : [1, 1 + %i, 1 - %i, -1, -1, 1 - %i, 1 + %i, 1] $
      (%i4) L1 : inverse_fft (L);
      (%o4) [4.0, 2.711L-19 %i + 4.0, 2.0 %i - 2.0,
      - 2.828 %i - 2.828, 0.0, 5.421L-20 %i + 4.0, - 2.0 %i - 2.0,
      2.828 %i + 2.828]
      (%i5) L2 : fft (L1);
      (%o5) [4.066E-20 %i + 1.0, 1.0 %i + 1.0, 1.0 - 1.0 %i,
      1.55L-19 %i - 1.0, - 4.066E-20 %i - 1.0, 1.0 - 1.0 %i,
      1.0 %i + 1.0, 1.0 - 7.368L-20 %i]
      (%i6) lmax (abs (L2 - L));
      (%o6)                       6.841L-17

  Categories: Package fft

Function: fft (x)

  Computes the complex fast Fourier transform. x is a list or array (named or unnamed) which contains the data to transform. The number of elements must be a power of 2. The elements must be literal numbers (integers, rationals, floats, or bigfloats) or symbolic constants, or expressions `a + b*%i` where `a` and `b` are literal numbers or symbolic constants.

  `fft` returns a new object of the same type as x, which is not modified. Results are always computed as floats or expressions `a + b*%i` where `a` and `b` are floats. If bigfloat precision is needed the function `bf_fft` can be used instead as a drop-in replacement of `fft` that is slower, but supports bfloats. In addition if it is known that the input consists of only real values (no imaginary parts), `real_fft` can be used which is potentially faster.

  The discrete Fourier transform is defined as follows. Let `y` be the output of the transform. Then for `k` from 0 through `n - 1`,

      y[k] = (1/n) sum(x[j] exp(+2 %i %pi j k / n), j, 0, n - 1)

  As there are various sign and normalization conventions possible, this definition of the transform may differ from that used by other mathematical software.

  When the data x are real, real coefficients `a` and `b` can be computed such that

      x[j] = sum(a[k]*cos(2*%pi*j*k/n)+b[k]*sin(2*%pi*j*k/n), k, 0, n/2)

  with

      a[0] = realpart (y[0])
      b[0] = 0

  and, for k from 1 through n/2 - 1,

      a[k] = realpart (y[k] + y[n - k])
      b[k] = imagpart (y[n - k] - y[k])

  and

      a[n/2] = realpart (y[n/2])
      b[n/2] = 0

  `load(fft)` loads this function.

  See also `inverse_fft` (inverse transform), `recttopolar`, and `polartorect`.. See `real_fft` for FFTs of a real-valued input, and `bf_fft` and `bf_real_fft` for operations on bigfloat values.

  Examples:

  Real data.

      (%i1) load ("fft") $
      (%i2) fpprintprec : 4 $
      (%i3) L : [1, 2, 3, 4, -1, -2, -3, -4] $
      (%i4) L1 : fft (L);
      (%o4) [0.0, - 1.811 %i - .1036, 0.0, .6036 - .3107 %i, 0.0,
                               .3107 %i + .6036, 0.0, 1.811 %i - .1036]
      (%i5) L2 : inverse_fft (L1);
      (%o5) [1.0, 2.168L-19 %i + 2.0, 7.525L-20 %i + 3.0,
      4.256L-19 %i + 4.0, - 1.0, - 2.168L-19 %i - 2.0,
      - 7.525L-20 %i - 3.0, - 4.256L-19 %i - 4.0]
      (%i6) lmax (abs (L2 - L));
      (%o6)                       3.545L-16

  Complex data.

      (%i1) load ("fft") $
      (%i2) fpprintprec : 4 $
      (%i3) L : [1, 1 + %i, 1 - %i, -1, -1, 1 - %i, 1 + %i, 1] $
      (%i4) L1 : fft (L);
      (%o4) [0.5, .3536 %i + .3536, - 0.25 %i - 0.25,
      0.5 - 6.776L-21 %i, 0.0, - .3536 %i - .3536, 0.25 %i - 0.25,
      0.5 - 3.388L-20 %i]
      (%i5) L2 : inverse_fft (L1);
      (%o5) [1.0 - 4.066E-20 %i, 1.0 %i + 1.0, 1.0 - 1.0 %i,
      - 1.008L-19 %i - 1.0, 4.066E-20 %i - 1.0, 1.0 - 1.0 %i,
      1.0 %i + 1.0, 1.947L-20 %i + 1.0]
      (%i6) lmax (abs (L2 - L));
      (%o6)                       6.83L-17

  Computation of sine and cosine coefficients.

      (%i1) load ("fft") $
      (%i2) fpprintprec : 4 $
      (%i3) L : [1, 2, 3, 4, 5, 6, 7, 8] $
      (%i4) n : length (L) $
      (%i5) x : make_array (any, n) $
      (%i6) fillarray (x, L) $
      (%i7) y : fft (x) $
      (%i8) a : make_array (any, n/2 + 1) $
      (%i9) b : make_array (any, n/2 + 1) $
      (%i10) a[0] : realpart (y[0]) $
      (%i11) b[0] : 0 $
      (%i12) for k : 1 thru n/2 - 1 do
         (a[k] : realpart (y[k] + y[n - k]),
          b[k] : imagpart (y[n - k] - y[k]));
      (%o12)                        done
      (%i13) a[n/2] : y[n/2] $
      (%i14) b[n/2] : 0 $
      (%i15) listarray (a);
      (%o15)          [4.5, - 1.0, - 1.0, - 1.0, - 0.5]
      (%i16) listarray (b);
      (%o16)           [0, - 2.414, - 1.0, - .4142, 0]
      (%i17) f(j) := sum (a[k]*cos(2*%pi*j*k/n) + b[k]*sin(2*%pi*j*k/n),
                          k, 0, n/2) $
      (%i18) makelist (float (f (j)), j, 0, n - 1);
      (%o18)      [1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0]

  Categories: Package fft

Function: real_fft (x)

  Computes the fast Fourier transform of a real-valued sequence x. This is equivalent to performing `fft(x)`, except that only the first `N/2+1` results are returned, where `N` is the length of x. `N` must be power of two.

  No check is made that x contains only real values.

  The symmetry properites of the Fourier transform of real sequences to reduce he complexity. In particular the first and last output values of `real_fft` are purely real. For larger sequencyes, `real_fft` may be computed more quickly than `fft`.

  Since the output length is short, the normal `inverse_fft` cannot be directly used. Use `inverse_real_fft` to compute the inverse.

  Package fft

Function: inverse_real_fft (y)

  Computes the inverse Fourier transfrom of y, which must have a length of `N/2+1` where `N` is a power of two. That is, the input x is expected to be the output of `real_fft`.

  No check is made to ensure that the input has the correct format. (The first and last elements must be purely real.)

  Categories: Package fft

Function: bf_inverse_fft (y)

  Computes the inverse complex fast Fourier transform. This is the bigfloat version of `inverse_fft` that converts the input to bigfloats and returns a bigfloat result.

  Package fft

Function: bf_fft (y)

  Computes the forward complex fast Fourier transform. This is the bigfloat version of `fft` that converts the input to bigfloats and returns a bigfloat result.

  Package fft

Function: bf_real_fft (x)

  Computes the forward fast Fourier transform of a real-valued input returning a bigfloat result. This is the bigfloat version of `real_fft`.

  Categories: Package fft

Function: bf_inverse_real_fft (y)

  Computes the inverse fast Fourier transfrom with a real-valued bigfloat output. This is the bigfloat version of `inverse_real_fft`.

  Categories: Package fft

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## 22.3 Functions for numerical solution of equations

Function: horner
`horner` (expr, x)
`horner` (expr)

  Returns a rearranged representation of expr as in Horner's rule, using x as the main variable if it is specified. `x` may be omitted in which case the main variable of the canonical rational expression form of expr is used.

  `horner` sometimes improves stability if `expr` is to be numerically evaluated. It is also useful if Maxima is used to generate programs to be run in Fortran. See also `stringout`.

      (%i1) expr: 1e-155*x^2 - 5.5*x + 5.2e155;
                                 2
      (%o1)             1.e-155 x  - 5.5 x + 5.2e+155
      (%i2) expr2: horner (%, x), keepfloat: true;
      (%o2)         1.0 ((1.e-155 x - 5.5) x + 5.2e+155)
      (%i3) ev (expr, x=1e155);
      Maxima encountered a Lisp error:

       arithmetic error FLOATING-POINT-OVERFLOW signalled

      Automatically continuing.
      To enable the Lisp debugger set *debugger-hook* to nil.
      (%i4) ev (expr2, x=1e155);
      (%o4)                 7.00000000000001e+154

  Categories: Numerical methods

Function: find_root (expr, x, a, b, [abserr, relerr])
Function: find_root (f, a, b, [abserr, relerr])
Function: bf_find_root (expr, x, a, b, [abserr, relerr])
Function: bf_find_root (f, a, b, [abserr, relerr])
Option variable: find_root_error
Option variable: find_root_abs
Option variable: find_root_rel

  Finds a root of the expression expr or the function f over the closed interval [a, b]. The expression expr may be an equation, in which case `find_root` seeks a root of `lhs(expr) - rhs(expr)`.

  Given that Maxima can evaluate expr or f over [a, b] and that expr or f is continuous, `find_root` is guaranteed to find the root, or one of the roots if there is more than one.

  `find_root` initially applies binary search. If the function in question appears to be smooth enough, `find_root` applies linear interpolation instead.

  `bf_find_root` is a bigfloat version of `find_root`. The function is computed using bigfloat arithmetic and a bigfloat result is returned. Otherwise, `bf_find_root` is identical to `find_root`, and the following description is equally applicable to `bf_find_root`.

  The accuracy of `find_root` is governed by `abserr` and `relerr`, which are optional keyword arguments to `find_root`. These keyword arguments take the form `key=val`. The keyword arguments are

  `abserr`

    Desired absolute error of function value at root. Default is `find_root_abs`.

  `relerr`

    Desired relative error of root. Default is `find_root_rel`.

  `find_root` stops when the function in question evaluates to something less than or equal to `abserr`, or if successive approximants x_0, x_1 differ by no more than `relerr * max(abs(x_0), abs(x_1))`. The default values of `find_root_abs` and `find_root_rel` are both zero.

  `find_root` expects the function in question to have a different sign at the endpoints of the search interval. When the function evaluates to a number at both endpoints and these numbers have the same sign, the behavior of `find_root` is governed by `find_root_error`. When `find_root_error` is `true`, `find_root` prints an error message. Otherwise `find_root` returns the value of `find_root_error`. The default value of `find_root_error` is `true`.

  If f evaluates to something other than a number at any step in the search algorithm, `find_root` returns a partially-evaluated `find_root` expression.

  The order of a and b is ignored; the region in which a root is sought is [min(a, b), max(a, b)].

  Examples:

      (%i1) f(x) := sin(x) - x/2;
                                              x
      (%o1)                  f(x) := sin(x) - -
                                              2
      (%i2) find_root (sin(x) - x/2, x, 0.1, %pi);
      (%o2)                   1.895494267033981
      (%i3) find_root (sin(x) = x/2, x, 0.1, %pi);
      (%o3)                   1.895494267033981
      (%i4) find_root (f(x), x, 0.1, %pi);
      (%o4)                   1.895494267033981
      (%i5) find_root (f, 0.1, %pi);
      (%o5)                   1.895494267033981
      (%i6) find_root (exp(x) = y, x, 0, 100);
                                  x
      (%o6)           find_root(%e  = y, x, 0.0, 100.0)
      (%i7) find_root (exp(x) = y, x, 0, 100), y = 10;
      (%o7)                   2.302585092994046
      (%i8) log (10.0);
      (%o8)                   2.302585092994046
      (%i9) fpprec:32;
      (%o9)                           32
      (%i10) bf_find_root (exp(x) = y, x, 0, 100), y = 10;
      (%o10)                  2.3025850929940456840179914546844b0
      (%i11) log(10b0);
      (%o11)                  2.3025850929940456840179914546844b0

  Categories: Algebraic equations · Numerical methods

Function: newton (expr, x, x_0, eps)

  Returns an approximate solution of `expr = 0` by Newton's method, considering expr to be a function of one variable, x. The search begins with `x = x_0` and proceeds until `abs(expr) < eps` (with expr evaluated at the current value of x).

  `newton` allows undefined variables to appear in expr, so long as the termination test `abs(expr) < eps` evaluates to `true` or `false`. Thus it is not necessary that expr evaluate to a number.

  `load(newton1)` loads this function.

  See also `realroots`, `allroots`, `find_root` and mnewton

  Examples:

      (%i1) load ("newton1");
      (%o1)  /maxima/share/numeric/newton1.mac
      (%i2) newton (cos (u), u, 1, 1/100);
      (%o2)                   1.570675277161251
      (%i3) ev (cos (u), u = %);
      (%o3)                 1.2104963335033529e-4
      (%i4) assume (a > 0);
      (%o4)                        [a > 0]
      (%i5) newton (x^2 - a^2, x, a/2, a^2/100);
      (%o5)                  1.00030487804878 a
      (%i6) ev (x^2 - a^2, x = %);
                                                 2
      (%o6)                6.098490481853958e-4 a

  Categories: Algebraic equations · Numerical methods

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## 22.4 Introduction to numerical solution of differential equations

The Ordinary Differential Equations (ODE) solved by the functions in this section should have the form,

           dy
           -- = F(x,y)
           dx

which is a first-order ODE. Higher order differential equations of order n must be written as a system of n first-order equations of that kind. For instance, a second-order ODE should be written as a system of two equations

           dx               dy
           -- = G(x,y,t)    -- = F(x,y,t)
           dt               dt

The first argument in the functions will be a list with the expressions on the right-side of the ODE's. The variables whose derivatives are represented by those expressions should be given in a second list. In the case above those variables are x and y. The independent variable, t in the examples above, might be given in a separated option. If the expressions given do not depend on that independent variable, the system is called autonomous.

Categories: Differential equations · Numerical methods · Plotting

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## 22.5 Functions for numerical solution of differential equations

Function: plotdf
`plotdf` (dydx, options…)
`plotdf` (dvdu, [u,v], options…)
`plotdf` ([dxdt,cdydt], options…)
`plotdf` ([dudt,cdvdt], [u,cv], options…)

  The function `plotdf` creates a two-dimensional plot of the direction field (also called slope field) for a first-order Ordinary Differential Equation (ODE) or a system of two autonomous first-order ODE's.

  Plotdf requires Xmaxima, even if its run from a Maxima session in a console, since the plot will be created by the Tk scripts in Xmaxima. If Xmaxima is not installed plotdf will not work.

  dydx, dxdt and dydt are expressions that depend on x and y. dvdu, dudt and dvdt are expressions that depend on u and v. In addition to those two variables, the expressions can also depend on a set of parameters, with numerical values given with the `parameters` option (the option syntax is given below), or with a range of allowed values specified by a sliders option.

  Several other options can be given within the command, or selected in the menu. Integral curves can be obtained by clicking on the plot, or with the option `trajectory_at`. The direction of the integration can be controlled with the `direction` option, which can have values of forward, backward or both. The number of integration steps is given by `nsteps`; at each integration step the time increment will be adjusted automatically to produce displacements much smaller than the size of the plot window. The numerical method used is 4th order Runge-Kutta with variable time steps.

  Plot window menu:

  The menu bar of the plot window has the following seven icons:

  An X. Can be used to close the plot window.

  A wrench and a screwdriver. Opens the configuration menu with several fields that show the ODE(s) in use and various other settings. If a pair of coordinates are entered in the field Trajectory at and the `enter` key is pressed, a new integral curve will be shown, in addition to the ones already shown.

  Two arrows following a circle. Replots the direction field with the new settings defined in the configuration menu and replots only the last integral curve that was previously plotted.

  Hard disk drive with an arrow. Used to save a copy of the plot, in Postscript format, in the file specified in a field of the box that appears when that icon is clicked.

  Magnifying glass with a plus sign. Zooms in the plot.

  Magnifying glass with a minus sign. Zooms out the plot. The plot can be displaced by holding down the right mouse button while the mouse is moved.

  Icon of a plot. Opens another window with a plot of the two variables in terms of time, for the last integral curve that was plotted.

  Plot options:

  Options can also be given within the `plotdf` itself, each one being a list of two or more elements. The first element in each option is the name of the option, and the remainder is the value or values assigned to the option.

  The options which are recognized by `plotdf` are the following:

  - nsteps defines the number of steps that will be used for the independent variable, to compute an integral curve. The default value is 100.
  - direction defines the direction of the independent variable that will be followed to compute an integral curve. Possible values are `forward`, to make the independent variable increase `nsteps` times, with increments `tstep`, `backward`, to make the independent variable decrease, or `both` that will lead to an integral curve that extends `nsteps` forward, and `nsteps` backward. The keywords `right` and `left` can be used as synonyms for `forward` and `backward`. The default value is `both`.
  - tinitial defines the initial value of variable t used to compute integral curves. Since the differential equations are autonomous, that setting will only appear in the plot of the curves as functions of t. The default value is 0.
  - versus_t is used to create a second plot window, with a plot of an integral curve, as two functions x, y, of the independent variable t. If `versus_t` is given any value different from 0, the second plot window will be displayed. The second plot window includes another menu, similar to the menu of the main plot window. The default value is 0.
  - trajectory_at defines the coordinates xinitial and yinitial for the starting point of an integral curve. The option is empty by default.
  - parameters defines a list of parameters, and their numerical values, used in the definition of the differential equations. The name and values of the parameters must be given in a string with a comma-separated sequence of pairs `name=value`.
  - sliders defines a list of parameters that will be changed interactively using slider buttons, and the range of variation of those parameters. The names and ranges of the parameters must be given in a string with a comma-separated sequence of elements `name=min:max`
  - xfun defines a string with semi-colon-separated sequence of functions of x to be displayed, on top of the direction field. Those functions will be parsed by Tcl and not by Maxima.
  - x should be followed by two numbers, which will set up the minimum and maximum values shown on the horizontal axis. If the variable on the horizontal axis is not x, then this option should have the name of the variable on the horizontal axis. The default horizontal range is from -10 to 10.
  - y should be followed by two numbers, which will set up the minimum and maximum values shown on the vertical axis. If the variable on the vertical axis is not y, then this option should have the name of the variable on the vertical axis. The default vertical range is from -10 to 10.
  - xaxislabel will be used to identify the horizontal axis. Its default value is the name of the first state variable.
  - yaxislabel will be used to identify the vertical axis. Its default value is the name of the second state variable.
  - number_of_arrows should be set to a square number and defines the approximate density of the arrows being drawn. The default value is 225.

  Examples:

  - To show the direction field of the differential equation y' = exp(-x) + y and the solution that goes through (2, -0.1):

        (%i1) plotdf(exp(-x)+y,[trajectory_at,2,-0.1])$

  - To obtain the direction field for the equation diff(y,x) = x - y^2 and the solution with initial condition y(-1) = 3, we can use the command:

        (%i1) plotdf(x-y^2,[xfun,"sqrt(x);-sqrt(x)"],
                 [trajectory_at,-1,3], [direction,forward],
                 [y,-5,5], [x,-4,16])$

    The graph also shows the function y = sqrt(x).

  - The following example shows the direction field of a harmonic oscillator, defined by the two equations dz/dt = v and dv/dt = -k*z/m, and the integral curve through (z,v) = (6,0), with a slider that will allow you to change the value of m interactively (k is fixed at 2):

        (%i1) plotdf([v,-k*z/m], [z,v], [parameters,"m=2,k=2"],
                   [sliders,"m=1:5"], [trajectory_at,6,0])$

  - To plot the direction field of the Duffing equation, m*x"+c*x'+k*x+b*x^3 = 0, we introduce the variable y=x' and use:

        (%i1) plotdf([y,-(k*x + c*y + b*x^3)/m],
                     [parameters,"k=-1,m=1.0,c=0,b=1"],
                     [sliders,"k=-2:2,m=-1:1"],[tstep,0.1])$

  - The direction field for a damped pendulum, including the solution for the given initial conditions, with a slider that can be used to change the value of the mass m, and with a plot of the two state variables as a function of time:

        (%i1) plotdf([w,-g*sin(a)/l - b*w/m/l], [a,w],
                [parameters,"g=9.8,l=0.5,m=0.3,b=0.05"],
                [trajectory_at,1.05,-9],[tstep,0.01],
                [a,-10,2], [w,-14,14], [direction,forward],
                [nsteps,300], [sliders,"m=0.1:1"], [versus_t,1])$

  Categories: Differential equations · Plotting · Numerical methods

Function: ploteq (exp, ...options...)

  Plots equipotential curves for exp, which should be an expression depending on two variables. The curves are obtained by integrating the differential equation that define the orthogonal trajectories to the solutions of the autonomous system obtained from the gradient of the expression given. The plot can also show the integral curves for that gradient system (option fieldlines).

  This program also requires Xmaxima, even if its run from a Maxima session in a console, since the plot will be created by the Tk scripts in Xmaxima. By default, the plot region will be empty until the user clicks in a point (or gives its coordinate with in the set-up menu or via the trajectory_at option).

  Most options accepted by plotdf can also be used for ploteq and the plot interface is the same that was described in plotdf.

  Example:

      (%i1) V: 900/((x+1)^2+y^2)^(1/2)-900/((x-1)^2+y^2)^(1/2)$
      (%i2) ploteq(V,[x,-2,2],[y,-2,2],[fieldlines,"blue"])$

  Clicking on a point will plot the equipotential curve that passes by that point (in red) and the orthogonal trajectory (in blue).

  Categories: Differential equations · Plotting · Numerical methods

Function: rk
`rk` (ODE, var, initial, domain)
`rk` ([ODE1, …, ODEm], [v1, …, vm], [init1, …, initm], domain)

  The first form solves numerically one first-order ordinary differential equation, and the second form solves a system of m of those equations, using the 4th order Runge-Kutta method. var represents the dependent variable. ODE must be an expression that depends only on the independent and dependent variables and defines the derivative of the dependent variable with respect to the independent variable.

  The independent variable is specified with `domain`, which must be a list of four elements as, for instance:

      [t, 0, 10, 0.1]

  the first element of the list identifies the independent variable, the second and third elements are the initial and final values for that variable, and the last element sets the increments that should be used within that interval.

  If m equations are going to be solved, there should be m dependent variables v1, v2, ..., vm. The initial values for those variables will be init1, init2, ..., initm. There will still be just one independent variable defined by `domain`, as in the previous case. ODE1, ..., ODEm are the expressions that define the derivatives of each dependent variable in terms of the independent variable. The only variables that may appear in those expressions are the independent variable and any of the dependent variables. It is important to give the derivatives ODE1, ..., ODEm in the list in exactly the same order used for the dependent variables; for instance, the third element in the list will be interpreted as the derivative of the third dependent variable.

  The program will try to integrate the equations from the initial value of the independent variable until its last value, using constant increments. If at some step one of the dependent variables takes an absolute value too large, the integration will be interrupted at that point. The result will be a list with as many elements as the number of iterations made. Each element in the results list is itself another list with m+1 elements: the value of the independent variable, followed by the values of the dependent variables corresponding to that point.

  To solve numerically the differential equation

                dx/dt = t - x^2

  With initial value x(t=0) = 1, in the interval of t from 0 to 8 and with increments of 0.1 for t, use:

      (%i1) results: rk(t-x^2,x,1,[t,0,8,0.1])$
      (%i2) plot2d ([discrete, results])$

  the results will be saved in the list `results` and the plot will show the solution obtained, with t on the horizontal axis and x on the vertical axis.

  To solve numerically the system:

              dx/dt = 4-x^2-4*y^2     dy/dt = y^2-x^2+1

  for t between 0 and 4, and with values of -1.25 and 0.75 for x and y at t=0:

      (%i1) sol: rk([4-x^2-4*y^2,y^2-x^2+1],[x,y],[-1.25,0.75],[t,0,4,0.02])$
      (%i2) plot2d ([discrete,makelist([p[1],p[3]],p,sol)], [xlabel,"t"],[ylabel,"y"])$

  The plot will show the solution for variable y as a function of t.

  Categories: Differential equations · Numerical methods

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# 23. Matrices and Linear Algebra

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## 23.1 Introduction to Matrices and Linear Algebra

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### 23.1.1 Dot

The operator `.` represents noncommutative multiplication and scalar product. When the operands are 1-column or 1-row matrices `a` and `b`, the expression `a.b` is equivalent to `sum (a[i]*b[i], i, 1, length(a))`. If `a` and `b` are not complex, this is the scalar product, also called the inner product or dot product, of `a` and `b`. The scalar product is defined as `conjugate(a).b` when `a` and `b` are complex; `innerproduct` in the `eigen` package provides the complex scalar product.

When the operands are more general matrices, the product is the matrix product `a` and `b`. The number of rows of `b` must equal the number of columns of `a`, and the result has number of rows equal to the number of rows of `a` and number of columns equal to the number of columns of `b`.

To distinguish `.` as an arithmetic operator from the decimal point in a floating point number, it may be necessary to leave spaces on either side. For example, `5.e3` is `5000.0` but `5 . e3` is `5` times `e3`.

There are several flags which govern the simplification of expressions involving `.`, namely `dot0nscsimp`, `dot0simp`, `dot1simp`, `dotassoc`, `dotconstrules`, `dotdistrib`, `dotexptsimp`, `dotident`, and `dotscrules`.

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### 23.1.2 Vectors

`vect` is a package of functions for vector analysis. `load ("vect")` loads this package, and `demo ("vect")` displays a demonstration.

The vector analysis package can combine and simplify symbolic expressions including dot products and cross products, together with the gradient, divergence, curl, and Laplacian operators. The distribution of these operators over sums or products is governed by several flags, as are various other expansions, including expansion into components in any specific orthogonal coordinate systems. There are also functions for deriving the scalar or vector potential of a field.

The `vect` package contains these functions: `vectorsimp`, `scalefactors`, `express`, `potential`, and `vectorpotential`.

By default the `vect` package does not declare the dot operator to be a commutative operator. To get a commutative dot operator `.`, the command `declare(".", commutative)` must be executed.

Categories: Vectors · Share packages · Package vect

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### 23.1.3 eigen

The package `eigen` contains several functions devoted to the symbolic computation of eigenvalues and eigenvectors. Maxima loads the package automatically if one of the functions `eigenvalues` or `eigenvectors` is invoked. The package may be loaded explicitly as `load ("eigen")`.

`demo ("eigen")` displays a demonstration of the capabilities of this package. `batch ("eigen")` executes the same demonstration, but without the user prompt between successive computations.

The functions in the `eigen` package are:
`innerproduct`, `unitvector`, `columnvector`, `gramschmidt`, `eigenvalues`,
`eigenvectors`, `uniteigenvectors`, and `similaritytransform`.

Categories: Vectors · Matrices · Share packages · Package eigen

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## 23.2 Functions and Variables for Matrices and Linear Algebra

Function: addcol (M, list_1, …, list_n)

  Appends the column(s) given by the one or more lists (or matrices) onto the matrix M.

  See also `addrow` and `append`.

  Categories: Matrices

Function: addrow (M, list_1, …, list_n)

  Appends the row(s) given by the one or more lists (or matrices) onto the matrix M.

  See also `addcol` and `append`.

  Categories: Matrices

Function: adjoint (M)

  Returns the adjoint of the matrix M. The adjoint matrix is the transpose of the matrix of cofactors of M.

  Categories: Matrices

Function: augcoefmatrix ([eqn_1, …, eqn_m], [x_1, …, x_n])

  Returns the augmented coefficient matrix for the variables x_1, …, x_n of the system of linear equations eqn_1, …, eqn_m. This is the coefficient matrix with a column adjoined for the constant terms in each equation (i.e., those terms not dependent upon x_1, …, x_n).

      (%i1) m: [2*x - (a - 1)*y = 5*b, c + b*y + a*x = 0]$
      (%i2) augcoefmatrix (m, [x, y]);
                             [ 2  1 - a  - 5 b ]
      (%o2)                  [                 ]
                             [ a    b      c   ]

  Categories: Linear equations · Matrices

Function: cauchy_matrix
`cauchy_matrix` ([x_1, x_2, …, x_m], [y_1, y_2, …, y_n])
`cauchy_matrix` ([x_1, x_2, …, x_n])

  Returns a `n` by m Cauchy matrix with the elements a[i,j] = 1/(x_i+y_i). The second argument of `cauchy_matrix` is optional. For this case the elements of the Cauchy matrix are a[i,j] = 1/(x_i+x_j).

  Remark: In the literature the Cauchy matrix can be found defined in two forms. A second definition is a[i,j] = 1/(x_i-y_i).

  Examples:

      (%i1) cauchy_matrix([x1,x2],[y1,y2]);
                            [    1        1    ]
                            [ -------  ------- ]
                            [ y1 + x1  y2 + x1 ]
      (%o1)                 [                  ]
                            [    1        1    ]
                            [ -------  ------- ]
                            [ y1 + x2  y2 + x2 ]

      (%i2) cauchy_matrix([x1,x2]);
                            [   1         1    ]
                            [  ----    ------- ]
                            [  2 x1    x2 + x1 ]
      (%o2)                 [                  ]
                            [    1       1     ]
                            [ -------   ----   ]
                            [ x2 + x1   2 x2   ]

  Categories: Matrices

Function: charpoly (M, x)

  Returns the characteristic polynomial for the matrix M with respect to variable x. That is, `determinant (M - diagmatrix (length (M), x))`.

      (%i1) a: matrix ([3, 1], [2, 4]);
                                  [ 3  1 ]
      (%o1)                       [      ]
                                  [ 2  4 ]
      (%i2) expand (charpoly (a, lambda));
                                 2
      (%o2)                lambda  - 7 lambda + 10
      (%i3) (programmode: true, solve (%));
      (%o3)               [lambda = 5, lambda = 2]
      (%i4) matrix ([x1], [x2]);
                                   [ x1 ]
      (%o4)                        [    ]
                                   [ x2 ]
      (%i5) ev (a . % - lambda*%, %th(2)[1]);
                                [ x2 - 2 x1 ]
      (%o5)                     [           ]
                                [ 2 x1 - x2 ]
      (%i6) %[1, 1] = 0;
      (%o6)                     x2 - 2 x1 = 0
      (%i7) x2^2 + x1^2 = 1;
                                  2     2
      (%o7)                     x2  + x1  = 1
      (%i8) solve ([%th(2), %], [x1, x2]);
                        1               2
      (%o8) [[x1 = - -------, x2 = - -------],
                     sqrt(5)         sqrt(5)

                                                   1             2
                                          [x1 = -------, x2 = -------]]
                                                sqrt(5)       sqrt(5)

  Categories: Matrices

Function: coefmatrix ([eqn_1, …, eqn_m], [x_1, …, x_n])

  Returns the coefficient matrix for the variables x_1, …, x_n of the system of linear equations eqn_1, …, eqn_m.

      (%i1) coefmatrix([2*x-(a-1)*y+5*b = 0, b*y+a*x = 3], [x,y]);
                                       [ 2  1 - a ]
      (%o1)                            [          ]
                                       [ a    b   ]

  Categories: Linear equations · Matrices

Function: col (M, i)

  Returns the i'th column of the matrix M. The return value is a matrix.

  Categories: Matrices

Function: columnvector (L)
Function: covect (L)

  Returns a matrix of one column and `length (L)` rows, containing the elements of the list L.

  `covect` is a synonym for `columnvector`.

  `load ("eigen")` loads this function.

  This is useful if you want to use parts of the outputs of the functions in this package in matrix calculations.

  Example:

      (%i1) load ("eigen")$
      Warning - you are redefining the Macsyma function eigenvalues
      Warning - you are redefining the Macsyma function eigenvectors
      (%i2) columnvector ([aa, bb, cc, dd]);
                                   [ aa ]
                                   [    ]
                                   [ bb ]
      (%o2)                        [    ]
                                   [ cc ]
                                   [    ]
                                   [ dd ]

  Categories: Matrices

Function: copymatrix (M)

  Returns a copy of the matrix M. This is the only way to make a copy aside from copying M element by element.

  Note that an assignment of one matrix to another, as in `m2: m1`, does not copy `m1`. An assignment `m2 [i,j]: x` or `setelmx(x, i, j, m2)` also modifies `m1 [i,j]`. Creating a copy with `copymatrix` and then using assignment creates a separate, modified copy.

  Categories: Matrices

Function: determinant (M)

  Computes the determinant of M by a method similar to Gaussian elimination.

  The form of the result depends upon the setting of the switch `ratmx`.

  There is a special routine for computing sparse determinants which is called when the switches `ratmx` and `sparse` are both `true`.

  Categories: Matrices

Option variable: detout

  Default value: `false`

  When `detout` is `true`, the determinant of a matrix whose inverse is computed is factored out of the inverse.

  For this switch to have an effect `doallmxops` and `doscmxops` should be `false` (see their descriptions). Alternatively this switch can be given to `ev` which causes the other two to be set correctly.

  Example:

      (%i1) m: matrix ([a, b], [c, d]);
                                  [ a  b ]
      (%o1)                       [      ]
                                  [ c  d ]
      (%i2) detout: true$
      (%i3) doallmxops: false$
      (%i4) doscmxops: false$
      (%i5) invert (m);
                                [  d   - b ]
                                [          ]
                                [ - c   a  ]
      (%o5)                     ------------
                                 a d - b c

  Categories: Matrices · Evaluation flags

Function: diagmatrix (n, x)

  Returns a diagonal matrix of size n by n with the diagonal elements all equal to x. `diagmatrix (n, 1)` returns an identity matrix (same as `ident (n)`).

  n must evaluate to an integer, otherwise `diagmatrix` complains with an error message.

  x can be any kind of expression, including another matrix. If x is a matrix, it is not copied; all diagonal elements refer to the same instance, x.

  Categories: Matrices

Option variable: doallmxops

  Default value: `true`

  When `doallmxops` is `true`, all operations relating to matrices are carried out. When it is `false` then the setting of the individual `dot` switches govern which operations are performed.

  Categories: Matrices

Option variable: domxexpt

  Default value: `true`

  When `domxexpt` is `true`, a matrix exponential, `exp (M)` where M is a matrix, is interpreted as a matrix with element `[i,j]` equal to `exp (m[i,j])`. Otherwise `exp (M)` evaluates to `exp (ev(M))`.

  `domxexpt` affects all expressions of the form `base^power` where base is an expression assumed scalar or constant, and power is a list or matrix.

  Example:

      (%i1) m: matrix ([1, %i], [a+b, %pi]);
                               [   1    %i  ]
      (%o1)                    [            ]
                               [ b + a  %pi ]
      (%i2) domxexpt: false$
      (%i3) (1 - c)^m;
                                   [   1    %i  ]
                                   [            ]
                                   [ b + a  %pi ]
      (%o3)                 (1 - c)
      (%i4) domxexpt: true$
      (%i5) (1 - c)^m;
                        [                      %i  ]
                        [    1 - c      (1 - c)    ]
      (%o5)             [                          ]
                        [        b + a         %pi ]
                        [ (1 - c)       (1 - c)    ]

  Categories: Matrices

Option variable: domxmxops

  Default value: `true`

  When `domxmxops` is `true`, all matrix-matrix or matrix-list operations are carried out (but not scalar-matrix operations); if this switch is `false` such operations are not carried out.

  Categories: Matrices

Option variable: domxnctimes

  Default value: `false`

  When `domxnctimes` is `true`, non-commutative products of matrices are carried out.

  Categories: Matrices

Option variable: dontfactor

  Default value: `[]`

  `dontfactor` may be set to a list of variables with respect to which factoring is not to occur. (The list is initially empty.) Factoring also will not take place with respect to any variables which are less important, according the variable ordering assumed for canonical rational expression (CRE) form, than those on the `dontfactor` list.

  Categories: Expressions

Option variable: doscmxops

  Default value: `false`

  When `doscmxops` is `true`, scalar-matrix operations are carried out.

  Categories: Matrices

Option variable: doscmxplus

  Default value: `false`

  When `doscmxplus` is `true`, scalar-matrix operations yield a matrix result. This switch is not subsumed under `doallmxops`.

  Categories: Matrices

Option variable: dot0nscsimp

  Default value: `true`

  When `dot0nscsimp` is `true`, a non-commutative product of zero and a nonscalar term is simplified to a commutative product.

  Categories: Simplification flags and variables

Option variable: dot0simp

  Default value: `true`

  When `dot0simp` is `true`, a non-commutative product of zero and a scalar term is simplified to a commutative product.

  Categories: Simplification flags and variables

Option variable: dot1simp

  Default value: `true`

  When `dot1simp` is `true`, a non-commutative product of one and another term is simplified to a commutative product.

  Categories: Simplification flags and variables

Option variable: dotassoc

  Default value: `true`

  When `dotassoc` is `true`, an expression `(A.B).C` simplifies to `A.(B.C)`.

  Categories: Simplification flags and variables

Option variable: dotconstrules

  Default value: `true`

  When `dotconstrules` is `true`, a non-commutative product of a constant and another term is simplified to a commutative product. Turning on this flag effectively turns on `dot0simp`, `dot0nscsimp`, and `dot1simp` as well.

  Categories: Simplification flags and variables

Option variable: dotdistrib

  Default value: `false`

  When `dotdistrib` is `true`, an expression `A.(B + C)` simplifies to `A.B + A.C`.

  Categories: Simplification flags and variables

Option variable: dotexptsimp

  Default value: `true`

  When `dotexptsimp` is `true`, an expression `A.A` simplifies to `A^^2`.

  Categories: Simplification flags and variables

Option variable: dotident

  Default value: 1

  `dotident` is the value returned by `X^^0`.

  Categories: Simplification flags and variables

Option variable: dotscrules

  Default value: `false`

  When `dotscrules` is `true`, an expression `A.SC` or `SC.A` simplifies to `SC*A` and `A.(SC*B)` simplifies to `SC*(A.B)`.

  Categories: Simplification flags and variables

Function: echelon (M)

  Returns the echelon form of the matrix M, as produced by Gaussian elimination. The echelon form is computed from M by elementary row operations such that the first non-zero element in each row in the resulting matrix is one and the column elements under the first one in each row are all zero.

  `triangularize` also carries out Gaussian elimination, but it does not normalize the leading non-zero element in each row.

  `lu_factor` and `cholesky` are other functions which yield triangularized matrices.

      (%i1) M: matrix ([3, 7, aa, bb], [-1, 8, 5, 2], [9, 2, 11, 4]);
                             [  3   7  aa  bb ]
                             [                ]
      (%o1)                  [ - 1  8  5   2  ]
                             [                ]
                             [  9   2  11  4  ]
      (%i2) echelon (M);
                        [ 1  - 8  - 5      - 2     ]
                        [                          ]
                        [         28       11      ]
                        [ 0   1   --       --      ]
      (%o2)             [         37       37      ]
                        [                          ]
                        [              37 bb - 119 ]
                        [ 0   0    1   ----------- ]
                        [              37 aa - 313 ]

  Categories: Linear equations · Matrices

Function: eigenvalues (M)
Function: eivals (M)

  Returns a list of two lists containing the eigenvalues of the matrix M. The first sublist of the return value is the list of eigenvalues of the matrix, and the second sublist is the list of the multiplicities of the eigenvalues in the corresponding order.

  `eivals` is a synonym for `eigenvalues`.

  `eigenvalues` calls the function `solve` to find the roots of the characteristic polynomial of the matrix. Sometimes `solve` may not be able to find the roots of the polynomial; in that case some other functions in this package (except `innerproduct`, `unitvector`, `columnvector` and `gramschmidt`) will not work. Sometimes `solve` may find only a subset of the roots of the polynomial. This may happen when the factoring of the polynomial contains polynomials of degree 5 or more. In such cases a warning message is displayed and the only the roots found and their corresponding multiplicities are returned.

  In some cases the eigenvalues found by `solve` may be complicated expressions. (This may happen when `solve` returns a not-so-obviously real expression for an eigenvalue which is known to be real.) It may be possible to simplify the eigenvalues using some other functions.

  The package `eigen.mac` is loaded automatically when `eigenvalues` or `eigenvectors` is referenced. If `eigen.mac` is not already loaded, `load ("eigen")` loads it. After loading, all functions and variables in the package are available.

  Categories: Package eigen

Function: eigenvectors (M)
Function: eivects (M)

  Computes eigenvectors of the matrix M. The return value is a list of two elements. The first is a list of the eigenvalues of M and a list of the multiplicities of the eigenvalues. The second is a list of lists of eigenvectors. There is one list of eigenvectors for each eigenvalue. There may be one or more eigenvectors in each list.

  `eivects` is a synonym for `eigenvectors`.

  The package `eigen.mac` is loaded automatically when `eigenvalues` or `eigenvectors` is referenced. If `eigen.mac` is not already loaded, `load ("eigen")` loads it. After loading, all functions and variables in the package are available.

  Note that `eigenvectors` internally calls `eigenvalues` to obtain eigenvalues. So, when `eigenvalues` returns a subset of all the eigenvalues, the `eigenvectors` returns the corresponding subset of the all the eigenvectors, with the same warning displayed as `eigenvalues`.

  The flags that affect this function are:

  `nondiagonalizable` is set to `true` or `false` depending on whether the matrix is nondiagonalizable or diagonalizable after `eigenvectors` returns.

  `hermitianmatrix` when `true`, causes the degenerate eigenvectors of the Hermitian matrix to be orthogonalized using the Gram-Schmidt algorithm.

  `knowneigvals` when `true` causes the `eigen` package to assume the eigenvalues of the matrix are known to the user and stored under the global name `listeigvals`. `listeigvals` should be set to a list similar to the output `eigenvalues`.

  The function `algsys` is used here to solve for the eigenvectors. Sometimes if the eigenvalues are messy, `algsys` may not be able to find a solution. In some cases, it may be possible to simplify the eigenvalues by first finding them using `eigenvalues` command and then using other functions to reduce them to something simpler. Following simplification, `eigenvectors` can be called again with the `knowneigvals` flag set to `true`.

  See also `eigenvalues`.

  Examples:

  A matrix which has just one eigenvector per eigenvalue.

      (%i1) M1 : matrix ([11, -1], [1, 7]);
                                 [ 11  - 1 ]
      (%o1)                      [         ]
                                 [ 1    7  ]
      (%i2) [vals, vecs] : eigenvectors (M1);
      (%o2) [[[9 - sqrt(3), sqrt(3) + 9], [1, 1]],
                              [[[1, sqrt(3) + 2]], [[1, 2 - sqrt(3)]]]]
      (%i3) for i thru length (vals[1]) do disp (val[i] = vals[1][i],
        mult[i] = vals[2][i], vec[i] = vecs[i]);
                             val  = 9 - sqrt(3)
                                1

                                  mult  = 1
                                      1

                          vec  = [[1, sqrt(3) + 2]]
                             1

                             val  = sqrt(3) + 9
                                2

                                  mult  = 1
                                      2

                          vec  = [[1, 2 - sqrt(3)]]
                             2

      (%o3)                         done

  A matrix which has two eigenvectors for one eigenvalue (namely 2).

      (%i1) M1 : matrix ([0, 1, 0, 0], [0, 0, 0, 0], [0, 0, 2, 0],
                         [0, 0, 0, 2]);
                               [ 0  1  0  0 ]
                               [            ]
                               [ 0  0  0  0 ]
      (%o1)                    [            ]
                               [ 0  0  2  0 ]
                               [            ]
                               [ 0  0  0  2 ]
      (%i2) [vals, vecs] : eigenvectors (M1);
      (%o2) [[[0, 2], [2, 2]], [[[1, 0, 0, 0]],
                                         [[0, 0, 1, 0], [0, 0, 0, 1]]]]
      (%i3) for i thru length (vals[1]) do disp (val[i] = vals[1][i],
        mult[i] = vals[2][i], vec[i] = vecs[i]);
                                  val  = 0
                                     1

                                  mult  = 2
                                      1

                            vec  = [[1, 0, 0, 0]]
                               1

                                  val  = 2
                                     2

                                  mult  = 2
                                      2

                     vec  = [[0, 0, 1, 0], [0, 0, 0, 1]]
                        2

      (%o3)                         done

  Categories: Package eigen

Function: ematrix (m, n, x, i, j)

  Returns an m by n matrix, all elements of which are zero except for the `[i, j]` element which is x.

  Categories: Matrices

Function: entermatrix (m, n)

  Returns an m by n matrix, reading the elements interactively.

  If n is equal to m, Maxima prompts for the type of the matrix (diagonal, symmetric, antisymmetric, or general) and for each element. Each response is terminated by a semicolon `;` or dollar sign `$`.

  If n is not equal to m, Maxima prompts for each element.

  The elements may be any expressions, which are evaluated. `entermatrix` evaluates its arguments.

      (%i1) n: 3$
      (%i2) m: entermatrix (n, n)$

      Is the matrix  1. Diagonal  2. Symmetric  3. Antisymmetric
      4. General
      Answer 1, 2, 3 or 4 :
      1$
      Row 1 Column 1:
      (a+b)^n$
      Row 2 Column 2:
      (a+b)^(n+1)$
      Row 3 Column 3:
      (a+b)^(n+2)$

      Matrix entered.
      (%i3) m;
                      [        3                     ]
                      [ (b + a)      0         0     ]
                      [                              ]
      (%o3)           [                  4           ]
                      [    0      (b + a)      0     ]
                      [                              ]
                      [                            5 ]
                      [    0         0      (b + a)  ]

  Categories: Console interaction · Matrices

Function: genmatrix
`genmatrix` (a, i_2, j_2, i_1, j_1)
`genmatrix` (a, i_2, j_2, i_1)
`genmatrix` (a, i_2, j_2)

  Returns a matrix generated from a, taking element `a[i_1, j_1]` as the upper-left element and `a[i_2, j_2]` as the lower-right element of the matrix. Here a is a declared array (created by `array` but not by `make_array`) or an undeclared array, or an array function, or a lambda expression of two arguments. (An array function is created like other functions with `:=` or `define`, but arguments are enclosed in square brackets instead of parentheses.)

  If j_1 is omitted, it is assumed equal to i_1. If both j_1 and i_1 are omitted, both are assumed equal to 1.

  If a selected element `i,j` of the array is undefined, the matrix will contain a symbolic element `a[i,j]`.

  Examples:

      (%i1) h [i, j] := 1 / (i + j - 1);
                                          1
      (%o1)                  h     := ---------
                              i, j    i + j - 1
      (%i2) genmatrix (h, 3, 3);
                                 [    1  1 ]
                                 [ 1  -  - ]
                                 [    2  3 ]
                                 [         ]
                                 [ 1  1  1 ]
      (%o2)                      [ -  -  - ]
                                 [ 2  3  4 ]
                                 [         ]
                                 [ 1  1  1 ]
                                 [ -  -  - ]
                                 [ 3  4  5 ]
      (%i3) array (a, fixnum, 2, 2);
      (%o3)                           a
      (%i4) a [1, 1] : %e;
      (%o4)                          %e
      (%i5) a [2, 2] : %pi;
      (%o5)                          %pi
      (%i6) genmatrix (a, 2, 2);
                                 [ %e   0  ]
      (%o6)                      [         ]
                                 [ 0   %pi ]
      (%i7) genmatrix (lambda ([i, j], j - i), 3, 3);
                               [  0    1   2 ]
                               [             ]
      (%o7)                    [ - 1   0   1 ]
                               [             ]
                               [ - 2  - 1  0 ]
      (%i8) genmatrix (B, 2, 2);
                              [ B      B     ]
                              [  1, 1   1, 2 ]
      (%o8)                   [              ]
                              [ B      B     ]
                              [  2, 1   2, 2 ]

  Categories: Matrices

Function: gramschmidt
`gramschmidt` (x)
`gramschmidt` (x, F)

  Carries out the Gram-Schmidt orthogonalization algorithm on x, which is either a matrix or a list of lists. x is not modified by `gramschmidt`. The inner product employed by `gramschmidt` is F, if present, otherwise the inner product is the function `innerproduct`.

  If x is a matrix, the algorithm is applied to the rows of x. If x is a list of lists, the algorithm is applied to the sublists, which must have equal numbers of elements. In either case, the return value is a list of lists, the sublists of which are orthogonal and span the same space as x. If the dimension of the span of x is less than the number of rows or sublists, some sublists of the return value are zero.

  `factor` is called at each stage of the algorithm to simplify intermediate results. As a consequence, the return value may contain factored integers.

  `load(eigen)` loads this function.

  Example:

  Gram-Schmidt algorithm using default inner product function.

      (%i1) load ("eigen")$
      (%i2) x: matrix ([1, 2, 3], [9, 18, 30], [12, 48, 60]);
                               [ 1   2   3  ]
                               [            ]
      (%o2)                    [ 9   18  30 ]
                               [            ]
                               [ 12  48  60 ]
      (%i3) y: gramschmidt (x);
                             2      2            4     3
                            3      3   3 5      2  3  2  3
      (%o3)  [[1, 2, 3], [- ---, - --, ---], [- ----, ----, 0]]
                            2 7    7   2 7       5     5
      (%i4) map (innerproduct, [y[1], y[2], y[3]], [y[2], y[3], y[1]]);
      (%o4)                       [0, 0, 0]

  Gram-Schmidt algorithm using a specified inner product function.

      (%i1) load ("eigen")$
      (%i2) ip (f, g) := integrate (f * g, u, a, b);
      (%o2)          ip(f, g) := integrate(f g, u, a, b)
      (%i3) y : gramschmidt([1, sin(u), cos(u)], ip), a= -%pi/2, b=%pi/2;
                                     %pi cos(u) - 2
      (%o3)              [1, sin(u), --------------]
                                          %pi
      (%i4) map (ip, [y[1], y[2], y[3]], [y[2], y[3], y[1]]),
                                                      a= -%pi/2, b=%pi/2;
      (%o4)                       [0, 0, 0]

  Categories: Package eigen

Function: ident (n)

  Returns an n by n identity matrix.

  Categories: Matrices

Function: innerproduct (x, y)
Function: inprod (x, y)

  Returns the inner product (also called the scalar product or dot product) of x and y, which are lists of equal length, or both 1-column or 1-row matrices of equal length. The return value is `conjugate (x) . y`, where `.` is the noncommutative multiplication operator.

  `load ("eigen")` loads this function.

  `inprod` is a synonym for `innerproduct`.

  Categories: Package eigen

Function: invert_by_adjoint (M)

  Returns the inverse of the matrix M. The inverse is computed by the adjoint method.

  `invert_by_adjoint` honors the `ratmx` and `detout` flags, the same as `invert`.

Function: invert (M)

  Returns the inverse of the matrix M. The inverse is computed via the LU decomposition.

  When `ratmx` is `true`, elements of M are converted to canonical rational expressions (CRE), and the elements of the return value are also CRE.

  When `ratmx` is `false`, elements of M are not converted to a common representation. In particular, float and bigfloat elements are not converted to rationals.

  When `detout` is `true`, the determinant is factored out of the inverse. The global flags `doallmxops` and `doscmxops` must be `false` to prevent the determinant from being absorbed into the inverse. `xthru` can multiply the determinant into the inverse.

  `invert` does not apply any simplifications to the elements of the inverse apart from the default arithmetic simplifications. `ratsimp` and `expand` can apply additional simplifications. In particular, when M has polynomial elements, `expand(invert(M))` might be preferable.

  `invert(M)` is equivalent to `M^^-1`.

  Categories: Matrices

Function: list_matrix_entries (M)

  Returns a list containing the elements of the matrix M.

  Example:

      (%i1) list_matrix_entries(matrix([a,b],[c,d]));
      (%o1)                     [a, b, c, d]

  Categories: Matrices

Option variable: lmxchar

  Default value: `[`

  `lmxchar` is the character displayed as the left delimiter of a matrix. See also `rmxchar`.

  Example:

      (%i1) lmxchar: "|"$
      (%i2) matrix ([a, b, c], [d, e, f], [g, h, i]);
                                 | a  b  c ]
                                 |         ]
      (%o2)                      | d  e  f ]
                                 |         ]
                                 | g  h  i ]

  Categories: Display flags and variables · Matrices

Function: matrix (row_1, …, row_n)

  Returns a rectangular matrix which has the rows row_1, …, row_n. Each row is a list of expressions. All rows must be the same length.

  The operations `+` (addition), `-` (subtraction), `*` (multiplication), and `/` (division), are carried out element by element when the operands are two matrices, a scalar and a matrix, or a matrix and a scalar. The operation `^` (exponentiation, equivalently `**`) is carried out element by element if the operands are a scalar and a matrix or a matrix and a scalar, but not if the operands are two matrices. All operations are normally carried out in full, including `.` (noncommutative multiplication).

  Matrix multiplication is represented by the noncommutative multiplication operator `.`. The corresponding noncommutative exponentiation operator is `^^`. For a matrix `A`, `A.A = A^^2` and `A^^-1` is the inverse of A, if it exists. `A^^-1` is equivalent to `invert(A)`.

  There are switches for controlling simplification of expressions involving dot and matrix-list operations. These are `doallmxops`, `domxexpt`, `domxmxops`, `doscmxops`, and `doscmxplus`.

  There are additional options which are related to matrices. These are: `lmxchar`, `rmxchar`, `ratmx`, `listarith`, `detout`, `scalarmatrix` and `sparse`.

  There are a number of functions which take matrices as arguments or yield matrices as return values. See `eigenvalues`, `eigenvectors`, `determinant`, `charpoly`, `genmatrix`, `addcol`, `addrow`, `copymatrix`, `transpose`, `echelon`, and `rank`.

  Examples:

  - Construction of matrices from lists.

      (%i1) x: matrix ([17, 3], [-8, 11]);
                                 [ 17   3  ]
      (%o1)                      [         ]
                                 [ - 8  11 ]
      (%i2) y: matrix ([%pi, %e], [a, b]);
                                 [ %pi  %e ]
      (%o2)                      [         ]
                                 [  a   b  ]

  - Addition, element by element.

      (%i3) x + y;
                            [ %pi + 17  %e + 3 ]
      (%o3)                 [                  ]
                            [  a - 8    b + 11 ]

  - Subtraction, element by element.

      (%i4) x - y;
                            [ 17 - %pi  3 - %e ]
      (%o4)                 [                  ]
                            [ - a - 8   11 - b ]

  - Multiplication, element by element.

      (%i5) x * y;
                              [ 17 %pi  3 %e ]
      (%o5)                   [              ]
                              [ - 8 a   11 b ]

  - Division, element by element.

      (%i6) x / y;
                              [ 17       - 1 ]
                              [ ---  3 %e    ]
                              [ %pi          ]
      (%o6)                   [              ]
                              [   8    11    ]
                              [ - -    --    ]
                              [   a    b     ]

  - Matrix to a scalar exponent, element by element.

      (%i7) x ^ 3;
                               [ 4913    27  ]
      (%o7)                    [             ]
                               [ - 512  1331 ]

  - Scalar base to a matrix exponent, element by element.

      (%i8) exp(y);
                               [   %pi    %e ]
                               [ %e     %e   ]
      (%o8)                    [             ]
                               [    a     b  ]
                               [  %e    %e   ]

  - Matrix base to a matrix exponent. This is not carried out element by element. See also `matrixexp`.

      (%i9) x ^ y;
                                      [ %pi  %e ]
                                      [         ]
                                      [  a   b  ]
                           [ 17   3  ]
      (%o9)                [         ]
                           [ - 8  11 ]

  - Noncommutative matrix multiplication.

      (%i10) x . y;
                        [ 3 a + 17 %pi  3 b + 17 %e ]
      (%o10)            [                           ]
                        [ 11 a - 8 %pi  11 b - 8 %e ]
      (%i11) y . x;
                      [ 17 %pi - 8 %e  3 %pi + 11 %e ]
      (%o11)          [                              ]
                      [  17 a - 8 b     11 b + 3 a   ]

  - Noncommutative matrix exponentiation. A scalar base b to a matrix power M is carried out element by element and so `b^^m` is the same as `b^m`.

      (%i12) x ^^ 3;
                              [  3833   1719 ]
      (%o12)                  [              ]
                              [ - 4584  395  ]
      (%i13) %e ^^ y;
                               [   %pi    %e ]
                               [ %e     %e   ]
      (%o13)                   [             ]
                               [    a     b  ]
                               [  %e    %e   ]

  - A matrix raised to a -1 exponent with noncommutative exponentiation is the matrix inverse, if it exists.

      (%i14) x ^^ -1;
                               [ 11      3  ]
                               [ ---  - --- ]
                               [ 211    211 ]
      (%o14)                   [            ]
                               [  8    17   ]
                               [ ---   ---  ]
                               [ 211   211  ]
      (%i15) x . (x ^^ -1);
                                  [ 1  0 ]
      (%o15)                      [      ]
                                  [ 0  1 ]

  Categories: Matrices

Function: matrixexp
`matrixexp` (M)
`matrixexp` (M, n)
`matrixexp` (M, V)

  Calculates the matrix exponential e^(M*V) . Instead of the vector V a number n can be specified as the second argument. If this argument is omitted `matrixexp` replaces it by `1`.

  The matrix exponential of a matrix M can be expressed as a power series: e^M=sum(M^k/k!,0,inf)

  Categories: Matrices

Function: matrixmap (f, M)

  Returns a matrix with element `i,j` equal to `f(M[i,j])`.

  See also `map`, `fullmap`, `fullmapl`, and `apply`.

  Categories: Matrices

Function: matrixp (expr)

  Returns `true` if expr is a matrix, otherwise `false`.

  Categories: Predicate functions · Matrices

Option variable: matrix_element_add

  Default value: `+`

  `matrix_element_add` is the operation invoked in place of addition in a matrix multiplication. `matrix_element_add` can be assigned any n-ary operator (that is, a function which handles any number of arguments). The assigned value may be the name of an operator enclosed in quote marks, the name of a function, or a lambda expression.

  See also `matrix_element_mult` and `matrix_element_transpose`.

  Example:

      (%i1) matrix_element_add: "*"$
      (%i2) matrix_element_mult: "^"$
      (%i3) aa: matrix ([a, b, c], [d, e, f]);
                                 [ a  b  c ]
      (%o3)                      [         ]
                                 [ d  e  f ]
      (%i4) bb: matrix ([u, v, w], [x, y, z]);
                                 [ u  v  w ]
      (%o4)                      [         ]
                                 [ x  y  z ]
      (%i5) aa . transpose (bb);
                           [  u  v  w   x  y  z ]
                           [ a  b  c   a  b  c  ]
      (%o5)                [                    ]
                           [  u  v  w   x  y  z ]
                           [ d  e  f   d  e  f  ]

  Categories: Matrices

Option variable: matrix_element_mult

  Default value: `*`

  `matrix_element_mult` is the operation invoked in place of multiplication in a matrix multiplication. `matrix_element_mult` can be assigned any binary operator. The assigned value may be the name of an operator enclosed in quote marks, the name of a function, or a lambda expression.

  The dot operator `.` is a useful choice in some contexts.

  See also `matrix_element_add` and `matrix_element_transpose`.

  Example:

      (%i1) matrix_element_add: lambda ([[x]], sqrt (apply ("+", x)))$
      (%i2) matrix_element_mult: lambda ([x, y], (x - y)^2)$
      (%i3) [a, b, c] . [x, y, z];
                                2          2          2
      (%o3)         sqrt((c - z)  + (b - y)  + (a - x) )
      (%i4) aa: matrix ([a, b, c], [d, e, f]);
                                 [ a  b  c ]
      (%o4)                      [         ]
                                 [ d  e  f ]
      (%i5) bb: matrix ([u, v, w], [x, y, z]);
                                 [ u  v  w ]
      (%o5)                      [         ]
                                 [ x  y  z ]
      (%i6) aa . transpose (bb);
                     [             2          2          2  ]
                     [ sqrt((c - w)  + (b - v)  + (a - u) ) ]
      (%o6)  Col 1 = [                                      ]
                     [             2          2          2  ]
                     [ sqrt((f - w)  + (e - v)  + (d - u) ) ]

                               [             2          2          2  ]
                               [ sqrt((c - z)  + (b - y)  + (a - x) ) ]
                       Col 2 = [                                      ]
                               [             2          2          2  ]
                               [ sqrt((f - z)  + (e - y)  + (d - x) ) ]

  Categories: Matrices

Option variable: matrix_element_transpose

  Default value: `false`

  `matrix_element_transpose` is the operation applied to each element of a matrix when it is transposed. `matrix_element_mult` can be assigned any unary operator. The assigned value may be the name of an operator enclosed in quote marks, the name of a function, or a lambda expression.

  When `matrix_element_transpose` equals `transpose`, the `transpose` function is applied to every element. When `matrix_element_transpose` equals `nonscalars`, the `transpose` function is applied to every nonscalar element. If some element is an atom, the `nonscalars` option applies `transpose` only if the atom is declared nonscalar, while the `transpose` option always applies `transpose`.

  The default value, `false`, means no operation is applied.

  See also `matrix_element_add` and `matrix_element_mult`.

  Examples:

      (%i1) declare (a, nonscalar)$
      (%i2) transpose ([a, b]);
                              [ transpose(a) ]
      (%o2)                   [              ]
                              [      b       ]
      (%i3) matrix_element_transpose: nonscalars$
      (%i4) transpose ([a, b]);
                              [ transpose(a) ]
      (%o4)                   [              ]
                              [      b       ]
      (%i5) matrix_element_transpose: transpose$
      (%i6) transpose ([a, b]);
                              [ transpose(a) ]
      (%o6)                   [              ]
                              [ transpose(b) ]
      (%i7) matrix_element_transpose: lambda ([x], realpart(x)
            - %i*imagpart(x))$
      (%i8) m: matrix ([1 + 5*%i, 3 - 2*%i], [7*%i, 11]);
                           [ 5 %i + 1  3 - 2 %i ]
      (%o8)                [                    ]
                           [   7 %i       11    ]
      (%i9) transpose (m);
                            [ 1 - 5 %i  - 7 %i ]
      (%o9)                 [                  ]
                            [ 2 %i + 3    11   ]

  Categories: Matrices

Function: mattrace (M)

  Returns the trace (that is, the sum of the elements on the main diagonal) of the square matrix M.

  `mattrace` is called by `ncharpoly`, an alternative to Maxima's `charpoly`.

  `load ("nchrpl")` loads this function.

  Categories: Matrices · Package nchrpl

Function: minor (M, i, j)

  Returns the i, j minor of the matrix M. That is, M with row i and column j removed.

  Categories: Matrices

Function: ncharpoly (M, x)

  Returns the characteristic polynomial of the matrix M with respect to x. This is an alternative to Maxima's `charpoly`.

  `ncharpoly` works by computing traces of powers of the given matrix, which are known to be equal to sums of powers of the roots of the characteristic polynomial. From these quantities the symmetric functions of the roots can be calculated, which are nothing more than the coefficients of the characteristic polynomial. `charpoly` works by forming the determinant of `x * ident [n] - a`. Thus `ncharpoly` wins, for example, in the case of large dense matrices filled with integers, since it avoids polynomial arithmetic altogether.

  `load ("nchrpl")` loads this file.

  Categories: Matrices · Package nchrpl

Function: newdet (M)

  Computes the determinant of the matrix M by the Johnson-Gentleman tree minor algorithm. `newdet` returns the result in CRE form.

  Categories: Matrices

Function: permanent (M)

  Computes the permanent of the matrix M by the Johnson-Gentleman tree minor algorithm. A permanent is like a determinant but with no sign changes. `permanent` returns the result in CRE form.

  See also `newdet`.

  Categories: Matrices

Function: rank (M)

  Computes the rank of the matrix M. That is, the order of the largest non-singular subdeterminant of M.

  rank may return the wrong answer if it cannot determine that a matrix element that is equivalent to zero is indeed so.

  Categories: Matrices

Option variable: ratmx

  Default value: `false`

  When `ratmx` is `false`, determinant and matrix addition, subtraction, and multiplication are performed in the representation of the matrix elements and cause the result of matrix inversion to be left in general representation.

  When `ratmx` is `true`, the 4 operations mentioned above are performed in CRE form and the result of matrix inverse is in CRE form. Note that this may cause the elements to be expanded (depending on the setting of `ratfac`) which might not always be desired.

  Categories: Matrices · Rational expressions

Function: row (M, i)

  Returns the i'th row of the matrix M. The return value is a matrix.

  Categories: Matrices

Option variable: rmxchar

  Default value: `]`

  `rmxchar` is the character drawn on the right-hand side of a matrix.

  See also `lmxchar`.

  Categories: Display flags and variables

Option variable: scalarmatrixp

  Default value: `true`

  When `scalarmatrixp` is `true`, then whenever a 1 x 1 matrix is produced as a result of computing the dot product of matrices it is simplified to a scalar, namely the sole element of the matrix.

  When `scalarmatrixp` is `all`, then all 1 x 1 matrices are simplified to scalars.

  When `scalarmatrixp` is `false`, 1 x 1 matrices are not simplified to scalars.

  Categories: Matrices · Simplification flags and variables

Function: scalefactors (coordinatetransform)

  Here the argument coordinatetransform evaluates to the form `[[expression1, expression2, ...], indeterminate1, indeterminat2, ...]`, where the variables indeterminate1, indeterminate2, etc. are the curvilinear coordinate variables and where a set of rectangular Cartesian components is given in terms of the curvilinear coordinates by `[expression1, expression2, ...]`. `coordinates` is set to the vector `[indeterminate1, indeterminate2,...]`, and `dimension` is set to the length of this vector. SF[1], SF[2], …, SF[DIMENSION] are set to the coordinate scale factors, and `sfprod` is set to the product of these scale factors. Initially, `coordinates` is `[X, Y, Z]`, `dimension` is 3, and SF[1]=SF[2]=SF[3]=SFPROD=1, corresponding to 3-dimensional rectangular Cartesian coordinates. To expand an expression into physical components in the current coordinate system, there is a function with usage of the form

  Categories: Package vect

Function: setelmx (x, i, j, M)

  Assigns x to the (i, j)'th element of the matrix M, and returns the altered matrix.

  `M [i, j]: x` has the same effect, but returns x instead of M.

  Categories: Matrices

Function: similaritytransform (M)
Function: simtran (M)

  `similaritytransform` computes a similarity transform of the matrix `M`. It returns a list which is the output of the `uniteigenvectors` command. In addition if the flag `nondiagonalizable` is `false` two global matrices `leftmatrix` and `rightmatrix` are computed. These matrices have the property that `leftmatrix . M . rightmatrix` is a diagonal matrix with the eigenvalues of M on the diagonal. If `nondiagonalizable` is `true` the left and right matrices are not computed.

  If the flag `hermitianmatrix` is `true` then `leftmatrix` is the complex conjugate of the transpose of `rightmatrix`. Otherwise `leftmatrix` is the inverse of `rightmatrix`.

  `rightmatrix` is the matrix the columns of which are the unit eigenvectors of M. The other flags (see `eigenvalues` and `eigenvectors`) have the same effects since `similaritytransform` calls the other functions in the package in order to be able to form `rightmatrix`.

  `load ("eigen")` loads this function.

  `simtran` is a synonym for `similaritytransform`.

  Categories: Package eigen

Option variable: sparse

  Default value: `false`

  When `sparse` is `true`, and if `ratmx` is `true`, then `determinant` will use special routines for computing sparse determinants.

  Categories: Matrices

Function: submatrix
`submatrix` (i_1, …, i_m, M, j_1, …, j_n)
`submatrix` (i_1, …, i_m, M)
`submatrix` (M, j_1, …, j_n)

  Returns a new matrix composed of the matrix M with rows i_1, …, i_m deleted, and columns j_1, …, j_n deleted.

  Categories: Matrices

Function: transpose (M)

  Returns the transpose of M.

  If M is a matrix, the return value is another matrix N such that `N[i,j] = M[j,i]`.

  If M is a list, the return value is a matrix N of `length (m)` rows and 1 column, such that `N[i,1] = M[i]`.

  Otherwise M is a symbol, and the return value is a noun expression `'transpose (M)`.

  Categories: Matrices

Function: triangularize (M)

  Returns the upper triangular form of the matrix `M`, as produced by Gaussian elimination. The return value is the same as `echelon`, except that the leading nonzero coefficient in each row is not normalized to 1.

  `lu_factor` and `cholesky` are other functions which yield triangularized matrices.

      (%i1) M: matrix ([3, 7, aa, bb], [-1, 8, 5, 2], [9, 2, 11, 4]);
                             [  3   7  aa  bb ]
                             [                ]
      (%o1)                  [ - 1  8  5   2  ]
                             [                ]
                             [  9   2  11  4  ]
      (%i2) triangularize (M);
                   [ - 1   8         5            2      ]
                   [                                     ]
      (%o2)        [  0   - 74     - 56         - 22     ]
                   [                                     ]
                   [  0    0    626 - 74 aa  238 - 74 bb ]

  Categories: Linear equations · Matrices

Function: uniteigenvectors (M)
Function: ueivects (M)

  Computes unit eigenvectors of the matrix M. The return value is a list of lists, the first sublist of which is the output of the `eigenvalues` command, and the other sublists of which are the unit eigenvectors of the matrix corresponding to those eigenvalues respectively.

  The flags mentioned in the description of the `eigenvectors` command have the same effects in this one as well.

  When `knowneigvects` is `true`, the `eigen` package assumes that the eigenvectors of the matrix are known to the user and are stored under the global name `listeigvects`. `listeigvects` should be set to a list similar to the output of the `eigenvectors` command.

  If `knowneigvects` is set to `true` and the list of eigenvectors is given the setting of the flag `nondiagonalizable` may not be correct. If that is the case please set it to the correct value. The author assumes that the user knows what he is doing and will not try to diagonalize a matrix the eigenvectors of which do not span the vector space of the appropriate dimension.

  `load ("eigen")` loads this function.

  `ueivects` is a synonym for `uniteigenvectors`.

  Categories: Package eigen

Function: unitvector (x)
Function: uvect (x)

  Returns x/norm(x); this is a unit vector in the same direction as x.

  `load ("eigen")` loads this function.

  `uvect` is a synonym for `unitvector`.

  Categories: Package eigen

Function: vectorpotential (givencurl)

  Returns the vector potential of a given curl vector, in the current coordinate system. `potentialzeroloc` has a similar role as for `potential`, but the order of the left-hand sides of the equations must be a cyclic permutation of the coordinate variables.

  Categories: Package vect

Function: vectorsimp (expr)

  Applies simplifications and expansions according to the following global flags:

  `expandall`, `expanddot`, `expanddotplus`, `expandcross`, `expandcrossplus`, `expandcrosscross`, `expandgrad`, `expandgradplus`, `expandgradprod`, `expanddiv`, `expanddivplus`, `expanddivprod`, `expandcurl`, `expandcurlplus`, `expandcurlcurl`, `expandlaplacian`, `expandlaplacianplus`, and `expandlaplacianprod`.

  All these flags have default value `false`. The `plus` suffix refers to employing additivity or distributivity. The `prod` suffix refers to the expansion for an operand that is any kind of product.

  `expandcrosscross`

    Simplifies p ~ (q ~ r) to (p . r)*q - (p . q)*r.

  `expandcurlcurl`

    Simplifies curl curl p to grad div p + div grad p.

  `expandlaplaciantodivgrad`

    Simplifies laplacian p to div grad p.

  `expandcross`

    Enables `expandcrossplus` and `expandcrosscross`.

  `expandplus`

    Enables `expanddotplus`, `expandcrossplus`, `expandgradplus`, `expanddivplus`, `expandcurlplus`, and `expandlaplacianplus`.

  `expandprod`

    Enables `expandgradprod`, `expanddivprod`, and `expandlaplacianprod`.

  These flags have all been declared `evflag`.

  Categories: Package vect · Simplification functions

Option variable: vect_cross

  Default value: `false`

  When `vect_cross` is `true`, it allows DIFF(X~Y,T) to work where ~ is defined in SHARE;VECT (where VECT_CROSS is set to `true`, anyway.)

  Categories: Package vect · Differential calculus

Function: zeromatrix (m, n)

  Returns an m by n matrix, all elements of which are zero.

  Categories: Matrices

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# 24. Affine

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## 24.1 Introduction to Affine

`affine` is a package to work with groups of polynomials.

Categories: Polynomials · Groebner bases · Share packages · Package affine

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## 24.2 Functions and Variables for Affine

Function: fast_linsolve ([expr_1, ..., expr_m], [x_1, ..., x_n])

  Solves the simultaneous linear equations expr_1, ..., expr_m for the variables x_1, ..., x_n. Each expr_i may be an equation or a general expression; if given as a general expression, it is treated as an equation of the form `expr_i = 0`.

  The return value is a list of equations of the form `[x_1 = a_1, ..., x_n = a_n]` where a_1, ..., a_n are all free of x_1, ..., x_n.

  `fast_linsolve` is faster than `linsolve` for system of equations which are sparse.

  `load(affine)` loads this function.

  Categories: Package affine

Function: grobner_basis ([expr_1, ..., expr_m])

  Returns a Groebner basis for the equations expr_1, ..., expr_m. The function `polysimp` can then be used to simplify other functions relative to the equations.

      grobner_basis ([3*x^2+1, y*x])$

      polysimp (y^2*x + x^3*9 + 2) ==> -3*x + 2

  `polysimp(f)` yields 0 if and only if f is in the ideal generated by expr_1, ..., expr_m, that is, if and only if f is a polynomial combination of the elements of expr_1, ..., expr_m.

  `load(affine)` loads this function.

  Categories: Package affine

Function: set_up_dot_simplifications
`set_up_dot_simplifications` (eqns, check_through_degree)
`set_up_dot_simplifications` (eqns)

  The eqns are polynomial equations in non commutative variables. The value of `current_variables` is the list of variables used for computing degrees. The equations must be homogeneous, in order for the procedure to terminate.

  If you have checked overlapping simplifications in `dot_simplifications` above the degree of f, then the following is true: `dotsimp (f)` yields 0 if and only if f is in the ideal generated by the equations, i.e., if and only if f is a polynomial combination of the elements of the equations.

  The degree is that returned by `nc_degree`. This in turn is influenced by the weights of individual variables.

  `load(affine)` loads this function.

  Categories: Package affine

Function: declare_weights (x_1, w_1, ..., x_n, w_n)

  Assigns weights w_1, ..., w_n to x_1, ..., x_n, respectively. These are the weights used in computing `nc_degree`.

  `load(affine)` loads this function.

  Categories: Package affine

Function: nc_degree (p)

  Returns the degree of a noncommutative polynomial p. See `declare_weights`.

  `load(affine)` loads this function.

  Categories: Package affine

Function: dotsimp (f)

  Returns 0 if and only if f is in the ideal generated by the equations, i.e., if and only if f is a polynomial combination of the elements of the equations.

  `load(affine)` loads this function.

  Categories: Package affine

Function: fast_central_elements ([x_1, ..., x_n], n)

  If `set_up_dot_simplifications` has been previously done, finds the central polynomials in the variables x_1, ..., x_n in the given degree, n.

  For example:

      set_up_dot_simplifications ([y.x + x.y], 3);
      fast_central_elements ([x, y], 2);
      [y.y, x.x];

  `load(affine)` loads this function.

  Categories: Package affine

Function: check_overlaps (n, add_to_simps)

  Checks the overlaps thru degree n, making sure that you have sufficient simplification rules in each degree, for `dotsimp` to work correctly. This process can be speeded up if you know before hand what the dimension of the space of monomials is. If it is of finite global dimension, then `hilbert` should be used. If you don't know the monomial dimensions, do not specify a `rank_function`. An optional third argument `reset`, `false` says don't bother to query about resetting things.

  `load(affine)` loads this function.

  Categories: Package affine

Function: mono ([x_1, ..., x_n], n)

  Returns the list of independent monomials relative to the current dot simplifications of degree n in the variables x_1, ..., x_n.

  `load(affine)` loads this function.

  Categories: Package affine

Function: monomial_dimensions (n)

  Compute the Hilbert series through degree n for the current algebra.

  `load(affine)` loads this function.

  Categories: Package affine

Function: extract_linear_equations ([p_1, ..., p_n], [m_1, ..., m_n])

  Makes a list of the coefficients of the noncommutative polynomials p_1, ..., p_n of the noncommutative monomials m_1, ..., m_n. The coefficients should be scalars. Use `list_nc_monomials` to build the list of monomials.

  `load(affine)` loads this function.

  Categories: Package affine

Function: list_nc_monomials
`list_nc_monomials` ([p_1, ..., p_n])
`list_nc_monomials` (p)

  Returns a list of the non commutative monomials occurring in a polynomial p or a list of polynomials p_1, ..., p_n.

  `load(affine)` loads this function.

  Categories: Package affine

Option variable: all_dotsimp_denoms

  Default value: `false`

  When `all_dotsimp_denoms` is a list, the denominators encountered by `dotsimp` are appended to the list. `all_dotsimp_denoms` may be initialized to an empty list `[]` before calling `dotsimp`.

  By default, denominators are not collected by `dotsimp`.

  Categories: Package affine

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# 25. itensor

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## 25.1 Introduction to itensor

Maxima implements symbolic tensor manipulation of two distinct types: component tensor manipulation (`ctensor` package) and indicial tensor manipulation (`itensor` package).

Nota bene: Please see the note on 'new tensor notation' below.

Component tensor manipulation means that geometrical tensor objects are represented as arrays or matrices. Tensor operations such as contraction or covariant differentiation are carried out by actually summing over repeated (dummy) indices with `do` statements. That is, one explicitly performs operations on the appropriate tensor components stored in an array or matrix.

Indicial tensor manipulation is implemented by representing tensors as functions of their covariant, contravariant and derivative indices. Tensor operations such as contraction or covariant differentiation are performed by manipulating the indices themselves rather than the components to which they correspond.

These two approaches to the treatment of differential, algebraic and analytic processes in the context of Riemannian geometry have various advantages and disadvantages which reveal themselves only through the particular nature and difficulty of the user's problem. However, one should keep in mind the following characteristics of the two implementations:

The representation of tensors and tensor operations explicitly in terms of their components makes `ctensor` easy to use. Specification of the metric and the computation of the induced tensors and invariants is straightforward. Although all of Maxima's powerful simplification capacity is at hand, a complex metric with intricate functional and coordinate dependencies can easily lead to expressions whose size is excessive and whose structure is hidden. In addition, many calculations involve intermediate expressions which swell causing programs to terminate before completion. Through experience, a user can avoid many of these difficulties.

Because of the special way in which tensors and tensor operations are represented in terms of symbolic operations on their indices, expressions which in the component representation would be unmanageable can sometimes be greatly simplified by using the special routines for symmetrical objects in `itensor`. In this way the structure of a large expression may be more transparent. On the other hand, because of the special indicial representation in `itensor`, in some cases the user may find difficulty with the specification of the metric, function definition, and the evaluation of differentiated "indexed" objects.

The `itensor` package can carry out differentiation with respect to an indexed variable, which allows one to use the package when dealing with Lagrangian and Hamiltonian formalisms. As it is possible to differentiate a field Lagrangian with respect to an (indexed) field variable, one can use Maxima to derive the corresponding Euler-Lagrange equations in indicial form. These equations can be translated into component tensor (`ctensor`) programs using the `ic_convert` function, allowing us to solve the field equations in a particular coordinate representation, or to recast the equations of motion in Hamiltonian form. See `einhil.dem` and `bradic.dem` for two comprehensive examples. The first, `einhil.dem`, uses the Einstein-Hilbert action to derive the Einstein field tensor in the homogeneous and isotropic case (Friedmann equations) and the spherically symmetric, static case (Schwarzschild solution.) The second, `bradic.dem`, demonstrates how to compute the Friedmann equations from the action of Brans-Dicke gravity theory, and also derives the Hamiltonian associated with the theory's scalar field.

Categories: Tensors · Share packages · Package itensor

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### 25.1.1 New tensor notation

Earlier versions of the `itensor` package in Maxima used a notation that sometimes led to incorrect index ordering. Consider the following, for instance:

    (%i2) imetric(g);
    (%o2)                                done
    (%i3) ishow(g([],[j,k])*g([],[i,l])*a([i,j],[]))$
                                     i l  j k
    (%t3)                           g    g    a
                                               i j
    (%i4) ishow(contract(%))$
                                          k l
    (%t4)                                a

This result is incorrect unless `a` happens to be a symmetric tensor. The reason why this happens is that although `itensor` correctly maintains the order within the set of covariant and contravariant indices, once an index is raised or lowered, its position relative to the other set of indices is lost.

To avoid this problem, a new notation has been developed that remains fully compatible with the existing notation and can be used interchangeably. In this notation, contravariant indices are inserted in the appropriate positions in the covariant index list, but with a minus sign prepended. Functions like `contract_Itensor` and `ishow` are now aware of this new index notation and can process tensors appropriately.

In this new notation, the previous example yields a correct result:

    (%i5) ishow(g([-j,-k],[])*g([-i,-l],[])*a([i,j],[]))$
                                     i l       j k
    (%t5)                           g    a    g
                                          i j
    (%i6) ishow(contract(%))$
                                          l k
    (%t6)                                a

Presently, the only code that makes use of this notation is the `lc2kdt` function. Through this notation, it achieves consistent results as it applies the metric tensor to resolve Levi-Civita symbols without resorting to numeric indices.

Since this code is brand new, it probably contains bugs. While it has been tested to make sure that it doesn't break anything using the "old" tensor notation, there is a considerable chance that "new" tensors will fail to interoperate with certain functions or features. These bugs will be fixed as they are encountered... until then, caveat emptor!

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### 25.1.2 Indicial tensor manipulation

The indicial tensor manipulation package may be loaded by `load(itensor)`. Demos are also available: try `demo(tensor)`.

In `itensor` a tensor is represented as an "indexed object" . This is a function of 3 groups of indices which represent the covariant, contravariant and derivative indices. The covariant indices are specified by a list as the first argument to the indexed object, and the contravariant indices by a list as the second argument. If the indexed object lacks either of these groups of indices then the empty list `[]` is given as the corresponding argument. Thus, `g([a,b],[c])` represents an indexed object called `g` which has two covariant indices `(a,b)`, one contravariant index (`c`) and no derivative indices.

The derivative indices, if they are present, are appended as additional arguments to the symbolic function representing the tensor. They can be explicitly specified by the user or be created in the process of differentiation with respect to some coordinate variable. Since ordinary differentiation is commutative, the derivative indices are sorted alphanumerically, unless `iframe_flag` is set to `true`, indicating that a frame metric is being used. This canonical ordering makes it possible for Maxima to recognize that, for example, `t([a],[b],i,j)` is the same as `t([a],[b],j,i)`. Differentiation of an indexed object with respect to some coordinate whose index does not appear as an argument to the indexed object would normally yield zero. This is because Maxima would not know that the tensor represented by the indexed object might depend implicitly on the corresponding coordinate. By modifying the existing Maxima function `diff` in `itensor`, Maxima now assumes that all indexed objects depend on any variable of differentiation unless otherwise stated. This makes it possible for the summation convention to be extended to derivative indices. It should be noted that `itensor` does not possess the capabilities of raising derivative indices, and so they are always treated as covariant.

The following functions are available in the tensor package for manipulating indexed objects. At present, with respect to the simplification routines, it is assumed that indexed objects do not by default possess symmetry properties. This can be overridden by setting the variable `allsym[false]` to `true`, which will result in treating all indexed objects completely symmetric in their lists of covariant indices and symmetric in their lists of contravariant indices.

The `itensor` package generally treats tensors as opaque objects. Tensorial equations are manipulated based on algebraic rules, specifically symmetry and contraction rules. In addition, the `itensor` package understands covariant differentiation, curvature, and torsion. Calculations can be performed relative to a metric of moving frame, depending on the setting of the `iframe_flag` variable.

A sample session below demonstrates how to load the `itensor` package, specify the name of the metric, and perform some simple calculations.

    (%i1) load(itensor);
    (%o1)      /share/tensor/itensor.lisp
    (%i2) imetric(g);
    (%o2)                                done
    (%i3) components(g([i,j],[]),p([i,j],[])*e([],[]))$
    (%i4) ishow(g([k,l],[]))$
    (%t4)                               e p
                                           k l
    (%i5) ishow(diff(v([i],[]),t))$
    (%t5)                                  0
    (%i6) depends(v,t);
    (%o6)                               [v(t)]
    (%i7) ishow(diff(v([i],[]),t))$
                                        d
    (%t7)                               -- (v )
                                        dt   i
    (%i8) ishow(idiff(v([i],[]),j))$
    (%t8)                                v
                                          i,j
    (%i9) ishow(extdiff(v([i],[]),j))$
    (%t9)                             v    - v
                                       j,i    i,j
                                      -----------
                                           2
    (%i10) ishow(liediff(v,w([i],[])))$
                                   %3          %3
    (%t10)                        v   w     + v   w
                                       i,%3    ,i  %3
    (%i11) ishow(covdiff(v([i],[]),j))$
                                                  %4
    (%t11)                        v    - v   ichr2
                                   i,j    %4      i j
    (%i12) ishow(ev(%,ichr2))$
                    %4 %5
    (%t12) v    - (g      v   (e p       + e   p     - e p       - e    p
            i,j            %4     j %5,i    ,i  j %5      i j,%5    ,%5  i j

                                             + e p       + e   p    ))/2
                                                  i %5,j    ,j  i %5
    (%i13) iframe_flag:true;
    (%o13)                               true
    (%i14) ishow(covdiff(v([i],[]),j))$
                                                 %6
    (%t14)                        v    - v   icc2
                                   i,j    %6     i j
    (%i15) ishow(ev(%,icc2))$
                                                 %6
    (%t15)                        v    - v   ifc2
                                   i,j    %6     i j
    (%i16) ishow(radcan(ev(%,ifc2,ifc1)))$
                 %6 %7                    %6 %7
    (%t16) - (ifg      v   ifb       + ifg      v   ifb       - 2 v
                        %6    j %7 i             %6    i j %7      i,j

                                                 %6 %7
                                            - ifg      v   ifb      )/2
                                                        %6    %7 i j
    (%i17) ishow(canform(s([i,j],[])-s([j,i])))$
    (%t17)                            s    - s
                                       i j    j i
    (%i18) decsym(s,2,0,[sym(all)],[]);
    (%o18)                               done
    (%i19) ishow(canform(s([i,j],[])-s([j,i])))$
    (%t19)                                 0
    (%i20) ishow(canform(a([i,j],[])+a([j,i])))$
    (%t20)                            a    + a
                                       j i    i j
    (%i21) decsym(a,2,0,[anti(all)],[]);
    (%o21)                               done
    (%i22) ishow(canform(a([i,j],[])+a([j,i])))$
    (%t22)                                 0

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## 25.2 Functions and Variables for itensor

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### 25.2.1 Managing indexed objects

Function: dispcon
`dispcon` (tensor_1, tensor_2, …)
`dispcon` (all)

  Displays the contraction properties of its arguments as were given to `defcon`. `dispcon (all)` displays all the contraction properties which were defined.

  Categories: Display functions

Function: entertensor (name)

  is a function which, by prompting, allows one to create an indexed object called name with any number of tensorial and derivative indices. Either a single index or a list of indices (which may be null) is acceptable input (see the example under `covdiff`).

  Categories: Package itensor

Function: changename (old, new, expr)

  will change the name of all indexed objects called old to new in expr. old may be either a symbol or a list of the form `[name, m, n]` in which case only those indexed objects called name with m covariant and n contravariant indices will be renamed to new.

  Categories: Package itensor

Function: listoftens

  Lists all tensors in a tensorial expression, complete with their indices. E.g.,

      (%i6) ishow(a([i,j],[k])*b([u],[],v)+c([x,y],[])*d([],[])*e)$
                                               k
      (%t6)                        d e c    + a    b
                                        x y    i j  u,v
      (%i7) ishow(listoftens(%))$
                                     k
      (%t7)                        [a   , b   , c   , d]
                                     i j   u,v   x y

  Categories: Package itensor

Function: ishow (expr)

  displays expr with the indexed objects in it shown having their covariant indices as subscripts and contravariant indices as superscripts. The derivative indices are displayed as subscripts, separated from the covariant indices by a comma (see the examples throughout this document).

  Categories: Package itensor

Function: indices (expr)

  Returns a list of two elements. The first is a list of the free indices in expr (those that occur only once). The second is the list of the dummy indices in expr (those that occur exactly twice) as the following example demonstrates.

      (%i1) load(itensor);
      (%o1)      /share/tensor/itensor.lisp
      (%i2) ishow(a([i,j],[k,l],m,n)*b([k,o],[j,m,p],q,r))$
                                      k l      j m p
      (%t2)                          a        b
                                      i j,m n  k o,q r
      (%i3) indices(%);
      (%o3)                 [[l, p, i, n, o, q, r], [k, j, m]]

  A tensor product containing the same index more than twice is syntactically illegal. `indices` attempts to deal with these expressions in a reasonable manner; however, when it is called to operate upon such an illegal expression, its behavior should be considered undefined.

  Categories: Package itensor

Function: rename
`rename` (expr)
`rename` (expr, count)

  Returns an expression equivalent to expr but with the dummy indices in each term chosen from the set `[%1, %2,...]`, if the optional second argument is omitted. Otherwise, the dummy indices are indexed beginning at the value of count. Each dummy index in a product will be different. For a sum, `rename` will operate upon each term in the sum resetting the counter with each term. In this way `rename` can serve as a tensorial simplifier. In addition, the indices will be sorted alphanumerically (if `allsym` is `true`) with respect to covariant or contravariant indices depending upon the value of `flipflag`. If `flipflag` is `false` then the indices will be renamed according to the order of the contravariant indices. If `flipflag` is `true` the renaming will occur according to the order of the covariant indices. It often happens that the combined effect of the two renamings will reduce an expression more than either one by itself.

      (%i1) load(itensor);
      (%o1)      /share/tensor/itensor.lisp
      (%i2) allsym:true;
      (%o2)                                true
      (%i3) g([],[%4,%5])*g([],[%6,%7])*ichr2([%1,%4],[%3])*
      ichr2([%2,%3],[u])*ichr2([%5,%6],[%1])*ichr2([%7,r],[%2])-
      g([],[%4,%5])*g([],[%6,%7])*ichr2([%1,%2],[u])*
      ichr2([%3,%5],[%1])*ichr2([%4,%6],[%3])*ichr2([%7,r],[%2]),noeval$
      (%i4) expr:ishow(%)$
             %4 %5  %6 %7      %3         u          %1         %2
      (%t4) g      g      ichr2      ichr2      ichr2      ichr2
                               %1 %4      %2 %3      %5 %6      %7 r

              %4 %5  %6 %7      u          %1         %3         %2
           - g      g      ichr2      ichr2      ichr2      ichr2
                                %1 %2      %3 %5      %4 %6      %7 r
      (%i5) flipflag:true;
      (%o5)                                true
      (%i6) ishow(rename(expr))$
             %2 %5  %6 %7      %4         u          %1         %3
      (%t6) g      g      ichr2      ichr2      ichr2      ichr2
                               %1 %2      %3 %4      %5 %6      %7 r

              %4 %5  %6 %7      u          %1         %3         %2
           - g      g      ichr2      ichr2      ichr2      ichr2
                                %1 %2      %3 %4      %5 %6      %7 r
      (%i7) flipflag:false;
      (%o7)                                false
      (%i8) rename(%th(2));
      (%o8)                                  0
      (%i9) ishow(rename(expr))$
             %1 %2  %3 %4      %5         %6         %7        u
      (%t9) g      g      ichr2      ichr2      ichr2     ichr2
                               %1 %6      %2 %3      %4 r      %5 %7

              %1 %2  %3 %4      %6         %5         %7        u
           - g      g      ichr2      ichr2      ichr2     ichr2
                                %1 %3      %2 %6      %4 r      %5 %7

  Categories: Package itensor

Function: show (expr)

  Displays `expr` with the indexed objects in it shown having covariant indices as subscripts, contravariant indices as superscripts. The derivative indices are displayed as subscripts, separated from the covariant indices by a comma.

  Categories: Package itensor · Display functions

Option variable: flipflag

  Default value: `false`

  If `false` then the indices will be renamed according to the order of the contravariant indices, otherwise according to the order of the covariant indices.

  If `flipflag` is `false` then `rename` forms a list of the contravariant indices as they are encountered from left to right (if `true` then of the covariant indices). The first dummy index in the list is renamed to `%1`, the next to `%2`, etc. Then sorting occurs after the `rename`-ing (see the example under `rename`).

  Categories: Package itensor

Function: defcon
`defcon` (tensor_1)
`defcon` (tensor_1, tensor_2, tensor_3)

  gives tensor_1 the property that the contraction of a product of tensor_1 and tensor_2 results in tensor_3 with the appropriate indices. If only one argument, tensor_1, is given, then the contraction of the product of tensor_1 with any indexed object having the appropriate indices (say `my_tensor`) will yield an indexed object with that name, i.e. `my_tensor`, and with a new set of indices reflecting the contractions performed. For example, if `imetric:g`, then `defcon(g)` will implement the raising and lowering of indices through contraction with the metric tensor. More than one `defcon` can be given for the same indexed object; the latest one given which applies in a particular contraction will be used. `contractions` is a list of those indexed objects which have been given contraction properties with `defcon`.

  Categories: Package itensor

Function: remcon
`remcon` (tensor_1, ..., tensor_n)
`remcon` (all)

  Removes all the contraction properties from the (tensor_1, ..., tensor_n). `remcon(all)` removes all contraction properties from all indexed objects.

  Categories: Package itensor

Function: contract (expr)

  Carries out the tensorial contractions in expr which may be any combination of sums and products. This function uses the information given to the `defcon` function. For best results, `expr` should be fully expanded. `ratexpand` is the fastest way to expand products and powers of sums if there are no variables in the denominators of the terms. The `gcd` switch should be `false` if GCD cancellations are unnecessary.

  Categories: Package itensor

Function: indexed_tensor (tensor)

  Must be executed before assigning components to a tensor for which a built in value already exists as with `ichr1`, `ichr2`, `icurvature`. See the example under `icurvature`.

  Categories: Package itensor

Function: components (tensor, expr)

  permits one to assign an indicial value to an expression expr giving the values of the components of tensor. These are automatically substituted for the tensor whenever it occurs with all of its indices. The tensor must be of the form `t([...],[...])` where either list may be empty. expr can be any indexed expression involving other objects with the same free indices as tensor. When used to assign values to the metric tensor wherein the components contain dummy indices one must be careful to define these indices to avoid the generation of multiple dummy indices. Removal of this assignment is given to the function `remcomps`.

  It is important to keep in mind that `components` cares only about the valence of a tensor, not about any particular index ordering. Thus assigning components to, say, `x([i,-j],[])`, `x([-j,i],[])`, or `x([i],[j])` all produce the same result, namely components being assigned to a tensor named `x` with valence `(1,1)`.

  Components can be assigned to an indexed expression in four ways, two of which involve the use of the `components` command:

  1) As an indexed expression. For instance:

      (%i2) components(g([],[i,j]),e([],[i])*p([],[j]))$
      (%i3) ishow(g([],[i,j]))$
                                            i  j
      (%t3)                                e  p

  2) As a matrix:

      (%i5) lg:-ident(4)$lg[1,1]:1$lg;
                                  [ 1   0    0    0  ]
                                  [                  ]
                                  [ 0  - 1   0    0  ]
      (%o5)                       [                  ]
                                  [ 0   0   - 1   0  ]
                                  [                  ]
                                  [ 0   0    0   - 1 ]
      (%i6) components(g([i,j],[]),lg);
      (%o6)                                done
      (%i7) ishow(g([i,j],[]))$
      (%t7)                                g
                                            i j
      (%i8) g([1,1],[]);
      (%o8)                                  1
      (%i9) g([4,4],[]);
      (%o9)                                 - 1

  3) As a function. You can use a Maxima function to specify the components of a tensor based on its indices. For instance, the following code assigns `kdelta` to `h` if `h` has the same number of covariant and contravariant indices and no derivative indices, and `g` otherwise:

      (%i4) h(l1,l2,[l3]):=if length(l1)=length(l2) and length(l3)=0
        then kdelta(l1,l2) else apply(g,append([l1,l2], l3))$
      (%i5) ishow(h([i],[j]))$
                                                j
      (%t5)                               kdelta
                                                i
      (%i6) ishow(h([i,j],[k],l))$
                                           k
      (%t6)                               g
                                           i j,l

  4) Using Maxima's pattern matching capabilities, specifically the `defrule` and `applyb1` commands:

      (%i1) load(itensor);
      (%o1)      /share/tensor/itensor.lisp
      (%i2) matchdeclare(l1,listp);
      (%o2)                                done
      (%i3) defrule(r1,m(l1,[]),(i1:idummy(),
            g([l1[1],l1[2]],[])*q([i1],[])*e([],[i1])))$

      (%i4) defrule(r2,m([],l1),(i1:idummy(),
            w([],[l1[1],l1[2]])*e([i1],[])*q([],[i1])))$

      (%i5) ishow(m([i,n],[])*m([],[i,m]))$
                                          i m
      (%t5)                              m    m
                                               i n
      (%i6) ishow(rename(applyb1(%,r1,r2)))$
                                 %1  %2  %3 m
      (%t6)                     e   q   w     q   e   g
                                               %1  %2  %3 n

  Categories: Package itensor

Function: remcomps (tensor)

  Unbinds all values from tensor which were assigned with the `components` function.

  Categories: Package itensor

Function: showcomps (tensor)

  Shows component assignments of a tensor, as made using the `components` command. This function can be particularly useful when a matrix is assigned to an indicial tensor using `components`, as demonstrated by the following example:

      (%i1) load(ctensor);
      (%o1)       /share/tensor/ctensor.mac
      (%i2) load(itensor);
      (%o2)      /share/tensor/itensor.lisp
      (%i3) lg:matrix([sqrt(r/(r-2*m)),0,0,0],[0,r,0,0],
                      [0,0,sin(theta)*r,0],[0,0,0,sqrt((r-2*m)/r)]);
                     [         r                                     ]
                     [ sqrt(-------)  0       0              0       ]
                     [      r - 2 m                                  ]
                     [                                               ]
                     [       0        r       0              0       ]
      (%o3)          [                                               ]
                     [       0        0  r sin(theta)        0       ]
                     [                                               ]
                     [                                      r - 2 m  ]
                     [       0        0       0        sqrt(-------) ]
                     [                                         r     ]
      (%i4) components(g([i,j],[]),lg);
      (%o4)                                done
      (%i5) showcomps(g([i,j],[]));
                        [         r                                     ]
                        [ sqrt(-------)  0       0              0       ]
                        [      r - 2 m                                  ]
                        [                                               ]
                        [       0        r       0              0       ]
      (%t5)      g    = [                                               ]
                  i j   [       0        0  r sin(theta)        0       ]
                        [                                               ]
                        [                                      r - 2 m  ]
                        [       0        0       0        sqrt(-------) ]
                        [                                         r     ]
      (%o5)                                false

  The `showcomps` command can also display components of a tensor of rank higher than 2.

  Categories: Package itensor

Function: idummy ()

  Increments `icounter` and returns as its value an index of the form `%n` where n is a positive integer. This guarantees that dummy indices which are needed in forming expressions will not conflict with indices already in use (see the example under `indices`).

  Categories: Package itensor

Option variable: idummyx

  Default value: `%`

  Is the prefix for dummy indices (see the example under `indices`).

  Categories: Package itensor

Option variable: icounter

  Default value: `1`

  Determines the numerical suffix to be used in generating the next dummy index in the tensor package. The prefix is determined by the option `idummy` (default: `%`).

  Categories: Package itensor

Function: kdelta (L1, L2)

  is the generalized Kronecker delta function defined in the `itensor` package with L1 the list of covariant indices and L2 the list of contravariant indices. `kdelta([i],[j])` returns the ordinary Kronecker delta. The command `ev(expr,kdelta)` causes the evaluation of an expression containing `kdelta([],[])` to the dimension of the manifold.

  In what amounts to an abuse of this notation, `itensor` also allows `kdelta` to have 2 covariant and no contravariant, or 2 contravariant and no covariant indices, in effect providing a co(ntra)variant "unit matrix" capability. This is strictly considered a programming aid and not meant to imply that `kdelta([i,j],[])` is a valid tensorial object.

  Categories: Package itensor

Function: kdels (L1, L2)

  Symmetrized Kronecker delta, used in some calculations. For instance:

      (%i1) load(itensor);
      (%o1)      /share/tensor/itensor.lisp
      (%i2) kdelta([1,2],[2,1]);
      (%o2)                                 - 1
      (%i3) kdels([1,2],[2,1]);
      (%o3)                                  1
      (%i4) ishow(kdelta([a,b],[c,d]))$
                                   c       d         d       c
      (%t4)                  kdelta  kdelta  - kdelta  kdelta
                                   a       b         a       b
      (%i4) ishow(kdels([a,b],[c,d]))$
                                   c       d         d       c
      (%t4)                  kdelta  kdelta  + kdelta  kdelta
                                   a       b         a       b

  Categories: Package itensor

Function: levi_civita (L)

  is the permutation (or Levi-Civita) tensor which yields 1 if the list L consists of an even permutation of integers, -1 if it consists of an odd permutation, and 0 if some indices in L are repeated.

  Categories: Package itensor

Function: lc2kdt (expr)

  Simplifies expressions containing the Levi-Civita symbol, converting these to Kronecker-delta expressions when possible. The main difference between this function and simply evaluating the Levi-Civita symbol is that direct evaluation often results in Kronecker expressions containing numerical indices. This is often undesirable as it prevents further simplification. The `lc2kdt` function avoids this problem, yielding expressions that are more easily simplified with `rename` or `contract`.

      (%i1) load(itensor);
      (%o1)      /share/tensor/itensor.lisp
      (%i2) expr:ishow('levi_civita([],[i,j])
                       *'levi_civita([k,l],[])*a([j],[k]))$
                                        i j  k
      (%t2)                  levi_civita    a  levi_civita
                                             j            k l
      (%i3) ishow(ev(expr,levi_civita))$
                                        i j  k       1 2
      (%t3)                       kdelta    a  kdelta
                                        1 2  j       k l
      (%i4) ishow(ev(%,kdelta))$
                   i       j         j       i   k
      (%t4) (kdelta  kdelta  - kdelta  kdelta ) a
                   1       2         1       2   j

                                     1       2         2       1
                              (kdelta  kdelta  - kdelta  kdelta )
                                     k       l         k       l
      (%i5) ishow(lc2kdt(expr))$
                           k       i       j    k       j       i
      (%t5)               a  kdelta  kdelta  - a  kdelta  kdelta
                           j       k       l    j       k       l
      (%i6) ishow(contract(expand(%)))$
                                       i           i
      (%t6)                           a  - a kdelta
                                       l           l

  The `lc2kdt` function sometimes makes use of the metric tensor. If the metric tensor was not defined previously with `imetric`, this results in an error.

      (%i7) expr:ishow('levi_civita([],[i,j])
                       *'levi_civita([],[k,l])*a([j,k],[]))$
                                       i j            k l
      (%t7)                 levi_civita    levi_civita    a
                                                           j k
      (%i8) ishow(lc2kdt(expr))$
      Maxima encountered a Lisp error:

       Error in $IMETRIC [or a callee]:
       $IMETRIC [or a callee] requires less than two arguments.

      Automatically continuing.
      To reenable the Lisp debugger set *debugger-hook* to nil.
      (%i9) imetric(g);
      (%o9)                                done
      (%i10) ishow(lc2kdt(expr))$
               %3 i       k   %4 j       l     %3 i       l   %4 j
      (%t10) (g     kdelta   g     kdelta   - g     kdelta   g
                          %3             %4               %3
                    k
              kdelta  ) a
                    %4   j k
      (%i11) ishow(contract(expand(%)))$
                                        l i    l i  j
      (%t11)                           a    - g    a
                                                    j

  Categories: Package itensor

Function: lc_l

  Simplification rule used for expressions containing the unevaluated Levi-Civita symbol (`levi_civita`). Along with `lc_u`, it can be used to simplify many expressions more efficiently than the evaluation of `levi_civita`. For example:

      (%i1) load(itensor);
      (%o1)      /share/tensor/itensor.lisp
      (%i2) el1:ishow('levi_civita([i,j,k],[])*a([],[i])*a([],[j]))$
                                   i  j
      (%t2)                       a  a  levi_civita
                                                   i j k
      (%i3) el2:ishow('levi_civita([],[i,j,k])*a([i])*a([j]))$
                                             i j k
      (%t3)                       levi_civita      a  a
                                                    i  j
      (%i4) canform(contract(expand(applyb1(el1,lc_l,lc_u))));
      (%t4)                                  0
      (%i5) canform(contract(expand(applyb1(el2,lc_l,lc_u))));
      (%t5)                                  0

  Categories: Package itensor

Function: lc_u

  Simplification rule used for expressions containing the unevaluated Levi-Civita symbol (`levi_civita`). Along with `lc_u`, it can be used to simplify many expressions more efficiently than the evaluation of `levi_civita`. For details, see `lc_l`.

  Categories: Package itensor

Function: canten (expr)

  Simplifies expr by renaming (see `rename`) and permuting dummy indices. `rename` is restricted to sums of tensor products in which no derivatives are present. As such it is limited and should only be used if `canform` is not capable of carrying out the required simplification.

  The `canten` function returns a mathematically correct result only if its argument is an expression that is fully symmetric in its indices. For this reason, `canten` returns an error if `allsym` is not set to `true`.

  Categories: Package itensor

Function: concan (expr)

  Similar to `canten` but also performs index contraction.

  Categories: Package itensor

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### 25.2.2 Tensor symmetries

Option variable: allsym

  Default value: `false`

  If `true` then all indexed objects are assumed symmetric in all of their covariant and contravariant indices. If `false` then no symmetries of any kind are assumed in these indices. Derivative indices are always taken to be symmetric unless `iframe_flag` is set to `true`.

  Categories: Package itensor

Function: decsym (tensor, m, n, [cov_1, cov_2, ...], [contr_1, contr_2, ...])

  Declares symmetry properties for tensor of m covariant and n contravariant indices. The cov_i and contr_i are pseudofunctions expressing symmetry relations among the covariant and contravariant indices respectively. These are of the form `symoper(index_1, index_2,...)` where `symoper` is one of `sym`, `anti` or `cyc` and the index_i are integers indicating the position of the index in the tensor. This will declare tensor to be symmetric, antisymmetric or cyclic respectively in the index_i. `symoper(all)` is also an allowable form which indicates all indices obey the symmetry condition. For example, given an object `b` with 5 covariant indices, `decsym(b,5,3,[sym(1,2),anti(3,4)],[cyc(all)])` declares `b` symmetric in its first and second and antisymmetric in its third and fourth covariant indices, and cyclic in all of its contravariant indices. Either list of symmetry declarations may be null. The function which performs the simplifications is `canform` as the example below illustrates.

      (%i1) load(itensor);
      (%o1)      /share/tensor/itensor.lisp
      (%i2) expr:contract( expand( a([i1, j1, k1], [])
                 *kdels([i, j, k], [i1, j1, k1])))$
      (%i3) ishow(expr)$
      (%t3)         a      + a      + a      + a      + a      + a
                     k j i    k i j    j k i    j i k    i k j    i j k
      (%i4) decsym(a,3,0,[sym(all)],[]);
      (%o4)                                done
      (%i5) ishow(canform(expr))$
      (%t5)                              6 a
                                            i j k
      (%i6) remsym(a,3,0);
      (%o6)                                done
      (%i7) decsym(a,3,0,[anti(all)],[]);
      (%o7)                                done
      (%i8) ishow(canform(expr))$
      (%t8)                                  0
      (%i9) remsym(a,3,0);
      (%o9)                                done
      (%i10) decsym(a,3,0,[cyc(all)],[]);
      (%o10)                               done
      (%i11) ishow(canform(expr))$
      (%t11)                        3 a      + 3 a
                                       i k j      i j k
      (%i12) dispsym(a,3,0);
      (%o12)                     [[cyc, [[1, 2, 3]], []]]

  Categories: Package itensor

Function: remsym (tensor, m, n)

  Removes all symmetry properties from tensor which has m covariant indices and n contravariant indices.

  Categories: Package itensor

Function: canform
`canform` (expr)
`canform` (expr, rename)

  Simplifies expr by renaming dummy indices and reordering all indices as dictated by symmetry conditions imposed on them. If `allsym` is `true` then all indices are assumed symmetric, otherwise symmetry information provided by `decsym` declarations will be used. The dummy indices are renamed in the same manner as in the `rename` function. When `canform` is applied to a large expression the calculation may take a considerable amount of time. This time can be shortened by calling `rename` on the expression first. Also see the example under `decsym`. Note: `canform` may not be able to reduce an expression completely to its simplest form although it will always return a mathematically correct result.

  The optional second parameter rename, if set to `false`, suppresses renaming.

  Categories: Package itensor

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### 25.2.3 Indicial tensor calculus

Function: diff (expr, v_1, [n_1, [v_2, n_2] ...])

  is the usual Maxima differentiation function which has been expanded in its abilities for `itensor`. It takes the derivative of expr with respect to v_1 n_1 times, with respect to v_2 n_2 times, etc. For the tensor package, the function has been modified so that the v_i may be integers from 1 up to the value of the variable `dim`. This will cause the differentiation to be carried out with respect to the v_ith member of the list `vect_coords`. If `vect_coords` is bound to an atomic variable, then that variable subscripted by v_i will be used for the variable of differentiation. This permits an array of coordinate names or subscripted names like `x[1]`, `x[2]`, ... to be used.

  A further extension adds the ability to `diff` to compute derivatives with respect to an indexed variable. In particular, the tensor package knows how to differentiate expressions containing combinations of the metric tensor and its derivatives with respect to the metric tensor and its first and second derivatives. This capability is particularly useful when considering Lagrangian formulations of a gravitational theory, allowing one to derive the Einstein tensor and field equations from the action principle.

  Categories: Package itensor

Function: idiff (expr, v_1, [n_1, [v_2, n_2] ...])

  Indicial differentiation. Unlike `diff`, which differentiates with respect to an independent variable, `idiff)` can be used to differentiate with respect to a coordinate. For an indexed object, this amounts to appending the v_i as derivative indices. Subsequently, derivative indices will be sorted, unless `iframe_flag` is set to `true`.

  `idiff` can also differentiate the determinant of the metric tensor. Thus, if `imetric` has been bound to `G` then `idiff(determinant(g),k)` will return `2 * determinant(g) * ichr2([%i,k],[%i])` where the dummy index `%i` is chosen appropriately.

  Categories: Package itensor

Function: liediff (v, ten)

  Computes the Lie-derivative of the tensorial expression ten with respect to the vector field v. ten should be any indexed tensor expression; v should be the name (without indices) of a vector field. For example:

      (%i1) load(itensor);
      (%o1)      /share/tensor/itensor.lisp
      (%i2) ishow(liediff(v,a([i,j],[])*b([],[k],l)))$
             k    %2            %2          %2
      (%t2) b   (v   a       + v   a     + v   a    )
             ,l       i j,%2    ,j  i %2    ,i  %2 j

                                %1  k        %1  k      %1  k
                            + (v   b      - b   v    + v   b   ) a
                                    ,%1 l    ,l  ,%1    ,l  ,%1   i j

  Categories: Package itensor

Function: rediff (ten)

  Evaluates all occurrences of the `idiff` command in the tensorial expression ten.

  Categories: Package itensor

Function: undiff (expr)

  Returns an expression equivalent to expr but with all derivatives of indexed objects replaced by the noun form of the `idiff` function. Its arguments would yield that indexed object if the differentiation were carried out. This is useful when it is desired to replace a differentiated indexed object with some function definition resulting in expr and then carry out the differentiation by saying `ev(expr, idiff)`.

  Categories: Package itensor

Function: evundiff (expr)

  Equivalent to the execution of `undiff`, followed by `ev` and `rediff`.

  The point of this operation is to easily evalute expressions that cannot be directly evaluated in derivative form. For instance, the following causes an error:

      (%i1) load(itensor);
      (%o1)      /share/tensor/itensor.lisp
      (%i2) icurvature([i,j,k],[l],m);
      Maxima encountered a Lisp error:

       Error in $ICURVATURE [or a callee]:
       $ICURVATURE [or a callee] requires less than three arguments.

      Automatically continuing.
      To reenable the Lisp debugger set *debugger-hook* to nil.

  However, if `icurvature` is entered in noun form, it can be evaluated using `evundiff`:

      (%i3) ishow('icurvature([i,j,k],[l],m))$
                                               l
      (%t3)                          icurvature
                                               i j k,m
      (%i4) ishow(evundiff(%))$
                   l              l         %1           l           %1
      (%t4) - ichr2        - ichr2     ichr2      - ichr2       ichr2
                   i k,j m        %1 j      i k,m        %1 j,m      i k

                   l              l         %1           l           %1
            + ichr2        + ichr2     ichr2      + ichr2       ichr2
                   i j,k m        %1 k      i j,m        %1 k,m      i j

  Note: In earlier versions of Maxima, derivative forms of the Christoffel-symbols also could not be evaluated. This has been fixed now, so `evundiff` is no longer necessary for expressions like this:

      (%i5) imetric(g);
      (%o5)                                done
      (%i6) ishow(ichr2([i,j],[k],l))$
             k %3
            g     (g         - g         + g        )
                    j %3,i l    i j,%3 l    i %3,j l
      (%t6) -----------------------------------------
                                2

                               k %3
                              g     (g       - g       + g      )
                               ,l     j %3,i    i j,%3    i %3,j
                            + -----------------------------------
                                               2

  Categories: Package itensor

Function: flush (expr, tensor_1, tensor_2, ...)

  Set to zero, in expr, all occurrences of the tensor_i that have no derivative indices.

  Categories: Package itensor

Function: flushd (expr, tensor_1, tensor_2, ...)

  Set to zero, in expr, all occurrences of the tensor_i that have derivative indices.

  Categories: Package itensor

Function: flushnd (expr, tensor, n)

  Set to zero, in expr, all occurrences of the differentiated object tensor that have n or more derivative indices as the following example demonstrates.

      (%i1) load(itensor);
      (%o1)      /share/tensor/itensor.lisp
      (%i2) ishow(a([i],[J,r],k,r)+a([i],[j,r,s],k,r,s))$
                                      J r      j r s
      (%t2)                          a      + a
                                      i,k r    i,k r s
      (%i3) ishow(flushnd(%,a,3))$
                                           J r
      (%t3)                               a
                                           i,k r

  Categories: Package itensor

Function: coord (tensor_1, tensor_2, ...)

  Gives tensor_i the coordinate differentiation property that the derivative of contravariant vector whose name is one of the tensor_i yields a Kronecker delta. For example, if `coord(x)` has been done then `idiff(x([],[i]),j)` gives `kdelta([i],[j])`. `coord` is a list of all indexed objects having this property.

  Categories: Package itensor

Function: remcoord
`remcoord` (tensor_1, tensor_2, ...)
`remcoord` (all)

  Removes the coordinate differentiation property from the `tensor_i` that was established by the function `coord`. `remcoord(all)` removes this property from all indexed objects.

  Categories: Package itensor

Function: makebox (expr)

  Display expr in the same manner as `show`; however, any tensor d'Alembertian occurring in expr will be indicated using the symbol `[]`. For example, `[]p([m],[n])` represents `g([],[i,j])*p([m],[n],i,j)`.

  Categories: Package itensor

Function: conmetderiv (expr, tensor)

  Simplifies expressions containing ordinary derivatives of both covariant and contravariant forms of the metric tensor (the current restriction). For example, `conmetderiv` can relate the derivative of the contravariant metric tensor with the Christoffel symbols as seen from the following:

      (%i1) load(itensor);
      (%o1)      /share/tensor/itensor.lisp
      (%i2) ishow(g([],[a,b],c))$
                                            a b
      (%t2)                                g
                                            ,c
      (%i3) ishow(conmetderiv(%,g))$
                               %1 b      a       %1 a      b
      (%t3)                 - g     ichr2     - g     ichr2
                                         %1 c              %1 c

  Categories: Package itensor

Function: simpmetderiv
`simpmetderiv` (expr)
`simpmetderiv` (expr[, stop])

  Simplifies expressions containing products of the derivatives of the metric tensor. Specifically, `simpmetderiv` recognizes two identities:

         ab        ab           ab                 a
        g   g   + g   g     = (g   g  )   = (kdelta )   = 0
         ,d  bc        bc,d         bc ,d          c ,d

  hence

         ab          ab
        g   g   = - g   g
         ,d  bc          bc,d

  and

        ab          ab
       g   g     = g   g
        ,j  ab,i    ,i  ab,j

  which follows from the symmetries of the Christoffel symbols.

  The `simpmetderiv` function takes one optional parameter which, when present, causes the function to stop after the first successful substitution in a product expression. The `simpmetderiv` function also makes use of the global variable flipflag which determines how to apply a "canonical" ordering to the product indices.

  Put together, these capabilities can be used to achieve powerful simplifications that are difficult or impossible to accomplish otherwise. This is demonstrated through the following example that explicitly uses the partial simplification features of `simpmetderiv` to obtain a contractible expression:

      (%i1) load(itensor);
      (%o1)      /share/tensor/itensor.lisp
      (%i2) imetric(g);
      (%o2)                                done
      (%i3) ishow(g([],[a,b])*g([],[b,c])*g([a,b],[],d)*g([b,c],[],e))$
                                   a b  b c
      (%t3)                       g    g    g      g
                                             a b,d  b c,e
      (%i4) ishow(canform(%))$

      errexp1 has improper indices
       -- an error.  Quitting.  To debug this try debugmode(true);
      (%i5) ishow(simpmetderiv(%))$
                                   a b  b c
      (%t5)                       g    g    g      g
                                             a b,d  b c,e
      (%i6) flipflag:not flipflag;
      (%o6)                                true
      (%i7) ishow(simpmetderiv(%th(2)))$
                                     a b  b c
      (%t7)                         g    g    g    g
                                     ,d   ,e   a b  b c
      (%i8) flipflag:not flipflag;
      (%o8)                                false
      (%i9) ishow(simpmetderiv(%th(2),stop))$
                                     a b  b c
      (%t9)                       - g    g    g      g
                                          ,e   a b,d  b c
      (%i10) ishow(contract(%))$
                                          b c
      (%t10)                           - g    g
                                          ,e   c b,d

  See also `weyl.dem` for an example that uses `simpmetderiv` and `conmetderiv` together to simplify contractions of the Weyl tensor.

  Categories: Package itensor

Function: flush1deriv (expr, tensor)

  Set to zero, in `expr`, all occurrences of `tensor` that have exactly one derivative index.

  Categories: Package itensor

[ < ] | [ > ] |  | [ << ] | [ Up ] | [ >> ] |  |  |  |  | [Top] | [Contents] | [Index] | [ ? ]

### 25.2.4 Tensors in curved spaces

Function: imetric (g)
System variable: imetric

  Specifies the metric by assigning the variable `imetric:g` in addition, the contraction properties of the metric g are set up by executing the commands `defcon(g), defcon(g, g, kdelta)`. The variable `imetric` (unbound by default), is bound to the metric, assigned by the `imetric(g)` command.

  Categories: Package itensor

Function: idim (n)

  Sets the dimensions of the metric. Also initializes the antisymmetry properties of the Levi-Civita symbols for the given dimension.

  Categories: Package itensor

Function: ichr1 ([i, j, k])

  Yields the Christoffel symbol of the first kind via the definition

             (g      + g      - g     )/2 .
               ik,j     jk,i     ij,k

  To evaluate the Christoffel symbols for a particular metric, the variable `imetric` must be assigned a name as in the example under `chr2`.

  Categories: Package itensor

Function: ichr2 ([i, j], [k])

  Yields the Christoffel symbol of the second kind defined by the relation

                             ks
         ichr2([i,j],[k]) = g    (g      + g      - g     )/2
                                   is,j     js,i     ij,s

  Categories: Package itensor

Function: icurvature ([i, j, k], [h])

  Yields the Riemann curvature tensor in terms of the Christoffel symbols of the second kind (`ichr2`). The following notation is used:

                  h             h            h         %1         h
        icurvature     = - ichr2      - ichr2     ichr2    + ichr2
                  i j k         i k,j        %1 j      i k        i j,k
                                  h          %1
                           + ichr2      ichr2
                                  %1 k       i j

  Categories: Package itensor

Function: covdiff (expr, v_1, v_2, ...)

  Yields the covariant derivative of expr with respect to the variables v_i in terms of the Christoffel symbols of the second kind (`ichr2`). In order to evaluate these, one should use `ev(expr,ichr2)`.

      (%i1) load(itensor);
      (%o1)      /share/tensor/itensor.lisp
      (%i2) entertensor()$
      Enter tensor name: a;
      Enter a list of the covariant indices: [i,j];
      Enter a list of the contravariant indices: [k];
      Enter a list of the derivative indices: [];
                                            k
      (%t2)                                a
                                            i j
      (%i3) ishow(covdiff(%,s))$
                   k         %1     k         %1     k
      (%t3)     - a     ichr2    - a     ichr2    + a
                   i %1      j s    %1 j      i s    i j,s

                   k     %1
            + ichr2     a
                   %1 s  i j
      (%i4) imetric:g;
      (%o4)                                  g
      (%i5) ishow(ev(%th(2),ichr2))$
               %1 %4  k
              g      a     (g       - g       + g      )
                      i %1   s %4,j    j s,%4    j %4,s
      (%t5) - ------------------------------------------
                                  2
          %1 %3  k
         g      a     (g       - g       + g      )
                 %1 j   s %3,i    i s,%3    i %3,s
       - ------------------------------------------
                             2
          k %2  %1
         g     a    (g        - g        + g       )
                i j   s %2,%1    %1 s,%2    %1 %2,s     k
       + ------------------------------------------- + a
                              2                         i j,s
      (%i6)

  Categories: Package itensor

Function: lorentz_gauge (expr)

  Imposes the Lorentz condition by substituting 0 for all indexed objects in expr that have a derivative index identical to a contravariant index.

  Categories: Package itensor

Function: igeodesic_coords (expr, name)

  Causes undifferentiated Christoffel symbols and first derivatives of the metric tensor vanish in expr. The name in the `igeodesic_coords` function refers to the metric name (if it appears in expr) while the connection coefficients must be called with the names `ichr1` and/or `ichr2`. The following example demonstrates the verification of the cyclic identity satisfied by the Riemann curvature tensor using the `igeodesic_coords` function.

      (%i1) load(itensor);
      (%o1)      /share/tensor/itensor.lisp
      (%i2) ishow(icurvature([r,s,t],[u]))$
                   u            u         %1         u
      (%t2) - ichr2      - ichr2     ichr2    + ichr2
                   r t,s        %1 s      r t        r s,t

                                                    u         %1
                                             + ichr2     ichr2
                                                    %1 t      r s
      (%i3) ishow(igeodesic_coords(%,ichr2))$
                                       u            u
      (%t3)                       ichr2      - ichr2
                                       r s,t        r t,s
      (%i4) ishow(igeodesic_coords(icurvature([r,s,t],[u]),ichr2)+
                  igeodesic_coords(icurvature([s,t,r],[u]),ichr2)+
                  igeodesic_coords(icurvature([t,r,s],[u]),ichr2))$
                   u            u            u            u
      (%t4) - ichr2      + ichr2      + ichr2      - ichr2
                   t s,r        t r,s        s t,r        s r,t

                                                   u            u
                                            - ichr2      + ichr2
                                                   r t,s        r s,t
      (%i5) canform(%);
      (%o5)                                  0

  Categories: Package itensor

[ < ] | [ > ] |  | [ << ] | [ Up ] | [ >> ] |  |  |  |  | [Top] | [Contents] | [Index] | [ ? ]

### 25.2.5 Moving frames

Maxima now has the ability to perform calculations using moving frames. These can be orthonormal frames (tetrads, vielbeins) or an arbitrary frame.

To use frames, you must first set `iframe_flag` to `true`. This causes the Christoffel-symbols, `ichr1` and `ichr2`, to be replaced by the more general frame connection coefficients `icc1` and `icc2` in calculations. Speficially, the behavior of `covdiff` and `icurvature` is changed.

The frame is defined by two tensors: the inverse frame field (`ifri`, the dual basis tetrad), and the frame metric `ifg`. The frame metric is the identity matrix for orthonormal frames, or the Lorentz metric for orthonormal frames in Minkowski spacetime. The inverse frame field defines the frame base (unit vectors). Contraction properties are defined for the frame field and the frame metric.

When `iframe_flag` is true, many `itensor` expressions use the frame metric `ifg` instead of the metric defined by `imetric` for raising and lowerind indices.

IMPORTANT: Setting the variable `iframe_flag` to `true` does NOT undefine the contraction properties of a metric defined by a call to `defcon` or `imetric`. If a frame field is used, it is best to define the metric by assigning its name to the variable `imetric` and NOT invoke the `imetric` function.

Maxima uses these two tensors to define the frame coefficients (`ifc1` and `ifc2`) which form part of the connection coefficients (`icc1` and `icc2`), as the following example demonstrates:

    (%i1) load(itensor);
    (%o1)      /share/tensor/itensor.lisp
    (%i2) iframe_flag:true;
    (%o2)                                true
    (%i3) ishow(covdiff(v([],[i]),j))$
                                   i        i     %1
    (%t3)                         v   + icc2     v
                                   ,j       %1 j
    (%i4) ishow(ev(%,icc2))$
                                   %1     i       i
    (%t4)                         v   ifc2     + v
                                          %1 j    ,j
    (%i5) ishow(ev(%,ifc2))$
                              %1    i %2                i
    (%t5)                    v   ifg     ifc1        + v
                                             %1 j %2    ,j
    (%i6) ishow(ev(%,ifc1))$
                %1    i %2
               v   ifg     (ifb        - ifb        + ifb       )
                               j %2 %1      %2 %1 j      %1 j %2     i
    (%t6)      -------------------------------------------------- + v
                                       2                             ,j
    (%i7) ishow(ifb([a,b,c]))$
                                                       %3    %4
    (%t7)               (ifri        - ifri       ) ifr   ifr
                             a %3,%4       a %4,%3     b     c

An alternate method is used to compute the frame bracket (`ifb`) if the `iframe_bracket_form` flag is set to `false`:

    (%i8) block([iframe_bracket_form:false],ishow(ifb([a,b,c])))$
                                    %6    %5        %5      %6
    (%t8)              ifri     (ifr   ifr     - ifr     ifr  )
                           a %5     b     c,%6      b,%6    c

Function: iframes ()

  Since in this version of Maxima, contraction identities for `ifr` and `ifri` are always defined, as is the frame bracket (`ifb`), this function does nothing.

  Categories: Package itensor

Variable: ifb

  The frame bracket. The contribution of the frame metric to the connection coefficients is expressed using the frame bracket:

                - ifb      + ifb      + ifb
                     c a b      b c a      a b c
      ifc1    = --------------------------------
          abc                  2

  The frame bracket itself is defined in terms of the frame field and frame metric. Two alternate methods of computation are used depending on the value of `frame_bracket_form`. If true (the default) or if the `itorsion_flag` is `true`:

                d      e                                      f
      ifb =  ifr    ifr   (ifri      - ifri      - ifri    itr   )
         abc    b      c       a d,e       a e,d       a f    d e

  Otherwise:

                   e      d        d      e
      ifb    = (ifr    ifr    - ifr    ifr   ) ifri
         abc       b      c,e      b,e    c        a d

  Categories: Package itensor

Variable: icc1

  Connection coefficients of the first kind. In `itensor`, defined as

      icc1    = ichr1    - ikt1    - inmc1
          abc        abc       abc        abc

  In this expression, if `iframe_flag` is true, the Christoffel-symbol `ichr1` is replaced with the frame connection coefficient `ifc1`. If `itorsion_flag` is `false`, `ikt1` will be omitted. It is also omitted if a frame base is used, as the torsion is already calculated as part of the frame bracket. Lastly, of `inonmet_flag` is `false`, `inmc1` will not be present.

  Categories: Package itensor

Variable: icc2

  Connection coefficients of the second kind. In `itensor`, defined as

          c         c        c         c
      icc2   = ichr2   - ikt2   - inmc2
          ab        ab       ab        ab

  In this expression, if `iframe_flag` is true, the Christoffel-symbol `ichr2` is replaced with the frame connection coefficient `ifc2`. If `itorsion_flag` is `false`, `ikt2` will be omitted. It is also omitted if a frame base is used, as the torsion is already calculated as part of the frame bracket. Lastly, of `inonmet_flag` is `false`, `inmc2` will not be present.

  Categories: Package itensor

Variable: ifc1

  Frame coefficient of the first kind (also known as Ricci-rotation coefficients.) This tensor represents the contribution of the frame metric to the connection coefficient of the first kind. Defined as:

                - ifb      + ifb      + ifb
                     c a b      b c a      a b c
      ifc1    = --------------------------------
          abc                   2

  Categories: Package itensor

Variable: ifc2

  Frame coefficient of the second kind. This tensor represents the contribution of the frame metric to the connection coefficient of the second kind. Defined as a permutation of the frame bracket (`ifb`) with the appropriate indices raised and lowered as necessary:

          c       cd
      ifc2   = ifg   ifc1
          ab             abd

  Categories: Package itensor

Variable: ifr

  The frame field. Contracts with the inverse frame field (`ifri`) to form the frame metric (`ifg`).

  Categories: Package itensor

Variable: ifri

  The inverse frame field. Specifies the frame base (dual basis vectors). Along with the frame metric, it forms the basis of all calculations based on frames.

  Categories: Package itensor

Variable: ifg

  The frame metric. Defaults to `kdelta`, but can be changed using `components`.

  Categories: Package itensor

Variable: ifgi

  The inverse frame metric. Contracts with the frame metric (`ifg`) to `kdelta`.

  Categories: Package itensor

Option variable: iframe_bracket_form

  Default value: `true`

  Specifies how the frame bracket (`ifb`) is computed.

  Categories: Package itensor

[ < ] | [ > ] |  | [ << ] | [ Up ] | [ >> ] |  |  |  |  | [Top] | [Contents] | [Index] | [ ? ]

### 25.2.6 Torsion and nonmetricity

Maxima can now take into account torsion and nonmetricity. When the flag `itorsion_flag` is set to `true`, the contribution of torsion is added to the connection coefficients. Similarly, when the flag `inonmet_flag` is true, nonmetricity components are included.

Variable: inm

  The nonmetricity vector. Conformal nonmetricity is defined through the covariant derivative of the metric tensor. Normally zero, the metric tensor's covariant derivative will evaluate to the following when `inonmet_flag` is set to `true`:

      g     =- g  inm
       ij;k     ij   k

  Categories: Package itensor

Variable: inmc1

  Covariant permutation of the nonmetricity vector components. Defined as

                 g   inm  - inm  g   - g   inm
                  ab    c      a  bc    ac    b
      inmc1    = ------------------------------
           abc                 2

  (Substitute `ifg` in place of `g` if a frame metric is used.)

  Categories: Package itensor

Variable: inmc2

  Contravariant permutation of the nonmetricity vector components. Used in the connection coefficients if `inonmet_flag` is `true`. Defined as:

                            c         c         cd
                -inm  kdelta  - kdelta  inm  + g   inm  g
           c        a       b         a    b          d  ab
      inmc2   = -------------------------------------------
           ab                        2

  (Substitute `ifg` in place of `g` if a frame metric is used.)

  Categories: Package itensor

Variable: ikt1

  Covariant permutation of the torsion tensor (also known as contorsion). Defined as:

                        d           d       d
                -g   itr  - g    itr   - itr   g
                  ad    cb    bd    ca      ab  cd
      ikt1    = ----------------------------------
          abc                   2

  (Substitute `ifg` in place of `g` if a frame metric is used.)

  Categories: Package itensor

Variable: ikt2

  Contravariant permutation of the torsion tensor (also known as contorsion). Defined as:

          c     cd
      ikt2   = g   ikt1
          ab           abd

  (Substitute `ifg` in place of `g` if a frame metric is used.)

  Categories: Package itensor

Variable: itr

  The torsion tensor. For a metric with torsion, repeated covariant differentiation on a scalar function will not commute, as demonstrated by the following example:

      (%i1) load(itensor);
      (%o1)      /share/tensor/itensor.lisp
      (%i2) imetric:g;
      (%o2)                                  g
      (%i3) covdiff( covdiff( f( [], []), i), j)
                            - covdiff( covdiff( f( [], []), j), i)$
      (%i4) ishow(%)$
                                         %4              %2
      (%t4)                    f    ichr2    - f    ichr2
                                ,%4      j i    ,%2      i j
      (%i5) canform(%);
      (%o5)                                  0
      (%i6) itorsion_flag:true;
      (%o6)                                true
      (%i7) covdiff( covdiff( f( [], []), i), j)
                            - covdiff( covdiff( f( [], []), j), i)$
      (%i8) ishow(%)$
                                 %8             %6
      (%t8)             f    icc2    - f    icc2    - f     + f
                         ,%8     j i    ,%6     i j    ,j i    ,i j
      (%i9) ishow(canform(%))$
                                         %1             %1
      (%t9)                     f    icc2    - f    icc2
                                 ,%1     j i    ,%1     i j
      (%i10) ishow(canform(ev(%,icc2)))$
                                         %1             %1
      (%t10)                    f    ikt2    - f    ikt2
                                 ,%1     i j    ,%1     j i
      (%i11) ishow(canform(ev(%,ikt2)))$
                            %2 %1                    %2 %1
      (%t11)          f    g      ikt1       - f    g      ikt1
                       ,%2            i j %1    ,%2            j i %1
      (%i12) ishow(factor(canform(rename(expand(ev(%,ikt1))))))$
                                 %3 %2            %1       %1
                           f    g      g      (itr    - itr   )
                            ,%3         %2 %1     j i      i j
      (%t12)               ------------------------------------
                                            2
      (%i13) decsym(itr,2,1,[anti(all)],[]);
      (%o13)                               done
      (%i14) defcon(g,g,kdelta);
      (%o14)                               done
      (%i15) subst(g,nounify(g),%th(3))$
      (%i16) ishow(canform(contract(%)))$
                                                 %1
      (%t16)                           - f    itr
                                          ,%1    i j

  Categories: Package itensor

[ < ] | [ > ] |  | [ << ] | [ Up ] | [ >> ] |  |  |  |  | [Top] | [Contents] | [Index] | [ ? ]

### 25.2.7 Exterior algebra

The `itensor` package can perform operations on totally antisymmetric covariant tensor fields. A totally antisymmetric tensor field of rank (0,L) corresponds with a differential L-form. On these objects, a multiplication operation known as the exterior product, or wedge product, is defined.

Unfortunately, not all authors agree on the definition of the wedge product. Some authors prefer a definition that corresponds with the notion of antisymmetrization: in these works, the wedge product of two vector fields, for instance, would be defined as

                a a  - a a
                 i j    j i
     a  /\ a  = -----------
      i     j        2

More generally, the product of a p-form and a q-form would be defined as

                           1     k1..kp l1..lq
    A       /\ B       = ------ D              A       B
     i1..ip     j1..jq   (p+q)!  i1..ip j1..jq  k1..kp  l1..lq

where `D` stands for the Kronecker-delta.

Other authors, however, prefer a "geometric" definition that corresponds with the notion of the volume element:

    a  /\ a  = a a  - a a
     i     j    i j    j i

and, in the general case

                           1    k1..kp l1..lq
    A       /\ B       = ----- D              A       B
     i1..ip     j1..jq   p! q!  i1..ip j1..jq  k1..kp  l1..lq

Since `itensor` is a tensor algebra package, the first of these two definitions appears to be the more natural one. Many applications, however, utilize the second definition. To resolve this dilemma, a flag has been implemented that controls the behavior of the wedge product: if `igeowedge_flag` is `false` (the default), the first, "tensorial" definition is used, otherwise the second, "geometric" definition will be applied.

Operator: ~

  The wedge product operator is denoted by the tilde `~`. This is a binary operator. Its arguments should be expressions involving scalars, covariant tensors of rank one, or covariant tensors of rank `l` that have been declared antisymmetric in all covariant indices.

  The behavior of the wedge product operator is controlled by the `igeowedge_flag` flag, as in the following example:

      (%i1) load(itensor);
      (%o1)      /share/tensor/itensor.lisp
      (%i2) ishow(a([i])~b([j]))$
                                       a  b  - b  a
                                        i  j    i  j
      (%t2)                            -------------
                                             2
      (%i3) decsym(a,2,0,[anti(all)],[]);
      (%o3)                                done
      (%i4) ishow(a([i,j])~b([k]))$
                                a    b  + b  a    - a    b
                                 i j  k    i  j k    i k  j
      (%t4)                     ---------------------------
                                             3
      (%i5) igeowedge_flag:true;
      (%o5)                                true
      (%i6) ishow(a([i])~b([j]))$
      (%t6)                            a  b  - b  a
                                        i  j    i  j
      (%i7) ishow(a([i,j])~b([k]))$
      (%t7)                     a    b  + b  a    - a    b
                                 i j  k    i  j k    i k  j

  Categories: Package itensor · Operators

Operator: |

  The vertical bar `|` denotes the "contraction with a vector" binary operation. When a totally antisymmetric covariant tensor is contracted with a contravariant vector, the result is the same regardless which index was used for the contraction. Thus, it is possible to define the contraction operation in an index-free manner.

  In the `itensor` package, contraction with a vector is always carried out with respect to the first index in the literal sorting order. This ensures better simplification of expressions involving the `|` operator. For instance:

      (%i1) load(itensor);
      (%o1)      /share/tensor/itensor.lisp
      (%i2) decsym(a,2,0,[anti(all)],[]);
      (%o2)                                done
      (%i3) ishow(a([i,j],[])|v)$
                                          %1
      (%t3)                              v   a
                                              %1 j
      (%i4) ishow(a([j,i],[])|v)$
                                           %1
      (%t4)                             - v   a
                                               %1 j

  Note that it is essential that the tensors used with the `|` operator be declared totally antisymmetric in their covariant indices. Otherwise, the results will be incorrect.

  Categories: Package itensor · Operators

Function: extdiff (expr, i)

  Computes the exterior derivative of expr with respect to the index i. The exterior derivative is formally defined as the wedge product of the partial derivative operator and a differential form. As such, this operation is also controlled by the setting of `igeowedge_flag`. For instance:

      (%i1) load(itensor);
      (%o1)      /share/tensor/itensor.lisp
      (%i2) ishow(extdiff(v([i]),j))$
                                        v    - v
                                         j,i    i,j
      (%t2)                             -----------
                                             2
      (%i3) decsym(a,2,0,[anti(all)],[]);
      (%o3)                                done
      (%i4) ishow(extdiff(a([i,j]),k))$
                                 a      - a      + a
                                  j k,i    i k,j    i j,k
      (%t4)                      ------------------------
                                            3
      (%i5) igeowedge_flag:true;
      (%o5)                                true
      (%i6) ishow(extdiff(v([i]),j))$
      (%t6)                             v    - v
                                         j,i    i,j
      (%i7) ishow(extdiff(a([i,j]),k))$
      (%t7)                    - (a      - a      + a     )
                                   k j,i    k i,j    j i,k

  Categories: Package itensor

Function: hodge (expr)

  Compute the Hodge-dual of expr. For instance:

      (%i1) load(itensor);
      (%o1)      /share/tensor/itensor.lisp
      (%i2) imetric(g);
      (%o2)                            done
      (%i3) idim(4);
      (%o3)                            done
      (%i4) icounter:100;
      (%o4)                             100
      (%i5) decsym(A,3,0,[anti(all)],[])$

      (%i6) ishow(A([i,j,k],[]))$
      (%t6)                           A
                                       i j k
      (%i7) ishow(canform(hodge(%)))$
                                %1 %2 %3 %4
                     levi_civita            g        A
                                             %1 %102  %2 %3 %4
      (%t7)          -----------------------------------------
                                         6
      (%i8) ishow(canform(hodge(%)))$
                       %1 %2 %3 %8            %4 %5 %6 %7
      (%t8) levi_civita            levi_civita            g
                                                           %1 %106
                                   g        g        g      A         /6
                                    %2 %107  %3 %108  %4 %8  %5 %6 %7
      (%i9) lc2kdt(%)$

      (%i10) %,kdelta$

      (%i11) ishow(canform(contract(expand(%))))$
      (%t11)                     - A
                                    %106 %107 %108

  Categories: Package itensor

Option variable: igeowedge_flag

  Default value: `false`

  Controls the behavior of the wedge product and exterior derivative. When set to `false` (the default), the notion of differential forms will correspond with that of a totally antisymmetric covariant tensor field. When set to `true`, differential forms will agree with the notion of the volume element.

  Categories: Package itensor

[ < ] | [ > ] |  | [ << ] | [ Up ] | [ >> ] |  |  |  |  | [Top] | [Contents] | [Index] | [ ? ]

### 25.2.8 Exporting TeX expressions

The `itensor` package provides limited support for exporting tensor expressions to TeX. Since `itensor` expressions appear as function calls, the regular Maxima `tex` command will not produce the expected output. You can try instead the `tentex` command, which attempts to translate tensor expressions into appropriately indexed TeX objects.

Function: tentex (expr)

  To use the `tentex` function, you must first load `tentex`, as in the following example:

      (%i1) load(itensor);
      (%o1)      /share/tensor/itensor.lisp
      (%i2) load(tentex);
      (%o2)       /share/tensor/tentex.lisp
      (%i3) idummyx:m;
      (%o3)                                  m
      (%i4) ishow(icurvature([j,k,l],[i]))$
                  m1       i           m1       i           i
      (%t4)  ichr2    ichr2     - ichr2    ichr2     - ichr2
                  j k      m1 l        j l      m1 k        j l,k

                                                            i
                                                     + ichr2
                                                            j k,l
      (%i5) tentex(%)$
      $$\Gamma_{j\,k}^{m_1}\,\Gamma_{l\,m_1}^{i}-\Gamma_{j\,l}^{m_1}\,
       \Gamma_{k\,m_1}^{i}-\Gamma_{j\,l,k}^{i}+\Gamma_{j\,k,l}^{i}$$

  Note the use of the `idummyx` assignment, to avoid the appearance of the percent sign in the TeX expression, which may lead to compile errors.

  NB: This version of the `tentex` function is somewhat experimental.

  Categories: Package itensor · TeX output

[ < ] | [ > ] |  | [ << ] | [ Up ] | [ >> ] |  |  |  |  | [Top] | [Contents] | [Index] | [ ? ]

### 25.2.9 Interfacing with ctensor

The `itensor` package has the ability to generate Maxima code that can then be executed in the context of the `ctensor` package. The function that performs this task is `ic_convert`.

Function: ic_convert (eqn)

  Converts the `itensor` equation eqn to a `ctensor` assignment statement. Implied sums over dummy indices are made explicit while indexed objects are transformed into arrays (the array subscripts are in the order of covariant followed by contravariant indices of the indexed objects). The derivative of an indexed object will be replaced by the noun form of `diff` taken with respect to `ct_coords` subscripted by the derivative index. The Christoffel symbols `ichr1` and `ichr2` will be translated to `lcs` and `mcs`, respectively and if `metricconvert` is `true` then all occurrences of the metric with two covariant (contravariant) indices will be renamed to `lg` (`ug`). In addition, `do` loops will be introduced summing over all free indices so that the transformed assignment statement can be evaluated by just doing `ev`. The following examples demonstrate the features of this function.

      (%i1) load(itensor);
      (%o1)      /share/tensor/itensor.lisp
      (%i2) eqn:ishow(t([i,j],[k])=f([],[])*g([l,m],[])*a([],[m],j)
            *b([i],[l,k]))$
                                   k        m   l k
      (%t2)                       t    = f a   b    g
                                   i j      ,j  i    l m
      (%i3) ic_convert(eqn);
      (%o3) for i thru dim do (for j thru dim do (
             for k thru dim do
              t        : f sum(sum(diff(a , ct_coords ) b
               i, j, k                   m           j   i, l, k

       g    , l, 1, dim), m, 1, dim)))
        l, m
      (%i4) imetric(g);
      (%o4)                                done
      (%i5) metricconvert:true;
      (%o5)                                true
      (%i6) ic_convert(eqn);
      (%o6) for i thru dim do (for j thru dim do (
             for k thru dim do
              t        : f sum(sum(diff(a , ct_coords ) b
               i, j, k                   m           j   i, l, k

       lg    , l, 1, dim), m, 1, dim)))
         l, m

  Categories: Package itensor · Package ctensor

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### 25.2.10 Reserved words

The following Maxima words are used by the `itensor` package internally and should not be redefined:

      Keyword    Comments
      ------------------------------------------
      indices2() Internal version of indices()
      conti      Lists contravariant indices
      covi       Lists covariant indices of a indexed object
      deri       Lists derivative indices of an indexed object
      name       Returns the name of an indexed object
      concan
      irpmon
      lc0
      _lc2kdt0
      _lcprod
      _extlc

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# 26. ctensor

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## 26.1 Introduction to ctensor

`ctensor` is a component tensor manipulation package. To use the `ctensor` package, type `load(ctensor)`. To begin an interactive session with `ctensor`, type `csetup()`. You are first asked to specify the dimension of the manifold. If the dimension is 2, 3 or 4 then the list of coordinates defaults to `[x,y]`, `[x,y,z]` or `[x,y,z,t]` respectively. These names may be changed by assigning a new list of coordinates to the variable `ct_coords` (described below) and the user is queried about this. Care must be taken to avoid the coordinate names conflicting with other object definitions.

Next, the user enters the metric either directly or from a file by specifying its ordinal position. The metric is stored in the matrix `lg`. Finally, the metric inverse is computed and stored in the matrix `ug`. One has the option of carrying out all calculations in a power series.

A sample protocol is begun below for the static, spherically symmetric metric (standard coordinates) which will be applied to the problem of deriving Einstein's vacuum equations (which lead to the Schwarzschild solution) as an example. Many of the functions in `ctensor` will be displayed for the standard metric as examples.

    (%i1) load(ctensor);
    (%o1)      /share/tensor/ctensor.mac
    (%i2) csetup();
    Enter the dimension of the coordinate system:
    4;
    Do you wish to change the coordinate names?
    n;
    Do you want to
    1. Enter a new metric?

    2. Enter a metric from a file?

    3. Approximate a metric with a Taylor series?
    1;

    Is the matrix  1. Diagonal  2. Symmetric  3. Antisymmetric  4. General
    Answer 1, 2, 3 or 4
    1;
    Row 1 Column 1:
    a;
    Row 2 Column 2:
    x^2;
    Row 3 Column 3:
    x^2*sin(y)^2;
    Row 4 Column 4:
    -d;

    Matrix entered.
    Enter functional dependencies with the DEPENDS function or 'N' if none
    depends([a,d],x);
    Do you wish to see the metric?
    y;
                              [ a  0       0        0  ]
                              [                        ]
                              [     2                  ]
                              [ 0  x       0        0  ]
                              [                        ]
                              [         2    2         ]
                              [ 0  0   x  sin (y)   0  ]
                              [                        ]
                              [ 0  0       0       - d ]
    (%o2)                                done
    (%i3) christof(mcs);
                                                a
                                                 x
    (%t3)                          mcs        = ---
                                      1, 1, 1   2 a

                                                 1
    (%t4)                           mcs        = -
                                       1, 2, 2   x

                                                 1
    (%t5)                           mcs        = -
                                       1, 3, 3   x

                                                d
                                                 x
    (%t6)                          mcs        = ---
                                      1, 4, 4   2 d

                                                  x
    (%t7)                          mcs        = - -
                                      2, 2, 1     a

                                               cos(y)
    (%t8)                         mcs        = ------
                                     2, 3, 3   sin(y)

                                                   2
                                              x sin (y)
    (%t9)                      mcs        = - ---------
                                  3, 3, 1         a

    (%t10)                   mcs        = - cos(y) sin(y)
                                3, 3, 2

                                                d
                                                 x
    (%t11)                         mcs        = ---
                                      4, 4, 1   2 a
    (%o11)                               done

Categories: Tensors · Share packages · Package ctensor

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## 26.2 Functions and Variables for ctensor

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### 26.2.1 Initialization and setup

Function: csetup ()

  A function in the `ctensor` (component tensor) package which initializes the package and allows the user to enter a metric interactively. See `ctensor` for more details.

  Categories: Package ctensor

Function: cmetric
`cmetric` (dis)
`cmetric` ()

  A function in the `ctensor` (component tensor) package that computes the metric inverse and sets up the package for further calculations.

  If `cframe_flag` is `false`, the function computes the inverse metric `ug` from the (user-defined) matrix `lg`. The metric determinant is also computed and stored in the variable `gdet`. Furthermore, the package determines if the metric is diagonal and sets the value of `diagmetric` accordingly. If the optional argument dis is present and not equal to `false`, the user is prompted to see the metric inverse.

  If `cframe_flag` is `true`, the function expects that the values of `fri` (the inverse frame matrix) and `lfg` (the frame metric) are defined. From these, the frame matrix `fr` and the inverse frame metric `ufg` are computed.

  Categories: Package ctensor

Function: ct_coordsys
`ct_coordsys` (coordinate_system, extra_arg)
`ct_coordsys` (coordinate_system)

  Sets up a predefined coordinate system and metric. The argument coordinate_system can be one of the following symbols:

       SYMBOL             Dim Coordinates     Description/comments
       ------------------------------------------------------------------
       cartesian2d           2  [x,y]             Cartesian 2D coordinate
                                                  system
       polar                 2  [r,phi]           Polar coordinate system
       elliptic              2  [u,v]             Elliptic coord. system
       confocalelliptic      2  [u,v]             Confocal elliptic
                                                  coordinates
       bipolar               2  [u,v]             Bipolar coord. system
       parabolic             2  [u,v]             Parabolic coord. system
       cartesian3d           3  [x,y,z]           Cartesian 3D coordinate
                                                  system
       polarcylindrical      3  [r,theta,z]       Polar 2D with
                                                  cylindrical z
       ellipticcylindrical   3  [u,v,z]           Elliptic 2D with
                                                  cylindrical z
       confocalellipsoidal   3  [u,v,w]           Confocal ellipsoidal
       bipolarcylindrical    3  [u,v,z]           Bipolar 2D with
                                                  cylindrical z
       paraboliccylindrical  3  [u,v,z]           Parabolic 2D with
                                                  cylindrical z
       paraboloidal          3  [u,v,phi]         Paraboloidal coords.
       conical               3  [u,v,w]           Conical coordinates
       toroidal              3  [phi,u,v]         Toroidal coordinates
       spherical             3  [r,theta,phi]     Spherical coord. system
       oblatespheroidal      3  [u,v,phi]         Oblate spheroidal
                                                  coordinates
       oblatespheroidalsqrt  3  [u,v,phi]
       prolatespheroidal     3  [u,v,phi]         Prolate spheroidal
                                                  coordinates
       prolatespheroidalsqrt 3  [u,v,phi]
       ellipsoidal           3  [r,theta,phi]     Ellipsoidal coordinates
       cartesian4d           4  [x,y,z,t]         Cartesian 4D coordinate
                                                  system
       spherical4d           4  [r,theta,eta,phi] Spherical 4D coordinate
                                                  system
       exteriorschwarzschild 4  [t,r,theta,phi]   Schwarzschild metric
       interiorschwarzschild 4  [t,z,u,v]         Interior Schwarzschild
                                                  metric
       kerr_newman           4  [t,r,theta,phi]   Charged axially
                                                  symmetric metric

  `coordinate_system` can also be a list of transformation functions, followed by a list containing the coordinate variables. For instance, you can specify a spherical metric as follows:

      (%i1) load(ctensor);
      (%o1)       /share/tensor/ctensor.mac
      (%i2) ct_coordsys([r*cos(theta)*cos(phi),r*cos(theta)*sin(phi),
            r*sin(theta),[r,theta,phi]]);
      (%o2)                                done
      (%i3) lg:trigsimp(lg);
                                 [ 1  0         0        ]
                                 [                       ]
                                 [     2                 ]
      (%o3)                      [ 0  r         0        ]
                                 [                       ]
                                 [         2    2        ]
                                 [ 0  0   r  cos (theta) ]
      (%i4) ct_coords;
      (%o4)                           [r, theta, phi]
      (%i5) dim;
      (%o5)                                  3

  Transformation functions can also be used when `cframe_flag` is `true`:

      (%i1) load(ctensor);
      (%o1)       /share/tensor/ctensor.mac
      (%i2) cframe_flag:true;
      (%o2)                                true
      (%i3) ct_coordsys([r*cos(theta)*cos(phi),r*cos(theta)*sin(phi),
            r*sin(theta),[r,theta,phi]]);
      (%o3)                                done
      (%i4) fri;
      (%o4)
       [cos(phi)cos(theta) -cos(phi) r sin(theta) -sin(phi) r cos(theta)]
       [                                                                ]
       [sin(phi)cos(theta) -sin(phi) r sin(theta)  cos(phi) r cos(theta)]
       [                                                                ]
       [    sin(theta)           r cos(theta)                0          ]

      (%i5) cmetric();
      (%o5)                                false
      (%i6) lg:trigsimp(lg);
                                 [ 1  0         0        ]
                                 [                       ]
                                 [     2                 ]
      (%o6)                      [ 0  r         0        ]
                                 [                       ]
                                 [         2    2        ]
                                 [ 0  0   r  cos (theta) ]

  The optional argument extra_arg can be any one of the following:

  `cylindrical` tells `ct_coordsys` to attach an additional cylindrical coordinate.

  `minkowski` tells `ct_coordsys` to attach an additional coordinate with negative metric signature.

  `all` tells `ct_coordsys` to call `cmetric` and `christof(false)` after setting up the metric.

  If the global variable `verbose` is set to `true`, `ct_coordsys` displays the values of `dim`, `ct_coords`, and either `lg` or `lfg` and `fri`, depending on the value of `cframe_flag`.

  Categories: Package ctensor

Function: init_ctensor ()

  Initializes the `ctensor` package.

  The `init_ctensor` function reinitializes the `ctensor` package. It removes all arrays and matrices used by `ctensor`, resets all flags, resets `dim` to 4, and resets the frame metric to the Lorentz-frame.

  Categories: Package ctensor

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### 26.2.2 The tensors of curved space

The main purpose of the `ctensor` package is to compute the tensors of curved space(time), most notably the tensors used in general relativity.

When a metric base is used, `ctensor` can compute the following tensors:

     lg  -- ug
       \      \
        lcs -- mcs -- ric -- uric
                  \      \       \
                   \      tracer - ein -- lein
                    \
                     riem -- lriem -- weyl
                         \
                          uriem

`ctensor` can also work using moving frames. When `cframe_flag` is set to `true`, the following tensors can be calculated:

     lfg -- ufg
         \
     fri -- fr -- lcs -- mcs -- lriem -- ric -- uric
          \                       |  \      \       \
           lg -- ug               |   weyl   tracer - ein -- lein
                                  |\
                                  | riem
                                  |
                                  \uriem

Function: christof (dis)

  A function in the `ctensor` (component tensor) package. It computes the Christoffel symbols of both kinds. The argument dis determines which results are to be immediately displayed. The Christoffel symbols of the first and second kinds are stored in the arrays `lcs[i,j,k]` and `mcs[i,j,k]` respectively and defined to be symmetric in the first two indices. If the argument to `christof` is `lcs` or `mcs` then the unique non-zero values of `lcs[i,j,k]` or `mcs[i,j,k]`, respectively, will be displayed. If the argument is `all` then the unique non-zero values of `lcs[i,j,k]` and `mcs[i,j,k]` will be displayed. If the argument is `false` then the display of the elements will not occur. The array elements `mcs[i,j,k]` are defined in such a manner that the final index is contravariant.

  Categories: Package ctensor

Function: ricci (dis)

  A function in the `ctensor` (component tensor) package. `ricci` computes the covariant (symmetric) components `ric[i,j]` of the Ricci tensor. If the argument dis is `true`, then the non-zero components are displayed.

  Categories: Package ctensor

Function: uricci (dis)

  This function first computes the covariant components `ric[i,j]` of the Ricci tensor. Then the mixed Ricci tensor is computed using the contravariant metric tensor. If the value of the argument dis is `true`, then these mixed components, `uric[i,j]` (the index `i` is covariant and the index `j` is contravariant), will be displayed directly. Otherwise, `ricci(false)` will simply compute the entries of the array `uric[i,j]` without displaying the results.

  Categories: Package ctensor

Function: scurvature ()

  Returns the scalar curvature (obtained by contracting the Ricci tensor) of the Riemannian manifold with the given metric.

  Categories: Package ctensor

Function: einstein (dis)

  A function in the `ctensor` (component tensor) package. `einstein` computes the mixed Einstein tensor after the Christoffel symbols and Ricci tensor have been obtained (with the functions `christof` and `ricci`). If the argument dis is `true`, then the non-zero values of the mixed Einstein tensor `ein[i,j]` will be displayed where `j` is the contravariant index. The variable `rateinstein` will cause the rational simplification on these components. If `ratfac` is `true` then the components will also be factored.

  Categories: Package ctensor

Function: leinstein (dis)

  Covariant Einstein-tensor. `leinstein` stores the values of the covariant Einstein tensor in the array `lein`. The covariant Einstein-tensor is computed from the mixed Einstein tensor `ein` by multiplying it with the metric tensor. If the argument dis is `true`, then the non-zero values of the covariant Einstein tensor are displayed.

  Categories: Package ctensor

Function: riemann (dis)

  A function in the `ctensor` (component tensor) package. `riemann` computes the Riemann curvature tensor from the given metric and the corresponding Christoffel symbols. The following index conventions are used:

                      l      _l       _l       _l   _m    _l   _m
       R[i,j,k,l] =  R    = |      - |      + |    |   - |    |
                      ijk     ij,k     ik,j     mk   ij    mj   ik

  This notation is consistent with the notation used by the `itensor` package and its `icurvature` function. If the optional argument dis is `true`, the unique non-zero components `riem[i,j,k,l]` will be displayed. As with the Einstein tensor, various switches set by the user control the simplification of the components of the Riemann tensor. If `ratriemann` is `true`, then rational simplification will be done. If `ratfac` is `true` then each of the components will also be factored.

  If the variable `cframe_flag` is `false`, the Riemann tensor is computed directly from the Christoffel-symbols. If `cframe_flag` is `true`, the covariant Riemann-tensor is computed first from the frame field coefficients.

  Categories: Package ctensor

Function: lriemann (dis)

  Covariant Riemann-tensor (`lriem[]`).

  Computes the covariant Riemann-tensor as the array `lriem`. If the argument dis is `true`, unique non-zero values are displayed.

  If the variable `cframe_flag` is `true`, the covariant Riemann tensor is computed directly from the frame field coefficients. Otherwise, the (3,1) Riemann tensor is computed first.

  For information on index ordering, see `riemann`.

  Categories: Package ctensor

Function: uriemann (dis)

  Computes the contravariant components of the Riemann curvature tensor as array elements `uriem[i,j,k,l]`. These are displayed if dis is `true`.

  Categories: Package ctensor

Function: rinvariant ()

  Forms the Kretchmann-invariant (`kinvariant`) obtained by contracting the tensors

      lriem[i,j,k,l]*uriem[i,j,k,l].

  This object is not automatically simplified since it can be very large.

  Categories: Package ctensor

Function: weyl (dis)

  Computes the Weyl conformal tensor. If the argument dis is `true`, the non-zero components `weyl[i,j,k,l]` will be displayed to the user. Otherwise, these components will simply be computed and stored. If the switch `ratweyl` is set to `true`, then the components will be rationally simplified; if `ratfac` is `true` then the results will be factored as well.

  Categories: Package ctensor

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### 26.2.3 Taylor series expansion

The `ctensor` package has the ability to truncate results by assuming that they are Taylor-series approximations. This behavior is controlled by the `ctayswitch` variable; when set to true, `ctensor` makes use internally of the function `ctaylor` when simplifying results.

The `ctaylor` function is invoked by the following `ctensor` functions:

        Function     Comments
        ---------------------------------
        christof()   For mcs only
        ricci()
        uricci()
        einstein()
        riemann()
        weyl()
        checkdiv()

Function: ctaylor ()

  The `ctaylor` function truncates its argument by converting it to a Taylor-series using `taylor`, and then calling `ratdisrep`. This has the combined effect of dropping terms higher order in the expansion variable `ctayvar`. The order of terms that should be dropped is defined by `ctaypov`; the point around which the series expansion is carried out is specified in `ctaypt`.

  As an example, consider a simple metric that is a perturbation of the Minkowski metric. Without further restrictions, even a diagonal metric produces expressions for the Einstein tensor that are far too complex:

      (%i1) load(ctensor);
      (%o1)       /share/tensor/ctensor.mac
      (%i2) ratfac:true;
      (%o2)                                true
      (%i3) derivabbrev:true;
      (%o3)                                true
      (%i4) ct_coords:[t,r,theta,phi];
      (%o4)                         [t, r, theta, phi]
      (%i5) lg:matrix([-1,0,0,0],[0,1,0,0],[0,0,r^2,0],
                      [0,0,0,r^2*sin(theta)^2]);
                              [ - 1  0  0         0        ]
                              [                            ]
                              [  0   1  0         0        ]
                              [                            ]
      (%o5)                   [          2                 ]
                              [  0   0  r         0        ]
                              [                            ]
                              [              2    2        ]
                              [  0   0  0   r  sin (theta) ]
      (%i6) h:matrix([h11,0,0,0],[0,h22,0,0],[0,0,h33,0],[0,0,0,h44]);
                                  [ h11   0    0    0  ]
                                  [                    ]
                                  [  0   h22   0    0  ]
      (%o6)                       [                    ]
                                  [  0    0   h33   0  ]
                                  [                    ]
                                  [  0    0    0   h44 ]
      (%i7) depends(l,r);
      (%o7)                               [l(r)]
      (%i8) lg:lg+l*h;
            [ h11 l - 1      0          0                 0            ]
            [                                                          ]
            [     0      h22 l + 1      0                 0            ]
            [                                                          ]
      (%o8) [                        2                                 ]
            [     0          0      r  + h33 l            0            ]
            [                                                          ]
            [                                    2    2                ]
            [     0          0          0       r  sin (theta) + h44 l ]
      (%i9) cmetric(false);
      (%o9)                                done
      (%i10) einstein(false);
      (%o10)                               done
      (%i11) ntermst(ein);
      [[1, 1], 62]
      [[1, 2], 0]
      [[1, 3], 0]
      [[1, 4], 0]
      [[2, 1], 0]
      [[2, 2], 24]
      [[2, 3], 0]
      [[2, 4], 0]
      [[3, 1], 0]
      [[3, 2], 0]
      [[3, 3], 46]
      [[3, 4], 0]
      [[4, 1], 0]
      [[4, 2], 0]
      [[4, 3], 0]
      [[4, 4], 46]
      (%o12)                               done

  However, if we recompute this example as an approximation that is linear in the variable `l`, we get much simpler expressions:

      (%i14) ctayswitch:true;
      (%o14)                               true
      (%i15) ctayvar:l;
      (%o15)                                 l
      (%i16) ctaypov:1;
      (%o16)                                 1
      (%i17) ctaypt:0;
      (%o17)                                 0
      (%i18) christof(false);
      (%o18)                               done
      (%i19) ricci(false);
      (%o19)                               done
      (%i20) einstein(false);
      (%o20)                               done
      (%i21) ntermst(ein);
      [[1, 1], 6]
      [[1, 2], 0]
      [[1, 3], 0]
      [[1, 4], 0]
      [[2, 1], 0]
      [[2, 2], 13]
      [[2, 3], 2]
      [[2, 4], 0]
      [[3, 1], 0]
      [[3, 2], 2]
      [[3, 3], 9]
      [[3, 4], 0]
      [[4, 1], 0]
      [[4, 2], 0]
      [[4, 3], 0]
      [[4, 4], 9]
      (%o21)                               done
      (%i22) ratsimp(ein[1,1]);
                               2      2  4               2     2
      (%o22) - (((h11 h22 - h11 ) (l )  r  - 2 h33 l    r ) sin (theta)
                                    r               r r

                                  2               2      4    2
                    - 2 h44 l    r  - h33 h44 (l ) )/(4 r  sin (theta))
                             r r                r

  This capability can be useful, for instance, when working in the weak field limit far from a gravitational source.

  Categories: Package ctensor

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### 26.2.4 Frame fields

When the variable `cframe_flag` is set to true, the `ctensor` package performs its calculations using a moving frame.

Function: frame_bracket (fr, fri, diagframe)

  The frame bracket (`fb[]`).

  Computes the frame bracket according to the following definition:

         c          c         c        d     e
      ifb   = ( ifri    - ifri    ) ifr   ifr
         ab         d,e       e,d      a     b

  Categories: Package ctensor

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### 26.2.5 Algebraic classification

A new feature (as of November, 2004) of `ctensor` is its ability to compute the Petrov classification of a 4-dimensional spacetime metric. For a demonstration of this capability, see the file `share/tensor/petrov.dem`.

Function: nptetrad ()

  Computes a Newman-Penrose null tetrad (`np`) and its raised-index counterpart (`npi`). See `petrov` for an example.

  The null tetrad is constructed on the assumption that a four-dimensional orthonormal frame metric with metric signature (-,+,+,+) is being used. The components of the null tetrad are related to the inverse frame matrix as follows:

      np  = (fri  + fri ) / sqrt(2)
        1       1      2

      np  = (fri  - fri ) / sqrt(2)
        2       1      2

      np  = (fri  + %i fri ) / sqrt(2)
        3       3         4

      np  = (fri  - %i fri ) / sqrt(2)
        4       3         4

  Categories: Package ctensor

Function: psi (dis)

  Computes the five Newman-Penrose coefficients `psi[0]`...`psi[4]`. If `dis` is set to `true`, the coefficients are displayed. See `petrov` for an example.

  These coefficients are computed from the Weyl-tensor in a coordinate base. If a frame base is used, the Weyl-tensor is first converted to a coordinate base, which can be a computationally expensive procedure. For this reason, in some cases it may be more advantageous to use a coordinate base in the first place before the Weyl tensor is computed. Note however, that constructing a Newman-Penrose null tetrad requires a frame base. Therefore, a meaningful computation sequence may begin with a frame base, which is then used to compute `lg` (computed automatically by `cmetric`) and then `ug`. See `petrov` for an example. At this point, you can switch back to a coordinate base by setting `cframe_flag` to false before beginning to compute the Christoffel symbols. Changing to a frame base at a later stage could yield inconsistent results, as you may end up with a mixed bag of tensors, some computed in a frame base, some in a coordinate base, with no means to distinguish between the two.

  Categories: Package ctensor

Function: petrov ()

  Computes the Petrov classification of the metric characterized by `psi[0]`...`psi[4]`.

  For example, the following demonstrates how to obtain the Petrov-classification of the Kerr metric:

      (%i1) load(ctensor);
      (%o1)       /share/tensor/ctensor.mac
      (%i2) (cframe_flag:true,gcd:spmod,ctrgsimp:true,ratfac:true);
      (%o2)                                true
      (%i3) ct_coordsys(exteriorschwarzschild,all);
      (%o3)                                done
      (%i4) ug:invert(lg)$
      (%i5) weyl(false);
      (%o5)                                done
      (%i6) nptetrad(true);
      (%t6) np =

      [ sqrt(r - 2 m)           sqrt(r)                                 ]
      [---------------   ---------------------    0            0        ]
      [sqrt(2) sqrt(r)   sqrt(2) sqrt(r - 2 m)                          ]
      [                                                                 ]
      [ sqrt(r - 2 m)            sqrt(r)                                ]
      [---------------  - ---------------------   0            0        ]
      [sqrt(2) sqrt(r)    sqrt(2) sqrt(r - 2 m)                         ]
      [                                                                 ]
      [                                          r      %i r sin(theta) ]
      [       0                    0          -------   --------------- ]
      [                                       sqrt(2)       sqrt(2)     ]
      [                                                                 ]
      [                                          r       %i r sin(theta)]
      [       0                    0          -------  - ---------------]
      [                                       sqrt(2)        sqrt(2)    ]

                                   sqrt(r)         sqrt(r - 2 m)
      (%t7) npi = matrix([- ---------------------,---------------, 0, 0],
                            sqrt(2) sqrt(r - 2 m) sqrt(2) sqrt(r)

                sqrt(r)            sqrt(r - 2 m)
      [- ---------------------, - ---------------, 0, 0],
         sqrt(2) sqrt(r - 2 m)    sqrt(2) sqrt(r)

                 1               %i
      [0, 0, ---------, --------------------],
             sqrt(2) r  sqrt(2) r sin(theta)

                 1                 %i
      [0, 0, ---------, - --------------------])
             sqrt(2) r    sqrt(2) r sin(theta)

      (%o7)                                done
      (%i7) psi(true);
      (%t8)                              psi  = 0
                                            0

      (%t9)                              psi  = 0
                                            1

                                                m
      (%t10)                             psi  = --
                                            2    3
                                                r

      (%t11)                             psi  = 0
                                            3

      (%t12)                             psi  = 0
                                            4
      (%o12)                               done
      (%i12) petrov();
      (%o12)                                 D

  The Petrov classification function is based on the algorithm published in "Classifying geometries in general relativity: III Classification in practice" by Pollney, Skea, and d'Inverno, Class. Quant. Grav. 17 2885-2902 (2000). Except for some simple test cases, the implementation is untested as of December 19, 2004, and is likely to contain errors.

  Categories: Package ctensor

[ < ] | [ > ] |  | [ << ] | [ Up ] | [ >> ] |  |  |  |  | [Top] | [Contents] | [Index] | [ ? ]

### 26.2.6 Torsion and nonmetricity

`ctensor` has the ability to compute and include torsion and nonmetricity coefficients in the connection coefficients.

The torsion coefficients are calculated from a user-supplied tensor `tr`, which should be a rank (2,1) tensor. From this, the torsion coefficients `kt` are computed according to the following formulae:

                  m          m      m
           - g  tr   - g   tr   - tr   g
              im  kj    jm   ki     ij  km
    kt   = -------------------------------
      ijk                 2

      k     km
    kt   = g   kt
      ij         ijm

Note that only the mixed-index tensor is calculated and stored in the array `kt`.

The nonmetricity coefficients are calculated from the user-supplied nonmetricity vector `nm`. From this, the nonmetricity coefficients `nmc` are computed as follows:

                 k    k        km
           -nm  D  - D  nm  + g   nm  g
       k      i  j    i   j         m  ij
    nmc  = ------------------------------
       ij                2

where D stands for the Kronecker-delta.

When `ctorsion_flag` is set to `true`, the values of `kt` are subtracted from the mixed-indexed connection coefficients computed by `christof` and stored in `mcs`. Similarly, if `cnonmet_flag` is set to `true`, the values of `nmc` are subtracted from the mixed-indexed connection coefficients.

If necessary, `christof` calls the functions `contortion` and `nonmetricity` in order to compute `kt` and `nm`.

Function: contortion (tr)

  Computes the (2,1) contortion coefficients from the torsion tensor tr.

  Categories: Package ctensor

Function: nonmetricity (nm)

  Computes the (2,1) nonmetricity coefficients from the nonmetricity vector nm.

  Categories: Package ctensor

[ < ] | [ > ] |  | [ << ] | [ Up ] | [ >> ] |  |  |  |  | [Top] | [Contents] | [Index] | [ ? ]

### 26.2.7 Miscellaneous features

Function: ctransform (M)

  A function in the `ctensor` (component tensor) package which will perform a coordinate transformation upon an arbitrary square symmetric matrix M. The user must input the functions which define the transformation. (Formerly called `transform`.)

  Categories: Package ctensor

Function: findde (A, n)

  returns a list of the unique differential equations (expressions) corresponding to the elements of the n dimensional square array A. Presently, n may be 2 or 3. `deindex` is a global list containing the indices of A corresponding to these unique differential equations. For the Einstein tensor (`ein`), which is a two dimensional array, if computed for the metric in the example below, `findde` gives the following independent differential equations:

      (%i1) load(ctensor);
      (%o1)       /share/tensor/ctensor.mac
      (%i2) derivabbrev:true;
      (%o2)                                true
      (%i3) dim:4;
      (%o3)                                  4
      (%i4) lg:matrix([a, 0, 0, 0], [ 0, x^2, 0, 0],
                                    [0, 0, x^2*sin(y)^2, 0], [0,0,0,-d]);
                                [ a  0       0        0  ]
                                [                        ]
                                [     2                  ]
                                [ 0  x       0        0  ]
      (%o4)                     [                        ]
                                [         2    2         ]
                                [ 0  0   x  sin (y)   0  ]
                                [                        ]
                                [ 0  0       0       - d ]
      (%i5) depends([a,d],x);
      (%o5)                            [a(x), d(x)]
      (%i6) ct_coords:[x,y,z,t];
      (%o6)                            [x, y, z, t]
      (%i7) cmetric();
      (%o7)                                done
      (%i8) einstein(false);
      (%o8)                                done
      (%i9) findde(ein,2);
                                                  2
      (%o9) [d  x - a d + d, 2 a d d    x - a (d )  x - a  d d  x
              x                     x x         x        x    x

                                                    2          2
                                + 2 a d d   - 2 a  d , a  x + a  - a]
                                         x       x      x
      (%i10) deindex;
      (%o10)                     [[1, 1], [2, 2], [4, 4]]

  Categories: Package ctensor

Function: cograd ()

  Computes the covariant gradient of a scalar function allowing the user to choose the corresponding vector name as the example under `contragrad` illustrates.

  Categories: Package ctensor

Function: contragrad ()

  Computes the contravariant gradient of a scalar function allowing the user to choose the corresponding vector name as the example below for the Schwarzschild metric illustrates:

      (%i1) load(ctensor);
      (%o1)       /share/tensor/ctensor.mac
      (%i2) derivabbrev:true;
      (%o2)                                true
      (%i3) ct_coordsys(exteriorschwarzschild,all);
      (%o3)                                done
      (%i4) depends(f,r);
      (%o4)                               [f(r)]
      (%i5) cograd(f,g1);
      (%o5)                                done
      (%i6) listarray(g1);
      (%o6)                            [0, f , 0, 0]
                                            r
      (%i7) contragrad(f,g2);
      (%o7)                                done
      (%i8) listarray(g2);
                                     f  r - 2 f  m
                                      r        r
      (%o8)                      [0, -------------, 0, 0]
                                           r

  Categories: Package ctensor

Function: dscalar ()

  computes the tensor d'Alembertian of the scalar function once dependencies have been declared upon the function. For example:

      (%i1) load(ctensor);
      (%o1)       /share/tensor/ctensor.mac
      (%i2) derivabbrev:true;
      (%o2)                                true
      (%i3) ct_coordsys(exteriorschwarzschild,all);
      (%o3)                                done
      (%i4) depends(p,r);
      (%o4)                               [p(r)]
      (%i5) factor(dscalar(p));
                                2
                          p    r  - 2 m p    r + 2 p  r - 2 m p
                           r r           r r        r          r
      (%o5)               --------------------------------------
                                             2
                                            r

  Categories: Package ctensor

Function: checkdiv ()

  computes the covariant divergence of the mixed second rank tensor (whose first index must be covariant) by printing the corresponding n components of the vector field (the divergence) where n = `dim`. If the argument to the function is `g` then the divergence of the Einstein tensor will be formed and must be zero. In addition, the divergence (vector) is given the array name `div`.

  Categories: Package ctensor

Function: cgeodesic (dis)

  A function in the `ctensor` (component tensor) package. `cgeodesic` computes the geodesic equations of motion for a given metric. They are stored in the array `geod[i]`. If the argument dis is `true` then these equations are displayed.

  Categories: Package ctensor

Function: bdvac (f)

  generates the covariant components of the vacuum field equations of the Brans- Dicke gravitational theory. The scalar field is specified by the argument f, which should be a (quoted) function name with functional dependencies, e.g., `'p(x)`.

  The components of the second rank covariant field tensor are represented by the array `bd`.

  Categories: Package ctensor

Function: invariant1 ()

  generates the mixed Euler- Lagrange tensor (field equations) for the invariant density of R^2. The field equations are the components of an array named `inv1`.

  Categories: Package ctensor

Function: invariant2 ()

  *** NOT YET IMPLEMENTED ***

  generates the mixed Euler- Lagrange tensor (field equations) for the invariant density of `ric[i,j]*uriem[i,j]`. The field equations are the components of an array named `inv2`.

  Categories: Package ctensor

Function: bimetric ()

  *** NOT YET IMPLEMENTED ***

  generates the field equations of Rosen's bimetric theory. The field equations are the components of an array named `rosen`.

  Categories: Package ctensor

[ < ] | [ > ] |  | [ << ] | [ Up ] | [ >> ] |  |  |  |  | [Top] | [Contents] | [Index] | [ ? ]

### 26.2.8 Utility functions

Function: diagmatrixp (M,n)

  Returns `true` if the first n rows and n columns of M form a diagonal matrix or (2D) array.

  Categories: Package ctensor · Predicate functions

Function: symmetricp (M, n)

  Returns `true` if M is a n by n symmetric matrix or two-dimensional array, otherwise `false`.

  If n is less than the size of M, `symmetricp` considers only the n by n submatrix (respectively, subarray) comprising rows 1 through n and columns 1 through n.

  Categories: Package ctensor · Predicate functions

Function: ntermst (f)

  gives the user a quick picture of the "size" of the doubly subscripted tensor (array) f. It prints two element lists where the second element corresponds to NTERMS of the components specified by the first elements. In this way, it is possible to quickly find the non-zero expressions and attempt simplification.

  Categories: Package ctensor

Function: cdisplay (ten)

  displays all the elements of the tensor ten, as represented by a multidimensional array. Tensors of rank 0 and 1, as well as other types of variables, are displayed as with `ldisplay`. Tensors of rank 2 are displayed as 2-dimensional matrices, while tensors of higher rank are displayed as a list of 2-dimensional matrices. For instance, the Riemann-tensor of the Schwarzschild metric can be viewed as:

      (%i1) load(ctensor);
      (%o1)       /share/tensor/ctensor.mac
      (%i2) ratfac:true;
      (%o2)                                true
      (%i3) ct_coordsys(exteriorschwarzschild,all);
      (%o3)                                done
      (%i4) riemann(false);
      (%o4)                                done
      (%i5) cdisplay(riem);
                [ 0               0                   0           0     ]
                [                                                       ]
                [                              2                        ]
                [      3 m (r - 2 m)   m    2 m                         ]
                [ 0  - ------------- + -- - ----      0           0     ]
                [            4          3     4                         ]
                [           r          r     r                          ]
                [                                                       ]
      riem    = [                                m (r - 2 m)            ]
          1, 1  [ 0               0              -----------      0     ]
                [                                     4                 ]
                [                                    r                  ]
                [                                                       ]
                [                                           m (r - 2 m) ]
                [ 0               0                   0     ----------- ]
                [                                                4      ]
                [                                               r       ]

                                      [    2 m (r - 2 m)       ]
                                      [ 0  -------------  0  0 ]
                                      [          4             ]
                                      [         r              ]
                           riem     = [                        ]
                               1, 2   [ 0        0        0  0 ]
                                      [                        ]
                                      [ 0        0        0  0 ]
                                      [                        ]
                                      [ 0        0        0  0 ]

                                      [         m (r - 2 m)    ]
                                      [ 0  0  - -----------  0 ]
                                      [              4         ]
                                      [             r          ]
                           riem     = [                        ]
                               1, 3   [ 0  0        0        0 ]
                                      [                        ]
                                      [ 0  0        0        0 ]
                                      [                        ]
                                      [ 0  0        0        0 ]

                                      [            m (r - 2 m) ]
                                      [ 0  0  0  - ----------- ]
                                      [                 4      ]
                                      [                r       ]
                           riem     = [                        ]
                               1, 4   [ 0  0  0        0       ]
                                      [                        ]
                                      [ 0  0  0        0       ]
                                      [                        ]
                                      [ 0  0  0        0       ]

                                     [       0         0  0  0 ]
                                     [                         ]
                                     [       2 m               ]
                                     [ - ------------  0  0  0 ]
                          riem     = [    2                    ]
                              2, 1   [   r  (r - 2 m)          ]
                                     [                         ]
                                     [       0         0  0  0 ]
                                     [                         ]
                                     [       0         0  0  0 ]

                   [     2 m                                         ]
                   [ ------------  0        0               0        ]
                   [  2                                              ]
                   [ r  (r - 2 m)                                    ]
                   [                                                 ]
                   [      0        0        0               0        ]
                   [                                                 ]
        riem     = [                         m                       ]
            2, 2   [      0        0  - ------------        0        ]
                   [                     2                           ]
                   [                    r  (r - 2 m)                 ]
                   [                                                 ]
                   [                                         m       ]
                   [      0        0        0         - ------------ ]
                   [                                     2           ]
                   [                                    r  (r - 2 m) ]

                                      [ 0  0       0        0 ]
                                      [                       ]
                                      [            m          ]
                                      [ 0  0  ------------  0 ]
                           riem     = [        2              ]
                               2, 3   [       r  (r - 2 m)    ]
                                      [                       ]
                                      [ 0  0       0        0 ]
                                      [                       ]
                                      [ 0  0       0        0 ]

                                      [ 0  0  0       0       ]
                                      [                       ]
                                      [               m       ]
                                      [ 0  0  0  ------------ ]
                           riem     = [           2           ]
                               2, 4   [          r  (r - 2 m) ]
                                      [                       ]
                                      [ 0  0  0       0       ]
                                      [                       ]
                                      [ 0  0  0       0       ]

                                            [ 0  0  0  0 ]
                                            [            ]
                                            [ 0  0  0  0 ]
                                            [            ]
                                 riem     = [ m          ]
                                     3, 1   [ -  0  0  0 ]
                                            [ r          ]
                                            [            ]
                                            [ 0  0  0  0 ]

                                            [ 0  0  0  0 ]
                                            [            ]
                                            [ 0  0  0  0 ]
                                            [            ]
                                 riem     = [    m       ]
                                     3, 2   [ 0  -  0  0 ]
                                            [    r       ]
                                            [            ]
                                            [ 0  0  0  0 ]

                                     [   m                      ]
                                     [ - -   0   0       0      ]
                                     [   r                      ]
                                     [                          ]
                                     [        m                 ]
                                     [  0   - -  0       0      ]
                          riem     = [        r                 ]
                              3, 3   [                          ]
                                     [  0    0   0       0      ]
                                     [                          ]
                                     [              2 m - r     ]
                                     [  0    0   0  ------- + 1 ]
                                     [                 r        ]

                                          [ 0  0  0    0   ]
                                          [                ]
                                          [ 0  0  0    0   ]
                                          [                ]
                               riem     = [            2 m ]
                                   3, 4   [ 0  0  0  - --- ]
                                          [             r  ]
                                          [                ]
                                          [ 0  0  0    0   ]

                                      [       0        0  0  0 ]
                                      [                        ]
                                      [       0        0  0  0 ]
                                      [                        ]
                           riem     = [       0        0  0  0 ]
                               4, 1   [                        ]
                                      [      2                 ]
                                      [ m sin (theta)          ]
                                      [ -------------  0  0  0 ]
                                      [       r                ]

                                      [ 0        0        0  0 ]
                                      [                        ]
                                      [ 0        0        0  0 ]
                                      [                        ]
                           riem     = [ 0        0        0  0 ]
                               4, 2   [                        ]
                                      [         2              ]
                                      [    m sin (theta)       ]
                                      [ 0  -------------  0  0 ]
                                      [          r             ]

                                    [ 0  0          0          0 ]
                                    [                            ]
                                    [ 0  0          0          0 ]
                                    [                            ]
                         riem     = [ 0  0          0          0 ]
                             4, 3   [                            ]
                                    [                2           ]
                                    [         2 m sin (theta)    ]
                                    [ 0  0  - ---------------  0 ]
                                    [                r           ]

                 [        2                                             ]
                 [   m sin (theta)                                      ]
                 [ - -------------         0                0         0 ]
                 [         r                                            ]
                 [                                                      ]
                 [                         2                            ]
                 [                    m sin (theta)                     ]
      riem     = [        0         - -------------         0         0 ]
          4, 4   [                          r                           ]
                 [                                                      ]
                 [                                          2           ]
                 [                                   2 m sin (theta)    ]
                 [        0                0         ---------------  0 ]
                 [                                          r           ]
                 [                                                      ]
                 [        0                0                0         0 ]

      (%o5)                                done

  Categories: Package ctensor

Function: deleten (L, n)

  Returns a new list consisting of L with the n'th element deleted.

  Categories: Package ctensor

[ < ] | [ > ] |  | [ << ] | [ Up ] | [ >> ] |  |  |  |  | [Top] | [Contents] | [Index] | [ ? ]

### 26.2.9 Variables used by `ctensor`

Option variable: dim

  Default value: 4

  An option in the `ctensor` (component tensor) package. `dim` is the dimension of the manifold with the default 4. The command `dim: n` will reset the dimension to any other value `n`.

  Categories: Package ctensor

Option variable: diagmetric

  Default value: `false`

  An option in the `ctensor` (component tensor) package. If `diagmetric` is `true` special routines compute all geometrical objects (which contain the metric tensor explicitly) by taking into consideration the diagonality of the metric. Reduced run times will, of course, result. Note: this option is set automatically by `csetup` if a diagonal metric is specified.

  Categories: Package ctensor

Option variable: ctrgsimp

  Causes trigonometric simplifications to be used when tensors are computed. Presently, `ctrgsimp` affects only computations involving a moving frame.

  Categories: Package ctensor · Simplification flags and variables

Option variable: cframe_flag

  Causes computations to be performed relative to a moving frame as opposed to a holonomic metric. The frame is defined by the inverse frame array `fri` and the frame metric `lfg`. For computations using a Cartesian frame, `lfg` should be the unit matrix of the appropriate dimension; for computations in a Lorentz frame, `lfg` should have the appropriate signature.

  Categories: Package ctensor

Option variable: ctorsion_flag

  Causes the contortion tensor to be included in the computation of the connection coefficients. The contortion tensor itself is computed by `contortion` from the user-supplied tensor `tr`.

  Categories: Package ctensor

Option variable: cnonmet_flag

  Causes the nonmetricity coefficients to be included in the computation of the connection coefficients. The nonmetricity coefficients are computed from the user-supplied nonmetricity vector `nm` by the function `nonmetricity`.

  Categories: Package ctensor

Option variable: ctayswitch

  If set to `true`, causes some `ctensor` computations to be carried out using Taylor-series expansions. Presently, `christof`, `ricci`, `uricci`, `einstein`, and `weyl` take into account this setting.

  Categories: Package ctensor

Option variable: ctayvar

  Variable used for Taylor-series expansion if `ctayswitch` is set to `true`.

  Categories: Package ctensor

Option variable: ctaypov

  Maximum power used in Taylor-series expansion when `ctayswitch` is set to `true`.

  Categories: Package ctensor

Option variable: ctaypt

  Point around which Taylor-series expansion is carried out when `ctayswitch` is set to `true`.

  Categories: Package ctensor

System variable: gdet

  The determinant of the metric tensor `lg`. Computed by `cmetric` when `cframe_flag` is set to `false`.

  Categories: Package ctensor

Option variable: ratchristof

  Causes rational simplification to be applied by `christof`.

  Categories: Package ctensor

Option variable: rateinstein

  Default value: `true`

  If `true` rational simplification will be performed on the non-zero components of Einstein tensors; if `ratfac` is `true` then the components will also be factored.

  Categories: Package ctensor

Option variable: ratriemann

  Default value: `true`

  One of the switches which controls simplification of Riemann tensors; if `true`, then rational simplification will be done; if `ratfac` is `true` then each of the components will also be factored.

  Categories: Package ctensor

Option variable: ratweyl

  Default value: `true`

  If `true`, this switch causes the `weyl` function to apply rational simplification to the values of the Weyl tensor. If `ratfac` is `true`, then the components will also be factored.

  Categories: Package ctensor

Variable: lfg

  The covariant frame metric. By default, it is initialized to the 4-dimensional Lorentz frame with signature (+,+,+,-). Used when `cframe_flag` is `true`.

  Categories: Package ctensor

Variable: ufg

  The inverse frame metric. Computed from `lfg` when `cmetric` is called while `cframe_flag` is set to `true`.

  Categories: Package ctensor

Variable: riem

  The (3,1) Riemann tensor. Computed when the function `riemann` is invoked. For information about index ordering, see the description of `riemann`.

  If `cframe_flag` is `true`, `riem` is computed from the covariant Riemann-tensor `lriem`.

  Categories: Package ctensor

Variable: lriem

  The covariant Riemann tensor. Computed by `lriemann`.

  Categories: Package ctensor

Variable: uriem

  The contravariant Riemann tensor. Computed by `uriemann`.

  Categories: Package ctensor

Variable: ric

  The mixed Ricci-tensor. Computed by `ricci`.

  Categories: Package ctensor

Variable: uric

  The contravariant Ricci-tensor. Computed by `uricci`.

  Categories: Package ctensor

Variable: lg

  The metric tensor. This tensor must be specified (as a `dim` by `dim` matrix) before other computations can be performed.

  Categories: Package ctensor

Variable: ug

  The inverse of the metric tensor. Computed by `cmetric`.

  Categories: Package ctensor

Variable: weyl

  The Weyl tensor. Computed by `weyl`.

  Categories: Package ctensor

Variable: fb

  Frame bracket coefficients, as computed by `frame_bracket`.

  Categories: Package ctensor

Variable: kinvariant

  The Kretchmann invariant. Computed by `rinvariant`.

  Categories: Package ctensor

Variable: np

  A Newman-Penrose null tetrad. Computed by `nptetrad`.

  Categories: Package ctensor

Variable: npi

  The raised-index Newman-Penrose null tetrad. Computed by `nptetrad`. Defined as `ug.np`. The product `np.transpose(npi)` is constant:

      (%i39) trigsimp(np.transpose(npi));
                                    [  0   - 1  0  0 ]
                                    [                ]
                                    [ - 1   0   0  0 ]
      (%o39)                        [                ]
                                    [  0    0   0  1 ]
                                    [                ]
                                    [  0    0   1  0 ]

  Categories: Package ctensor

Variable: tr

  User-supplied rank-3 tensor representing torsion. Used by `contortion`.

  Categories: Package ctensor

Variable: kt

  The contortion tensor, computed from `tr` by `contortion`.

  Categories: Package ctensor

Variable: nm

  User-supplied nonmetricity vector. Used by `nonmetricity`.

  Categories: Package ctensor

Variable: nmc

  The nonmetricity coefficients, computed from `nm` by `nonmetricity`.

  Categories: Package ctensor

System variable: tensorkill

  Variable indicating if the tensor package has been initialized. Set and used by `csetup`, reset by `init_ctensor`.

  Categories: Package ctensor

Option variable: ct_coords

  Default value: `[]`

  An option in the `ctensor` (component tensor) package. `ct_coords` contains a list of coordinates. While normally defined when the function `csetup` is called, one may redefine the coordinates with the assignment `ct_coords: [j1, j2, ..., jn]` where the j's are the new coordinate names. See also `csetup`.

  Categories: Package ctensor

[ < ] | [ > ] |  | [ << ] | [ Up ] | [ >> ] |  |  |  |  | [Top] | [Contents] | [Index] | [ ? ]

### 26.2.10 Reserved names

The following names are used internally by the `ctensor` package and should not be redefined:

      Name         Description
      ---------------------------------------------------------------------
      _lg()        Evaluates to lfg if frame metric used, lg otherwise
      _ug()        Evaluates to ufg if frame metric used, ug otherwise
      cleanup()    Removes items drom the deindex list
      contract4()  Used by psi()
      filemet()    Used by csetup() when reading the metric from a file
      findde1()    Used by findde()
      findde2()    Used by findde()
      findde3()    Used by findde()
      kdelt()      Kronecker-delta (not generalized)
      newmet()     Used by csetup() for setting up a metric interactively
      setflags()   Used by init_ctensor()
      readvalue()
      resimp()
      sermet()     Used by csetup() for entering a metric as Taylor-series
      txyzsum()
      tmetric()    Frame metric, used by cmetric() when cframe_flag:true
      triemann()   Riemann-tensor in frame base, used when cframe_flag:true
      tricci()     Ricci-tensor in frame base, used when cframe_flag:true
      trrc()       Ricci rotation coefficients, used by christof()
      yesp()

[ < ] | [ > ] |  | [ << ] | [ Up ] | [ >> ] |  |  |  |  | [Top] | [Contents] | [Index] | [ ? ]

### 26.2.11 Changes

In November, 2004, the `ctensor` package was extensively rewritten. Many functions and variables have been renamed in order to make the package compatible with the commercial version of Macsyma.

      New Name     Old Name        Description
      ---------------------------------------------------------------------
      ctaylor()    DLGTAYLOR()     Taylor-series expansion of an expression
      lgeod[]      EM              Geodesic equations
      ein[]        G[]             Mixed Einstein-tensor
      ric[]        LR[]            Mixed Ricci-tensor
      ricci()      LRICCICOM()     Compute the mixed Ricci-tensor
      ctaypov      MINP            Maximum power in Taylor-series expansion
      cgeodesic()  MOTION          Compute geodesic equations
      ct_coords    OMEGA           Metric coordinates
      ctayvar      PARAM           Taylor-series expansion variable
      lriem[]      R[]             Covariant Riemann-tensor
      uriemann()   RAISERIEMANN()  Compute the contravariant Riemann-tensor
      ratriemann   RATRIEMAN       Rational simplif. of the Riemann-tensor
      uric[]       RICCI[]         Contravariant Ricci-tensor
      uricci()     RICCICOM()      Compute the contravariant Ricci-tensor
      cmetric()    SETMETRIC()     Set up the metric
      ctaypt       TAYPT           Point for Taylor-series expansion
      ctayswitch   TAYSWITCH       Taylor-series setting switch
      csetup()     TSETUP()        Start interactive setup session
      ctransform() TTRANSFORM()    Interactive coordinate transformation
      uriem[]      UR[]            Contravariant Riemann-tensor
      weyl[]       W[]             (3,1) Weyl-tensor

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# 27. atensor

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## 27.1 Introduction to atensor

`atensor` is an algebraic tensor manipulation package. To use `atensor`, type `load(atensor)`, followed by a call to the `init_atensor` function.

The essence of `atensor` is a set of simplification rules for the noncommutative (dot) product operator ("`.`"). `atensor` recognizes several algebra types; the corresponding simplification rules are put into effect when the `init_atensor` function is called.

The capabilities of `atensor` can be demonstrated by defining the algebra of quaternions as a Clifford-algebra Cl(0,2) with two basis vectors. The three quaternionic imaginary units are then the two basis vectors and their product, i.e.:

        i = v     j = v     k = v  . v
             1         2         1    2

Although the `atensor` package has a built-in definition for the quaternion algebra, it is not used in this example, in which we endeavour to build the quaternion multiplication table as a matrix:

    (%i1) load(atensor);
    (%o1)       /share/tensor/atensor.mac
    (%i2) init_atensor(clifford,0,0,2);
    (%o2)                                done
    (%i3) atensimp(v[1].v[1]);
    (%o3)                                 - 1
    (%i4) atensimp((v[1].v[2]).(v[1].v[2]));
    (%o4)                                 - 1
    (%i5) q:zeromatrix(4,4);
                                    [ 0  0  0  0 ]
                                    [            ]
                                    [ 0  0  0  0 ]
    (%o5)                           [            ]
                                    [ 0  0  0  0 ]
                                    [            ]
                                    [ 0  0  0  0 ]
    (%i6) q[1,1]:1;
    (%o6)                                  1
    (%i7) for i thru adim do q[1,i+1]:q[i+1,1]:v[i];
    (%o7)                                done
    (%i8) q[1,4]:q[4,1]:v[1].v[2];
    (%o8)                               v  . v
                                         1    2
    (%i9) for i from 2 thru 4 do for j from 2 thru 4 do
          q[i,j]:atensimp(q[i,1].q[1,j]);
    (%o9)                                done
    (%i10) q;
                       [    1        v         v      v  . v  ]
                       [              1         2      1    2 ]
                       [                                      ]
                       [   v         - 1     v  . v    - v    ]
                       [    1                 1    2      2   ]
    (%o10)             [                                      ]
                       [   v      - v  . v     - 1      v     ]
                       [    2        1    2              1    ]
                       [                                      ]
                       [ v  . v      v        - v       - 1   ]
                       [  1    2      2          1            ]

`atensor` recognizes as base vectors indexed symbols, where the symbol is that stored in `asymbol` and the index runs between 1 and `adim`. For indexed symbols, and indexed symbols only, the bilinear forms `sf`, `af`, and `av` are evaluated. The evaluation substitutes the value of `aform[i,j]` in place of `fun(v[i],v[j])` where `v` represents the value of `asymbol` and `fun` is either `af` or `sf`; or, it substitutes `v[aform[i,j]]` in place of `av(v[i],v[j])`.

Needless to say, the functions `sf`, `af` and `av` can be redefined.

When the `atensor` package is loaded, the following flags are set:

    dotscrules:true;
    dotdistrib:true;
    dotexptsimp:false;

If you wish to experiment with a nonassociative algebra, you may also consider setting `dotassoc` to `false`. In this case, however, `atensimp` will not always be able to obtain the desired simplifications.

Categories: Tensors · Share packages · Package atensor

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## 27.2 Functions and Variables for atensor

Function: init_atensor
`init_atensor` (alg_type, opt_dims)
`init_atensor` (alg_type)

  Initializes the `atensor` package with the specified algebra type. alg_type can be one of the following:

  `universal`: The universal algebra has no commutation rules.

  `grassmann`: The Grassman algebra is defined by the commutation relation `u.v+v.u=0`.

  `clifford`: The Clifford algebra is defined by the commutation relation `u.v+v.u=-2*sf(u,v)` where `sf` is a symmetric scalar-valued function. For this algebra, opt_dims can be up to three nonnegative integers, representing the number of positive, degenerate, and negative dimensions of the algebra, respectively. If any opt_dims values are supplied, `atensor` will configure the values of `adim` and `aform` appropriately. Otherwise, `adim` will default to 0 and `aform` will not be defined.

  `symmetric`: The symmetric algebra is defined by the commutation relation `u.v-v.u=0`.

  `symplectic`: The symplectic algebra is defined by the commutation relation `u.v-v.u=2*af(u,v)` where `af` is an antisymmetric scalar-valued function. For the symplectic algebra, opt_dims can be up to two nonnegative integers, representing the nondegenerate and degenerate dimensions, respectively. If any opt_dims values are supplied, `atensor` will configure the values of `adim` and `aform` appropriately. Otherwise, `adim` will default to 0 and `aform` will not be defined.

  `lie_envelop`: The algebra of the Lie envelope is defined by the commutation relation `u.v-v.u=2*av(u,v)` where `av` is an antisymmetric function.

  The `init_atensor` function also recognizes several predefined algebra types:

  `complex` implements the algebra of complex numbers as the Clifford algebra Cl(0,1). The call `init_atensor(complex)` is equivalent to `init_atensor(clifford,0,0,1)`.

  `quaternion` implements the algebra of quaternions. The call `init_atensor (quaternion)` is equivalent to `init_atensor (clifford,0,0,2)`.

  `pauli` implements the algebra of Pauli-spinors as the Clifford-algebra Cl(3,0). A call to `init_atensor(pauli)` is equivalent to `init_atensor(clifford,3)`.

  `dirac` implements the algebra of Dirac-spinors as the Clifford-algebra Cl(3,1). A call to `init_atensor(dirac)` is equivalent to `init_atensor(clifford,3,0,1)`.

  Categories: Package atensor

Function: atensimp (expr)

  Simplifies an algebraic tensor expression expr according to the rules configured by a call to `init_atensor`. Simplification includes recursive application of commutation relations and resolving calls to `sf`, `af`, and `av` where applicable. A safeguard is used to ensure that the function always terminates, even for complex expressions.

  Categories: Package atensor · Simplification functions

Function: alg_type

  The algebra type. Valid values are `universal`, `grassmann`, `clifford`, `symmetric`, `symplectic` and `lie_envelop`.

  Categories: Package atensor

Variable: adim

  Default value: 0

  The dimensionality of the algebra. `atensor` uses the value of `adim` to determine if an indexed object is a valid base vector. See `abasep`.

  Categories: Package atensor · Global variables

Variable: aform

  Default value: `ident(3)`

  Default values for the bilinear forms `sf`, `af`, and `av`. The default is the identity matrix `ident(3)`.

  Categories: Package atensor · Global variables

Variable: asymbol

  Default value: `v`

  The symbol for base vectors.

  Categories: Package atensor · Global variables

Function: sf (u, v)

  A symmetric scalar function that is used in commutation relations. The default implementation checks if both arguments are base vectors using `abasep` and if that is the case, substitutes the corresponding value from the matrix `aform`.

  Categories: Package atensor

Function: af (u, v)

  An antisymmetric scalar function that is used in commutation relations. The default implementation checks if both arguments are base vectors using `abasep` and if that is the case, substitutes the corresponding value from the matrix `aform`.

  Categories: Package atensor

Function: av (u, v)

  An antisymmetric function that is used in commutation relations. The default implementation checks if both arguments are base vectors using `abasep` and if that is the case, substitutes the corresponding value from the matrix `aform`.

  For instance:

      (%i1) load(atensor);
      (%o1)       /share/tensor/atensor.mac
      (%i2) adim:3;
      (%o2)                                  3
      (%i3) aform:matrix([0,3,-2],[-3,0,1],[2,-1,0]);
                                     [  0    3   - 2 ]
                                     [               ]
      (%o3)                          [ - 3   0    1  ]
                                     [               ]
                                     [  2   - 1   0  ]
      (%i4) asymbol:x;
      (%o4)                                  x
      (%i5) av(x[1],x[2]);
      (%o5)                                 x
                                             3

  Categories: Package atensor

Function: abasep (v)

  Checks if its argument is an `atensor` base vector. That is, if it is an indexed symbol, with the symbol being the same as the value of `asymbol`, and the index having a numeric value between 1 and `adim`.

  Categories: Package atensor · Predicate functions

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# 28. Sums, Products, and Series

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## 28.1 Functions and Variables for Sums and Products

Function: bashindices (expr)

  Transforms the expression expr by giving each summation and product a unique index. This gives `changevar` greater precision when it is working with summations or products. The form of the unique index is `jnumber`. The quantity number is determined by referring to `gensumnum`, which can be changed by the user. For example, `gensumnum:0$` resets it.

  Categories: Sums and products

Function: lsum (expr, x, L)

  Represents the sum of expr for each element x in L. A noun form `'lsum` is returned if the argument L does not evaluate to a list.

  Examples:

      (%i1) lsum (x^i, i, [1, 2, 7]);
                                  7    2
      (%o1)                      x  + x  + x
      (%i2) lsum (i^2, i, rootsof (x^3 - 1, x));
                           ====
                           \      2
      (%o2)                 >    i
                           /
                           ====
                                         3
                           i in rootsof(x  - 1, x)

  Categories: Sums and products

Function: intosum (expr)

  Moves multiplicative factors outside a summation to inside. If the index is used in the outside expression, then the function tries to find a reasonable index, the same as it does for `sumcontract`. This is essentially the reverse idea of the `outative` property of summations, but note that it does not remove this property, it only bypasses it.

  In some cases, a `scanmap (multthru, expr)` may be necessary before the `intosum`.

  Categories: Expressions

Option variable: simpproduct

  Default value: `false`

  When `simpproduct` is `true`, the result of a `product` is simplified. This simplification may sometimes be able to produce a closed form. If `simpproduct` is `false` or if the quoted form `'product` is used, the value is a product noun form which is a representation of the pi notation used in mathematics.

  Categories: Sums and products · Simplification flags and variables

Function: product (expr, i, i_0, i_1)

  Represents a product of the values of expr as the index i varies from i_0 to i_1. The noun form `'product` is displayed as an uppercase letter pi.

  `product` evaluates expr and lower and upper limits i_0 and i_1, `product` quotes (does not evaluate) the index i.

  If the upper and lower limits differ by an integer, expr is evaluated for each value of the index i, and the result is an explicit product.

  Otherwise, the range of the index is indefinite. Some rules are applied to simplify the product. When the global variable `simpproduct` is `true`, additional rules are applied. In some cases, simplification yields a result which is not a product; otherwise, the result is a noun form `'product`.

  See also `nouns` and `evflag`.

  Examples:

      (%i1) product (x + i*(i+1)/2, i, 1, 4);
      (%o1)           (x + 1) (x + 3) (x + 6) (x + 10)
      (%i2) product (i^2, i, 1, 7);
      (%o2)                       25401600
      (%i3) product (a[i], i, 1, 7);
      (%o3)                 a  a  a  a  a  a  a
                             1  2  3  4  5  6  7
      (%i4) product (a(i), i, 1, 7);
      (%o4)          a(1) a(2) a(3) a(4) a(5) a(6) a(7)
      (%i5) product (a(i), i, 1, n);
                                   n
                                 /===\
                                  ! !
      (%o5)                       ! !  a(i)
                                  ! !
                                 i = 1
      (%i6) product (k, k, 1, n);
                                     n
                                   /===\
                                    ! !
      (%o6)                         ! !  k
                                    ! !
                                   k = 1
      (%i7) product (k, k, 1, n), simpproduct;
      (%o7)                          n!
      (%i8) product (integrate (x^k, x, 0, 1), k, 1, n);
                                   n
                                 /===\
                                  ! !    1
      (%o8)                       ! !  -----
                                  ! !  k + 1
                                 k = 1
      (%i9) product (if k <= 5 then a^k else b^k, k, 1, 10);
                                    15  40
      (%o9)                        a   b

  Categories: Sums and products

Option variable: simpsum

  Default value: `false`

  When `simpsum` is `true`, the result of a `sum` is simplified. This simplification may sometimes be able to produce a closed form. If `simpsum` is `false` or if the quoted form `'sum` is used, the value is a sum noun form which is a representation of the sigma notation used in mathematics.

  Categories: Sums and products · Simplification flags and variables

Function: sum (expr, i, i_0, i_1)

  Represents a summation of the values of expr as the index i varies from i_0 to i_1. The noun form `'sum` is displayed as an uppercase letter sigma.

  `sum` evaluates its summand expr and lower and upper limits i_0 and i_1, `sum` quotes (does not evaluate) the index i.

  If the upper and lower limits differ by an integer, the summand expr is evaluated for each value of the summation index i, and the result is an explicit sum.

  Otherwise, the range of the index is indefinite. Some rules are applied to simplify the summation. When the global variable `simpsum` is `true`, additional rules are applied. In some cases, simplification yields a result which is not a summation; otherwise, the result is a noun form `'sum`.

  When the `evflag` (evaluation flag) `cauchysum` is `true`, a product of summations is expressed as a Cauchy product, in which the index of the inner summation is a function of the index of the outer one, rather than varying independently.

  The global variable `genindex` is the alphabetic prefix used to generate the next index of summation, when an automatically generated index is needed.

  `gensumnum` is the numeric suffix used to generate the next index of summation, when an automatically generated index is needed. When `gensumnum` is `false`, an automatically-generated index is only `genindex` with no numeric suffix.

  See also `lsum`, `sumcontract`, `intosum`, `bashindices`, `niceindices`, `nouns`, `evflag`, and zeilberger

  Examples:

      (%i1) sum (i^2, i, 1, 7);
      (%o1)                          140
      (%i2) sum (a[i], i, 1, 7);
      (%o2)           a  + a  + a  + a  + a  + a  + a
                       7    6    5    4    3    2    1
      (%i3) sum (a(i), i, 1, 7);
      (%o3)    a(7) + a(6) + a(5) + a(4) + a(3) + a(2) + a(1)
      (%i4) sum (a(i), i, 1, n);
                                  n
                                 ====
                                 \
      (%o4)                       >    a(i)
                                 /
                                 ====
                                 i = 1
      (%i5) sum (2^i + i^2, i, 0, n);
                                n
                               ====
                               \       i    2
      (%o5)                     >    (2  + i )
                               /
                               ====
                               i = 0
      (%i6) sum (2^i + i^2, i, 0, n), simpsum;
                                    3      2
                         n + 1   2 n  + 3 n  + n
      (%o6)             2      + --------------- - 1
                                        6
      (%i7) sum (1/3^i, i, 1, inf);
                                  inf
                                  ====
                                  \     1
      (%o7)                        >    --
                                  /      i
                                  ====  3
                                  i = 1
      (%i8) sum (1/3^i, i, 1, inf), simpsum;
                                      1
      (%o8)                           -
                                      2
      (%i9) sum (i^2, i, 1, 4) * sum (1/i^2, i, 1, inf);
                                    inf
                                    ====
                                    \     1
      (%o9)                      30  >    --
                                    /      2
                                    ====  i
                                    i = 1
      (%i10) sum (i^2, i, 1, 4) * sum (1/i^2, i, 1, inf), simpsum;
                                        2
      (%o10)                       5 %pi
      (%i11) sum (integrate (x^k, x, 0, 1), k, 1, n);
                                  n
                                 ====
                                 \       1
      (%o11)                      >    -----
                                 /     k + 1
                                 ====
                                 k = 1
      (%i12) sum (if k <= 5 then a^k else b^k, k, 1, 10);
                10    9    8    7    6    5    4    3    2
      (%o12)   b   + b  + b  + b  + b  + a  + a  + a  + a  + a

  Categories: Sums and products

Function: sumcontract (expr)

  Combines all sums of an addition that have upper and lower bounds that differ by constants. The result is an expression containing one summation for each set of such summations added to all appropriate extra terms that had to be extracted to form this sum. `sumcontract` combines all compatible sums and uses one of the indices from one of the sums if it can, and then try to form a reasonable index if it cannot use any supplied.

  It may be necessary to do an `intosum (expr)` before the `sumcontract`.

  Categories: Sums and products

Option variable: sumexpand

  Default value: `false`

  When `sumexpand` is `true`, products of sums and exponentiated sums simplify to nested sums.

  See also `cauchysum`.

  Examples:

      (%i1) sumexpand: true$
      (%i2) sum (f (i), i, 0, m) * sum (g (j), j, 0, n);
                           m      n
                          ====   ====
                          \      \
      (%o2)                >      >     f(i1) g(i2)
                          /      /
                          ====   ====
                          i1 = 0 i2 = 0
      (%i3) sum (f (i), i, 0, m)^2;
                           m      m
                          ====   ====
                          \      \
      (%o3)                >      >     f(i3) f(i4)
                          /      /
                          ====   ====
                          i3 = 0 i4 = 0

  Categories: Sums and products · Simplification flags and variables

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## 28.2 Introduction to Series

Maxima contains functions `taylor` and `powerseries` for finding the series of differentiable functions. It also has tools such as `nusum` capable of finding the closed form of some series. Operations such as addition and multiplication work as usual on series. This section presents the global variables which control the expansion.

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## 28.3 Functions and Variables for Series

Option variable: cauchysum

  Default value: `false`

  When multiplying together sums with `inf` as their upper limit, if `sumexpand` is `true` and `cauchysum` is `true` then the Cauchy product will be used rather than the usual product. In the Cauchy product the index of the inner summation is a function of the index of the outer one rather than varying independently.

  Example:

      (%i1) sumexpand: false$
      (%i2) cauchysum: false$
      (%i3) s: sum (f(i), i, 0, inf) * sum (g(j), j, 0, inf);
                            inf         inf
                            ====        ====
                            \           \
      (%o3)                ( >    f(i))  >    g(j)
                            /           /
                            ====        ====
                            i = 0       j = 0
      (%i4) sumexpand: true$
      (%i5) cauchysum: true$
      (%i6) expand(s,0,0);
                       inf     i1
                       ====   ====
                       \      \
      (%o6)             >      >     g(i1 - i2) f(i2)
                       /      /
                       ====   ====
                       i1 = 0 i2 = 0

  Categories: Sums and products

Function: deftaylor (f_1(x_1), expr_1, …, f_n(x_n), expr_n)

  For each function f_i of one variable x_i, `deftaylor` defines expr_i as the Taylor series about zero. expr_i is typically a polynomial in x_i or a summation; more general expressions are accepted by `deftaylor` without complaint.

  `powerseries (f_i(x_i), x_i, 0)` returns the series defined by `deftaylor`.

  `deftaylor` returns a list of the functions f_1, …, f_n. `deftaylor` evaluates its arguments.

  Example:

      (%i1) deftaylor (f(x), x^2 + sum(x^i/(2^i*i!^2), i, 4, inf));
      (%o1)                          [f]
      (%i2) powerseries (f(x), x, 0);
                            inf
                            ====      i1
                            \        x         2
      (%o2)                  >     -------- + x
                            /       i1    2
                            ====   2   i1!
                            i1 = 4
      (%i3) taylor (exp (sqrt (f(x))), x, 0, 4);
                            2         3          4
                           x    3073 x    12817 x
      (%o3)/T/     1 + x + -- + ------- + -------- + . . .
                           2     18432     307200

  Categories: Power series

Option variable: maxtayorder

  Default value: `true`

  When `maxtayorder` is `true`, then during algebraic manipulation of (truncated) Taylor series, `taylor` tries to retain as many terms as are known to be correct.

  Categories: Power series

Function: niceindices (expr)

  Renames the indices of sums and products in expr. `niceindices` attempts to rename each index to the value of `niceindicespref[1]`, unless that name appears in the summand or multiplicand, in which case `niceindices` tries the succeeding elements of `niceindicespref` in turn, until an unused variable is found. If the entire list is exhausted, additional indices are constructed by appending integers to the value of `niceindicespref[1]`, e.g., `i0`, `i1`, `i2`, …

  `niceindices` returns an expression. `niceindices` evaluates its argument.

  Example:

      (%i1) niceindicespref;
      (%o1)                  [i, j, k, l, m, n]
      (%i2) product (sum (f (foo + i*j*bar), foo, 1, inf), bar, 1, inf);
                       inf    inf
                      /===\   ====
                       ! !    \
      (%o2)            ! !     >      f(bar i j + foo)
                       ! !    /
                      bar = 1 ====
                              foo = 1
      (%i3) niceindices (%);
                           inf  inf
                          /===\ ====
                           ! !  \
      (%o3)                ! !   >    f(i j l + k)
                           ! !  /
                          l = 1 ====
                                k = 1

  Categories: Sums and products

Option variable: niceindicespref

  Default value: `[i, j, k, l, m, n]`

  `niceindicespref` is the list from which `niceindices` takes the names of indices for sums and products.

  The elements of `niceindicespref` are typically names of variables, although that is not enforced by `niceindices`.

  Example:

      (%i1) niceindicespref: [p, q, r, s, t, u]$
      (%i2) product (sum (f (foo + i*j*bar), foo, 1, inf), bar, 1, inf);
                       inf    inf
                      /===\   ====
                       ! !    \
      (%o2)            ! !     >      f(bar i j + foo)
                       ! !    /
                      bar = 1 ====
                              foo = 1
      (%i3) niceindices (%);
                           inf  inf
                          /===\ ====
                           ! !  \
      (%o3)                ! !   >    f(i j q + p)
                           ! !  /
                          q = 1 ====
                                p = 1

  Categories: Sums and products

Function: nusum (expr, x, i_0, i_1)

  Carries out indefinite hypergeometric summation of expr with respect to x using a decision procedure due to R.W. Gosper. expr and the result must be expressible as products of integer powers, factorials, binomials, and rational functions.

  The terms "definite" and "indefinite summation" are used analogously to "definite" and "indefinite integration". To sum indefinitely means to give a symbolic result for the sum over intervals of variable length, not just e.g. 0 to inf. Thus, since there is no formula for the general partial sum of the binomial series, `nusum` can't do it.

  `nusum` and `unsum` know a little about sums and differences of finite products. See also `unsum`.

  Examples:

      (%i1) nusum (n*n!, n, 0, n);

      Dependent equations eliminated:  (1)
      (%o1)                     (n + 1)! - 1
      (%i2) nusum (n^4*4^n/binomial(2*n,n), n, 0, n);
                           4        3       2              n
            2 (n + 1) (63 n  + 112 n  + 18 n  - 22 n + 3) 4      2
      (%o2) ------------------------------------------------ - ------
                          693 binomial(2 n, n)                 3 11 7
      (%i3) unsum (%, n);
                                    4  n
                                   n  4
      (%o3)                   ----------------
                              binomial(2 n, n)
      (%i4) unsum (prod (i^2, i, 1, n), n);
                          n - 1
                          /===\
                           ! !   2
      (%o4)              ( ! !  i ) (n - 1) (n + 1)
                           ! !
                          i = 1
      (%i5) nusum (%, n, 1, n);

      Dependent equations eliminated:  (2 3)
                                  n
                                /===\
                                 ! !   2
      (%o5)                      ! !  i  - 1
                                 ! !
                                i = 1

  Categories: Sums and products

Function: pade (taylor_series, numer_deg_bound, denom_deg_bound)

  Returns a list of all rational functions which have the given Taylor series expansion where the sum of the degrees of the numerator and the denominator is less than or equal to the truncation level of the power series, i.e. are "best" approximants, and which additionally satisfy the specified degree bounds.

  taylor_series is a univariate Taylor series. numer_deg_bound and denom_deg_bound are positive integers specifying degree bounds on the numerator and denominator.

  taylor_series can also be a Laurent series, and the degree bounds can be `inf` which causes all rational functions whose total degree is less than or equal to the length of the power series to be returned. Total degree is defined as `numer_deg_bound + denom_deg_bound`. Length of a power series is defined as `"truncation level" + 1 - min(0, "order of series")`.

      (%i1) taylor (1 + x + x^2 + x^3, x, 0, 3);
                                    2    3
      (%o1)/T/             1 + x + x  + x  + . . .
      (%i2) pade (%, 1, 1);
                                       1
      (%o2)                       [- -----]
                                     x - 1
      (%i3) t: taylor(-(83787*x^10 - 45552*x^9 - 187296*x^8
                         + 387072*x^7 + 86016*x^6 - 1507328*x^5
                         + 1966080*x^4 + 4194304*x^3 - 25165824*x^2
                         + 67108864*x - 134217728)
             /134217728, x, 0, 10);
                          2    3       4       5       6        7
                   x   3 x    x    15 x    23 x    21 x    189 x
      (%o3)/T/ 1 - - + ---- - -- - ----- + ----- - ----- - ------
                   2    16    32   1024    2048    32768   65536

                                        8         9          10
                                  5853 x    2847 x    83787 x
                                + ------- + ------- - --------- + . . .
                                  4194304   8388608   134217728
      (%i4) pade (t, 4, 4);
      (%o4)                          []

  There is no rational function of degree 4 numerator/denominator, with this power series expansion. You must in general have degree of the numerator and degree of the denominator adding up to at least the degree of the power series, in order to have enough unknown coefficients to solve.

      (%i5) pade (t, 5, 5);
                           5                4                 3
      (%o5) [- (520256329 x  - 96719020632 x  - 489651410240 x

                        2
       - 1619100813312 x  - 2176885157888 x - 2386516803584)

                     5                 4                  3
      /(47041365435 x  + 381702613848 x  + 1360678489152 x

                        2
       + 2856700692480 x  + 3370143559680 x + 2386516803584)]

  Categories: Power series

Function: powerseries (expr, x, a)

  Returns the general form of the power series expansion for expr in the variable x about the point a (which may be `inf` for infinity):

                 inf
                 ====
                 \               n
                  >    b  (x - a)
                 /      n
                 ====
                 n = 0

  If `powerseries` is unable to expand expr, `taylor` may give the first several terms of the series.

  When `verbose` is `true`, `powerseries` prints progress messages.

      (%i1) verbose: true$
      (%i2) powerseries (log(sin(x)/x), x, 0);
      can't expand
                                       log(sin(x))
      so we'll try again after applying the rule:
                                              d
                                            / -- (sin(x))
                                            [ dx
                              log(sin(x)) = i ----------- dx
                                            ]   sin(x)
                                            /
      in the first simplification we have returned:
                                   /
                                   [
                                   i cot(x) dx - log(x)
                                   ]
                                   /
                          inf
                          ====        i1  2 i1             2 i1
                          \      (- 1)   2     bern(2 i1) x
                           >     ------------------------------
                          /                i1 (2 i1)!
                          ====
                          i1 = 1
      (%o2)                -------------------------------------
                                            2

  Categories: Power series

Option variable: psexpand

  Default value: `false`

  When `psexpand` is `true`, an extended rational function expression is displayed fully expanded. The switch `ratexpand` has the same effect.

  When `psexpand` is `false`, a multivariate expression is displayed just as in the rational function package.

  When `psexpand` is `multi`, then terms with the same total degree in the variables are grouped together.

  Categories: Display flags and variables

Function: revert (expr, x)
Function: revert2 (expr, x, n)

  These functions return the reversion of expr, a Taylor series about zero in the variable x. `revert` returns a polynomial of degree equal to the highest power in expr. `revert2` returns a polynomial of degree n, which may be greater than, equal to, or less than the degree of expr.

  `load ("revert")` loads these functions.

  Examples:

      (%i1) load ("revert")$
      (%i2) t: taylor (exp(x) - 1, x, 0, 6);
                         2    3    4    5     6
                        x    x    x    x     x
      (%o2)/T/      x + -- + -- + -- + --- + --- + . . .
                        2    6    24   120   720
      (%i3) revert (t, x);
                     6       5       4       3       2
                 10 x  - 12 x  + 15 x  - 20 x  + 30 x  - 60 x
      (%o3)/R/ - --------------------------------------------
                                      60
      (%i4) ratexpand (%);
                           6    5    4    3    2
                          x    x    x    x    x
      (%o4)             - -- + -- - -- + -- - -- + x
                          6    5    4    3    2
      (%i5) taylor (log(x+1), x, 0, 6);
                          2    3    4    5    6
                         x    x    x    x    x
      (%o5)/T/       x - -- + -- - -- + -- - -- + . . .
                         2    3    4    5    6
      (%i6) ratsimp (revert (t, x) - taylor (log(x+1), x, 0, 6));
      (%o6)                           0
      (%i7) revert2 (t, x, 4);
                                4    3    2
                               x    x    x
      (%o7)                  - -- + -- - -- + x
                               4    3    2

  Categories: Power series

Function: taylor
`taylor` (expr, x, a, n)
`taylor` (expr, [x_1, x_2, …], a, n)
`taylor` (expr, [x, a, n, 'asymp])
`taylor` (expr, [x_1, x_2, …], [a_1, a_2, …], [n_1, n_2, …])
`taylor` (expr, [x_1, a_1, n_1], [x_2, a_2, n_2], …)

  `taylor (expr, x, a, n)` expands the expression expr in a truncated Taylor or Laurent series in the variable x around the point a, containing terms through `(x - a)^n`.

  If expr is of the form `f(x)/g(x)` and `g(x)` has no terms up to degree n then `taylor` attempts to expand `g(x)` up to degree `2 n`. If there are still no nonzero terms, `taylor` doubles the degree of the expansion of `g(x)` so long as the degree of the expansion is less than or equal to `n 2^taylordepth`.

  `taylor (expr, [x_1, x_2, ...], a, n)` returns a truncated power series of degree n in all variables x_1, x_2, … about the point `(a, a, ...)`.

  `taylor (expr, [x_1, a_1, n_1], [x_2, a_2, n_2], ...)` returns a truncated power series in the variables x_1, x_2, … about the point `(a_1, a_2, ...)`, truncated at n_1, n_2, …

  `taylor (expr, [x_1, x_2, ...], [a_1, a_2, ...], [n_1, n_2, ...])` returns a truncated power series in the variables x_1, x_2, … about the point `(a_1, a_2, ...)`, truncated at n_1, n_2, …

  `taylor (expr, [x, a, n, 'asymp])` returns an expansion of expr in negative powers of `x - a`. The highest order term is `(x - a)^-n`.

  When `maxtayorder` is `true`, then during algebraic manipulation of (truncated) Taylor series, `taylor` tries to retain as many terms as are known to be correct.

  When `psexpand` is `true`, an extended rational function expression is displayed fully expanded. The switch `ratexpand` has the same effect. When `psexpand` is `false`, a multivariate expression is displayed just as in the rational function package. When `psexpand` is `multi`, then terms with the same total degree in the variables are grouped together.

  See also the `taylor_logexpand` switch for controlling expansion.

  Examples:

      (%i1) taylor (sqrt (sin(x) + a*x + 1), x, 0, 3);
                                 2             2
                   (a + 1) x   (a  + 2 a + 1) x
      (%o1)/T/ 1 + --------- - -----------------
                       2               8

                                         3      2             3
                                     (3 a  + 9 a  + 9 a - 1) x
                                   + -------------------------- + . . .
                                                 48
      (%i2) %^2;
                                          3
                                         x
      (%o2)/T/           1 + (a + 1) x - -- + . . .
                                         6
      (%i3) taylor (sqrt (x + 1), x, 0, 5);
                             2    3      4      5
                        x   x    x    5 x    7 x
      (%o3)/T/      1 + - - -- + -- - ---- + ---- + . . .
                        2   8    16   128    256
      (%i4) %^2;
      (%o4)/T/                  1 + x + . . .
      (%i5) product ((1 + x^i)^2.5, i, 1, inf)/(1 + x^2);
                               inf
                              /===\
                               ! !    i     2.5
                               ! !  (x  + 1)
                               ! !
                              i = 1
      (%o5)                   -----------------
                                    2
                                   x  + 1
      (%i6) ev (taylor(%, x,  0, 3), keepfloat);
                                     2           3
      (%o6)/T/    1 + 2.5 x + 3.375 x  + 6.5625 x  + . . .
      (%i7) taylor (1/log (x + 1), x, 0, 3);
                                     2       3
                       1   1   x    x    19 x
      (%o7)/T/         - + - - -- + -- - ----- + . . .
                       x   2   12   24    720
      (%i8) taylor (cos(x) - sec(x), x, 0, 5);
                                      4
                                 2   x
      (%o8)/T/                - x  - -- + . . .
                                     6
      (%i9) taylor ((cos(x) - sec(x))^3, x, 0, 5);
      (%o9)/T/                    0 + . . .
      (%i10) taylor (1/(cos(x) - sec(x))^3, x, 0, 5);
                                                     2          4
                  1     1       11      347    6767 x    15377 x
      (%o10)/T/ - -- + ---- + ------ - ----- - ------- - --------
                   6      4        2   15120   604800    7983360
                  x    2 x    120 x

                                                                + . . .
      (%i11) taylor (sqrt (1 - k^2*sin(x)^2), x, 0, 6);
                     2  2       4      2   4
                    k  x    (3 k  - 4 k ) x
      (%o11)/T/ 1 - ----- - ----------------
                      2            24

                                          6       4       2   6
                                     (45 k  - 60 k  + 16 k ) x
                                   - -------------------------- + . . .
                                                720
      (%i12) taylor ((x + 1)^n, x, 0, 4);
                            2       2     3      2         3
                          (n  - n) x    (n  - 3 n  + 2 n) x
      (%o12)/T/ 1 + n x + ----------- + --------------------
                               2                 6

                                     4      3       2         4
                                   (n  - 6 n  + 11 n  - 6 n) x
                                 + ---------------------------- + . . .
                                                24
      (%i13) taylor (sin (y + x), x, 0, 3, y, 0, 3);
                     3                 2
                    y                 y
      (%o13)/T/ y - -- + . . . + (1 - -- + . . .) x
                    6                 2

                          3                       2
                     y   y            2      1   y            3
                + (- - + -- + . . .) x  + (- - + -- + . . .) x  + . . .
                     2   12                  6   12
      (%i14) taylor (sin (y + x), [x, y], 0, 3);
                           3        2      2      3
                          x  + 3 y x  + 3 y  x + y
      (%o14)/T/   y + x - ------------------------- + . . .
                                      6
      (%i15) taylor (1/sin (y + x), x, 0, 3, y, 0, 3);
                1   y              1    1               1            2
      (%o15)/T/ - + - + . . . + (- -- + - + . . .) x + (-- + . . .) x
                y   6               2   6                3
                                   y                    y

                                                 1            3
                                            + (- -- + . . .) x  + . . .
                                                  4
                                                 y
      (%i16) taylor (1/sin (y + x), [x, y], 0, 3);
                                   3         2       2        3
                  1     x + y   7 x  + 21 y x  + 21 y  x + 7 y
      (%o16)/T/ ----- + ----- + ------------------------------- + . . .
                x + y     6                   360

  Categories: Power series

Option variable: taylordepth

  Default value: 3

  If there are still no nonzero terms, `taylor` doubles the degree of the expansion of `g(x)` so long as the degree of the expansion is less than or equal to `n 2^taylordepth`.

  Categories: Power series

Function: taylorinfo (expr)

  Returns information about the Taylor series expr. The return value is a list of lists. Each list comprises the name of a variable, the point of expansion, and the degree of the expansion.

  `taylorinfo` returns `false` if expr is not a Taylor series.

  Example:

      (%i1) taylor ((1 - y^2)/(1 - x), x, 0, 3, [y, a, inf]);
                        2                       2
      (%o1)/T/ - (y - a)  - 2 a (y - a) + (1 - a )

               2                        2
       + (1 - a  - 2 a (y - a) - (y - a) ) x

               2                        2   2
       + (1 - a  - 2 a (y - a) - (y - a) ) x

               2                        2   3
       + (1 - a  - 2 a (y - a) - (y - a) ) x  + . . .
      (%i2) taylorinfo(%);
      (%o2)               [[y, a, inf], [x, 0, 3]]

  Categories: Power series

Function: taylorp (expr)

  Returns `true` if expr is a Taylor series, and `false` otherwise.

  Categories: Predicate functions · Power series

Option variable: taylor_logexpand

  Default value: `true`

  `taylor_logexpand` controls expansions of logarithms in `taylor` series.

  When `taylor_logexpand` is `true`, all logarithms are expanded fully so that zero-recognition problems involving logarithmic identities do not disturb the expansion process. However, this scheme is not always mathematically correct since it ignores branch information.

  When `taylor_logexpand` is set to `false`, then the only expansion of logarithms that occur is that necessary to obtain a formal power series.

  Categories: Power series · Exponential and logarithm functions

Option variable: taylor_order_coefficients

  Default value: `true`

  `taylor_order_coefficients` controls the ordering of coefficients in a Taylor series.

  When `taylor_order_coefficients` is `true`, coefficients of taylor series are ordered canonically.

  Categories: Power series

Function: taylor_simplifier (expr)

  Simplifies coefficients of the power series expr. `taylor` calls this function.

  Categories: Power series

Option variable: taylor_truncate_polynomials

  Default value: `true`

  When `taylor_truncate_polynomials` is `true`, polynomials are truncated based upon the input truncation levels.

  Otherwise, polynomials input to `taylor` are considered to have infinite precison.

  Categories: Power series

Function: taytorat (expr)

  Converts expr from `taylor` form to canonical rational expression (CRE) form. The effect is the same as `rat (ratdisrep (expr))`, but faster.

  Categories: Power series · Rational expressions

Function: trunc (expr)

  Annotates the internal representation of the general expression expr so that it is displayed as if its sums were truncated Taylor series. expr is not otherwise modified.

  Example:

      (%i1) expr: x^2 + x + 1;
                                  2
      (%o1)                      x  + x + 1
      (%i2) trunc (expr);
                                      2
      (%o2)                  1 + x + x  + . . .
      (%i3) is (expr = trunc (expr));
      (%o3)                         true

  Categories: Power series

Function: unsum (f, n)

  Returns the first backward difference `f(n) - f(n - 1)`. Thus `unsum` in a sense is the inverse of `sum`.

  See also `nusum`.

  Examples:

      (%i1) g(p) := p*4^n/binomial(2*n,n);
                                           n
                                        p 4
      (%o1)               g(p) := ----------------
                                  binomial(2 n, n)
      (%i2) g(n^4);
                                    4  n
                                   n  4
      (%o2)                   ----------------
                              binomial(2 n, n)
      (%i3) nusum (%, n, 0, n);
                           4        3       2              n
            2 (n + 1) (63 n  + 112 n  + 18 n  - 22 n + 3) 4      2
      (%o3) ------------------------------------------------ - ------
                          693 binomial(2 n, n)                 3 11 7
      (%i4) unsum (%, n);
                                    4  n
                                   n  4
      (%o4)                   ----------------
                              binomial(2 n, n)

  Categories: Sums and products

Option variable: verbose

  Default value: `false`

  When `verbose` is `true`, `powerseries` prints progress messages.

  Categories: Power series

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## 28.4 Introduction to Fourier series

The `fourie` package comprises functions for the symbolic computation of Fourier series. There are functions in the `fourie` package to calculate Fourier integral coefficients and some functions for manipulation of expressions.

Categories: Fourier transform · Share packages · Package fourie

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## 28.5 Functions and Variables for Fourier series

Function: equalp (x, y)

  Returns `true` if `equal (x, y)` otherwise `false` (doesn't give an error message like `equal (x, y)` would do in this case).

  Categories: Package fourie

Function: remfun
`remfun` (f, expr)
`remfun` (f, expr, x)

  `remfun (f, expr)` replaces all occurrences of `f (arg)` by arg in expr.

  `remfun (f, expr, x)` replaces all occurrences of `f (arg)` by arg in expr only if arg contains the variable x.

  Categories: Package fourie

Function: funp
`funp` (f, expr)
`funp` (f, expr, x)

  `funp (f, expr)` returns `true` if expr contains the function f.

  `funp (f, expr, x)` returns `true` if expr contains the function f and the variable x is somewhere in the argument of one of the instances of f.

  Categories: Package fourie

Function: absint
`absint` (f, x, halfplane)
`absint` (f, x)
`absint` (f, x, a, b)

  `absint (f, x, halfplane)` returns the indefinite integral of f with respect to x in the given halfplane (`pos`, `neg`, or `both`). f may contain expressions of the form `abs (x)`, `abs (sin (x))`, `abs (a) * exp (-abs (b) * abs (x))`.

  `absint (f, x)` is equivalent to `absint (f, x, pos)`.

  `absint (f, x, a, b)` returns the definite integral of f with respect to x from a to b. f may include absolute values.

  Categories: Package fourie · Integral calculus

Function: fourier (f, x, p)

  Returns a list of the Fourier coefficients of `f(x)` defined on the interval `[-p, p]`.

  Categories: Package fourie

Function: foursimp (l)

  Simplifies `sin (n %pi)` to 0 if `sinnpiflag` is `true` and `cos (n %pi)` to `(-1)^n` if `cosnpiflag` is `true`.

  Categories: Package fourie · Trigonometric functions · Simplification functions

Option variable: sinnpiflag

  Default value: `true`

  See `foursimp`.

  Categories: Package fourie

Option variable: cosnpiflag

  Default value: `true`

  See `foursimp`.

  Categories: Package fourie

Function: fourexpand (l, x, p, limit)

  Constructs and returns the Fourier series from the list of Fourier coefficients l up through limit terms (limit may be `inf`). x and p have same meaning as in `fourier`.

  Categories: Package fourie

Function: fourcos (f, x, p)

  Returns the Fourier cosine coefficients for `f(x)` defined on `[0, p]`.

  Categories: Package fourie

Function: foursin (f, x, p)

  Returns the Fourier sine coefficients for `f(x)` defined on `[0, p]`.

  Categories: Package fourie

Function: totalfourier (f, x, p)

  Returns `fourexpand (foursimp (fourier (f, x, p)), x, p, 'inf)`.

  Categories: Package fourie

Function: fourint (f, x)

  Constructs and returns a list of the Fourier integral coefficients of `f(x)` defined on `[minf, inf]`.

  Categories: Package fourie

Function: fourintcos (f, x)

  Returns the Fourier cosine integral coefficients for `f(x)` on `[0, inf]`.

  Categories: Package fourie

Function: fourintsin (f, x)

  Returns the Fourier sine integral coefficients for `f(x)` on `[0, inf]`.

  Categories: Package fourie

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## 28.6 Functions and Variables for Poisson series

Function: intopois (a)

  Converts a into a Poisson encoding.

  Categories: Poisson series

Function: outofpois (a)

  Converts a from Poisson encoding to general representation. If a is not in Poisson form, `outofpois` carries out the conversion, i.e., the return value is `outofpois (intopois (a))`. This function is thus a canonical simplifier for sums of powers of sine and cosine terms of a particular type.

  Categories: Poisson series

Function: poisdiff (a, b)

  Differentiates a with respect to b. b must occur only in the trig arguments or only in the coefficients.

  Categories: Poisson series

Function: poisexpt (a, b)

  Functionally identical to `intopois (a^b)`. b must be a positive integer.

  Categories: Poisson series

Function: poisint (a, b)

  Integrates in a similarly restricted sense (to `poisdiff`). Non-periodic terms in b are dropped if b is in the trig arguments.

  Categories: Poisson series

Option variable: poislim

  Default value: 5

  `poislim` determines the domain of the coefficients in the arguments of the trig functions. The initial value of 5 corresponds to the interval [-2^(5-1)+1,2^(5-1)], or [-15,16], but it can be set to [-2^(n-1)+1, 2^(n-1)].

  Categories: Poisson series

Function: poismap (series, sinfn, cosfn)

  will map the functions sinfn on the sine terms and cosfn on the cosine terms of the Poisson series given. sinfn and cosfn are functions of two arguments which are a coefficient and a trigonometric part of a term in series respectively.

  Categories: Poisson series

Function: poisplus (a, b)

  Is functionally identical to `intopois (a + b)`.

  Categories: Poisson series

Function: poissimp (a)

  Converts a into a Poisson series for a in general representation.

  Categories: Poisson series

Special symbol: poisson

  The symbol `/P/` follows the line label of Poisson series expressions.

  Categories: Poisson series

Function: poissubst (a, b, c)

  Substitutes a for b in c. c is a Poisson series.

  (1) Where B is a variable u, v, w, x, y, or z, then a must be an expression linear in those variables (e.g., `6*u + 4*v`).

  (2) Where b is other than those variables, then a must also be free of those variables, and furthermore, free of sines or cosines.

  `poissubst (a, b, c, d, n)` is a special type of substitution which operates on a and b as in type (1) above, but where d is a Poisson series, expands `cos(d)` and `sin(d)` to order n so as to provide the result of substituting `a + d` for b in c. The idea is that d is an expansion in terms of a small parameter. For example, `poissubst (u, v, cos(v), %e, 3)` yields `cos(u)*(1 - %e^2/2) - sin(u)*(%e - %e^3/6)`.

  Categories: Poisson series

Function: poistimes (a, b)

  Is functionally identical to `intopois (a*b)`.

  Categories: Poisson series

Function: poistrim ()

  is a reserved function name which (if the user has defined it) gets applied during Poisson multiplication. It is a predicate function of 6 arguments which are the coefficients of the u, v, ..., z in a term. Terms for which `poistrim` is `true` (for the coefficients of that term) are eliminated during multiplication.

  Categories: Poisson series

Function: printpois (a)

  Prints a Poisson series in a readable format. In common with `outofpois`, it will convert a into a Poisson encoding first, if necessary.

  Categories: Poisson series · Display functions

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# 29. Number Theory

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## 29.1 Functions and Variables for Number Theory

Function: bern (n)

  Returns the n'th Bernoulli number for integer n. Bernoulli numbers equal to zero are suppressed if `zerobern` is `false`.

  See also `burn`.

      (%i1) zerobern: true$
      (%i2) map (bern, [0, 1, 2, 3, 4, 5, 6, 7, 8]);
                            1  1       1      1        1
      (%o2)           [1, - -, -, 0, - --, 0, --, 0, - --]
                            2  6       30     42       30
      (%i3) zerobern: false$
      (%i4) map (bern, [0, 1, 2, 3, 4, 5, 6, 7, 8]);
                            1  1    1   1     1   5     691   7
      (%o4)           [1, - -, -, - --, --, - --, --, - ----, -]
                            2  6    30  42    30  66    2730  6

  Categories: Number theory

Function: bernpoly (x, n)

  Returns the n'th Bernoulli polynomial in the variable x.

  Categories: Number theory

Function: bfzeta (s, n)

  Returns the Riemann zeta function for the argument s. The return value is a big float (bfloat); n is the number of digits in the return value.

  Categories: Number theory · Numerical evaluation

Function: bfhzeta (s, h, n)

  Returns the Hurwitz zeta function for the arguments s and h. The return value is a big float (bfloat); n is the number of digits in the return value.

  The Hurwitz zeta function is defined as

                              inf
                              ====
                              \        1
               zeta (s,h)  =   >    --------
                              /            s
                              ====  (k + h)
                              k = 0

  `load ("bffac")` loads this function.

  Categories: Number theory · Numerical evaluation

Function: burn (n)

  Returns a rational number, which is an approximation of the n'th Bernoulli number for integer n. `burn` exploits the observation that (rational) Bernoulli numbers can be approximated by (transcendental) zetas with tolerable efficiency:

                         n - 1  1 - 2 n
                    (- 1)      2        zeta(2 n) (2 n)!
           B(2 n) = ------------------------------------
                                      2 n
                                   %pi

  `burn` may be more efficient than `bern` for large, isolated n as `bern` computes all the Bernoulli numbers up to index n before returning. `burn` invokes the approximation for even integers n > 255. For odd integers and n <= 255 the function `bern` is called.

  `load ("bffac")` loads this function. See also `bern`.

  Categories: Number theory

Function: chinese ([r_1, …, r_n], [m_1, …, m_n])

  Solves the system of congruences `x = r_1 mod m_1`, …, `x = r_n mod m_n`. The remainders r_n may be arbitrary integers while the moduli m_n have to be positive and pairwise coprime integers.

      (%i1) mods : [1000, 1001, 1003, 1007];
      (%o1)                   [1000, 1001, 1003, 1007]
      (%i2) lreduce('gcd, mods);
      (%o2)                               1
      (%i3) x : random(apply("*", mods));
      (%o3)                         685124877004
      (%i4) rems : map(lambda([z], mod(x, z)), mods);
      (%o4)                       [4, 568, 54, 624]
      (%i5) chinese(rems, mods);
      (%o5)                         685124877004
      (%i6) chinese([1, 2], [3, n]);
      (%o6)                    chinese([1, 2], [3, n])
      (%i7) %, n = 4;
      (%o7)                              10

  Categories: Number theory

Function: cf (expr)

  Computes a continued fraction approximation. expr is an expression comprising continued fractions, square roots of integers, and literal real numbers (integers, rational numbers, ordinary floats, and bigfloats). `cf` computes exact expansions for rational numbers, but expansions are truncated at `ratepsilon` for ordinary floats and `10^(-fpprec)` for bigfloats.

  Operands in the expression may be combined with arithmetic operators. Maxima does not know about operations on continued fractions outside of `cf`.

  `cf` evaluates its arguments after binding `listarith` to `false`. `cf` returns a continued fraction, represented as a list.

  A continued fraction `a + 1/(b + 1/(c + ...))` is represented by the list `[a, b, c, ...]`. The list elements `a`, `b`, `c`, … must evaluate to integers. expr may also contain `sqrt (n)` where `n` is an integer. In this case `cf` will give as many terms of the continued fraction as the value of the variable `cflength` times the period.

  A continued fraction can be evaluated to a number by evaluating the arithmetic representation returned by `cfdisrep`. See also `cfexpand` for another way to evaluate a continued fraction.

  See also `cfdisrep`, `cfexpand`, and `cflength`.

  Examples:

  - expr is an expression comprising continued fractions and square roots of integers.

        (%i1) cf ([5, 3, 1]*[11, 9, 7] + [3, 7]/[4, 3, 2]);
        (%o1)               [59, 17, 2, 1, 1, 1, 27]
        (%i2) cf ((3/17)*[1, -2, 5]/sqrt(11) + (8/13));
        (%o2)        [0, 1, 1, 1, 3, 2, 1, 4, 1, 9, 1, 9, 2]

  - `cflength` controls how many periods of the continued fraction are computed for algebraic, irrational numbers.

        (%i1) cflength: 1$
        (%i2) cf ((1 + sqrt(5))/2);
        (%o2)                    [1, 1, 1, 1, 2]
        (%i3) cflength: 2$
        (%i4) cf ((1 + sqrt(5))/2);
        (%o4)               [1, 1, 1, 1, 1, 1, 1, 2]
        (%i5) cflength: 3$
        (%i6) cf ((1 + sqrt(5))/2);
        (%o6)           [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2]

  - A continued fraction can be evaluated by evaluating the arithmetic representation returned by `cfdisrep`.

        (%i1) cflength: 3$
        (%i2) cfdisrep (cf (sqrt (3)))$
        (%i3) ev (%, numer);
        (%o3)                   1.731707317073171

  - Maxima does not know about operations on continued fractions outside of `cf`.

        (%i1) cf ([1,1,1,1,1,2] * 3);
        (%o1)                     [4, 1, 5, 2]
        (%i2) cf ([1,1,1,1,1,2]) * 3;
        (%o2)                  [3, 3, 3, 3, 3, 6]

  Categories: Continued fractions

Function: cfdisrep (list)

  Constructs and returns an ordinary arithmetic expression of the form `a + 1/(b + 1/(c + ...))` from the list representation of a continued fraction `[a, b, c, ...]`.

      (%i1) cf ([1, 2, -3] + [1, -2, 1]);
      (%o1)                     [1, 1, 1, 2]
      (%i2) cfdisrep (%);
                                        1
      (%o2)                     1 + ---------
                                          1
                                    1 + -----
                                            1
                                        1 + -
                                            2

  Categories: Continued fractions

Function: cfexpand (x)

  Returns a matrix of the numerators and denominators of the last (column 1) and next-to-last (column 2) convergents of the continued fraction x.

      (%i1) cf (rat (ev (%pi, numer)));

      `rat' replaced 3.141592653589793 by 103993/33102 =3.141592653011902
      (%o1)                  [3, 7, 15, 1, 292]
      (%i2) cfexpand (%);
                               [ 103993  355 ]
      (%o2)                    [             ]
                               [ 33102   113 ]
      (%i3) %[1,1]/%[2,1], numer;
      (%o3)                   3.141592653011902

  Categories: Continued fractions

Option variable: cflength

  Default value: 1

  `cflength` controls the number of terms of the continued fraction the function `cf` will give, as the value `cflength` times the period. Thus the default is to give one period.

      (%i1) cflength: 1$
      (%i2) cf ((1 + sqrt(5))/2);
      (%o2)                    [1, 1, 1, 1, 2]
      (%i3) cflength: 2$
      (%i4) cf ((1 + sqrt(5))/2);
      (%o4)               [1, 1, 1, 1, 1, 1, 1, 2]
      (%i5) cflength: 3$
      (%i6) cf ((1 + sqrt(5))/2);
      (%o6)           [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2]

  Categories: Continued fractions

Function: divsum
`divsum` (n, k)
`divsum` (n)

  `divsum (n, k)` returns the sum of the divisors of n raised to the k'th power.

  `divsum (n)` returns the sum of the divisors of n.

      (%i1) divsum (12);
      (%o1)                          28
      (%i2) 1 + 2 + 3 + 4 + 6 + 12;
      (%o2)                          28
      (%i3) divsum (12, 2);
      (%o3)                          210
      (%i4) 1^2 + 2^2 + 3^2 + 4^2 + 6^2 + 12^2;
      (%o4)                          210

  Categories: Number theory

Function: euler (n)

  Returns the n'th Euler number for nonnegative integer n. Euler numbers equal to zero are suppressed if `zerobern` is `false`.

  For the Euler-Mascheroni constant, see `%gamma`.

      (%i1) zerobern: true$
      (%i2) map (euler, [0, 1, 2, 3, 4, 5, 6]);
      (%o2)               [1, 0, - 1, 0, 5, 0, - 61]
      (%i3) zerobern: false$
      (%i4) map (euler, [0, 1, 2, 3, 4, 5, 6]);
      (%o4)               [1, - 1, 5, - 61, 1385, - 50521, 2702765]

  Categories: Number theory

Option variable: factors_only

  Default value: `false`

  Controls the value returned by `ifactors`. The default `false` causes `ifactors` to provide information about multiplicities of the computed prime factors. If `factors_only` is set to `true`, `ifactors` returns nothing more than a list of prime factors.

  Example: See `ifactors`.

  Categories: Number theory

Function: fib (n)

  Returns the n'th Fibonacci number. `fib(0)` is equal to 0 and `fib(1)` equal to 1, and `fib (-n)` equal to `(-1)^(n + 1) * fib(n)`.

  After calling `fib`, `prevfib` is equal to `fib(n - 1)`, the Fibonacci number preceding the last one computed.

      (%i1) map (fib, [-4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8]);
      (%o1)           [- 3, 2, - 1, 1, 0, 1, 1, 2, 3, 5, 8, 13, 21]

  Categories: Number theory

Function: fibtophi (expr)

  Expresses Fibonacci numbers in expr in terms of the constant `%phi`, which is `(1 + sqrt(5))/2`, approximately 1.61803399.

  Examples:

      (%i1) fibtophi (fib (n));
                                 n             n
                             %phi  - (1 - %phi)
      (%o1)                  -------------------
                                 2 %phi - 1
      (%i2) fib (n-1) + fib (n) - fib (n+1);
      (%o2)          - fib(n + 1) + fib(n) + fib(n - 1)
      (%i3) fibtophi (%);
                  n + 1             n + 1       n             n
              %phi      - (1 - %phi)        %phi  - (1 - %phi)
      (%o3) - --------------------------- + -------------------
                      2 %phi - 1                2 %phi - 1
                                                n - 1             n - 1
                                            %phi      - (1 - %phi)
                                          + ---------------------------
                                                    2 %phi - 1
      (%i4) ratsimp (%);
      (%o4)                           0

  Categories: Number theory

Function: ifactors (n)

  For a positive integer n returns the factorization of n. If `n=p1^e1..pk^nk` is the decomposition of n into prime factors, ifactors returns `[[p1, e1], ... , [pk, ek]]`.

  Factorization methods used are trial divisions by primes up to 9973, Pollard's rho and p-1 method and elliptic curves.

  If the variable `ifactor_verbose` is set to `true` ifactor produces detailed output about what it is doing including immediate feedback as soon as a factor has been found.

  The value returned by `ifactors` is controlled by the option variable `factors_only`. The default `false` causes `ifactors` to provide information about the multiplicities of the computed prime factors. If `factors_only` is set to `true`, `ifactors` simply returns the list of prime factors.

      (%i1) ifactors(51575319651600);
      (%o1)     [[2, 4], [3, 2], [5, 2], [1583, 1], [9050207, 1]]
      (%i2) apply("*", map(lambda([u], u[1]^u[2]), %));
      (%o2)                        51575319651600
      (%i3) ifactors(51575319651600), factors_only : true;
      (%o3)                   [2, 3, 5, 1583, 9050207]

  Categories: Number theory

Function: igcdex (n, k)

  Returns a list `[a, b, u]` where u is the greatest common divisor of n and k, and u is equal to `a n + b k`. The arguments n and k must be integers.

  `igcdex` implements the Euclidean algorithm. See also `gcdex`.

  The command `load(gcdex)` loads the function.

  Examples:

      (%i1) load(gcdex)$

      (%i2) igcdex(30,18);
      (%o2)                      [- 1, 2, 6]
      (%i3) igcdex(1526757668, 7835626735736);
      (%o3)            [845922341123, - 164826435, 4]
      (%i4) igcdex(fib(20), fib(21));
      (%o4)                   [4181, - 2584, 1]

  Categories: Number theory

Function: inrt (x, n)

  Returns the integer n'th root of the absolute value of x.

      (%i1) l: [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]$
      (%i2) map (lambda ([a], inrt (10^a, 3)), l);
      (%o2) [2, 4, 10, 21, 46, 100, 215, 464, 1000, 2154, 4641, 10000]

  Categories: Number theory

Function: inv_mod (n, m)

  Computes the inverse of n modulo m. `inv_mod (n,m)` returns `false`, if n is a zero divisor modulo m.

      (%i1) inv_mod(3, 41);
      (%o1)                           14
      (%i2) ratsimp(3^-1), modulus = 41;
      (%o2)                           14
      (%i3) inv_mod(3, 42);
      (%o3)                          false

  Categories: Number theory

Function: isqrt (x)

  Returns the "integer square root" of the absolute value of x, which is an integer.

  Categories: Mathematical functions

Function: jacobi (p, q)

  Returns the Jacobi symbol of p and q.

      (%i1) l: [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]$
      (%i2) map (lambda ([a], jacobi (a, 9)), l);
      (%o2)         [1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0]

  Categories: Number theory

Function: lcm (expr_1, …, expr_n)

  Returns the least common multiple of its arguments. The arguments may be general expressions as well as integers.

  `load ("functs")` loads this function.

  Categories: Number theory

Function: lucas (n)

  Returns the n'th Lucas number. `lucas(0)` is equal to 2 and `lucas(1)` equal to 1, and `lucas(-n)` equal to `(-1)^(-n) * lucas(n)`.

      (%i1) map (lucas, [-4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8]);
      (%o1)             [7, - 4, 3, - 1, 2, 1, 3, 4, 7, 11, 18, 29, 47]

  After calling `lucas`, the global variable `next_lucas` is equal to `lucas (n + 1)`, the Lucas number following the last returned. The example shows how Fibonacci numbers can be computed via `lucas` and `next_lucas`.

      (%i1) fib_via_lucas(n) :=
               block([lucas : lucas(n)],
               signum(n) * (2*next_lucas - lucas)/5 )$
      (%i2) map (fib_via_lucas, [-4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8]);
      (%o2)             [- 3, 2, - 1, 1, 0, 1, 1, 2, 3, 5, 8, 13, 21]

  Categories: Number theory

Function: mod (x, y)

  If x and y are real numbers and y is nonzero, return `x - y * floor(x / y)`. Further for all real x, we have `mod (x, 0) = x`. For a discussion of the definition `mod (x, 0) = x`, see Section 3.4, of "Concrete Mathematics," by Graham, Knuth, and Patashnik. The function `mod (x, 1)` is a sawtooth function with period 1 with `mod (1, 1) = 0` and `mod (0, 1) = 0`.

  To find the principal argument (a number in the interval `(-%pi, %pi]`) of a complex number, use the function `x |-> %pi - mod (%pi - x, 2*%pi)`, where x is an argument.

  When x and y are constant expressions (`10 * %pi`, for example), `mod` uses the same big float evaluation scheme that `floor` and `ceiling` uses. Again, it's possible, although unlikely, that `mod` could return an erroneous value in such cases.

  For nonnumerical arguments x or y, `mod` knows several simplification rules:

      (%i1) mod (x, 0);
      (%o1)                           x
      (%i2) mod (a*x, a*y);
      (%o2)                      a mod(x, y)
      (%i3) mod (0, x);
      (%o3)                           0

  Categories: Mathematical functions

Function: next_prime (n)

  Returns the smallest prime bigger than n.

      (%i1) next_prime(27);
      (%o1)                       29

  Categories: Number theory

Function: partfrac (expr, var)

  Expands the expression expr in partial fractions with respect to the main variable var. `partfrac` does a complete partial fraction decomposition. The algorithm employed is based on the fact that the denominators of the partial fraction expansion (the factors of the original denominator) are relatively prime. The numerators can be written as linear combinations of denominators, and the expansion falls out.

      (%i1) 1/(1+x)^2 - 2/(1+x) + 2/(2+x);
                            2       2        1
      (%o1)               ----- - ----- + --------
                          x + 2   x + 1          2
                                          (x + 1)
      (%i2) ratsimp (%);
                                       x
      (%o2)                 - -------------------
                               3      2
                              x  + 4 x  + 5 x + 2
      (%i3) partfrac (%, x);
                            2       2        1
      (%o3)               ----- - ----- + --------
                          x + 2   x + 1          2
                                          (x + 1)

Function: power_mod (a, n, m)

  Uses a modular algorithm to compute `a^n mod m` where a and n are integers and m is a positive integer. If n is negative, `inv_mod` is used to find the modular inverse.

      (%i1) power_mod(3, 15, 5);
      (%o1)                          2
      (%i2) mod(3^15,5);
      (%o2)                          2
      (%i3) power_mod(2, -1, 5);
      (%o3)                          3
      (%i4) inv_mod(2,5);
      (%o4)                          3

  Categories: Number theory

Function: primep (n)

  Primality test. If `primep (n)` returns `false`, n is a composite number and if it returns `true`, n is a prime number with very high probability.

  For n less than 341550071728321 a deterministic version of Miller-Rabin's test is used. If `primep (n)` returns `true`, then n is a prime number.

  For n bigger than 341550071728321 `primep` uses `primep_number_of_tests` Miller-Rabin's pseudo-primality tests and one Lucas pseudo-primality test. The probability that a non-prime n will pass one Miller-Rabin test is less than 1/4. Using the default value 25 for `primep_number_of_tests`, the probability of n being composite is much smaller that 10^-15.

  Categories: Predicate functions · Number theory

Option variable: primep_number_of_tests

  Default value: 25

  Number of Miller-Rabin's tests used in `primep`.

  Categories: Number theory

Function: primes (start, end)

  Returns the list of all primes from start to end.

      (%i1) primes(3, 7);
      (%o1)                     [3, 5, 7]

  Categories: Number theory

Function: prev_prime (n)

  Returns the greatest prime smaller than n.

      (%i1) prev_prime(27);
      (%o1)                       23

  Categories: Number theory

Function: qunit (n)

  Returns the principal unit of the real quadratic number field `sqrt (n)` where n is an integer, i.e., the element whose norm is unity. This amounts to solving Pell's equation `a^2 - n b^2 = 1`.

      (%i1) qunit (17);
      (%o1)                     sqrt(17) + 4
      (%i2) expand (% * (sqrt(17) - 4));
      (%o2)                           1

  Categories: Number theory

Function: totient (n)

  Returns the number of integers less than or equal to n which are relatively prime to n.

  Categories: Number theory

Option variable: zerobern

  Default value: `true`

  When `zerobern` is `false`, `bern` excludes the Bernoulli numbers and `euler` excludes the Euler numbers which are equal to zero. See `bern` and `euler`.

  Categories: Number theory

Function: zeta (n)

  Returns the Riemann zeta function. If n is a negative integer, 0, or a positive even integer, the Riemann zeta function simplifies to an exact value. For a positive even integer the option variable `zeta%pi` has to be `true` in addition (See `zeta%pi`). For a floating point or bigfloat number the Riemann zeta function is evaluated numerically. Maxima returns a noun form `zeta (n)` for all other arguments, including rational noninteger, and complex arguments, or for even integers, if `zeta%pi` has the value `false`.

  `zeta(1)` is undefined, but Maxima knows the limit `limit(zeta(x), x, 1)` from above and below.

  The Riemann zeta function distributes over lists, matrices, and equations.

  See also `bfzeta` and `zeta%pi`.

  Examples:

      (%i1) zeta([-2, -1, 0, 0.5, 2, 3, 1+%i]);
                                                   2
                  1     1                       %pi
      (%o1) [0, - --, - -, - 1.460354508809586, ----, zeta(3),
                  12    2                        6
                                                          zeta(%i + 1)]
      (%i2) limit(zeta(x),x,1,plus);
      (%o2)                          inf
      (%i3) limit(zeta(x),x,1,minus);
      (%o3)                         minf

  Categories: Number theory

Option variable: zeta%pi

  Default value: `true`

  When `zeta%pi` is `true`, `zeta` returns an expression proportional to `%pi^n` for even integer `n`. Otherwise, `zeta` returns a noun form `zeta (n)` for even integer `n`.

  Examples:

      (%i1) zeta%pi: true$
      (%i2) zeta (4);
                                       4
                                    %pi
      (%o2)                         ----
                                     90
      (%i3) zeta%pi: false$
      (%i4) zeta (4);
      (%o4)                        zeta(4)

  Categories: Number theory

Function: zn_add_table (n)

  Shows an addition table of all elements in (Z/nZ).

  See also `zn_mult_table`, `zn_power_table`.

  Categories: Number theory

Function: zn_characteristic_factors (n)

  Returns a list containing the characteristic factors of the totient of n.

  Using the characteristic factors a multiplication group modulo n can be expressed as a group direct product of cyclic subgroups.

  In case the group itself is cyclic the list only contains the totient and using `zn_primroot` a generator can be computed. If the totient splits into more than one characteristic factors `zn_factor_generators` finds generators of the corresponding subgroups.

  Each of the `r` factors in the list divides the right following factors. For the last factor `f_r` therefore holds `a^f_r = 1 (mod n)` for all `a` coprime to n. This factor is also known as Carmichael function or Carmichael lambda.

  If `n > 2`, then `totient(n)/2^r` is the number of quadratic residues, and each of these has `2^r` square roots.

  See also `totient`, `zn_primroot`, `zn_factor_generators`.

  Examples:

  The multiplication group modulo `14` is cyclic and its `6` elements can be generated by a primitive root.

      (%i1) [zn_characteristic_factors(14), phi: totient(14)];
      (%o1)                              [[6], 6]
      (%i2) [zn_factor_generators(14), g: zn_primroot(14)];
      (%o2)                              [[3], 3]
      (%i3) M14: makelist(power_mod(g,i,14), i,0,phi-1);
      (%o3)                         [1, 3, 9, 13, 11, 5]

  The multiplication group modulo `15` is not cyclic and its `8` elements can be generated by two factor generators.

      (%i1) [[f1,f2]: zn_characteristic_factors(15), totient(15)];
      (%o1)                             [[2, 4], 8]
      (%i2) [[g1,g2]: zn_factor_generators(15), zn_primroot(15)];
      (%o2)                           [[11, 7], false]
      (%i3) UG1: makelist(power_mod(g1,i,15), i,0,f1-1);
      (%o3)                               [1, 11]
      (%i4) UG2: makelist(power_mod(g2,i,15), i,0,f2-1);
      (%o4)                            [1, 7, 4, 13]
      (%i5) M15: create_list(mod(i*j,15), i,UG1, j,UG2);
      (%o5)                      [1, 7, 4, 13, 11, 2, 14, 8]

  For the last characteristic factor `4` it holds that `a^4 = 1 (mod 15)` for all `a` in `M15`.

  `M15` has two characteristic factors and therefore `8/2^2` quadratic residues, and each of these has `2^2` square roots.

      (%i6) zn_power_table(15);
                                     [ 1   1  1   1 ]
                                     [              ]
                                     [ 2   4  8   1 ]
                                     [              ]
                                     [ 4   1  4   1 ]
                                     [              ]
                                     [ 7   4  13  1 ]
      (%o6)                          [              ]
                                     [ 8   4  2   1 ]
                                     [              ]
                                     [ 11  1  11  1 ]
                                     [              ]
                                     [ 13  4  7   1 ]
                                     [              ]
                                     [ 14  1  14  1 ]
      (%i7) map(lambda([i], zn_nth_root(i,2,15)), [1,4]);
      (%o7)                   [[1, 4, 11, 14], [2, 7, 8, 13]]

  Categories: Number theory

Function: zn_carmichael_lambda (n)

  Returns `1` if n is `1` and otherwise the greatest characteristic factor of the totient of n.

  For remarks and examples see `zn_characteristic_factors`.

  Categories: Number theory

Function: zn_determinant (matrix, p)

  Uses the technique of LU-decomposition to compute the determinant of matrix over (Z/pZ). p must be a prime.

  However if the determinant is equal to zero the LU-decomposition might fail. In that case `zn_determinant` computes the determinant non-modular and reduces thereafter.

  See also `zn_invert_by_lu`.

  Examples:

      (%i1) m : matrix([1,3],[2,4]);
                                      [ 1  3 ]
      (%o1)                           [      ]
                                      [ 2  4 ]
      (%i2) zn_determinant(m, 5);
      (%o2)                               3
      (%i3) m : matrix([2,4,1],[3,1,4],[4,3,2]);
                                     [ 2  4  1 ]
                                     [         ]
      (%o3)                          [ 3  1  4 ]
                                     [         ]
                                     [ 4  3  2 ]
      (%i4) zn_determinant(m, 5);
      (%o4)                               0

  Categories: Number theory

Function: zn_factor_generators (n)

  Returns a list containing factor generators corresponding to the characteristic factors of the totient of n.

  For remarks and examples see `zn_characteristic_factors`.

  Categories: Number theory

Function: zn_invert_by_lu (matrix, p)

  Uses the technique of LU-decomposition to compute the modular inverse of matrix over (Z/pZ). p must be a prime and matrix invertible. `zn_invert_by_lu` returns `false` if matrix is not invertible.

  See also `zn_determinant`.

  Example:

      (%i1) m : matrix([1,3],[2,4]);
                                      [ 1  3 ]
      (%o1)                           [      ]
                                      [ 2  4 ]
      (%i2) zn_determinant(m, 5);
      (%o2)                               3
      (%i3) mi : zn_invert_by_lu(m, 5);
                                      [ 3  4 ]
      (%o3)                           [      ]
                                      [ 1  2 ]
      (%i4) matrixmap(lambda([a], mod(a, 5)), m . mi);
                                      [ 1  0 ]
      (%o4)                           [      ]
                                      [ 0  1 ]

  Categories: Number theory

Function: zn_log
`zn_log` (a, g, n)
`zn_log` (a, g, n, [[p1, e1], …, [pk, ek]])

  Computes the discrete logarithm. Let (Z/nZ)* be a cyclic group, g a primitive root modulo n and let a be a member of this group. `zn_log (a, g, n)` then solves the congruence `g^x = a mod n`.

  The applied algorithm needs a prime factorization of `totient(n)`. This factorization might be time consuming as well and in some cases it can be useful to factor first and then to pass the list of factors to `zn_log` as the fourth argument. The list must be of the same form as the list returned by `ifactors(totient(n))` using the default option `factors_only : false`.

  The algorithm uses a Pohlig-Hellman-reduction and Pollard's Rho-method for discrete logarithms. The run time of `zn_log` primarily depends on the bitlength of the totient's greatest prime factor.

  See also `zn_primroot`, `zn_order`, `ifactors`, `totient`.

  Examples:

  `zn_log (a, g, n)` solves the congruence `g^x = a mod n`.

      (%i1) n : 22$
      (%i2) g : zn_primroot(n);
      (%o2)                               7
      (%i3) ord_7 : zn_order(7, n);
      (%o3)                              10
      (%i4) powers_7 : makelist(power_mod(g, x, n), x, 0, ord_7 - 1);
      (%o4)              [1, 7, 5, 13, 3, 21, 15, 17, 9, 19]
      (%i5) zn_log(21, g, n);
      (%o5)                               5
      (%i6) map(lambda([x], zn_log(x, g, n)), powers_7);
      (%o6)                [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]

  The optional fourth argument must be of the same form as the list returned by `ifactors(totient(n))`. The run time primarily depends on the bitlength of the totient's greatest prime factor.

      (%i1) (p : 2^127-1, primep(p));
      (%o1)                             true
      (%i2) ifs : ifactors(p - 1)$
      (%i3) g : zn_primroot(p, ifs);
      (%o3)                              43
      (%i4) a : power_mod(g, 1234567890, p)$
      (%i5) zn_log(a, g, p, ifs);
      (%o5)                          1234567890
      (%i6) time(%o5);
      (%o6)                            [1.204]
      (%i7) f_max : last(ifs);
      (%o7)                       [77158673929, 1]
      (%i8) slength( printf(false, "~b", f_max[1]) );
      (%o8)                              37

  Categories: Number theory

Function: zn_mult_table
`zn_mult_table` (n)
`zn_mult_table` (n, gcd)

  Without the optional argument gcd `zn_mult_table(n)` shows a multiplication table of all elements in (Z/nZ)* which are all elements coprime to n.

  The optional second argument gcd allows to select a specific subset of (Z/nZ). If gcd is an integer, a multiplication table of all residues `x` with `gcd(x,n) =` gcd are returned. Additionally row and column headings are added for better readability. If necessary, these can be easily removed by `submatrix(1, table, 1)`.

  If gcd is set to `all`, the table is printed for all non-zero elements in (Z/nZ).

  The second example shows an alternative way to create a multiplication table for subgroups.

  See also `zn_add_table`, `zn_power_table`.

  Examples:

  The default table shows all elements in (Z/nZ)* and allows to demonstrate and study basic properties of modular multiplication groups. E.g. the principal diagonal contains all quadratic residues, each row and column contains every element, the tables are symmetric, etc..

  If gcd is set to `all`, the table is printed for all non-zero elements in (Z/nZ).

      (%i1) zn_mult_table(8);
                                      [ 1  3  5  7 ]
                                      [            ]
                                      [ 3  1  7  5 ]
      (%o1)                           [            ]
                                      [ 5  7  1  3 ]
                                      [            ]
                                      [ 7  5  3  1 ]
      (%i2) zn_mult_table(8, all);
                                  [ 1  2  3  4  5  6  7 ]
                                  [                     ]
                                  [ 2  4  6  0  2  4  6 ]
                                  [                     ]
                                  [ 3  6  1  4  7  2  5 ]
                                  [                     ]
      (%o2)                       [ 4  0  4  0  4  0  4 ]
                                  [                     ]
                                  [ 5  2  7  4  1  6  3 ]
                                  [                     ]
                                  [ 6  4  2  0  6  4  2 ]
                                  [                     ]
                                  [ 7  6  5  4  3  2  1 ]

  If gcd is an integer, row and column headings are added for better readability.

  If the subset chosen by gcd is a group there is another way to create a multiplication table. An isomorphic mapping from a group with `1` as identity builds a table which is easy to read. The mapping is accomplished via CRT.

  In the second version of `T36_4` the identity, here `28`, is placed in the top left corner, just like in table `T9`.

      (%i1) T36_4: zn_mult_table(36,4);
                              [ *   4   8   16  20  28  32 ]
                              [                            ]
                              [ 4   16  32  28  8   4   20 ]
                              [                            ]
                              [ 8   32  28  20  16  8   4  ]
                              [                            ]
      (%o1)                   [ 16  28  20  4   32  16  8  ]
                              [                            ]
                              [ 20  8   16  32  4   20  28 ]
                              [                            ]
                              [ 28  4   8   16  20  28  32 ]
                              [                            ]
                              [ 32  20  4   8   28  32  16 ]
      (%i2) T9: zn_mult_table(36/4);
                                   [ 1  2  4  5  7  8 ]
                                   [                  ]
                                   [ 2  4  8  1  5  7 ]
                                   [                  ]
                                   [ 4  8  7  2  1  5 ]
      (%o2)                        [                  ]
                                   [ 5  1  2  7  8  4 ]
                                   [                  ]
                                   [ 7  5  1  8  4  2 ]
                                   [                  ]
                                   [ 8  7  5  4  2  1 ]
      (%i3) T36_4: matrixmap(lambda([x], chinese([0,x],[4,9])), T9);
                                [ 28  20  4   32  16  8  ]
                                [                        ]
                                [ 20  4   8   28  32  16 ]
                                [                        ]
                                [ 4   8   16  20  28  32 ]
      (%o3)                     [                        ]
                                [ 32  28  20  16  8   4  ]
                                [                        ]
                                [ 16  32  28  8   4   20 ]
                                [                        ]
                                [ 8   16  32  4   20  28 ]

  Categories: Number theory

Function: zn_nth_root
`zn_nth_root` (x, n, m)
`zn_nth_root` (x, n, m, [[p1, e1], …, [pk, ek]])

  Returns a list with all n-th roots of x from the multiplication subgroup of (Z/mZ) which contains x, or `false`, if x is no n-th power modulo m or not contained in any multiplication subgroup of (Z/mZ).

  x is an element of a multiplication subgroup modulo m, if the greatest common divisor `g = gcd(x,m)` is coprime to `m/g`.

  `zn_nth_root` is based on an algorithm by Adleman, Manders and Miller and on theorems about modulo multiplication groups by Daniel Shanks.

  The algorithm needs a prime factorization of the modulus m. So in case the factorization of m is known, the list of factors can be passed as the fourth argument. This optional argument must be of the same form as the list returned by `ifactors(m)` using the default option `factors_only: false`.

  Examples:

  A power table of the multiplication group modulo `14` followed by a list of lists containing all n-th roots of `1` with n from `1` to `6`.

      (%i1) zn_power_table(14);
                               [ 1   1   1   1   1   1 ]
                               [                       ]
                               [ 3   9   13  11  5   1 ]
                               [                       ]
                               [ 5   11  13  9   3   1 ]
      (%o1)                    [                       ]
                               [ 9   11  1   9   11  1 ]
                               [                       ]
                               [ 11  9   1   11  9   1 ]
                               [                       ]
                               [ 13  1   13  1   13  1 ]
      (%i2) makelist(zn_nth_root(1,n,14), n,1,6);
      (%o2)  [[1], [1, 13], [1, 9, 11], [1, 13], [1], [1, 3, 5, 9, 11, 13]]

  In the following example x is not coprime to m, but is a member of a multiplication subgroup of (Z/mZ) and any n-th root is a member of the same subgroup.

  The residue class `3` is no member of any multiplication subgroup of (Z/63Z) and is therefore not returned as a third root of `27`.

  Here `zn_power_table` shows all residues `x` in (Z/63Z) with `gcd(x,63) = 9`. This subgroup is isomorphic to (Z/7Z)* and its identity `36` is computed via CRT.

      (%i1) m: 7*9$

      (%i2) zn_power_table(m,9);
                               [ 9   18  36  9   18  36 ]
                               [                        ]
                               [ 18  9   36  18  9   36 ]
                               [                        ]
                               [ 27  36  27  36  27  36 ]
      (%o2)                    [                        ]
                               [ 36  36  36  36  36  36 ]
                               [                        ]
                               [ 45  9   27  18  54  36 ]
                               [                        ]
                               [ 54  18  27  9   45  36 ]
      (%i3) zn_nth_root(27,3,m);
      (%o3)                           [27, 45, 54]
      (%i4) id7:1$  id63_9: chinese([id7,0],[7,9]);
      (%o5)                                36

  In the following RSA-like example, where the modulus `N` is squarefree, i.e. it splits into exclusively first power factors, every `x` from `0` to `N-1` is contained in a multiplication subgroup.

  The process of decryption needs the `e`-th root. `e` is coprime to `totient(N)` and therefore the `e`-th root is unique. In this case `zn_nth_root` effectively performs CRT-RSA. (Please note that `flatten` removes braces but no solutions.)

      (%i1) [p,q,e]: [5,7,17]$  N: p*q$

      (%i3) xs: makelist(x,x,0,N-1)$

      (%i4) ys: map(lambda([x],power_mod(x,e,N)),xs)$

      (%i5) zs: flatten(map(lambda([y], zn_nth_root(y,e,N)), ys))$

      (%i6) is(zs = xs);
      (%o6)                             true

  In the following example the factorization of the modulus is known and passed as the fourth argument.

      (%i1) p: 2^107-1$  q: 2^127-1$  N: p*q$

      (%i4) ibase: obase: 16$

      (%i5) msg: 11223344556677889900aabbccddeeff$

      (%i6) enc: power_mod(msg, 10001, N);
      (%o6)    1a8db7892ae588bdc2be25dd5107a425001fe9c82161abc673241c8b383
      (%i7) zn_nth_root(enc, 10001, N, [[p,1],[q,1]]);
      (%o7)               [11223344556677889900aabbccddeeff]

  Categories: Number theory

Function: zn_order
`zn_order` (x, n)
`zn_order` (x, n, [[p1, e1], …, [pk, ek]])

  Returns the order of x if it is a unit of the finite group (Z/nZ)* or returns `false`. x is a unit modulo n if it is coprime to n.

  The applied algorithm needs a prime factorization of `totient(n)`. This factorization might be time consuming in some cases and it can be useful to factor first and then to pass the list of factors to `zn_log` as the third argument. The list must be of the same form as the list returned by `ifactors(totient(n))` using the default option `factors_only : false`.

  See also `zn_primroot`, `ifactors`, `totient`.

  Examples:

  `zn_order` computes the order of the unit x in (Z/nZ)*.

      (%i1) n : 22$
      (%i2) g : zn_primroot(n);
      (%o2)                               7
      (%i3) units_22 : sublist(makelist(i,i,1,21), lambda([x], gcd(x, n) = 1));
      (%o3)              [1, 3, 5, 7, 9, 13, 15, 17, 19, 21]
      (%i4) (ord_7 : zn_order(7, n)) = totient(n);
      (%o4)                            10 = 10
      (%i5) powers_7 : makelist(power_mod(g,i,n), i,0,ord_7 - 1);
      (%o5)              [1, 7, 5, 13, 3, 21, 15, 17, 9, 19]
      (%i6) map(lambda([x], zn_order(x, n)), powers_7);
      (%o6)              [1, 10, 5, 10, 5, 2, 5, 10, 5, 10]
      (%i7) map(lambda([x], ord_7/gcd(x, ord_7)), makelist(i, i,0,ord_7 - 1));
      (%o7)              [1, 10, 5, 10, 5, 2, 5, 10, 5, 10]
      (%i8) totient(totient(n));
      (%o8)                               4

  The optional third argument must be of the same form as the list returned by `ifactors(totient(n))`.

      (%i1) (p : 2^142 + 217, primep(p));
      (%o1)                             true
      (%i2) ifs : ifactors( totient(p) )$
      (%i3) g : zn_primroot(p, ifs);
      (%o3)                               3
      (%i4) is( (ord_3 : zn_order(g, p, ifs)) = totient(p) );
      (%o4)                             true
      (%i5) map(lambda([x], ord_3/zn_order(x, p, ifs)), makelist(i,i,2,15));
      (%o5)        [22, 1, 44, 10, 5, 2, 22, 2, 8, 2, 1, 1, 20, 1]

  Categories: Number theory

Function: zn_power_table
`zn_power_table` (n)
`zn_power_table` (n, gcd)
`zn_power_table` (n, gcd, max_exp)

  Without any optional argument `zn_power_table(n)` shows a power table of all elements in (Z/nZ)* which are all residue classes coprime to n. The exponent loops from `1` to the greatest characteristic factor of `totient(n)` (also known as Carmichael function or Carmichael lambda) and the table ends with a column of ones on the right side.

  The optional second argument gcd allows to select powers of a specific subset of (Z/nZ). If gcd is an integer, powers of all residue classes `x` with `gcd(x,n) =` gcd are returned, i.e. the default value for gcd is `1`. If gcd is set to `all`, the table contains powers of all elements in (Z/nZ).

  If the optional third argument max_exp is given, the exponent loops from `1` to max_exp.

  See also `zn_add_table`, `zn_mult_table`.

  Examples:

  The default which is gcd`= 1` allows to demonstrate and study basic theorems of e.g. Fermat and Euler.

  The argument gcd allows to select subsets of (Z/nZ) and to study multiplication subgroups and isomorphisms. E.g. the groups `G10` and `G10_2` are under multiplication both isomorphic to `G5`. `1` is the identity in `G5`. So are `1` resp. `6` the identities in `G10` resp. `G10_2`. There are corresponding mappings for primitive roots, n-th roots, etc..

      (%i1) zn_power_table(10);
                                    [ 1  1  1  1 ]
                                    [            ]
                                    [ 3  9  7  1 ]
      (%o1)                         [            ]
                                    [ 7  9  3  1 ]
                                    [            ]
                                    [ 9  1  9  1 ]
      (%i2) zn_power_table(10,2);
                                    [ 2  4  8  6 ]
                                    [            ]
                                    [ 4  6  4  6 ]
      (%o2)                         [            ]
                                    [ 6  6  6  6 ]
                                    [            ]
                                    [ 8  4  2  6 ]
      (%i3) zn_power_table(10,5);
      (%o3)                         [ 5  5  5  5 ]
      (%i4) zn_power_table(10,10);
      (%o4)                         [ 0  0  0  0 ]
      (%i5) G5: [1,2,3,4];
      (%o6)                          [1, 2, 3, 4]
      (%i6) G10_2: map(lambda([x], chinese([0,x],[2,5])), G5);
      (%o6)                          [6, 2, 8, 4]
      (%i7) G10: map(lambda([x], power_mod(3, zn_log(x,2,5), 10)), G5);
      (%o7)                          [1, 3, 7, 9]

  If gcd is set to `all`, the table contains powers of all elements in (Z/nZ).

  The third argument max_exp allows to set the highest exponent. The following table shows a very small example of RSA.

      (%i1) N:2*5$ phi:totient(N)$ e:7$ d:inv_mod(e,phi)$

      (%i5) zn_power_table(N, all, e*d);
             [ 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ]
             [                                                               ]
             [ 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 ]
             [                                                               ]
             [ 2  4  8  6  2  4  8  6  2  4  8  6  2  4  8  6  2  4  8  6  2 ]
             [                                                               ]
             [ 3  9  7  1  3  9  7  1  3  9  7  1  3  9  7  1  3  9  7  1  3 ]
             [                                                               ]
             [ 4  6  4  6  4  6  4  6  4  6  4  6  4  6  4  6  4  6  4  6  4 ]
      (%o5)  [                                                               ]
             [ 5  5  5  5  5  5  5  5  5  5  5  5  5  5  5  5  5  5  5  5  5 ]
             [                                                               ]
             [ 6  6  6  6  6  6  6  6  6  6  6  6  6  6  6  6  6  6  6  6  6 ]
             [                                                               ]
             [ 7  9  3  1  7  9  3  1  7  9  3  1  7  9  3  1  7  9  3  1  7 ]
             [                                                               ]
             [ 8  4  2  6  8  4  2  6  8  4  2  6  8  4  2  6  8  4  2  6  8 ]
             [                                                               ]
             [ 9  1  9  1  9  1  9  1  9  1  9  1  9  1  9  1  9  1  9  1  9 ]

  Categories: Number theory

Function: zn_primroot
`zn_primroot` (n)
`zn_primroot` (n, [[p1, e1], …, [pk, ek]])

  If the multiplicative group (Z/nZ)* is cyclic, `zn_primroot` computes the smallest primitive root modulo n. (Z/nZ)* is cyclic if n is equal to `2`, `4`, `p^k` or `2*p^k`, where `p` is prime and greater than `2` and `k` is a natural number. `zn_primroot` performs an according pretest if the option variable `zn_primroot_pretest` (default: `false`) is set to `true`. In any case the computation is limited by the upper bound `zn_primroot_limit`.

  If (Z/nZ)* is not cyclic or if there is no primitive root up to `zn_primroot_limit`, `zn_primroot` returns `false`.

  The applied algorithm needs a prime factorization of `totient(n)`. This factorization might be time consuming in some cases and it can be useful to factor first and then to pass the list of factors to `zn_log` as an additional argument. The list must be of the same form as the list returned by `ifactors(totient(n))` using the default option `factors_only : false`.

  See also `zn_primroot_p`, `zn_order`, `ifactors`, `totient`.

  Examples:

  `zn_primroot` computes the smallest primitive root modulo n or returns `false`.

      (%i1) n : 14$
      (%i2) g : zn_primroot(n);
      (%o2)                               3
      (%i3) zn_order(g, n) = totient(n);
      (%o3)                             6 = 6
      (%i4) n : 15$
      (%i5) zn_primroot(n);
      (%o5)                             false

  The optional second argument must be of the same form as the list returned by `ifactors(totient(n))`.

      (%i1) (p : 2^142 + 217, primep(p));
      (%o1)                             true
      (%i2) ifs : ifactors( totient(p) )$
      (%i3) g : zn_primroot(p, ifs);
      (%o3)                               3
      (%i4) [time(%o2), time(%o3)];
      (%o4)                    [[15.556972], [0.004]]
      (%i5) is(zn_order(g, p, ifs) = p - 1);
      (%o5)                             true
      (%i6) n : 2^142 + 216$
      (%i7) ifs : ifactors(totient(n))$
      (%i8) zn_primroot(n, ifs),
            zn_primroot_limit : 200, zn_primroot_verbose : true;
      `zn_primroot' stopped at zn_primroot_limit = 200
      (%o8)                             false

  Categories: Number theory

Option variable: zn_primroot_limit

  Default value: `1000`

  If `zn_primroot` cannot find a primitve root, it stops at this upper bound. If the option variable `zn_primroot_verbose` (default: `false`) is set to `true`, a message will be printed when `zn_primroot_limit` is reached.

  Categories: Number theory

Function: zn_primroot_p
`zn_primroot_p` (x, n)
`zn_primroot_p` (x, n, [[p1, e1], …, [pk, ek]])

  Checks whether x is a primitive root in the multiplicative group (Z/nZ)*.

  The applied algorithm needs a prime factorization of `totient(n)`. This factorization might be time consuming and in case `zn_primroot_p` will be consecutively applied to a list of candidates it can be useful to factor first and then to pass the list of factors to `zn_log` as a third argument. The list must be of the same form as the list returned by `ifactors(totient(n))` using the default option `factors_only : false`.

  See also `zn_primroot`, `zn_order`, `ifactors`, `totient`.

  Examples:

  `zn_primroot_p` as a predicate function.

      (%i1) n : 14$
      (%i2) units_14 : sublist(makelist(i,i,1,13), lambda([i], gcd(i, n) = 1));
      (%o2)                     [1, 3, 5, 9, 11, 13]
      (%i3) zn_primroot_p(13, n);
      (%o3)                            false
      (%i4) sublist(units_14, lambda([x], zn_primroot_p(x, n)));
      (%o4)                            [3, 5]
      (%i5) map(lambda([x], zn_order(x, n)), units_14);
      (%o5)                      [1, 6, 6, 3, 3, 2]

  The optional third argument must be of the same form as the list returned by `ifactors(totient(n))`.

      (%i1) (p : 2^142 + 217, primep(p));
      (%o1)                             true
      (%i2) ifs : ifactors( totient(p) )$
      (%i3) sublist(makelist(i,i,1,50), lambda([x], zn_primroot_p(x, p, ifs)));
      (%o3)      [3, 12, 13, 15, 21, 24, 26, 27, 29, 33, 38, 42, 48]
      (%i4) [time(%o2), time(%o3)];
      (%o4)                   [[7.748484], [0.036002]]

  Categories: Predicate functions · Number theory

Option variable: zn_primroot_pretest

  Default value: `false`

  The multiplicative group (Z/nZ)* is cyclic if n is equal to `2`, `4`, `p^k` or `2*p^k`, where `p` is prime and greater than `2` and `k` is a natural number.

  `zn_primroot_pretest` controls whether `zn_primroot` will check if one of these cases occur before it computes the smallest primitive root. Only if `zn_primroot_pretest` is set to `true` this pretest will be performed.

  Categories: Number theory

Option variable: zn_primroot_verbose

  Default value: `false`

  Controls whether `zn_primroot` prints a message when reaching `zn_primroot_limit`.

  Categories: Number theory

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# 30. Symmetries

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## 30.1 Introduction to Symmetries

`sym` is a package for working with symmetric groups of polynomials.

It was written for Macsyma-Symbolics by Annick Valibouze (http://www-calfor.lip6.fr/~avb/). The algorithms are described in the following papers:

1. Fonctions symétriques et changements de bases. Annick Valibouze. EUROCAL'87 (Leipzig, 1987), 323-332, Lecture Notes in Comput. Sci 378. Springer, Berlin, 1989.
   http://www.stix.polytechnique.fr/publications/1984-1994.html
2. Résolvantes et fonctions symétriques. Annick Valibouze. Proceedings of the ACM-SIGSAM 1989 International Symposium on Symbolic and Algebraic Computation, ISSAC'89 (Portland, Oregon). ACM Press, 390-399, 1989.
   http://www-calfor.lip6.fr/~avb/DonneesTelechargeables/MesArticles/issac89ACMValibouze.pdf
3. Symbolic computation with symmetric polynomials, an extension to Macsyma. Annick Valibouze. Computers and Mathematics (MIT, USA, June 13-17, 1989), Springer-Verlag, New York Berlin, 308-320, 1989.
   http://www.stix.polytechnique.fr/publications/1984-1994.html
4. Théorie de Galois Constructive. Annick Valibouze. Mémoire d'habilitation à diriger les recherches (HDR), Université P. et M. Curie (Paris VI), 1994.

Categories: Group theory · Polynomials · Share packages · Package sym

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## 30.2 Functions and Variables for Symmetries

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### 30.2.1 Changing bases

Function: comp2pui (n, L)

  implements passing from the complete symmetric functions given in the list L to the elementary symmetric functions from 0 to n. If the list L contains fewer than n+1 elements, it will be completed with formal values of the type h1, h2, etc. If the first element of the list L exists, it specifies the size of the alphabet, otherwise the size is set to n.

      (%i1) comp2pui (3, [4, g]);
                              2                    2
      (%o1)    [4, g, 2 h2 - g , 3 h3 - g h2 + g (g  - 2 h2)]

  Categories: Package sym

Function: ele2pui (m, L)

  goes from the elementary symmetric functions to the complete functions. Similar to `comp2ele` and `comp2pui`.

  Other functions for changing bases: `comp2ele`.

  Categories: Package sym

Function: ele2comp (m, L)

  Goes from the elementary symmetric functions to the compete functions. Similar to `comp2ele` and `comp2pui`.

  Other functions for changing bases: `comp2ele`.

  Categories: Package sym

Function: elem (ele, sym, lvar)

  decomposes the symmetric polynomial sym, in the variables contained in the list lvar, in terms of the elementary symmetric functions given in the list ele. If the first element of ele is given, it will be the size of the alphabet, otherwise the size will be the degree of the polynomial sym. If values are missing in the list ele, formal values of the type e1, e2, etc. will be added. The polynomial sym may be given in three different forms: contracted (`elem` should then be 1, its default value), partitioned (`elem` should be 3), or extended (i.e. the entire polynomial, and `elem` should then be 2). The function `pui` is used in the same way.

  On an alphabet of size 3 with e1, the first elementary symmetric function, with value 7, the symmetric polynomial in 3 variables whose contracted form (which here depends on only two of its variables) is x^4-2*x*y decomposes as follows in elementary symmetric functions:

      (%i1) elem ([3, 7], x^4 - 2*x*y, [x, y]);
      (%o1) 7 (e3 - 7 e2 + 7 (49 - e2)) + 21 e3

                                               + (- 2 (49 - e2) - 2) e2
      (%i2) ratsimp (%);
                                    2
      (%o2)             28 e3 + 2 e2  - 198 e2 + 2401

  Other functions for changing bases: `comp2ele`.

  Categories: Package sym

Function: mon2schur (L)

  The list L represents the Schur function S_L: we have L = [i_1, i_2, ..., i_q], with i_1 <= i_2 <= ... <= i_q. The Schur function S_[i_1, i_2, ..., i_q] is the minor of the infinite matrix h_[i-j], i <= 1, j <= 1, consisting of the q first rows and the columns 1 + i_1, 2 + i_2, ..., q + i_q.

  This Schur function can be written in terms of monomials by using `treinat` and `kostka`. The form returned is a symmetric polynomial in a contracted representation in the variables x_1,x_2,...

      (%i1) mon2schur ([1, 1, 1]);
      (%o1)                       x1 x2 x3
      (%i2) mon2schur ([3]);
                                        2        3
      (%o2)                x1 x2 x3 + x1  x2 + x1
      (%i3) mon2schur ([1, 2]);
                                            2
      (%o3)                  2 x1 x2 x3 + x1  x2

  which means that for 3 variables this gives:

         2 x1 x2 x3 + x1^2 x2 + x2^2 x1 + x1^2 x3 + x3^2 x1
          + x2^2 x3 + x3^2 x2

  Other functions for changing bases: `comp2ele`.

  Categories: Package sym

Function: multi_elem (l_elem, multi_pc, l_var)

  decomposes a multi-symmetric polynomial in the multi-contracted form multi_pc in the groups of variables contained in the list of lists l_var in terms of the elementary symmetric functions contained in l_elem.

      (%i1) multi_elem ([[2, e1, e2], [2, f1, f2]], a*x + a^2 + x^3,
            [[x, y], [a, b]]);
                                                        3
      (%o1)         - 2 f2 + f1 (f1 + e1) - 3 e1 e2 + e1
      (%i2) ratsimp (%);
                               2                       3
      (%o2)         - 2 f2 + f1  + e1 f1 - 3 e1 e2 + e1

  Other functions for changing bases: `comp2ele`.

  Categories: Package sym

Function: multi_pui

  is to the function `pui` what the function `multi_elem` is to the function `elem`.

      (%i1) multi_pui ([[2, p1, p2], [2, t1, t2]], a*x + a^2 + x^3,
            [[x, y], [a, b]]);
                                                  3
                                      3 p1 p2   p1
      (%o1)              t2 + p1 t1 + ------- - ---
                                         2       2

  Categories: Package sym

Function: pui (L, sym, lvar)

  decomposes the symmetric polynomial sym, in the variables in the list lvar, in terms of the power functions in the list L. If the first element of L is given, it will be the size of the alphabet, otherwise the size will be the degree of the polynomial sym. If values are missing in the list L, formal values of the type p1, p2 , etc. will be added. The polynomial sym may be given in three different forms: contracted (`elem` should then be 1, its default value), partitioned (`elem` should be 3), or extended (i.e. the entire polynomial, and `elem` should then be 2). The function `pui` is used in the same way.

      (%i1) pui;
      (%o1)                           1
      (%i2) pui ([3, a, b], u*x*y*z, [x, y, z]);
                             2
                         a (a  - b) u   (a b - p3) u
      (%o2)              ------------ - ------------
                              6              3
      (%i3) ratsimp (%);
                                             3
                            (2 p3 - 3 a b + a ) u
      (%o3)                 ---------------------
                                      6

  Other functions for changing bases: `comp2ele`.

  Categories: Package sym

Function: pui2comp (n, lpui)

  renders the list of the first n complete functions (with the length first) in terms of the power functions given in the list lpui. If the list lpui is empty, the cardinal is n, otherwise it is its first element (as in `comp2ele` and `comp2pui`).

      (%i1) pui2comp (2, []);
                                             2
                                      p2 + p1
      (%o1)                   [2, p1, --------]
                                         2
      (%i2) pui2comp (3, [2, a1]);
                                                  2
                                       a1 (p2 + a1 )
                               2  p3 + ------------- + a1 p2
                        p2 + a1              2
      (%o2)     [2, a1, --------, --------------------------]
                           2                  3
      (%i3) ratsimp (%);
                                  2                     3
                           p2 + a1   2 p3 + 3 a1 p2 + a1
      (%o3)        [2, a1, --------, --------------------]
                              2               6

  Other functions for changing bases: `comp2ele`.

  Categories: Package sym

Function: pui2ele (n, lpui)

  effects the passage from power functions to the elementary symmetric functions. If the flag `pui2ele` is `girard`, it will return the list of elementary symmetric functions from 1 to n, and if the flag is `close`, it will return the n-th elementary symmetric function.

  Other functions for changing bases: `comp2ele`.

  Categories: Package sym

Function: puireduc (n, lpui)

  lpui is a list whose first element is an integer m. `puireduc` gives the first n power functions in terms of the first m.

      (%i1) puireduc (3, [2]);
                                               2
                                         p1 (p1  - p2)
      (%o1)          [2, p1, p2, p1 p2 - -------------]
                                               2
      (%i2) ratsimp (%);
                                                 3
                                     3 p1 p2 - p1
      (%o2)              [2, p1, p2, -------------]
                                           2

  Categories: Package sym

Function: schur2comp (P, l_var)

  P is a polynomial in the variables of the list l_var. Each of these variables represents a complete symmetric function. In l_var the i-th complete symmetric function is represented by the concatenation of the letter `h` and the integer i: `hi`. This function expresses P in terms of Schur functions.

      (%i1) schur2comp (h1*h2 - h3, [h1, h2, h3]);
      (%o1)                         s
                                     1, 2
      (%i2) schur2comp (a*h3, [h3]);
      (%o2)                         s  a
                                     3

  Categories: Package sym

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### 30.2.2 Changing representations

Function: cont2part (pc, lvar)

  returns the partitioned polynomial associated to the contracted form pc whose variables are in lvar.

      (%i1) pc: 2*a^3*b*x^4*y + x^5;
                                 3    4      5
      (%o1)                   2 a  b x  y + x
      (%i2) cont2part (pc, [x, y]);
                                         3
      (%o2)              [[1, 5, 0], [2 a  b, 4, 1]]

  Categories: Package sym

Function: contract (psym, lvar)

  returns a contracted form (i.e. a monomial orbit under the action of the symmetric group) of the polynomial psym in the variables contained in the list lvar. The function `explose` performs the inverse operation. The function `tcontract` tests the symmetry of the polynomial.

      (%i1) psym: explose (2*a^3*b*x^4*y, [x, y, z]);
               3      4      3      4      3    4        3    4
      (%o1) 2 a  b y z  + 2 a  b x z  + 2 a  b y  z + 2 a  b x  z

                                                 3      4      3    4
                                            + 2 a  b x y  + 2 a  b x  y
      (%i2) contract (psym, [x, y, z]);
                                    3    4
      (%o2)                      2 a  b x  y

  Categories: Package sym

Function: explose (pc, lvar)

  returns the symmetric polynomial associated with the contracted form pc. The list lvar contains the variables.

      (%i1) explose (a*x + 1, [x, y, z]);
      (%o1)                  a z + a y + a x + 1

  Categories: Package sym

Function: part2cont (ppart, lvar)

  goes from the partitioned form to the contracted form of a symmetric polynomial. The contracted form is rendered with the variables in lvar.

      (%i1) part2cont ([[2*a^3*b, 4, 1]], [x, y]);
                                    3    4
      (%o1)                      2 a  b x  y

  Categories: Package sym

Function: partpol (psym, lvar)

  psym is a symmetric polynomial in the variables of the list lvar. This function retturns its partitioned representation.

      (%i1) partpol (-a*(x + y) + 3*x*y, [x, y]);
      (%o1)               [[3, 1, 1], [- a, 1, 0]]

  Categories: Package sym

Function: tcontract (pol, lvar)

  tests if the polynomial pol is symmetric in the variables of the list lvar. If so, it returns a contracted representation like the function `contract`.

  Categories: Package sym

Function: tpartpol (pol, lvar)

  tests if the polynomial pol is symmetric in the variables of the list lvar. If so, it returns its partitioned representation like the function `partpol`.

  Categories: Package sym

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### 30.2.3 Groups and orbits

Function: direct ([p_1, ..., p_n], y, f, [lvar_1, ..., lvar_n])

  calculates the direct image (see M. Giusti, D. Lazard et A. Valibouze, ISSAC 1988, Rome) associated to the function f, in the lists of variables lvar_1, ..., lvar_n, and in the polynomials p_1, ..., p_n in a variable y. The arity of the function f is important for the calulation. Thus, if the expression for f does not depend on some variable, it is useless to include this variable, and not including it will also considerably reduce the amount of computation.

      (%i1) direct ([z^2  - e1* z + e2, z^2  - f1* z + f2],
                    z, b*v + a*u, [[u, v], [a, b]]);
             2
      (%o1) y  - e1 f1 y

                                       2            2             2   2
                        - 4 e2 f2 - (e1  - 2 e2) (f1  - 2 f2) + e1  f1
                      + -----------------------------------------------
                                               2
      (%i2) ratsimp (%);
                    2                2                   2
      (%o2)        y  - e1 f1 y + (e1  - 4 e2) f2 + e2 f1
      (%i3) ratsimp (direct ([z^3-e1*z^2+e2*z-e3,z^2  - f1* z + f2],
                    z, b*v + a*u, [[u, v], [a, b]]));
             6            5         2                        2    2   4
      (%o3) y  - 2 e1 f1 y  + ((2 e1  - 6 e2) f2 + (2 e2 + e1 ) f1 ) y

                                3                               3   3
       + ((9 e3 + 5 e1 e2 - 2 e1 ) f1 f2 + (- 2 e3 - 2 e1 e2) f1 ) y

               2       2        4    2
       + ((9 e2  - 6 e1  e2 + e1 ) f2

                          2       2       2                   2    4
       + (- 9 e1 e3 - 6 e2  + 3 e1  e2) f1  f2 + (2 e1 e3 + e2 ) f1 )

        2          2                      2     3          2
       y  + (((9 e1  - 27 e2) e3 + 3 e1 e2  - e1  e2) f1 f2

                       2            2    3                5
       + ((15 e2 - 2 e1 ) e3 - e1 e2 ) f1  f2 - 2 e2 e3 f1 ) y

                 2                   3           3     2   2    3
       + (- 27 e3  + (18 e1 e2 - 4 e1 ) e3 - 4 e2  + e1  e2 ) f2

               2      3                   3    2   2
       + (27 e3  + (e1  - 9 e1 e2) e3 + e2 ) f1  f2

                         2    4        2   6
       + (e1 e2 e3 - 9 e3 ) f1  f2 + e3  f1

  Finding the polynomial whose roots are the sums a+u where a is a root of z^2 - e_1 z + e_2 and u is a root of z^2 - f_1 z + f_2.

      (%i1) ratsimp (direct ([z^2 - e1* z + e2, z^2 - f1* z + f2],
                                z, a + u, [[u], [a]]));
             4                    3             2
      (%o1) y  + (- 2 f1 - 2 e1) y  + (2 f2 + f1  + 3 e1 f1 + 2 e2

           2   2                              2               2
       + e1 ) y  + ((- 2 f1 - 2 e1) f2 - e1 f1  + (- 2 e2 - e1 ) f1

                        2                     2            2
       - 2 e1 e2) y + f2  + (e1 f1 - 2 e2 + e1 ) f2 + e2 f1  + e1 e2 f1

           2
       + e2

  `direct` accepts two flags: `elementaires` and `puissances` (default) which allow decomposing the symmetric polynomials appearing in the calculation into elementary symmetric functions, or power functions, respectively.

  Functions of `sym` used in this function:

  `multi_orbit` (so `orbit`), `pui_direct`, `multi_elem` (so `elem`), `multi_pui` (so `pui`), `pui2ele`, `ele2pui` (if the flag `direct` is in `puissances`).

  Categories: Package sym

Function: multi_orbit (P, [lvar_1, lvar_2,..., lvar_p])

  P is a polynomial in the set of variables contained in the lists lvar_1, lvar_2, ..., lvar_p. This function returns the orbit of the polynomial P under the action of the product of the symmetric groups of the sets of variables represented in these p lists.

      (%i1) multi_orbit (a*x + b*y, [[x, y], [a, b]]);
      (%o1)                [b y + a x, a y + b x]
      (%i2) multi_orbit (x + y + 2*a, [[x, y], [a, b, c]]);
      (%o2)        [y + x + 2 c, y + x + 2 b, y + x + 2 a]

  Also see: `orbit` for the action of a single symmetric group.

  Categories: Package sym

Function: multsym (ppart_1, ppart_2, n)

  returns the product of the two symmetric polynomials in n variables by working only modulo the action of the symmetric group of order n. The polynomials are in their partitioned form.

  Given the 2 symmetric polynomials in x, y: `3*(x + y) + 2*x*y` and `5*(x^2 + y^2)` whose partitioned forms are `[[3, 1], [2, 1, 1]]` and `[[5, 2]]`, their product will be

      (%i1) multsym ([[3, 1], [2, 1, 1]], [[5, 2]], 2);
      (%o1)         [[10, 3, 1], [15, 3, 0], [15, 2, 1]]

  that is `10*(x^3*y + y^3*x) + 15*(x^2*y + y^2*x) + 15*(x^3 + y^3)`.

  Functions for changing the representations of a symmetric polynomial:

  `contract`, `cont2part`, `explose`, `part2cont`, `partpol`, `tcontract`, `tpartpol`.

  Categories: Package sym

Function: orbit (P, lvar)

  computes the orbit of the polynomial P in the variables in the list lvar under the action of the symmetric group of the set of variables in the list lvar.

      (%i1) orbit (a*x + b*y, [x, y]);
      (%o1)                [a y + b x, b y + a x]
      (%i2) orbit (2*x + x^2, [x, y]);
                              2         2
      (%o2)                 [y  + 2 y, x  + 2 x]

  See also `multi_orbit` for the action of a product of symmetric groups on a polynomial.

  Categories: Package sym

Function: pui_direct (orbite, [lvar_1, ..., lvar_n], [d_1, d_2, ..., d_n])

  Let f be a polynomial in n blocks of variables lvar_1, ..., lvar_n. Let c_i be the number of variables in lvar_i, and SC be the product of n symmetric groups of degree c_1, ..., c_n. This group acts naturally on f. The list orbite is the orbit, denoted `SC(f)`, of the function f under the action of SC. (This list may be obtained by the function `multi_orbit`.) The di are integers s.t. c_1 <= d_1, c_2 <= d_2, ..., c_n <= d_n.

  Let SD be the product of the symmetric groups S_[d_1] x S_[d_2] x ... x S_[d_n]. The function `pui_direct` returns the first n power functions of `SD(f)` deduced from the power functions of `SC(f)`, where n is the size of `SD(f)`.

  The result is in multi-contracted form w.r.t. SD, i.e. only one element is kept per orbit, under the action of SD.

      (%i1) l: [[x, y], [a, b]];
      (%o1)                   [[x, y], [a, b]]
      (%i2) pui_direct (multi_orbit (a*x + b*y, l), l, [2, 2]);
                                             2  2
      (%o2)               [a x, 4 a b x y + a  x ]
      (%i3) pui_direct (multi_orbit (a*x + b*y, l), l, [3, 2]);
                                   2  2     2    2        3  3
      (%o3) [2 a x, 4 a b x y + 2 a  x , 3 a  b x  y + 2 a  x ,

          2  2  2  2      3    3        4  4
      12 a  b  x  y  + 4 a  b x  y + 2 a  x ,

          3  2  3  2      4    4        5  5
      10 a  b  x  y  + 5 a  b x  y + 2 a  x ,

          3  3  3  3       4  2  4  2      5    5        6  6
      40 a  b  x  y  + 15 a  b  x  y  + 6 a  b x  y + 2 a  x ]
      (%i4) pui_direct ([y + x + 2*c, y + x + 2*b, y + x + 2*a],
            [[x, y], [a, b, c]], [2, 3]);
                                   2              2
      (%o4) [3 x + 2 a, 6 x y + 3 x  + 4 a x + 4 a ,

                       2                   3        2       2        3
                    9 x  y + 12 a x y + 3 x  + 6 a x  + 12 a  x + 8 a ]

  Categories: Package sym

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### 30.2.4 Partitions

Function: kostka (part_1, part_2)

  written by P. Esperet, calculates the Kostka number of the partition part_1 and part_2.

      (%i1) kostka ([3, 3, 3], [2, 2, 2, 1, 1, 1]);
      (%o1)                           6

  Categories: Package sym

Function: lgtreillis (n, m)

  returns the list of partitions of weight n and length m.

      (%i1) lgtreillis (4, 2);
      (%o1)                   [[3, 1], [2, 2]]

  Also see: `ltreillis`, `treillis` and `treinat`.

  Categories: Package sym

Function: ltreillis (n, m)

  returns the list of partitions of weight n and length less than or equal to m.

      (%i1) ltreillis (4, 2);
      (%o1)               [[4, 0], [3, 1], [2, 2]]

  Also see: `lgtreillis`, `treillis` and `treinat`.

  Categories: Package sym

Function: treillis (n)

  returns all partitions of weight n.

      (%i1) treillis (4);
      (%o1)    [[4], [3, 1], [2, 2], [2, 1, 1], [1, 1, 1, 1]]

  See also: `lgtreillis`, `ltreillis` and `treinat`.

  Categories: Package sym

Function: treinat (part)

  retruns the list of partitions inferior to the partition part w.r.t. the natural order.

      (%i1) treinat ([5]);
      (%o1)                         [[5]]
      (%i2) treinat ([1, 1, 1, 1, 1]);
      (%o2) [[5], [4, 1], [3, 2], [3, 1, 1], [2, 2, 1], [2, 1, 1, 1],

                                                       [1, 1, 1, 1, 1]]
      (%i3) treinat ([3, 2]);
      (%o3)                 [[5], [4, 1], [3, 2]]

  See also: `lgtreillis`, `ltreillis` and `treillis`.

  Categories: Package sym

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### 30.2.5 Polynomials and their roots

Function: ele2polynome (L, z)

  returns the polynomial in z s.t. the elementary symmetric functions of its roots are in the list `L = [n, e_1, ..., e_n]`, where n is the degree of the polynomial and e_i the i-th elementary symmetric function.

      (%i1) ele2polynome ([2, e1, e2], z);
                                2
      (%o1)                    z  - e1 z + e2
      (%i2) polynome2ele (x^7 - 14*x^5 + 56*x^3  - 56*x + 22, x);
      (%o2)          [7, 0, - 14, 0, 56, 0, - 56, - 22]
      (%i3) ele2polynome ([7, 0, -14, 0, 56, 0, -56, -22], x);
                        7       5       3
      (%o3)            x  - 14 x  + 56 x  - 56 x + 22

  The inverse: `polynome2ele (P, z)`.

  Also see: `polynome2ele`, `pui2polynome`.

  Categories: Package sym

Function: polynome2ele (P, x)

  gives the list `l = [n, e_1, ..., e_n]` where n is the degree of the polynomial P in the variable x and e_i is the i-the elementary symmetric function of the roots of P.

      (%i1) polynome2ele (x^7 - 14*x^5 + 56*x^3 - 56*x + 22, x);
      (%o1)          [7, 0, - 14, 0, 56, 0, - 56, - 22]
      (%i2) ele2polynome ([7, 0, -14, 0, 56, 0, -56, -22], x);
                        7       5       3
      (%o2)            x  - 14 x  + 56 x  - 56 x + 22

  The inverse: `ele2polynome (l, x)`

  Categories: Package sym

Function: prodrac (L, k)

  L is a list containing the elementary symmetric functions on a set A. `prodrac` returns the polynomial whose roots are the k by k products of the elements of A.

  Also see `somrac`.

  Categories: Package sym

Function: pui2polynome (x, lpui)

  calculates the polynomial in x whose power functions of the roots are given in the list lpui.

      (%i1) pui;
      (%o1)                           1
      (%i2) kill(labels);
      (%o0)                         done
      (%i1) polynome2ele (x^3 - 4*x^2 + 5*x - 1, x);
      (%o1)                     [3, 4, 5, 1]
      (%i2) ele2pui (3, %);
      (%o2)                     [3, 4, 6, 7]
      (%i3) pui2polynome (x, %);
                              3      2
      (%o3)                  x  - 4 x  + 5 x - 1

  See also: `polynome2ele`, `ele2polynome`.

  Categories: Package sym

Function: somrac (L, k)

  The list L contains elementary symmetric functions of a polynomial P . The function computes the polynomial whose roots are the k by k distinct sums of the roots of P.

  Also see `prodrac`.

  Categories: Package sym

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### 30.2.6 Resolvents

Function: resolvante (P, x, f, [x_1,..., x_d])

  calculates the resolvent of the polynomial P in x of degree n >= d by the function f expressed in the variables x_1, ..., x_d. For efficiency of computation it is important to not include in the list `[x_1, ..., x_d]` variables which do not appear in the transformation function f.

  To increase the efficiency of the computation one may set flags in `resolvante` so as to use appropriate algorithms:

  If the function f is unitary:

  - A polynomial in a single variable,
  - linear,
  - alternating,
  - a sum,
  - symmetric,
  - a product,
  - the function of the Cayley resolvent (usable up to degree 5)

        (x1*x2 + x2*x3 + x3*x4 + x4*x5 + x5*x1 -
             (x1*x3 + x3*x5 + x5*x2 + x2*x4 + x4*x1))^2

    general,

  the flag of `resolvante` may be, respectively:

  - unitaire,
  - lineaire,
  - alternee,
  - somme,
  - produit,
  - cayley,
  - generale.

      (%i1) resolvante: unitaire$
      (%i2) resolvante (x^7 - 14*x^5 + 56*x^3 - 56*x + 22, x, x^3 - 1,
            [x]);

      " resolvante unitaire " [7, 0, 28, 0, 168, 0, 1120, - 154, 7840,
                               - 2772, 56448, - 33880,

      413952, - 352352, 3076668, - 3363360, 23114112, - 30494464,

      175230832, - 267412992, 1338886528, - 2292126760]
        3       6      3       9      6      3
      [x  - 1, x  - 2 x  + 1, x  - 3 x  + 3 x  - 1,

       12      9      6      3       15      12       9       6      3
      x   - 4 x  + 6 x  - 4 x  + 1, x   - 5 x   + 10 x  - 10 x  + 5 x

             18      15       12       9       6      3
       - 1, x   - 6 x   + 15 x   - 20 x  + 15 x  - 6 x  + 1,

       21      18       15       12       9       6      3
      x   - 7 x   + 21 x   - 35 x   + 35 x  - 21 x  + 7 x  - 1]
      [- 7, 1127, - 6139, 431767, - 5472047, 201692519, - 3603982011]
             7      6        5         4          3           2
      (%o2) y  + 7 y  - 539 y  - 1841 y  + 51443 y  + 315133 y

                                                    + 376999 y + 125253
      (%i3) resolvante: lineaire$
      (%i4) resolvante (x^4 - 1, x, x1 + 2*x2 + 3*x3, [x1, x2, x3]);

      " resolvante lineaire "
             24       20         16            12             8
      (%o4) y   + 80 y   + 7520 y   + 1107200 y   + 49475840 y

                                                          4
                                             + 344489984 y  + 655360000
      (%i5) resolvante: general$
      (%i6) resolvante (x^4 - 1, x, x1 + 2*x2 + 3*x3, [x1, x2, x3]);

      " resolvante generale "
             24       20         16            12             8
      (%o6) y   + 80 y   + 7520 y   + 1107200 y   + 49475840 y

                                                          4
                                             + 344489984 y  + 655360000
      (%i7) resolvante (x^4 - 1, x, x1 + 2*x2 + 3*x3, [x1, x2, x3, x4]);

      " resolvante generale "
             24       20         16            12             8
      (%o7) y   + 80 y   + 7520 y   + 1107200 y   + 49475840 y

                                                          4
                                             + 344489984 y  + 655360000
      (%i8) direct ([x^4 - 1], x, x1 + 2*x2 + 3*x3, [[x1, x2, x3]]);
             24       20         16            12             8
      (%o8) y   + 80 y   + 7520 y   + 1107200 y   + 49475840 y

                                                          4
                                             + 344489984 y  + 655360000
      (%i9) resolvante :lineaire$
      (%i10) resolvante (x^4 - 1, x, x1 + x2 + x3, [x1, x2, x3]);

      " resolvante lineaire "
                                    4
      (%o10)                       y  - 1
      (%i11) resolvante: symetrique$
      (%i12) resolvante (x^4 - 1, x, x1 + x2 + x3, [x1, x2, x3]);

      " resolvante symetrique "
                                    4
      (%o12)                       y  - 1
      (%i13) resolvante (x^4 + x + 1, x, x1 - x2, [x1, x2]);

      " resolvante symetrique "
                                 6      2
      (%o13)                    y  - 4 y  - 1
      (%i14) resolvante: alternee$
      (%i15) resolvante (x^4 + x + 1, x, x1 - x2, [x1, x2]);

      " resolvante alternee "
                  12      8       6        4        2
      (%o15)     y   + 8 y  + 26 y  - 112 y  + 216 y  + 229
      (%i16) resolvante: produit$
      (%i17) resolvante (x^7 - 7*x + 3, x, x1*x2*x3, [x1, x2, x3]);

      " resolvante produit "
              35      33         29        28         27        26
      (%o17) y   - 7 y   - 1029 y   + 135 y   + 7203 y   - 756 y

               24           23          22            21           20
       + 1323 y   + 352947 y   - 46305 y   - 2463339 y   + 324135 y

                19           18             17              15
       - 30618 y   - 453789 y   - 40246444 y   + 282225202 y

                   14              12             11            10
       - 44274492 y   + 155098503 y   + 12252303 y   + 2893401 y

                    9            8            7             6
       - 171532242 y  + 6751269 y  + 2657205 y  - 94517766 y

                  5             3
       - 3720087 y  + 26040609 y  + 14348907
      (%i18) resolvante: symetrique$
      (%i19) resolvante (x^7 - 7*x + 3, x, x1*x2*x3, [x1, x2, x3]);

      " resolvante symetrique "
              35      33         29        28         27        26
      (%o19) y   - 7 y   - 1029 y   + 135 y   + 7203 y   - 756 y

               24           23          22            21           20
       + 1323 y   + 352947 y   - 46305 y   - 2463339 y   + 324135 y

                19           18             17              15
       - 30618 y   - 453789 y   - 40246444 y   + 282225202 y

                   14              12             11            10
       - 44274492 y   + 155098503 y   + 12252303 y   + 2893401 y

                    9            8            7             6
       - 171532242 y  + 6751269 y  + 2657205 y  - 94517766 y

                  5             3
       - 3720087 y  + 26040609 y  + 14348907
      (%i20) resolvante: cayley$
      (%i21) resolvante (x^5 - 4*x^2 + x + 1, x, a, []);

      " resolvante de Cayley "
              6       5         4          3            2
      (%o21) x  - 40 x  + 4080 x  - 92928 x  + 3772160 x  + 37880832 x

                                                             + 93392896

  For the Cayley resolvent, the 2 last arguments are neutral and the input polynomial must necessarily be of degree 5.

  See also:

  `resolvante_bipartite`, `resolvante_produit_sym`, `resolvante_unitaire`, `resolvante_alternee1`, `resolvante_klein`, `resolvante_klein3`, `resolvante_vierer`, `resolvante_diedrale`.

  Categories: Package sym

Function: resolvante_alternee1 (P, x)

  calculates the transformation `P(x)` of degree n by the function product(x_i - x_j, 1 <= i < j <= n - 1).

  See also:

  `resolvante_produit_sym`, `resolvante_unitaire`, `resolvante`, `resolvante_klein`, `resolvante_klein3`, `resolvante_vierer`, `resolvante_diedrale`, `resolvante_bipartite`.

  Categories: Package sym

Function: resolvante_bipartite (P, x)

  calculates the transformation of `P(x)` of even degree n by the function x_1 x_2 ... x_[n/2] + x_[n/2 + 1] ... x_n.

      (%i1) resolvante_bipartite (x^6 + 108, x);
                    10        8           6             4
      (%o1)        y   - 972 y  + 314928 y  - 34012224 y

  See also:

  `resolvante_produit_sym`, `resolvante_unitaire`, `resolvante`, `resolvante_klein`, `resolvante_klein3`, `resolvante_vierer`, `resolvante_diedrale`, `resolvante_alternee1`.

  Categories: Package sym

Function: resolvante_diedrale (P, x)

  calculates the transformation of `P(x)` by the function `x_1 x_2 + x_3 x_4`.

      (%i1) resolvante_diedrale (x^5 - 3*x^4 + 1, x);
             15       12       11       10        9         8         7
      (%o1) x   - 21 x   - 81 x   - 21 x   + 207 x  + 1134 x  + 2331 x

              6         5          4          3          2
       - 945 x  - 4970 x  - 18333 x  - 29079 x  - 20745 x  - 25326 x

       - 697

  See also:

  `resolvante_produit_sym`, `resolvante_unitaire`, `resolvante_alternee1`, `resolvante_klein`, `resolvante_klein3`, `resolvante_vierer`, `resolvante`.

  Categories: Package sym

Function: resolvante_klein (P, x)

  calculates the transformation of `P(x)` by the function `x_1 x_2 x_4 + x_4`.

  See also:

  `resolvante_produit_sym`, `resolvante_unitaire`, `resolvante_alternee1`, `resolvante`, `resolvante_klein3`, `resolvante_vierer`, `resolvante_diedrale`.

  Categories: Package sym

Function: resolvante_klein3 (P, x)

  calculates the transformation of `P(x)` by the function `x_1 x_2 x_4 + x_4`.

  See also:

  `resolvante_produit_sym`, `resolvante_unitaire`, `resolvante_alternee1`, `resolvante_klein`, `resolvante`, `resolvante_vierer`, `resolvante_diedrale`.

  Categories: Package sym

Function: resolvante_produit_sym (P, x)

  calculates the list of all product resolvents of the polynomial `P(x)`.

      (%i1) resolvante_produit_sym (x^5 + 3*x^4 + 2*x - 1, x);
              5      4             10      8       7       6       5
      (%o1) [y  + 3 y  + 2 y - 1, y   - 2 y  - 21 y  - 31 y  - 14 y

          4       3      2       10      8       7    6       5       4
       - y  + 14 y  + 3 y  + 1, y   + 3 y  + 14 y  - y  - 14 y  - 31 y

             3      2       5      4
       - 21 y  - 2 y  + 1, y  - 2 y  - 3 y - 1, y - 1]
      (%i2) resolvante: produit$
      (%i3) resolvante (x^5 + 3*x^4 + 2*x - 1, x, a*b*c, [a, b, c]);

      " resolvante produit "
             10      8       7    6        5       4       3     2
      (%o3) y   + 3 y  + 14 y  - y  - 14 y  - 31 y  - 21 y  - 2 y  + 1

  See also:

  `resolvante`, `resolvante_unitaire`, `resolvante_alternee1`, `resolvante_klein`, `resolvante_klein3`, `resolvante_vierer`, `resolvante_diedrale`.

  Categories: Package sym

Function: resolvante_unitaire (P, Q, x)

  computes the resolvent of the polynomial `P(x)` by the polynomial `Q(x)`.

  See also:

  `resolvante_produit_sym`, `resolvante`, `resolvante_alternee1`, `resolvante_klein`, `resolvante_klein3`, `resolvante_vierer`, `resolvante_diedrale`.

  Categories: Package sym

Function: resolvante_vierer (P, x)

  computes the transformation of `P(x)` by the function `x_1 x_2 - x_3 x_4`.

  See also:

  `resolvante_produit_sym`, `resolvante_unitaire`, `resolvante_alternee1`, `resolvante_klein`, `resolvante_klein3`, `resolvante`, `resolvante_diedrale`.

  Categories: Package sym

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### 30.2.7 Miscellaneous

Function: multinomial (r, part)

  where r is the weight of the partition part. This function returns the associate multinomial coefficient: if the parts of part are i_1, i_2, ..., i_k, the result is `r!/(i_1! i_2! ... i_k!)`.

  Categories: Package sym

Function: permut (L)

  returns the list of permutations of the list L.

  Categories: Package sym · Lists

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# 31. Groups

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## 31.1 Functions and Variables for Groups

Function: todd_coxeter
`todd_coxeter` (relations, subgroup)
`todd_coxeter` (relations)

  Find the order of G/H where G is the Free Group modulo relations, and H is the subgroup of G generated by subgroup. subgroup is an optional argument, defaulting to []. In doing this it produces a multiplication table for the right action of G on G/H, where the cosets are enumerated [H,Hg2,Hg3,...]. This can be seen internally in the variable `todd_coxeter_state`.

  Example:

      (%i1) symet(n):=create_list(
              if (j - i) = 1 then (p(i,j))^^3 else
                  if (not i = j) then (p(i,j))^^2 else
                      p(i,i) , j, 1, n-1, i, 1, j);
                                                             <3>
      (%o1) symet(n) := create_list(if j - i = 1 then p(i, j)

                                      <2>
       else (if not i = j then p(i, j)    else p(i, i)), j, 1, n - 1,

      i, 1, j)
      (%i2) p(i,j) := concat(x,i).concat(x,j);
      (%o2)        p(i, j) := concat(x, i) . concat(x, j)
      (%i3) symet(5);
               <2>           <3>    <2>           <2>           <3>
      (%o3) [x1   , (x1 . x2)   , x2   , (x1 . x3)   , (x2 . x3)   ,

                  <2>           <2>           <2>           <3>    <2>
                x3   , (x1 . x4)   , (x2 . x4)   , (x3 . x4)   , x4   ]
      (%i4) todd_coxeter(%o3);

      Rows tried 426
      (%o4)                          120
      (%i5) todd_coxeter(%o3,[x1]);

      Rows tried 213
      (%o5)                          60
      (%i6) todd_coxeter(%o3,[x1,x2]);

      Rows tried 71
      (%o6)                          20

  Categories: Group theory

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# 32. Runtime Environment

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## 32.1 Introduction for Runtime Environment

`maxima-init.mac` is a file which is loaded automatically when Maxima starts. You can use `maxima-init.mac` to customize your Maxima environment. `maxima-init.mac`, if it exists, is typically placed in the directory named by `maxima_userdir`, although it can be in any directory searched by the function `file_search`.

Here is an example `maxima-init.mac` file:

    setup_autoload ("specfun.mac", ultraspherical, assoc_legendre_p);
    showtime:all;

In this example, `setup_autoload` tells Maxima to load the specified file (`specfun.mac`) if any of the functions (`ultraspherical`, `assoc_legendre_p`) are called but not yet defined. Thus you needn't remember to load the file before calling the functions.

The statement `showtime: all` tells Maxima to set the `showtime` variable. The `maxima-init.mac` file can contain any other assignments or other Maxima statements.

Categories: Session management

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## 32.2 Interrupts

The user can stop a time-consuming computation with the ^C (control-C) character. The default action is to stop the computation and print another user prompt. In this case, it is not possible to restart a stopped computation.

If the Lisp variable `*debugger-hook*` is set to `nil`, by executing

    :lisp (setq *debugger-hook* nil)

then upon receiving ^C, Maxima will enter the Lisp debugger, and the user may use the debugger to inspect the Lisp environment. The stopped computation can be restarted by entering `continue` in the Lisp debugger. The means of returning to Maxima from the Lisp debugger (other than running the computation to completion) is different for each version of Lisp.

On Unix systems, the character ^Z (control-Z) causes Maxima to stop altogether, and control is returned to the shell prompt. The `fg` command causes Maxima to resume from the point at which it was stopped.

Categories: Console interaction

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## 32.3 Functions and Variables for Runtime Environment

System variable: maxima_tempdir

  `maxima_tempdir` names the directory in which Maxima creates some temporary files. In particular, temporary files for plotting are created in `maxima_tempdir`.

  The initial value of `maxima_tempdir` is the user's home directory, if Maxima can locate it; otherwise Maxima makes a guess about a suitable directory.

  `maxima_tempdir` may be assigned a string which names a directory.

  Categories: Global variables

System variable: maxima_userdir

  `maxima_userdir` names a directory which Maxima searches to find Maxima and Lisp files. (Maxima searches some other directories as well; `file_search_maxima` and `file_search_lisp` are the complete lists.)

  The initial value of `maxima_userdir` is a subdirectory of the user's home directory, if Maxima can locate it; otherwise Maxima makes a guess about a suitable directory.

  `maxima_userdir` may be assigned a string which names a directory. However, assigning to `maxima_userdir` does not automatically change `file_search_maxima` and `file_search_lisp`; those variables must be changed separately.

  Categories: Global variables

Function: room
`room` ()
`room` (true)
`room` (false)

  Prints out a description of the state of storage and stack management in Maxima. `room` calls the Lisp function of the same name.

  - `room ()` prints out a moderate description.
  - `room (true)` prints out a verbose description.
  - `room (false)` prints out a terse description.

  Categories: Debugging

Function: sstatus (keyword, item)

  When keyword is the symbol `feature`, item is put on the list of system features. After `sstatus (keyword, item)` is executed, `status (feature, item)` returns `true`. If keyword is the symbol `nofeature`, item is deleted from the list of system features. This can be useful for package writers, to keep track of what features they have loaded in.

  See also `status`.

  Categories: Programming

Function: status
`status` (`feature`)
`status` (`feature`, item)

  Returns information about the presence or absence of certain system-dependent features.

  - `status (feature)` returns a list of system features. These include Lisp version, operating system type, etc. The list may vary from one Lisp type to another.
  - `status (feature, item)` returns `true` if item is on the list of items returned by `status (feature)` and `false` otherwise. `status` quotes the argument item. The quote-quote operator `''` defeats quotation. A feature whose name contains a special character, such as a hyphen, must be given as a string argument. For example, `status (feature, "ansi-cl")`.

  See also `sstatus`.

  The variable `features` contains a list of features which apply to mathematical expressions. See `features` and `featurep` for more information.

  Categories: Programming

Function: system (command)

  Executes command as a separate process. The command is passed to the default shell for execution. `system` is not supported by all operating systems, but generally exists in Unix and Unix-like environments.

  Supposing `_hist.out` is a list of frequencies which you wish to plot as a bar graph using `xgraph`.

      (%i1) (with_stdout("_hist.out",
                 for i:1 thru length(hist) do (
                   print(i,hist[i]))),
             system("xgraph -bar -brw .7 -nl < _hist.out"));

  In order to make the plot be done in the background (returning control to Maxima) and remove the temporary file after it is done do:

      system("(xgraph -bar -brw .7 -nl < _hist.out;  rm -f _hist.out)&")

Function: time (%o1, %o2, %o3, …)

  Returns a list of the times, in seconds, taken to compute the output lines `%o1`, `%o2`, `%o3`, … The time returned is Maxima's estimate of the internal computation time, not the elapsed time. `time` can only be applied to output line variables; for any other variables, `time` returns `unknown`.

  Set `showtime: true` to make Maxima print out the computation time and elapsed time with each output line.

  Categories: Debugging

Function: timedate
`timedate` (T, tz_offset)

  `timedate` (T)
  `timedate` ()

  `timedate(T, tz_offset)` returns a string representing the time T in the time zone tz_offset. The string format is `YYYY-MM-DD HH:MM:SS.NNN[+|-]ZZ:ZZ` (using as many digits as necessary to represent the fractional part) if T has a nonzero fractional part, or `YYYY-MM-DD HH:MM:SS[+|-]ZZ:ZZ` if its fractional part is zero.

  T measures time, in seconds, since midnight, January 1, 1900, in the GMT time zone.

  tz_offset measures the offset of the time zone, in hours, east (positive) or west (negative) of GMT. tz_offset must be an integer, rational, or float between -24 and 24, inclusive. If tz_offset is not a multiple of 1/60, it is rounded to the nearest multiple of 1/60.

  `timedate(T)` is equivalent to `timedate(T, tz_offset)` with tz_offset equal to the offset of the local time zone.

  `timedate()` is equivalent to `timedate(absolute_real_time())`. That is, it returns the current time in the local time zone.

  Example:

  `timedate` with no argument returns a string representing the current time and date.

      (%i1) d : timedate ();
      (%o1)                      2010-06-08 04:08:09+01:00
      (%i2) print ("timedate reports current time", d) $
      timedate reports current time 2010-06-08 04:08:09+01:00

  `timedate` with an argument returns a string representing the argument.

      (%i1) timedate (0);
      (%o1)                      1900-01-01 01:00:00+01:00
      (%i2) timedate (absolute_real_time () - 7*24*3600);
      (%o2)                      2010-06-01 04:19:51+01:00

  `timedate` with optional timezone offset.

      (%i1) timedate (1000000000, -9.5);
      (%o1)               1931-09-09 16:16:40-09:30

  Categories: Time and date functions

Function: parse_timedate
`parse_timedate` (S)

  Parses a string S representing a date or date and time of day and returns the number of seconds since midnight, January 1, 1900 GMT. If there is a nonzero fractional part, the value returned is a rational number, otherwise, it is an integer. `parse_timedate` returns `false` if it cannot parse S according to any of the allowed formats.

  The string S must have one of the following formats, optionally followed by a timezone designation:

  - `YYYY-MM-DD[ T]hh:mm:ss[,.]nnn`
  - `YYYY-MM-DD[ T]hh:mm:ss`
  - `YYYY-MM-DD`

  where the fields are year, month, day, hours, minutes, seconds, and fraction of a second, and square brackets indicate acceptable alternatives. The fraction may contain one or more digits.

  Except for the fraction of a second, each field must have exactly the number of digits indicated: four digits for the year, and two for the month, day of the month, hours, minutes, and seconds.

  A timezone designation must have one of the following forms:

  - `[+-]hh:mm`
  - `[+-]hhmm`
  - `[+-]hh`
  - `Z`

  where `hh` and `mm` indicate hours and minutes east (`+`) or west (`-`) of GMT. The timezone may be from +24 hours (inclusive) to -24 hours (inclusive).

  A literal character `Z` is equivalent to `+00:00` and its variants, indicating GMT.

  If no timezone is indicated, the time is assumed to be in the local time zone.

  Any leading or trailing whitespace (space, tab, newline, and carriage return) is ignored, but any other leading or trailing characters cause `parse_timedate` to fail and return `false`.

  See also `timedate` and `absolute_real_time`.

  Examples:

  Midnight, January 1, 1900, in the local time zone, in each acceptable format. The result is the number of seconds the local time zone is ahead (negative result) or behind (positive result) GMT. In this example, the local time zone is 8 hours behind GMT.

      (%i1) parse_timedate ("1900-01-01 00:00:00,000");
      (%o1)                         28800
      (%i2) parse_timedate ("1900-01-01 00:00:00.000");
      (%o2)                         28800
      (%i3) parse_timedate ("1900-01-01T00:00:00,000");
      (%o3)                         28800
      (%i4) parse_timedate ("1900-01-01T00:00:00.000");
      (%o4)                         28800
      (%i5) parse_timedate ("1900-01-01 00:00:00");
      (%o5)                         28800
      (%i6) parse_timedate ("1900-01-01T00:00:00");
      (%o6)                         28800
      (%i7) parse_timedate ("1900-01-01");
      (%o7)                         28800

  Midnight, January 1, 1900, GMT, in different indicated time zones.

      (%i1) parse_timedate ("1900-01-01 19:00:00+19:00");
      (%o1)                           0
      (%i2) parse_timedate ("1900-01-01 07:00:00+07:00");
      (%o2)                           0
      (%i3) parse_timedate ("1900-01-01 01:00:00+01:00");
      (%o3)                           0
      (%i4) parse_timedate ("1900-01-01Z");
      (%o4)                           0
      (%i5) parse_timedate ("1899-12-31 21:00:00-03:00");
      (%o5)                           0
      (%i6) parse_timedate ("1899-12-31 13:00:00-11:00");
      (%o6)                           0
      (%i7) parse_timedate ("1899-12-31 08:00:00-16:00");
      (%o7)                           0

  Categories: Time and date functions

Function: encode_time
`encode_time` (year, month, day, hours, minutes, seconds, tz_offset)
`encode_time` (year, month, day, hours, minutes, seconds)

  Given a time and date specified by year, month, day, hours, minutes, and seconds, `encode_time` returns the number of seconds (possibly including a fractional part) since midnight, January 1, 1900 GMT.

  year must be an integer greater than or equal to 1899. However, 1899 is allowed only if the resulting encoded time is greater than or equal to 0.

  month must be an integer from 1 to 12, inclusive.

  day must be an integer from 1 to n, inclusive, where n is the number of days in the month specified by month.

  hours must be an integer from 0 to 23, inclusive.

  minutes must be an integer from 0 to 59, inclusive.

  seconds must be an integer, rational, or float greater than or equal to 0 and less than 60. When seconds is not an integer, `encode_time` returns a rational, such that the fractional part of the return value is equal to the fractional part of seconds. Otherwise, seconds is an integer, and the return value is likewise an integer.

  tz_offset measures the offset of the time zone, in hours, east (positive) or west (negative) of GMT. tz_offset must be an integer, rational, or float between -24 and 24, inclusive. If tz_offset is not a multiple of 1/3600, it is rounded to the nearest multiple of 1/3600.

  If tz_offset is not present, the offset of the local time zone is assumed.

  See also `decode_time`.

  Examples:

      (%i1) encode_time (1900, 1, 1, 0, 0, 0, 0);
      (%o1)                           0
      (%i2) encode_time (1970, 1, 1, 0, 0, 0, 0);
      (%o2)                      2208988800
      (%i3) encode_time (1970, 1, 1, 8, 30, 0, 8.5);
      (%o3)                      2208988800
      (%i4) encode_time (1969, 12, 31, 16, 0, 0, -8);
      (%o4)                      2208988800
      (%i5) encode_time (1969, 12, 31, 16, 0, 1/1000, -8);
                                2208988800001
      (%o5)                     -------------
                                    1000
      (%i6) % - 2208988800;
                                     1
      (%o6)                         ----
                                    1000

  Categories: Time and date functions

Function: decode_time
`decode_time` (T, tz_offset)
`decode_time` (T)

  Given the number of seconds (possibly including a fractional part) since midnight, January 1, 1900 GMT, returns the date and time as represented by a list comprising the year, month, day of the month, hours, minutes, seconds, and time zone offset.

  tz_offset measures the offset of the time zone, in hours, east (positive) or west (negative) of GMT. tz_offset must be an integer, rational, or float between -24 and 24, inclusive. If tz_offset is not a multiple of 1/3600, it is rounded to the nearest multiple of 1/3600.

  If tz_offset is not present, the offset of the local time zone is assumed.

  See also `encode_time`.

  Examples:

      (%i1) decode_time (0, 0);
      (%o1)               [1900, 1, 1, 0, 0, 0, 0]
      (%i2) decode_time (0);
      (%o2)             [1899, 12, 31, 16, 0, 0, - 8]
      (%i3) decode_time (2208988800, 9.25);
                                                37
      (%o3)              [1970, 1, 1, 9, 15, 0, --]
                                                4
      (%i4) decode_time (2208988800);
      (%o4)             [1969, 12, 31, 16, 0, 0, - 8]
      (%i5) decode_time (2208988800 + 1729/1000, -6);
                                            1729
      (%o5)           [1969, 12, 31, 18, 0, ----, - 6]
                                            1000
      (%i6) decode_time (2208988800 + 1729/1000);
                                            1729
      (%o6)           [1969, 12, 31, 16, 0, ----, - 8]
                                            1000

  Categories: Time and date functions

Function: absolute_real_time ()

  Returns the number of seconds since midnight, January 1, 1900 GMT. The return value is an integer.

  See also `elapsed_real_time` and `elapsed_run_time`.

  Example:

      (%i1) absolute_real_time ();
      (%o1)                      3385045277
      (%i2) 1900 + absolute_real_time () / (365.25 * 24 * 3600);
      (%o2)                   2007.265612087104

  Categories: Time and date functions

Function: elapsed_real_time ()

  Returns the number of seconds (including fractions of a second) since Maxima was most recently started or restarted. The return value is a floating-point number.

  See also `absolute_real_time` and `elapsed_run_time`.

  Example:

      (%i1) elapsed_real_time ();
      (%o1)                       2.559324
      (%i2) expand ((a + b)^500)$
      (%i3) elapsed_real_time ();
      (%o3)                       7.552087

  Categories: Time and date functions

Function: elapsed_run_time ()

  Returns an estimate of the number of seconds (including fractions of a second) which Maxima has spent in computations since Maxima was most recently started or restarted. The return value is a floating-point number.

  See also `absolute_real_time` and `elapsed_real_time`.

  Example:

      (%i1) elapsed_run_time ();
      (%o1)                         0.04
      (%i2) expand ((a + b)^500)$
      (%i3) elapsed_run_time ();
      (%o3)                         1.26

  Categories: Time and date functions

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# 33. Miscellaneous Options

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## 33.1 Introduction to Miscellaneous Options

In this section various options are discussed which have a global effect on the operation of Maxima. Also various lists such as the list of all user defined functions, are discussed.

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## 33.2 Share

The Maxima "share" directory contains programs and other files of interest to Maxima users, but not part of the core implementation of Maxima. These programs are typically loaded via `load` or `setup_autoload`.

`:lisp *maxima-sharedir*` displays the location of the share directory within the user's file system.

`printfile ("share.usg")` prints an out-of-date list of share packages. Users may find it more informative to browse the share directory using a file system browser.

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## 33.3 Functions and Variables for Miscellaneous Options

System variable: askexp

  When `asksign` is called, `askexp` is the expression `asksign` is testing.

  At one time, it was possible for a user to inspect `askexp` by entering a Maxima break with control-A.

  Categories: Declarations and inferences

Option variable: genindex

  Default value: `i`

  `genindex` is the alphabetic prefix used to generate the next variable of summation when necessary.

  Categories: Sums and products

Option variable: gensumnum

  Default value: 0

  `gensumnum` is the numeric suffix used to generate the next variable of summation. If it is set to `false` then the index will consist only of `genindex` with no numeric suffix.

  Categories: Sums and products

Function: gensym
`gensym` ()
`gensym` (x)

  `gensym()` creates and returns a fresh symbol.

  The name of the new symbol is the concatenation of a prefix, which defaults to "g", and a suffix, which is an integer that defaults to the value of an internal counter.

  If x is supplied, and is a string, then that string is used as a prefix instead of "g" for this call to gensym only.

  If x is supplied, and is a nonnegative integer, then that integer, instead of the value of the internal counter, is used as the suffix for this call to gensym only.

  If and only if no explicit suffix is supplied, the internal counter is incremented after it is used.

  Examples:

      (%i1) gensym();
      (%o1)                         g887
      (%i2) gensym("new");
      (%o2)                        new888
      (%i3) gensym(123);
      (%o3)                         g123

Option variable: packagefile

  Default value: `false`

  Package designers who use `save` or `translate` to create packages (files) for others to use may want to set `packagefile: true` to prevent information from being added to Maxima's information-lists (e.g. `values`, `functions`) except where necessary when the file is loaded in. In this way, the contents of the package will not get in the user's way when he adds his own data. Note that this will not solve the problem of possible name conflicts. Also note that the flag simply affects what is output to the package file. Setting the flag to `true` is also useful for creating Maxima init files.

  Categories: Translation flags and variables

Function: remvalue
`remvalue` (name_1, …, name_n)
`remvalue` remvalue (all)

  Removes the values of user variables name_1, …, name_n (which can be subscripted) from the system.

  `remvalue (all)` removes the values of all variables in `values`, the list of all variables given names by the user (as opposed to those which are automatically assigned by Maxima).

  See also `values`.

  Categories: Evaluation

Function: rncombine (expr)

  Transforms expr by combining all terms of expr that have identical denominators or denominators that differ from each other by numerical factors only. This is slightly different from the behavior of `combine`, which collects terms that have identical denominators.

  Setting `pfeformat: true` and using `combine` yields results similar to those that can be obtained with `rncombine`, but `rncombine` takes the additional step of cross-multiplying numerical denominator factors. This results in neater forms, and the possibility of recognizing some cancellations.

  `load(rncomb)` loads this function.

  Categories: Expressions

Function: setup_autoload (filename, function_1, …, function_n)

  Specifies that if any of function_1, …, function_n are referenced and not yet defined, filename is loaded via `load`. filename usually contains definitions for the functions specified, although that is not enforced.

  `setup_autoload` does not work for array functions.

  `setup_autoload` quotes its arguments.

  Example:

      (%i1) legendre_p (1, %pi);
      (%o1)                  legendre_p(1, %pi)
      (%i2) setup_autoload ("specfun.mac", legendre_p, ultraspherical);
      (%o2)                         done
      (%i3) ultraspherical (2, 1/2, %pi);
      Warning - you are redefining the Macsyma function ultraspherical
      Warning - you are redefining the Macsyma function legendre_p
                                  2
                       3 (%pi - 1)
      (%o3)            ------------ + 3 (%pi - 1) + 1
                            2
      (%i4) legendre_p (1, %pi);
      (%o4)                          %pi
      (%i5) legendre_q (1, %pi);
                                    %pi + 1
                            %pi log(-------)
                                    1 - %pi
      (%o5)                 ---------------- - 1
                                   2

  Categories: Function definition · File input

Function: tcl_output
`tcl_output` (list, i0, skip)
`tcl_output` (list, i0)
`tcl_output` ([list_1, …, list_n], i)

  Prints elements of a list enclosed by curly braces `{ }`, suitable as part of a program in the Tcl/Tk language.

  `tcl_output (list, i0, skip)` prints list, beginning with element i0 and printing elements `i0 + skip`, `i0 + 2 skip`, etc.

  `tcl_output (list, i0)` is equivalent to `tcl_output (list, i0, 2)`.

  `tcl_output ([list_1, ..., list_n], i)` prints the i'th elements of list_1, …, list_n.

  Examples:

      (%i1) tcl_output ([1, 2, 3, 4, 5, 6], 1, 3)$

       {1.000000000     4.000000000
       }
      (%i2) tcl_output ([1, 2, 3, 4, 5, 6], 2, 3)$

       {2.000000000     5.000000000
       }
      (%i3) tcl_output ([3/7, 5/9, 11/13, 13/17], 1)$

       {((RAT SIMP) 3 7) ((RAT SIMP) 11 13)
       }
      (%i4) tcl_output ([x1, y1, x2, y2, x3, y3], 2)$

       {$Y1 $Y2 $Y3
       }
      (%i5) tcl_output ([[1, 2, 3], [11, 22, 33]], 1)$

       {SIMP 1.000000000     11.00000000
       }

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# 34. Rules and Patterns

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## 34.1 Introduction to Rules and Patterns

This section describes user-defined pattern matching and simplification rules. There are two groups of functions which implement somewhat different pattern matching schemes. In one group are `tellsimp`, `tellsimpafter`, `defmatch`, `defrule`, `apply1`, `applyb1`, and `apply2`. In the other group are `let` and `letsimp`. Both schemes define patterns in terms of pattern variables declared by `matchdeclare`.

Pattern-matching rules defined by `tellsimp` and `tellsimpafter` are applied automatically by the Maxima simplifier. Rules defined by `defmatch`, `defrule`, and `let` are applied by an explicit function call.

There are additional mechanisms for rules applied to polynomials by `tellrat`, and for commutative and noncommutative algebra in `affine` package.

Categories: Simplification · Rules and patterns

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## 34.2 Functions and Variables for Rules and Patterns

Function: apply1 (expr, rule_1, …, rule_n)

  Repeatedly applies rule_1 to expr until it fails, then repeatedly applies the same rule to all subexpressions of expr, left to right, until rule_1 has failed on all subexpressions. Call the result of transforming expr in this manner expr_2. Then rule_2 is applied in the same fashion starting at the top of expr_2. When rule_n fails on the final subexpression, the result is returned.

  `maxapplydepth` is the depth of the deepest subexpressions processed by `apply1` and `apply2`.

  See also `applyb1`, `apply2` and `let`.

  Categories: Rules and patterns

Function: apply2 (expr, rule_1, …, rule_n)

  If rule_1 fails on a given subexpression, then rule_2 is repeatedly applied, etc. Only if all rules fail on a given subexpression is the whole set of rules repeatedly applied to the next subexpression. If one of the rules succeeds, then the same subexpression is reprocessed, starting with the first rule.

  `maxapplydepth` is the depth of the deepest subexpressions processed by `apply1` and `apply2`.

  See also `apply1` and `let`.

  Categories: Rules and patterns

Function: applyb1 (expr, rule_1, …, rule_n)

  Repeatedly applies rule_1 to the deepest subexpression of expr until it fails, then repeatedly applies the same rule one level higher (i.e., larger subexpressions), until rule_1 has failed on the top-level expression. Then rule_2 is applied in the same fashion to the result of rule_1. After rule_n has been applied to the top-level expression, the result is returned.

  `applyb1` is similar to `apply1` but works from the bottom up instead of from the top down.

  `maxapplyheight` is the maximum height which `applyb1` reaches before giving up.

  See also `apply1`, `apply2` and `let`.

  Categories: Rules and patterns

Option variable: current_let_rule_package

  Default value: `default_let_rule_package`

  `current_let_rule_package` is the name of the rule package that is used by functions in the `let` package (`letsimp`, etc.) if no other rule package is specified. This variable may be assigned the name of any rule package defined via the `let` command.

  If a call such as `letsimp (expr, rule_pkg_name)` is made, the rule package `rule_pkg_name` is used for that function call only, and the value of `current_let_rule_package` is not changed.

  Categories: Rules and patterns

Option variable: default_let_rule_package

  Default value: `default_let_rule_package`

  `default_let_rule_package` is the name of the rule package used when one is not explicitly set by the user with `let` or by changing the value of `current_let_rule_package`.

  Categories: Rules and patterns

Function: defmatch
`defmatch` (progname, pattern, x_1, …, x_n)
`defmatch` (progname, pattern)

  Defines a function `progname(expr, x_1, ..., x_n)` which tests expr to see if it matches pattern.

  pattern is an expression containing the pattern arguments x_1, …, x_n (if any) and some pattern variables (if any). The pattern arguments are given explicitly as arguments to `defmatch` while the pattern variables are declared by the `matchdeclare` function. Any variable not declared as a pattern variable in `matchdeclare` or as a pattern argument in `defmatch` matches only itself.

  The first argument to the created function progname is an expression to be matched against the pattern and the other arguments are the actual arguments which correspond to the dummy variables x_1, …, x_n in the pattern.

  If the match is successful, progname returns a list of equations whose left sides are the pattern arguments and pattern variables, and whose right sides are the subexpressions which the pattern arguments and variables matched. The pattern variables, but not the pattern arguments, are assigned the subexpressions they match. If the match fails, progname returns `false`.

  A literal pattern (that is, a pattern which contains neither pattern arguments nor pattern variables) returns `true` if the match succeeds.

  See also `matchdeclare`, `defrule`, `tellsimp` and `tellsimpafter`.

  Examples:

  Define a function `linearp(expr, x)` which tests `expr` to see if it is of the form `a*x + b` such that `a` and `b` do not contain `x` and `a` is nonzero. This match function matches expressions which are linear in any variable, because the pattern argument `x` is given to `defmatch`.

      (%i1) matchdeclare (a, lambda ([e], e#0 and freeof(x, e)), b,
                          freeof(x));
      (%o1)                         done
      (%i2) defmatch (linearp, a*x + b, x);
      (%o2)                        linearp
      (%i3) linearp (3*z + (y + 1)*z + y^2, z);
                               2
      (%o3)              [b = y , a = y + 4, x = z]
      (%i4) a;
      (%o4)                         y + 4
      (%i5) b;
                                      2
      (%o5)                          y
      (%i6) x;
      (%o6)                           x

  Define a function `linearp(expr)` which tests `expr` to see if it is of the form `a*x + b` such that `a` and `b` do not contain `x` and `a` is nonzero. This match function only matches expressions linear in `x`, not any other variable, because no pattern argument is given to `defmatch`.

      (%i1) matchdeclare (a, lambda ([e], e#0 and freeof(x, e)), b,
                          freeof(x));
      (%o1)                         done
      (%i2) defmatch (linearp, a*x + b);
      (%o2)                        linearp
      (%i3) linearp (3*z + (y + 1)*z + y^2);
      (%o3)                         false
      (%i4) linearp (3*x + (y + 1)*x + y^2);
                                   2
      (%o4)                  [b = y , a = y + 4]

  Define a function `checklimits(expr)` which tests `expr` to see if it is a definite integral.

      (%i1) matchdeclare ([a, f], true);
      (%o1)                         done
      (%i2) constinterval (l, h) := constantp (h - l);
      (%o2)        constinterval(l, h) := constantp(h - l)
      (%i3) matchdeclare (b, constinterval (a));
      (%o3)                         done
      (%i4) matchdeclare (x, atom);
      (%o4)                         done
      (%i5) simp : false;
      (%o5)                         false
      (%i6) defmatch (checklimits, 'integrate (f, x, a, b));
      (%o6)                      checklimits
      (%i7) simp : true;
      (%o7)                         true
      (%i8) 'integrate (sin(t), t, %pi + x, 2*%pi + x);
                             x + 2 %pi
                            /
                            [
      (%o8)                 I          sin(t) dt
                            ]
                            /
                             x + %pi
      (%i9) checklimits (%);
      (%o9)    [b = x + 2 %pi, a = x + %pi, x = t, f = sin(t)]

  Categories: Rules and patterns

Function: defrule (rulename, pattern, replacement)

  Defines and names a replacement rule for the given pattern. If the rule named rulename is applied to an expression (by `apply1`, `applyb1`, or `apply2`), every subexpression matching the pattern will be replaced by the replacement. All variables in the replacement which have been assigned values by the pattern match are assigned those values in the replacement which is then simplified.

  The rules themselves can be treated as functions which transform an expression by one operation of the pattern match and replacement. If the match fails, the rule function returns `false`.

  Categories: Rules and patterns

Function: disprule
`disprule` (rulename_1, …, rulename_2)
`disprule` (all)

  Display rules with the names rulename_1, …, rulename_n, as returned by `defrule`, `tellsimp`, or `tellsimpafter`, or a pattern defined by `defmatch`. Each rule is displayed with an intermediate expression label (`%t`).

  `disprule (all)` displays all rules.

  `disprule` quotes its arguments. `disprule` returns the list of intermediate expression labels corresponding to the displayed rules.

  See also `letrules`, which displays rules defined by `let`.

  Examples:

      (%i1) tellsimpafter (foo (x, y), bar (x) + baz (y));
      (%o1)                   [foorule1, false]
      (%i2) tellsimpafter (x + y, special_add (x, y));
      (%o2)                   [+rule1, simplus]
      (%i3) defmatch (quux, mumble (x));
      (%o3)                         quux
      (%i4) disprule (foorule1, ?\+rule1, quux);
      (%t4)        foorule1 : foo(x, y) -> baz(y) + bar(x)

      (%t5)          +rule1 : y + x -> special_add(x, y)

      (%t6)                quux : mumble(x) -> []

      (%o6)                    [%t4, %t5, %t6]
      (%i6) ev(%);
      (%o6) [foorule1 : foo(x, y) -> baz(y) + bar(x),
           +rule1 : y + x -> special_add(x, y), quux : mumble(x) -> []]

  Categories: Rules and patterns · Display functions

Function: let
`let` (prod, repl, predname, arg_1, …, arg_n)
`let` ([prod, repl, predname, arg_1, …, arg_n], package_name)

  Defines a substitution rule for `letsimp` such that prod is replaced by repl. prod is a product of positive or negative powers of the following terms:

  - Atoms which `letsimp` will search for literally unless previous to calling `letsimp` the `matchdeclare` function is used to associate a predicate with the atom. In this case `letsimp` will match the atom to any term of a product satisfying the predicate.
  - Kernels such as `sin(x)`, `n!`, `f(x,y)`, etc. As with atoms above `letsimp` will look for a literal match unless `matchdeclare` is used to associate a predicate with the argument of the kernel.

  A term to a positive power will only match a term having at least that power. A term to a negative power on the other hand will only match a term with a power at least as negative. In the case of negative powers in prod the switch `letrat` must be set to `true`. See also `letrat`.

  If a predicate is included in the `let` function followed by a list of arguments, a tentative match (i.e. one that would be accepted if the predicate were omitted) is accepted only if `predname (arg_1', ..., arg_n')` evaluates to `true` where arg_i' is the value matched to arg_i. The arg_i may be the name of any atom or the argument of any kernel appearing in prod. repl may be any rational expression. If any of the atoms or arguments from prod appear in repl the appropriate substitutions are made.

  The global flag `letrat` controls the simplification of quotients by `letsimp`. When `letrat` is `false`, `letsimp` simplifies the numerator and denominator of expr separately, and does not simplify the quotient. Substitutions such as `n!/n` goes to `(n-1)!` then fail. When `letrat` is `true`, then the numerator, denominator, and the quotient are simplified in that order.

  These substitution functions allow you to work with several rule packages at once. Each rule package can contain any number of `let` rules and is referenced by a user-defined name. The command `let ([prod, repl, predname, arg_1, ..., arg_n], package_name)` adds the rule predname to the rule package package_name. The command `letsimp (expr, package_name)` applies the rules in package_name. `letsimp (expr, package_name1, package_name2, ...)` is equivalent to `letsimp (expr, package_name1)` followed by `letsimp (%, package_name2)`, …

  `current_let_rule_package` is the name of the rule package that is presently being used. This variable may be assigned the name of any rule package defined via the `let` command. Whenever any of the functions comprising the `let` package are called with no package name, the package named by `current_let_rule_package` is used. If a call such as `letsimp (expr, rule_pkg_name)` is made, the rule package rule_pkg_name is used for that `letsimp` command only, and `current_let_rule_package` is not changed. If not otherwise specified, `current_let_rule_package` defaults to `default_let_rule_package`.

      (%i1) matchdeclare ([a, a1, a2], true)$
      (%i2) oneless (x, y) := is (x = y-1)$
      (%i3) let (a1*a2!, a1!, oneless, a2, a1);
      (%o3)         a1 a2! --> a1! where oneless(a2, a1)
      (%i4) letrat: true$
      (%i5) let (a1!/a1, (a1-1)!);
                              a1!
      (%o5)                   --- --> (a1 - 1)!
                              a1
      (%i6) letsimp (n*m!*(n-1)!/m);
      (%o6)                      (m - 1)! n!
      (%i7) let (sin(a)^2, 1 - cos(a)^2);
                              2               2
      (%o7)                sin (a) --> 1 - cos (a)
      (%i8) letsimp (sin(x)^4);
                              4           2
      (%o8)                cos (x) - 2 cos (x) + 1

  Categories: Rules and patterns

Option variable: letrat

  Default value: `false`

  When `letrat` is `false`, `letsimp` simplifies the numerator and denominator of a ratio separately, and does not simplify the quotient.

  When `letrat` is `true`, the numerator, denominator, and their quotient are simplified in that order.

      (%i1) matchdeclare (n, true)$
      (%i2) let (n!/n, (n-1)!);
                               n!
      (%o2)                    -- --> (n - 1)!
                               n
      (%i3) letrat: false$
      (%i4) letsimp (a!/a);
                                     a!
      (%o4)                          --
                                     a
      (%i5) letrat: true$
      (%i6) letsimp (a!/a);
      (%o6)                       (a - 1)!

  Categories: Rules and patterns

Function: letrules
`letrules` ()
`letrules` (package_name)

  Displays the rules in a rule package. `letrules ()` displays the rules in the current rule package. `letrules (package_name)` displays the rules in package_name.

  The current rule package is named by `current_let_rule_package`. If not otherwise specified, `current_let_rule_package` defaults to `default_let_rule_package`.

  See also `disprule`, which displays rules defined by `tellsimp` and `tellsimpafter`.

  Categories: Rules and patterns

Function: letsimp
`letsimp` (expr)
`letsimp` (expr, package_name)
`letsimp` (expr, package_name_1, …, package_name_n)

  Repeatedly applies the substitution rules defined by `let` until no further change is made to expr.

  `letsimp (expr)` uses the rules from `current_let_rule_package`.

  `letsimp (expr, package_name)` uses the rules from package_name without changing `current_let_rule_package`.

  `letsimp (expr, package_name_1, ..., package_name_n)` is equivalent to `letsimp (expr, package_name_1)`, followed by `letsimp (%, package_name_2)`, and so on.

  Categories: Rules and patterns

Option variable: let_rule_packages

  Default value: `[default_let_rule_package]`

  `let_rule_packages` is a list of all user-defined let rule packages plus the default package `default_let_rule_package`.

  Categories: Rules and patterns

Function: matchdeclare (a_1, pred_1, …, a_n, pred_n)

  Associates a predicate pred_k with a variable or list of variables a_k so that a_k matches expressions for which the predicate returns anything other than `false`.

  A predicate is the name of a function, or a lambda expression, or a function call or lambda call missing the last argument, or `true` or `all`. Any expression matches `true` or `all`. If the predicate is specified as a function call or lambda call, the expression to be tested is appended to the list of arguments; the arguments are evaluated at the time the match is evaluated. Otherwise, the predicate is specified as a function name or lambda expression, and the expression to be tested is the sole argument. A predicate function need not be defined when `matchdeclare` is called; the predicate is not evaluated until a match is attempted.

  A predicate may return a Boolean expression as well as `true` or `false`. Boolean expressions are evaluated by `is` within the constructed rule function, so it is not necessary to call `is` within the predicate.

  If an expression satisfies a match predicate, the match variable is assigned the expression, except for match variables which are operands of addition `+` or multiplication `*`. Only addition and multiplication are handled specially; other n-ary operators (both built-in and user-defined) are treated like ordinary functions.

  In the case of addition and multiplication, the match variable may be assigned a single expression which satisfies the match predicate, or a sum or product (respectively) of such expressions. Such multiple-term matching is greedy: predicates are evaluated in the order in which their associated variables appear in the match pattern, and a term which satisfies more than one predicate is taken by the first predicate which it satisfies. Each predicate is tested against all operands of the sum or product before the next predicate is evaluated. In addition, if 0 or 1 (respectively) satisfies a match predicate, and there are no other terms which satisfy the predicate, 0 or 1 is assigned to the match variable associated with the predicate.

  The algorithm for processing addition and multiplication patterns makes some match results (for example, a pattern in which a "match anything" variable appears) dependent on the ordering of terms in the match pattern and in the expression to be matched. However, if all match predicates are mutually exclusive, the match result is insensitive to ordering, as one match predicate cannot accept terms matched by another.

  Calling `matchdeclare` with a variable a as an argument changes the `matchdeclare` property for a, if one was already declared; only the most recent `matchdeclare` is in effect when a rule is defined. Later changes to the `matchdeclare` property (via `matchdeclare` or `remove`) do not affect existing rules.

  `propvars (matchdeclare)` returns the list of all variables for which there is a `matchdeclare` property. `printprops (a, matchdeclare)` returns the predicate for variable `a`. `printprops (all, matchdeclare)` returns the list of predicates for all `matchdeclare` variables. `remove (a, matchdeclare)` removes the `matchdeclare` property from a.

  The functions `defmatch`, `defrule`, `tellsimp`, `tellsimpafter`, and `let` construct rules which test expressions against patterns.

  `matchdeclare` quotes its arguments. `matchdeclare` always returns `done`.

  Examples:

  A predicate is the name of a function, or a lambda expression, or a function call or lambda call missing the last argument, or `true` or `all`.

      (%i1) matchdeclare (aa, integerp);
      (%o1)                         done
      (%i2) matchdeclare (bb, lambda ([x], x > 0));
      (%o2)                         done
      (%i3) matchdeclare (cc, freeof (%e, %pi, %i));
      (%o3)                         done
      (%i4) matchdeclare (dd, lambda ([x, y], gcd (x, y) = 1) (1728));
      (%o4)                         done
      (%i5) matchdeclare (ee, true);
      (%o5)                         done
      (%i6) matchdeclare (ff, all);
      (%o6)                         done

  If an expression satisfies a match predicate, the match variable is assigned the expression.

      (%i1) matchdeclare (aa, integerp, bb, atom);
      (%o1)                         done
      (%i2) defrule (r1, bb^aa, ["integer" = aa, "atom" = bb]);
                          aa
      (%o2)        r1 : bb   -> [integer = aa, atom = bb]
      (%i3) r1 (%pi^8);
      (%o3)               [integer = 8, atom = %pi]

  In the case of addition and multiplication, the match variable may be assigned a single expression which satisfies the match predicate, or a sum or product (respectively) of such expressions.

      (%i1) matchdeclare (aa, atom, bb, lambda ([x], not atom(x)));
      (%o1)                         done
      (%i2) defrule (r1, aa + bb, ["all atoms" = aa, "all nonatoms" =
                     bb]);
      bb + aa partitions `sum'
      (%o2)  r1 : bb + aa -> [all atoms = aa, all nonatoms = bb]
      (%i3) r1 (8 + a*b + sin(x));
      (%o3)     [all atoms = 8, all nonatoms = sin(x) + a b]
      (%i4) defrule (r2, aa * bb, ["all atoms" = aa, "all nonatoms" =
                     bb]);
      bb aa partitions `product'
      (%o4)   r2 : aa bb -> [all atoms = aa, all nonatoms = bb]
      (%i5) r2 (8 * (a + b) * sin(x));
      (%o5)    [all atoms = 8, all nonatoms = (b + a) sin(x)]

  When matching arguments of `+` and `*`, if all match predicates are mutually exclusive, the match result is insensitive to ordering, as one match predicate cannot accept terms matched by another.

      (%i1) matchdeclare (aa, atom, bb, lambda ([x], not atom(x)));
      (%o1)                         done
      (%i2) defrule (r1, aa + bb, ["all atoms" = aa, "all nonatoms" =
                     bb]);
      bb + aa partitions `sum'
      (%o2)  r1 : bb + aa -> [all atoms = aa, all nonatoms = bb]
      (%i3) r1 (8 + a*b + %pi + sin(x) - c + 2^n);
                                                           n
      (%o3) [all atoms = %pi + 8, all nonatoms = sin(x) + 2  - c + a b]
      (%i4) defrule (r2, aa * bb, ["all atoms" = aa, "all nonatoms" =
                     bb]);
      bb aa partitions `product'
      (%o4)   r2 : aa bb -> [all atoms = aa, all nonatoms = bb]
      (%i5) r2 (8 * (a + b) * %pi * sin(x) / c * 2^n);
                                                        n
                                               (b + a) 2  sin(x)
      (%o5) [all atoms = 8 %pi, all nonatoms = -----------------]
                                                       c

  The functions `propvars` and `printprops` return information about match variables.

      (%i1) matchdeclare ([aa, bb, cc], atom, [dd, ee], integerp);
      (%o1)                         done
      (%i2) matchdeclare (ff, floatnump, gg, lambda ([x], x > 100));
      (%o2)                         done
      (%i3) propvars (matchdeclare);
      (%o3)             [aa, bb, cc, dd, ee, ff, gg]
      (%i4) printprops (ee, matchdeclare);
      (%o4)                    [integerp(ee)]
      (%i5) printprops (gg, matchdeclare);
      (%o5)              [lambda([x], x > 100, gg)]
      (%i6) printprops (all, matchdeclare);
      (%o6) [lambda([x], x > 100, gg), floatnump(ff), integerp(ee),
                            integerp(dd), atom(cc), atom(bb), atom(aa)]

  Categories: Rules and patterns · Declarations and inferences

Option variable: maxapplydepth

  Default value: 10000

  `maxapplydepth` is the maximum depth to which `apply1` and `apply2` will delve.

  Categories: Function application

Option variable: maxapplyheight

  Default value: 10000

  `maxapplyheight` is the maximum height to which `applyb1` will reach before giving up.

  Categories: Function application

Function: remlet
`remlet` (prod, name)
`remlet` ()
`remlet` (all)
`remlet` (all, name)

  Deletes the substitution rule, prod -> repl, most recently defined by the `let` function. If name is supplied the rule is deleted from the rule package name.

  `remlet()` and `remlet(all)` delete all substitution rules from the current rule package. If the name of a rule package is supplied, e.g. `remlet (all, name)`, the rule package name is also deleted.

  If a substitution is to be changed using the same product, `remlet` need not be called, just redefine the substitution using the same product (literally) with the `let` function and the new replacement and/or predicate name. Should `remlet (prod)` now be called the original substitution rule is revived.

  See also `remrule`, which removes a rule defined by `tellsimp` or `tellsimpafter`.

  Categories: Rules and patterns

Function: remrule
`remrule` (op, rulename)
`remrule` (op, all)

  Removes rules defined by `tellsimp` or `tellsimpafter`.

  `remrule (op, rulename)` removes the rule with the name rulename from the operator op. When op is a built-in or user-defined operator (as defined by `infix`, `prefix`, etc.), op and rulename must be enclosed in double quote marks.

  `remrule (op, all)` removes all rules for the operator op.

  See also `remlet`, which removes a rule defined by `let`.

  Examples:

      (%i1) tellsimp (foo (aa, bb), bb - aa);
      (%o1)                   [foorule1, false]
      (%i2) tellsimpafter (aa + bb, special_add (aa, bb));
      (%o2)                   [+rule1, simplus]
      (%i3) infix ("@@");
      (%o3)                          @@
      (%i4) tellsimp (aa @@ bb, bb/aa);
      (%o4)                   [@@rule1, false]
      (%i5) tellsimpafter (quux (%pi, %e), %pi - %e);
      (%o5)                  [quuxrule1, false]
      (%i6) tellsimpafter (quux (%e, %pi), %pi + %e);
      (%o6)             [quuxrule2, quuxrule1, false]
      (%i7) [foo (aa, bb), aa + bb, aa @@ bb, quux (%pi, %e),
             quux (%e, %pi)];
                                           bb
      (%o7) [bb - aa, special_add(aa, bb), --, %pi - %e, %pi + %e]
                                           aa
      (%i8) remrule (foo, foorule1);
      (%o8)                          foo
      (%i9) remrule ("+", ?\+rule1);
      (%o9)                           +
      (%i10) remrule ("@@", ?\@\@rule1);
      (%o10)                         @@
      (%i11) remrule (quux, all);
      (%o11)                        quux
      (%i12) [foo (aa, bb), aa + bb, aa @@ bb, quux (%pi, %e),
              quux (%e, %pi)];
      (%o12) [foo(aa, bb), bb + aa, aa @@ bb, quux(%pi, %e),
                                               quux(%e, %pi)]

  Categories: Rules and patterns

Function: tellsimp (pattern, replacement)

  is similar to `tellsimpafter` but places new information before old so that it is applied before the built-in simplification rules.

  `tellsimp` is used when it is important to modify the expression before the simplifier works on it, for instance if the simplifier "knows" something about the expression, but what it returns is not to your liking. If the simplifier "knows" something about the main operator of the expression, but is simply not doing enough for you, you probably want to use `tellsimpafter`.

  The pattern may not be a sum, product, single variable, or number.

  The system variable `rules` is the list of rules defined by `defrule`, `defmatch`, `tellsimp`, and `tellsimpafter`.

  Examples:

      (%i1) matchdeclare (x, freeof (%i));
      (%o1)                         done
      (%i2) %iargs: false$
      (%i3) tellsimp (sin(%i*x), %i*sinh(x));
      (%o3)                 [sinrule1, simp-%sin]
      (%i4) trigexpand (sin (%i*y + x));
      (%o4)         sin(x) cos(%i y) + %i cos(x) sinh(y)
      (%i5) %iargs:true$
      (%i6) errcatch(0^0);
       0
      0  has been generated
      (%o6)                          []
      (%i7) ev (tellsimp (0^0, 1), simp: false);
      (%o7)                  [^rule1, simpexpt]
      (%i8) 0^0;
      (%o8)                           1
      (%i9) remrule ("^", %th(2)[1]);
      (%o9)                           ^
      (%i10) tellsimp (sin(x)^2, 1 - cos(x)^2);
      (%o10)                 [^rule2, simpexpt]
      (%i11) (1 + sin(x))^2;
                                            2
      (%o11)                    (sin(x) + 1)
      (%i12) expand (%);
                                         2
      (%o12)               2 sin(x) - cos (x) + 2
      (%i13) sin(x)^2;
                                        2
      (%o13)                     1 - cos (x)
      (%i14) kill (rules);
      (%o14)                        done
      (%i15) matchdeclare (a, true);
      (%o15)                        done
      (%i16) tellsimp (sin(a)^2, 1 - cos(a)^2);
      (%o16)                 [^rule3, simpexpt]
      (%i17) sin(y)^2;
                                        2
      (%o17)                     1 - cos (y)

  Categories: Rules and patterns

Function: tellsimpafter (pattern, replacement)

  Defines a simplification rule which the Maxima simplifier applies after built-in simplification rules. pattern is an expression, comprising pattern variables (declared by `matchdeclare`) and other atoms and operators, considered literals for the purpose of pattern matching. replacement is substituted for an actual expression which matches pattern; pattern variables in replacement are assigned the values matched in the actual expression.

  pattern may be any nonatomic expression in which the main operator is not a pattern variable; the simplification rule is associated with the main operator. The names of functions (with one exception, described below), lists, and arrays may appear in pattern as the main operator only as literals (not pattern variables); this rules out expressions such as `aa(x)` and `bb[y]` as patterns, if `aa` and `bb` are pattern variables. Names of functions, lists, and arrays which are pattern variables may appear as operators other than the main operator in pattern.

  There is one exception to the above rule concerning names of functions. The name of a subscripted function in an expression such as `aa[x](y)` may be a pattern variable, because the main operator is not `aa` but rather the Lisp atom `mqapply`. This is a consequence of the representation of expressions involving subscripted functions.

  Simplification rules are applied after evaluation (if not suppressed through quotation or the flag `noeval`). Rules established by `tellsimpafter` are applied in the order they were defined, and after any built-in rules. Rules are applied bottom-up, that is, applied first to subexpressions before application to the whole expression. It may be necessary to repeatedly simplify a result (for example, via the quote-quote operator `''` or the flag `infeval`) to ensure that all rules are applied.

  Pattern variables are treated as local variables in simplification rules. Once a rule is defined, the value of a pattern variable does not affect the rule, and is not affected by the rule. An assignment to a pattern variable which results from a successful rule match does not affect the current assignment (or lack of it) of the pattern variable. However, as with all atoms in Maxima, the properties of pattern variables (as declared by `put` and related functions) are global.

  The rule constructed by `tellsimpafter` is named after the main operator of pattern. Rules for built-in operators, and user-defined operators defined by `infix`, `prefix`, `postfix`, `matchfix`, and `nofix`, have names which are Lisp identifiers. Rules for other functions have names which are Maxima identifiers.

  The treatment of noun and verb forms is slightly confused. If a rule is defined for a noun (or verb) form and a rule for the corresponding verb (or noun) form already exists, the newly-defined rule applies to both forms (noun and verb). If a rule for the corresponding verb (or noun) form does not exist, the newly-defined rule applies only to the noun (or verb) form.

  The rule constructed by `tellsimpafter` is an ordinary Lisp function. If the name of the rule is `$foorule1`, the construct `:lisp (trace $foorule1)` traces the function, and `:lisp (symbol-function '$foorule1)` displays its definition.

  `tellsimpafter` quotes its arguments. `tellsimpafter` returns the list of rules for the main operator of pattern, including the newly established rule.

  See also `matchdeclare`, `defmatch`, `defrule`, `tellsimp`, `let`, `kill`, `remrule` and `clear_rules`.

  Examples:

  pattern may be any nonatomic expression in which the main operator is not a pattern variable.

      (%i1) matchdeclare (aa, atom, [ll, mm], listp, xx, true)$
      (%i2) tellsimpafter (sin (ll), map (sin, ll));
      (%o2)                 [sinrule1, simp-%sin]
      (%i3) sin ([1/6, 1/4, 1/3, 1/2, 1]*%pi);
                          1  sqrt(2)  sqrt(3)
      (%o3)              [-, -------, -------, 1, 0]
                          2     2        2
      (%i4) tellsimpafter (ll^mm, map ("^", ll, mm));
      (%o4)                  [^rule1, simpexpt]
      (%i5) [a, b, c]^[1, 2, 3];
                                      2   3
      (%o5)                      [a, b , c ]
      (%i6) tellsimpafter (foo (aa (xx)), aa (foo (xx)));
      (%o6)                   [foorule1, false]
      (%i7) foo (bar (u - v));
      (%o7)                    bar(foo(u - v))

  Rules are applied in the order they were defined. If two rules can match an expression, the rule which was defined first is applied.

      (%i1) matchdeclare (aa, integerp);
      (%o1)                         done
      (%i2) tellsimpafter (foo (aa), bar_1 (aa));
      (%o2)                   [foorule1, false]
      (%i3) tellsimpafter (foo (aa), bar_2 (aa));
      (%o3)              [foorule2, foorule1, false]
      (%i4) foo (42);
      (%o4)                       bar_1(42)

  Pattern variables are treated as local variables in simplification rules. (Compare to `defmatch`, which treats pattern variables as global variables.)

      (%i1) matchdeclare (aa, integerp, bb, atom);
      (%o1)                         done
      (%i2) tellsimpafter (foo(aa, bb), bar('aa=aa, 'bb=bb));
      (%o2)                   [foorule1, false]
      (%i3) bb: 12345;
      (%o3)                         12345
      (%i4) foo (42, %e);
      (%o4)                 bar(aa = 42, bb = %e)
      (%i5) bb;
      (%o5)                         12345

  As with all atoms, properties of pattern variables are global even though values are local. In this example, an assignment property is declared via `define_variable`. This is a property of the atom `bb` throughout Maxima.

      (%i1) matchdeclare (aa, integerp, bb, atom);
      (%o1)                         done
      (%i2) tellsimpafter (foo(aa, bb), bar('aa=aa, 'bb=bb));
      (%o2)                   [foorule1, false]
      (%i3) foo (42, %e);
      (%o3)                 bar(aa = 42, bb = %e)
      (%i4) define_variable (bb, true, boolean);
      (%o4)                         true
      (%i5) foo (42, %e);
      Error: bb was declared mode boolean, has value: %e
       -- an error.  Quitting.  To debug this try debugmode(true);

  Rules are named after main operators. Names of rules for built-in and user-defined operators are Lisp identifiers, while names for other functions are Maxima identifiers.

      (%i1) tellsimpafter (foo (%pi + %e), 3*%pi);
      (%o1)                   [foorule1, false]
      (%i2) tellsimpafter (foo (%pi * %e), 17*%e);
      (%o2)              [foorule2, foorule1, false]
      (%i3) tellsimpafter (foo (%i ^ %e), -42*%i);
      (%o3)         [foorule3, foorule2, foorule1, false]
      (%i4) tellsimpafter (foo (9) + foo (13), quux (22));
      (%o4)                   [+rule1, simplus]
      (%i5) tellsimpafter (foo (9) * foo (13), blurf (22));
      (%o5)                  [*rule1, simptimes]
      (%i6) tellsimpafter (foo (9) ^ foo (13), mumble (22));
      (%o6)                  [^rule1, simpexpt]
      (%i7) rules;
      (%o7) [foorule1, foorule2, foorule3, +rule1, *rule1, ^rule1]
      (%i8) foorule_name: first (%o1);
      (%o8)                       foorule1
      (%i9) plusrule_name: first (%o4);
      (%o9)                        +rule1
      (%i10) remrule (foo, foorule1);
      (%o10)                         foo
      (%i11) remrule ("^", ?\^rule1);
      (%o11)                          ^
      (%i12) rules;
      (%o12)        [foorule2, foorule3, +rule1, *rule1]

  A worked example: anticommutative multiplication.

      (%i1) gt (i, j) := integerp(j) and i < j;
      (%o1)           gt(i, j) := integerp(j) and i < j
      (%i2) matchdeclare (i, integerp, j, gt(i));
      (%o2)                         done
      (%i3) tellsimpafter (s[i]^^2, 1);
      (%o3)                 [^^rule1, simpncexpt]
      (%i4) tellsimpafter (s[i] . s[j], -s[j] . s[i]);
      (%o4)                   [.rule1, simpnct]
      (%i5) s[1] . (s[1] + s[2]);
      (%o5)                    s  . (s  + s )
                                1     2    1
      (%i6) expand (%);
      (%o6)                      1 - s  . s
                                      2    1
      (%i7) factor (expand (sum (s[i], i, 0, 9)^^5));
      (%o7) 100 (s  + s  + s  + s  + s  + s  + s  + s  + s  + s )
                  9    8    7    6    5    4    3    2    1    0

  Categories: Rules and patterns

Function: clear_rules ()

  Executes `kill (rules)` and then resets the next rule number to 1 for addition `+`, multiplication `*`, and exponentiation `^`.

  Categories: Rules and patterns

[ < ] | [ > ] |  | [ << ] | [ Up ] | [ >> ] |  |  |  |  | [Top] | [Contents] | [Index] | [ ? ]

# 35. Sets

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## 35.1 Introduction to Sets

Maxima provides set functions, such as intersection and union, for finite sets that are defined by explicit enumeration. Maxima treats lists and sets as distinct objects. This feature makes it possible to work with sets that have members that are either lists or sets.

In addition to functions for finite sets, Maxima provides some functions related to combinatorics; these include the Stirling numbers of the first and second kind, the Bell numbers, multinomial coefficients, partitions of nonnegative integers, and a few others. Maxima also defines a Kronecker delta function.

[ < ] | [ > ] |  | [ << ] | [ Up ] | [ >> ] |  |  |  |  | [Top] | [Contents] | [Index] | [ ? ]

### 35.1.1 Usage

To construct a set with members `a_1, ..., a_n`, write `set(a_1, ..., a_n)` or `{a_1, ..., a_n}`; to construct the empty set, write `set()` or `{}`. In input, `set(...)` and `{ ... }` are equivalent. Sets are always displayed with curly braces.

If a member is listed more than once, simplification eliminates the redundant member.

    (%i1) set();
    (%o1)                          {}
    (%i2) set(a, b, a);
    (%o2)                        {a, b}
    (%i3) set(a, set(b));
    (%o3)                       {a, {b}}
    (%i4) set(a, [b]);
    (%o4)                       {a, [b]}
    (%i5) {};
    (%o5)                          {}
    (%i6) {a, b, a};
    (%o6)                        {a, b}
    (%i7) {a, {b}};
    (%o7)                       {a, {b}}
    (%i8) {a, [b]};
    (%o8)                       {a, [b]}

Two would-be elements x and y are redundant (i.e., considered the same for the purpose of set construction) if and only if `is(x = y)` yields `true`. Note that `is(equal(x, y))` can yield `true` while `is(x = y)` yields `false`; in that case the elements x and y are considered distinct.

    (%i1) x: a/c + b/c;
                                  b   a
    (%o1)                         - + -
                                  c   c
    (%i2) y: a/c + b/c;
                                  b   a
    (%o2)                         - + -
                                  c   c
    (%i3) z: (a + b)/c;
                                  b + a
    (%o3)                         -----
                                    c
    (%i4) is (x = y);
    (%o4)                         true
    (%i5) is (y = z);
    (%o5)                         false
    (%i6) is (equal (y, z));
    (%o6)                         true
    (%i7) y - z;
                               b + a   b   a
    (%o7)                    - ----- + - + -
                                 c     c   c
    (%i8) ratsimp (%);
    (%o8)                           0
    (%i9) {x, y, z};
                              b + a  b   a
    (%o9)                    {-----, - + -}
                                c    c   c

To construct a set from the elements of a list, use `setify`.

    (%i1) setify ([b, a]);
    (%o1)                        {a, b}

Set members `x` and `y` are equal provided `is(x = y)` evaluates to `true`. Thus `rat(x)` and `x` are equal as set members; consequently,

    (%i1) {x, rat(x)};
    (%o1)                          {x}

Further, since `is((x - 1)*(x + 1) = x^2 - 1)` evaluates to `false`, `(x - 1)*(x + 1)` and `x^2 - 1` are distinct set members; thus

    (%i1) {(x - 1)*(x + 1), x^2 - 1};
                                           2
    (%o1)               {(x - 1) (x + 1), x  - 1}

To reduce this set to a singleton set, apply `rat` to each set member:

    (%i1) {(x - 1)*(x + 1), x^2 - 1};
                                           2
    (%o1)               {(x - 1) (x + 1), x  - 1}
    (%i2) map (rat, %);
                                  2
    (%o2)/R/                    {x  - 1}

To remove redundancies from other sets, you may need to use other simplification functions. Here is an example that uses `trigsimp`:

    (%i1) {1, cos(x)^2 + sin(x)^2};
                                2         2
    (%o1)                {1, sin (x) + cos (x)}
    (%i2) map (trigsimp, %);
    (%o2)                          {1}

A set is simplified when its members are non-redundant and sorted. The current version of the set functions uses the Maxima function `orderlessp` to order sets; however, future versions of the set functions might use a different ordering function.

Some operations on sets, such as substitution, automatically force a re-simplification; for example,

    (%i1) s: {a, b, c}$
    (%i2) subst (c=a, s);
    (%o2)                        {a, b}
    (%i3) subst ([a=x, b=x, c=x], s);
    (%o3)                          {x}
    (%i4) map (lambda ([x], x^2), set (-1, 0, 1));
    (%o4)                        {0, 1}

Maxima treats lists and sets as distinct objects; functions such as `union` and `intersection` complain if any argument is not a set. If you need to apply a set function to a list, use the `setify` function to convert it to a set. Thus

    (%i1) union ([1, 2], {a, b});
    Function union expects a set, instead found [1,2]
     -- an error.  Quitting.  To debug this try debugmode(true);
    (%i2) union (setify ([1, 2]), {a, b});
    (%o2)                     {1, 2, a, b}

To extract all set elements of a set `s` that satisfy a predicate `f`, use `subset(s, f)`. (A predicate is a boolean-valued function.) For example, to find the equations in a given set that do not depend on a variable `z`, use

    (%i1) subset ({x + y + z, x - y + 4, x + y - 5},
                                        lambda ([e], freeof (z, e)));
    (%o1)               {- y + x + 4, y + x - 5}

The section Functions and Variables for Sets has a complete list of the set functions in Maxima.

Categories: Sets

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### 35.1.2 Set Member Iteration

There two ways to to iterate over set members. One way is the use `map`; for example:

    (%i1) map (f, {a, b, c});
    (%o1)                  {f(a), f(b), f(c)}

The other way is to use `for x in s do`

    (%i1) s: {a, b, c};
    (%o1)                       {a, b, c}
    (%i2) for si in s do print (concat (si, 1));
    a1
    b1
    c1
    (%o2)                         done

The Maxima functions `first` and `rest` work correctly on sets. Applied to a set, `first` returns the first displayed element of a set; which element that is may be implementation-dependent. If `s` is a set, then `rest(s)` is equivalent to `disjoin(first(s), s)`. Currently, there are other Maxima functions that work correctly on sets. In future versions of the set functions, `first` and `rest` may function differently or not at all.

Maxima's `orderless` and `ordergreat` mechanisms are incompatible with the set functions. If you need to use either `orderless` or `ordergreat`, call those functions before constructing any sets, and do not call `unorder`.

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### 35.1.3 Authors

Stavros Macrakis of Cambridge, Massachusetts and Barton Willis of the University of Nebraska at Kearney (UNK) wrote the Maxima set functions and their documentation.

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## 35.2 Functions and Variables for Sets

Function: adjoin (x, a)

  Returns the union of the set a with `{x}`.

  `adjoin` complains if a is not a literal set.

  `adjoin(x, a)` and `union(set(x), a)` are equivalent; however, `adjoin` may be somewhat faster than `union`.

  See also `disjoin`.

  Examples:

      (%i1) adjoin (c, {a, b});
      (%o1)                       {a, b, c}
      (%i2) adjoin (a, {a, b});
      (%o2)                        {a, b}

  Categories: Sets

Function: belln (n)

  Represents the n-th Bell number. `belln(n)` is the number of partitions of a set with n members.

  For nonnegative integers n, `belln(n)` simplifies to the n-th Bell number. `belln` does not simplify for any other arguments.

  `belln` distributes over equations, lists, matrices, and sets.

  Examples:

  `belln` applied to nonnegative integers.

      (%i1) makelist (belln (i), i, 0, 6);
      (%o1)               [1, 1, 2, 5, 15, 52, 203]
      (%i2) is (cardinality (set_partitions ({})) = belln (0));
      (%o2)                         true
      (%i3) is (cardinality (set_partitions ({1, 2, 3, 4, 5, 6})) =
                             belln (6));
      (%o3)                         true

  `belln` applied to arguments which are not nonnegative integers.

      (%i1) [belln (x), belln (sqrt(3)), belln (-9)];
      (%o1)        [belln(x), belln(sqrt(3)), belln(- 9)]

  Categories: Sets

Function: cardinality (a)

  Returns the number of distinct elements of the set a.

  `cardinality` ignores redundant elements even when simplification is disabled.

  Examples:

      (%i1) cardinality ({});
      (%o1)                           0
      (%i2) cardinality ({a, a, b, c});
      (%o2)                           3
      (%i3) simp : false;
      (%o3)                         false
      (%i4) cardinality ({a, a, b, c});
      (%o4)                           3

  Categories: Sets

Function: cartesian_product (b_1, ... , b_n)

  Returns a set of lists of the form `[x_1, ..., x_n]`, where x_1, ..., x_n are elements of the sets b_1, ... , b_n, respectively.

  `cartesian_product` complains if any argument is not a literal set.

  Examples:

      (%i1) cartesian_product ({0, 1});
      (%o1)                      {[0], [1]}
      (%i2) cartesian_product ({0, 1}, {0, 1});
      (%o2)           {[0, 0], [0, 1], [1, 0], [1, 1]}
      (%i3) cartesian_product ({x}, {y}, {z});
      (%o3)                      {[x, y, z]}
      (%i4) cartesian_product ({x}, {-1, 0, 1});
      (%o4)              {[x, - 1], [x, 0], [x, 1]}

  Categories: Sets

Function: disjoin (x, a)

  Returns the set a without the member x. If x is not a member of a, return a unchanged.

  `disjoin` complains if a is not a literal set.

  `disjoin(x, a)`, `delete(x, a)`, and `setdifference(a, set(x))` are all equivalent. Of these, `disjoin` is generally faster than the others.

  Examples:

      (%i1) disjoin (a, {a, b, c, d});
      (%o1)                       {b, c, d}
      (%i2) disjoin (a + b, {5, z, a + b, %pi});
      (%o2)                      {5, %pi, z}
      (%i3) disjoin (a - b, {5, z, a + b, %pi});
      (%o3)                  {5, %pi, b + a, z}

  Categories: Sets

Function: disjointp (a, b)

  Returns `true` if and only if the sets a and b are disjoint.

  `disjointp` complains if either a or b is not a literal set.

  Examples:

      (%i1) disjointp ({a, b, c}, {1, 2, 3});
      (%o1)                         true
      (%i2) disjointp ({a, b, 3}, {1, 2, 3});
      (%o2)                         false

  Categories: Sets · Predicate functions

Function: divisors (n)

  Represents the set of divisors of n.

  `divisors(n)` simplifies to a set of integers when n is a nonzero integer. The set of divisors includes the members 1 and n. The divisors of a negative integer are the divisors of its absolute value.

  `divisors` distributes over equations, lists, matrices, and sets.

  Examples:

  We can verify that 28 is a perfect number: the sum of its divisors (except for itself) is 28.

      (%i1) s: divisors(28);
      (%o1)                 {1, 2, 4, 7, 14, 28}
      (%i2) lreduce ("+", args(s)) - 28;
      (%o2)                          28

  `divisors` is a simplifying function. Substituting 8 for `a` in `divisors(a)` yields the divisors without reevaluating `divisors(8)`.

      (%i1) divisors (a);
      (%o1)                      divisors(a)
      (%i2) subst (8, a, %);
      (%o2)                     {1, 2, 4, 8}

  `divisors` distributes over equations, lists, matrices, and sets.

      (%i1) divisors (a = b);
      (%o1)               divisors(a) = divisors(b)
      (%i2) divisors ([a, b, c]);
      (%o2)        [divisors(a), divisors(b), divisors(c)]
      (%i3) divisors (matrix ([a, b], [c, d]));
                        [ divisors(a)  divisors(b) ]
      (%o3)             [                          ]
                        [ divisors(c)  divisors(d) ]
      (%i4) divisors ({a, b, c});
      (%o4)        {divisors(a), divisors(b), divisors(c)}

  Categories: Integers

Function: elementp (x, a)

  Returns `true` if and only if x is a member of the set a.

  `elementp` complains if a is not a literal set.

  Examples:

      (%i1) elementp (sin(1), {sin(1), sin(2), sin(3)});
      (%o1)                         true
      (%i2) elementp (sin(1), {cos(1), cos(2), cos(3)});
      (%o2)                         false

  Categories: Sets · Predicate functions

Function: emptyp (a)

  Return `true` if and only if a is the empty set or the empty list.

  Examples:

      (%i1) map (emptyp, [{}, []]);
      (%o1)                     [true, true]
      (%i2) map (emptyp, [a + b, {{}}, %pi]);
      (%o2)                 [false, false, false]

  Categories: Sets · Predicate functions

Function: equiv_classes (s, F)

  Returns a set of the equivalence classes of the set s with respect to the equivalence relation F.

  F is a function of two variables defined on the Cartesian product of s with s. The return value of F is either `true` or `false`, or an expression expr such that `is(expr)` is either `true` or `false`.

  When F is not an equivalence relation, `equiv_classes` accepts it without complaint, but the result is generally incorrect in that case.

  Examples:

  The equivalence relation is a lambda expression which returns `true` or `false`.

      (%i1) equiv_classes ({1, 1.0, 2, 2.0, 3, 3.0},
                              lambda ([x, y], is (equal (x, y))));
      (%o1)            {{1, 1.0}, {2, 2.0}, {3, 3.0}}

  The equivalence relation is the name of a relational function which `is` evaluates to `true` or `false`.

      (%i1) equiv_classes ({1, 1.0, 2, 2.0, 3, 3.0}, equal);
      (%o1)            {{1, 1.0}, {2, 2.0}, {3, 3.0}}

  The equivalence classes are numbers which differ by a multiple of 3.

      (%i1) equiv_classes ({1, 2, 3, 4, 5, 6, 7},
                           lambda ([x, y], remainder (x - y, 3) = 0));
      (%o1)              {{1, 4, 7}, {2, 5}, {3, 6}}

  Categories: Sets

Function: every
`every` (f, s)
`every` (f, L_1, ..., L_n)

  Returns `true` if the predicate f is `true` for all given arguments.

  Given one set as the second argument, `every(f, s)` returns `true` if `is(f(a_i))` returns `true` for all a_i in s. `every` may or may not evaluate f for all a_i in s. Since sets are unordered, `every` may evaluate `f(a_i)` in any order.

  Given one or more lists as arguments, `every(f, L_1, ..., L_n)` returns `true` if `is(f(x_1, ..., x_n))` returns `true` for all x_1, ..., x_n in L_1, ..., L_n, respectively. `every` may or may not evaluate f for every combination x_1, ..., x_n. `every` evaluates lists in the order of increasing index.

  Given an empty set `{}` or empty lists `[]` as arguments, `every` returns `true`.

  When the global flag `maperror` is `true`, all lists L_1, ..., L_n must have equal lengths. When `maperror` is `false`, list arguments are effectively truncated to the length of the shortest list.

  Return values of the predicate f which evaluate (via `is`) to something other than `true` or `false` are governed by the global flag `prederror`. When `prederror` is `true`, such values are treated as `false`, and the return value from `every` is `false`. When `prederror` is `false`, such values are treated as `unknown`, and the return value from `every` is `unknown`.

  Examples:

  `every` applied to a single set. The predicate is a function of one argument.

      (%i1) every (integerp, {1, 2, 3, 4, 5, 6});
      (%o1)                         true
      (%i2) every (atom, {1, 2, sin(3), 4, 5 + y, 6});
      (%o2)                         false

  `every` applied to two lists. The predicate is a function of two arguments.

      (%i1) every ("=", [a, b, c], [a, b, c]);
      (%o1)                         true
      (%i2) every ("#", [a, b, c], [a, b, c]);
      (%o2)                         false

  Return values of the predicate f which evaluate to something other than `true` or `false` are governed by the global flag `prederror`.

      (%i1) prederror : false;
      (%o1)                         false
      (%i2) map (lambda ([a, b], is (a < b)), [x, y, z],
                         [x^2, y^2, z^2]);
      (%o2)              [unknown, unknown, unknown]
      (%i3) every ("<", [x, y, z], [x^2, y^2, z^2]);
      (%o3)                        unknown
      (%i4) prederror : true;
      (%o4)                         true
      (%i5) every ("<", [x, y, z], [x^2, y^2, z^2]);
      (%o5)                         false

  Categories: Sets

Function: extremal_subset
`extremal_subset` (s, f, max)
`extremal_subset` (s, f, min)

  Returns the subset of s for which the function f takes on maximum or minimum values.

  `extremal_subset(s, f, max)` returns the subset of the set or list s for which the real-valued function f takes on its maximum value.

  `extremal_subset(s, f, min)` returns the subset of the set or list s for which the real-valued function f takes on its minimum value.

  Examples:

      (%i1) extremal_subset ({-2, -1, 0, 1, 2}, abs, max);
      (%o1)                       {- 2, 2}
      (%i2) extremal_subset ({sqrt(2), 1.57, %pi/2}, sin, min);
      (%o2)                       {sqrt(2)}

  Categories: Sets

Function: flatten (expr)

  Collects arguments of subexpressions which have the same operator as expr and constructs an expression from these collected arguments.

  Subexpressions in which the operator is different from the main operator of `expr` are copied without modification, even if they, in turn, contain some subexpressions in which the operator is the same as for `expr`.

  It may be possible for `flatten` to construct expressions in which the number of arguments differs from the declared arguments for an operator; this may provoke an error message from the simplifier or evaluator. `flatten` does not try to detect such situations.

  Expressions with special representations, for example, canonical rational expressions (CRE), cannot be flattened; in such cases, `flatten` returns its argument unchanged.

  Examples:

  Applied to a list, `flatten` gathers all list elements that are lists.

      (%i1) flatten ([a, b, [c, [d, e], f], [[g, h]], i, j]);
      (%o1)            [a, b, c, d, e, f, g, h, i, j]

  Applied to a set, `flatten` gathers all members of set elements that are sets.

      (%i1) flatten ({a, {b}, {{c}}});
      (%o1)                       {a, b, c}
      (%i2) flatten ({a, {[a], {a}}});
      (%o2)                       {a, [a]}

  `flatten` is similar to the effect of declaring the main operator n-ary. However, `flatten` has no effect on subexpressions which have an operator different from the main operator, while an n-ary declaration affects those.

      (%i1) expr: flatten (f (g (f (f (x)))));
      (%o1)                     f(g(f(f(x))))
      (%i2) declare (f, nary);
      (%o2)                         done
      (%i3) ev (expr);
      (%o3)                      f(g(f(x)))

  `flatten` treats subscripted functions the same as any other operator.

      (%i1) flatten (f[5] (f[5] (x, y), z));
      (%o1)                      f (x, y, z)
                                  5

  It may be possible for `flatten` to construct expressions in which the number of arguments differs from the declared arguments for an operator;

      (%i1) 'mod (5, 'mod (7, 4));
      (%o1)                   mod(5, mod(7, 4))
      (%i2) flatten (%);
      (%o2)                     mod(5, 7, 4)
      (%i3) ''%, nouns;
      Wrong number of arguments to mod
       -- an error.  Quitting.  To debug this try debugmode(true);

  Categories: Sets · Lists

Function: full_listify (a)

  Replaces every set operator in a by a list operator, and returns the result. `full_listify` replaces set operators in nested subexpressions, even if the main operator is not `set`.

  `listify` replaces only the main operator.

  Examples:

      (%i1) full_listify ({a, b, {c, {d, e, f}, g}});
      (%o1)               [a, b, [c, [d, e, f], g]]
      (%i2) full_listify (F (G ({a, b, H({c, d, e})})));
      (%o2)              F(G([a, b, H([c, d, e])]))

  Categories: Sets

Function: fullsetify (a)

  When a is a list, replaces the list operator with a set operator, and applies `fullsetify` to each member which is a set. When a is not a list, it is returned unchanged.

  `setify` replaces only the main operator.

  Examples:

  In line `(%o2)`, the argument of `f` isn't converted to a set because the main operator of `f([b])` isn't a list.

      (%i1) fullsetify ([a, [a]]);
      (%o1)                       {a, {a}}
      (%i2) fullsetify ([a, f([b])]);
      (%o2)                      {a, f([b])}

  Categories: Lists

Function: identity (x)

  Returns x for any argument x.

  Examples:

  `identity` may be used as a predicate when the arguments are already Boolean values.

      (%i1) every (identity, [true, true]);
      (%o1)                         true

Function: integer_partitions
`integer_partitions` (n)
`integer_partitions` (n, len)

  Returns integer partitions of n, that is, lists of integers which sum to n.

  `integer_partitions(n)` returns the set of all partitions of the integer n. Each partition is a list sorted from greatest to least.

  `integer_partitions(n, len)` returns all partitions that have length len or less; in this case, zeros are appended to each partition with fewer than len terms to make each partition have exactly len terms. Each partition is a list sorted from greatest to least.

  A list [a_1, ..., a_m] is a partition of a nonnegative integer n when (1) each a_i is a nonzero integer, and (2) a_1 + ... + a_m = n. Thus 0 has no partitions.

  Examples:

      (%i1) integer_partitions (3);
      (%o1)               {[1, 1, 1], [2, 1], [3]}
      (%i2) s: integer_partitions (25)$
      (%i3) cardinality (s);
      (%o3)                         1958
      (%i4) map (lambda ([x], apply ("+", x)), s);
      (%o4)                         {25}
      (%i5) integer_partitions (5, 3);
      (%o5) {[2, 2, 1], [3, 1, 1], [3, 2, 0], [4, 1, 0], [5, 0, 0]}
      (%i6) integer_partitions (5, 2);
      (%o6)               {[3, 2], [4, 1], [5, 0]}

  To find all partitions that satisfy a condition, use the function `subset`; here is an example that finds all partitions of 10 that consist of prime numbers.

      (%i1) s: integer_partitions (10)$
      (%i2) cardinality (s);
      (%o2)                          42
      (%i3) xprimep(x) := integerp(x) and (x > 1) and primep(x)$
      (%i4) subset (s, lambda ([x], every (xprimep, x)));
      (%o4) {[2, 2, 2, 2, 2], [3, 3, 2, 2], [5, 3, 2], [5, 5], [7, 3]}

  Categories: Integers

Function: intersect (a_1, ..., a_n)

  `intersect` is the same as `intersection`, which see.

  Categories: Sets

Function: intersection (a_1, ..., a_n)

  Returns a set containing the elements that are common to the sets a_1 through a_n.

  `intersection` complains if any argument is not a literal set.

  Examples:

      (%i1) S_1 : {a, b, c, d};
      (%o1)                     {a, b, c, d}
      (%i2) S_2 : {d, e, f, g};
      (%o2)                     {d, e, f, g}
      (%i3) S_3 : {c, d, e, f};
      (%o3)                     {c, d, e, f}
      (%i4) S_4 : {u, v, w};
      (%o4)                       {u, v, w}
      (%i5) intersection (S_1, S_2);
      (%o5)                          {d}
      (%i6) intersection (S_2, S_3);
      (%o6)                       {d, e, f}
      (%i7) intersection (S_1, S_2, S_3);
      (%o7)                          {d}
      (%i8) intersection (S_1, S_2, S_3, S_4);
      (%o8)                          {}

  Categories: Sets

Function: kron_delta (x1, x2, …, xp)

  Represents the Kronecker delta function.

  `kron_delta` simplifies to 1 when xi and yj are equal for all pairs of arguments, and it simplifies to 0 when xi and yj are not equal for some pair of arguments. Equality is determined using `is(equal(xi,xj))` and inequality by `is(notequal(xi,xj))`. For exactly one argument, `kron_delta` signals an error.

  Examples:

      (%i1) kron_delta(a,a);
      (%o1)                                  1
      (%i2) kron_delta(a,b,a,b);
      (%o2)                          kron_delta(a, b)
      (%i3) kron_delta(a,a,b,a+1);
      (%o3)                                  0
      (%i4) assume(equal(x,y));
      (%o4)                            [equal(x, y)]
      (%i5) kron_delta(x,y);
      (%o5)                                  1

Function: listify (a)

  Returns a list containing the members of a when a is a set. Otherwise, `listify` returns a.

  `full_listify` replaces all set operators in a by list operators.

  Examples:

      (%i1) listify ({a, b, c, d});
      (%o1)                     [a, b, c, d]
      (%i2) listify (F ({a, b, c, d}));
      (%o2)                    F({a, b, c, d})

  Categories: Sets

Function: makeset (expr, x, s)

  Returns a set with members generated from the expression expr, where x is a list of variables in expr, and s is a set or list of lists. To generate each set member, expr is evaluated with the variables x bound in parallel to a member of s.

  Each member of s must have the same length as x. The list of variables x must be a list of symbols, without subscripts. Even if there is only one symbol, x must be a list of one element, and each member of s must be a list of one element.

  See also `makelist`.

  Examples:

      (%i1) makeset (i/j, [i, j], [[1, a], [2, b], [3, c], [4, d]]);
                                 1  2  3  4
      (%o1)                     {-, -, -, -}
                                 a  b  c  d
      (%i2) S : {x, y, z}$
      (%i3) S3 : cartesian_product (S, S, S);
      (%o3) {[x, x, x], [x, x, y], [x, x, z], [x, y, x], [x, y, y],
      [x, y, z], [x, z, x], [x, z, y], [x, z, z], [y, x, x],
      [y, x, y], [y, x, z], [y, y, x], [y, y, y], [y, y, z],
      [y, z, x], [y, z, y], [y, z, z], [z, x, x], [z, x, y],
      [z, x, z], [z, y, x], [z, y, y], [z, y, z], [z, z, x],
      [z, z, y], [z, z, z]}
      (%i4) makeset (i + j + k, [i, j, k], S3);
      (%o4) {3 x, 3 y, y + 2 x, 2 y + x, 3 z, z + 2 x, z + y + x,
                                             z + 2 y, 2 z + x, 2 z + y}
      (%i5) makeset (sin(x), [x], {[1], [2], [3]});
      (%o5)               {sin(1), sin(2), sin(3)}

  Categories: Sets

Function: moebius (n)

  Represents the Moebius function.

  When n is product of k distinct primes, `moebius(n)` simplifies to (-1)^k; when n = 1, it simplifies to 1; and it simplifies to 0 for all other positive integers.

  `moebius` distributes over equations, lists, matrices, and sets.

  Examples:

      (%i1) moebius (1);
      (%o1)                           1
      (%i2) moebius (2 * 3 * 5);
      (%o2)                          - 1
      (%i3) moebius (11 * 17 * 29 * 31);
      (%o3)                           1
      (%i4) moebius (2^32);
      (%o4)                           0
      (%i5) moebius (n);
      (%o5)                      moebius(n)
      (%i6) moebius (n = 12);
      (%o6)                    moebius(n) = 0
      (%i7) moebius ([11, 11 * 13, 11 * 13 * 15]);
      (%o7)                      [- 1, 1, 1]
      (%i8) moebius (matrix ([11, 12], [13, 14]));
                                 [ - 1  0 ]
      (%o8)                      [        ]
                                 [ - 1  1 ]
      (%i9) moebius ({21, 22, 23, 24});
      (%o9)                      {- 1, 0, 1}

  Categories: Integers

Function: multinomial_coeff
`multinomial_coeff` (a_1, ..., a_n)
`multinomial_coeff` ()

  Returns the multinomial coefficient.

  When each a_k is a nonnegative integer, the multinomial coefficient gives the number of ways of placing `a_1 + ... + a_n` distinct objects into n boxes with a_k elements in the k'th box. In general, `multinomial_coeff (a_1, ..., a_n)` evaluates to `(a_1 + ... + a_n)!/(a_1! ... a_n!)`.

  `multinomial_coeff()` (with no arguments) evaluates to 1.

  `minfactorial` may be able to simplify the value returned by `multinomial_coeff`.

  Examples:

      (%i1) multinomial_coeff (1, 2, x);
                                  (x + 3)!
      (%o1)                       --------
                                    2 x!
      (%i2) minfactorial (%);
                           (x + 1) (x + 2) (x + 3)
      (%o2)                -----------------------
                                      2
      (%i3) multinomial_coeff (-6, 2);
                                   (- 4)!
      (%o3)                       --------
                                  2 (- 6)!
      (%i4) minfactorial (%);
      (%o4)                          10

  Categories: Integers

Function: num_distinct_partitions
`num_distinct_partitions` (n)
`num_distinct_partitions` (n, list)

  Returns the number of distinct integer partitions of n when n is a nonnegative integer. Otherwise, `num_distinct_partitions` returns a noun expression.

  `num_distinct_partitions(n, list)` returns a list of the number of distinct partitions of 1, 2, 3, ..., n.

  A distinct partition of n is a list of distinct positive integers k_1, ..., k_m such that n = k_1 + ... + k_m.

  Examples:

      (%i1) num_distinct_partitions (12);
      (%o1)                          15
      (%i2) num_distinct_partitions (12, list);
      (%o2)      [1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15]
      (%i3) num_distinct_partitions (n);
      (%o3)              num_distinct_partitions(n)

  Categories: Integers

Function: num_partitions
`num_partitions` (n)
`num_partitions` (n, list)

  Returns the number of integer partitions of n when n is a nonnegative integer. Otherwise, `num_partitions` returns a noun expression.

  `num_partitions(n, list)` returns a list of the number of integer partitions of 1, 2, 3, ..., n.

  For a nonnegative integer n, `num_partitions(n)` is equal to `cardinality(integer_partitions(n))`; however, `num_partitions` does not actually construct the set of partitions, so it is much faster.

  Examples:

      (%i1) num_partitions (5) = cardinality (integer_partitions (5));
      (%o1)                         7 = 7
      (%i2) num_partitions (8, list);
      (%o2)            [1, 1, 2, 3, 5, 7, 11, 15, 22]
      (%i3) num_partitions (n);
      (%o3)                   num_partitions(n)

  Categories: Integers

Function: partition_set (a, f)

  Partitions the set a according to the predicate f.

  `partition_set` returns a list of two sets. The first set comprises the elements of a for which f evaluates to `false`, and the second comprises any other elements of a. `partition_set` does not apply `is` to the return value of f.

  `partition_set` complains if a is not a literal set.

  See also `subset`.

  Examples:

      (%i1) partition_set ({2, 7, 1, 8, 2, 8}, evenp);
      (%o1)                   [{1, 7}, {2, 8}]
      (%i2) partition_set ({x, rat(y), rat(y) + z, 1},
                           lambda ([x], ratp(x)));
      (%o2)/R/              [{1, x}, {y, y + z}]

  Categories: Sets

Function: permutations (a)

  Returns a set of all distinct permutations of the members of the list or set a. Each permutation is a list, not a set.

  When a is a list, duplicate members of a are included in the permutations.

  `permutations` complains if a is not a literal list or set.

  See also `random_permutation`.

  Examples:

      (%i1) permutations ([a, a]);
      (%o1)                       {[a, a]}
      (%i2) permutations ([a, a, b]);
      (%o2)           {[a, a, b], [a, b, a], [b, a, a]}

  Categories: Sets · Lists

Function: powerset
`powerset` (a)
`powerset` (a, n)

  Returns the set of all subsets of a, or a subset of that set.

  `powerset(a)` returns the set of all subsets of the set a. `powerset(a)` has `2^cardinality(a)` members.

  `powerset(a, n)` returns the set of all subsets of a that have cardinality n.

  `powerset` complains if a is not a literal set, or if n is not a nonnegative integer.

  Examples:

      (%i1) powerset ({a, b, c});
      (%o1) {{}, {a}, {a, b}, {a, b, c}, {a, c}, {b}, {b, c}, {c}}
      (%i2) powerset ({w, x, y, z}, 4);
      (%o2)                    {{w, x, y, z}}
      (%i3) powerset ({w, x, y, z}, 3);
      (%o3)     {{w, x, y}, {w, x, z}, {w, y, z}, {x, y, z}}
      (%i4) powerset ({w, x, y, z}, 2);
      (%o4)   {{w, x}, {w, y}, {w, z}, {x, y}, {x, z}, {y, z}}
      (%i5) powerset ({w, x, y, z}, 1);
      (%o5)                 {{w}, {x}, {y}, {z}}
      (%i6) powerset ({w, x, y, z}, 0);
      (%o6)                         {{}}

  Categories: Sets

Function: random_permutation (a)

  Returns a random permutation of the set or list a, as constructed by the Knuth shuffle algorithm.

  The return value is a new list, which is distinct from the argument even if all elements happen to be the same. However, the elements of the argument are not copied.

  Examples:

      (%i1) random_permutation ([a, b, c, 1, 2, 3]);
      (%o1)                  [c, 1, 2, 3, a, b]
      (%i2) random_permutation ([a, b, c, 1, 2, 3]);
      (%o2)                  [b, 3, 1, c, a, 2]
      (%i3) random_permutation ({x + 1, y + 2, z + 3});
      (%o3)                 [y + 2, z + 3, x + 1]
      (%i4) random_permutation ({x + 1, y + 2, z + 3});
      (%o4)                 [x + 1, y + 2, z + 3]

  Categories: Sets · Lists

Function: setdifference (a, b)

  Returns a set containing the elements in the set a that are not in the set b.

  `setdifference` complains if either a or b is not a literal set.

  Examples:

      (%i1) S_1 : {a, b, c, x, y, z};
      (%o1)                  {a, b, c, x, y, z}
      (%i2) S_2 : {aa, bb, c, x, y, zz};
      (%o2)                 {aa, bb, c, x, y, zz}
      (%i3) setdifference (S_1, S_2);
      (%o3)                       {a, b, z}
      (%i4) setdifference (S_2, S_1);
      (%o4)                     {aa, bb, zz}
      (%i5) setdifference (S_1, S_1);
      (%o5)                          {}
      (%i6) setdifference (S_1, {});
      (%o6)                  {a, b, c, x, y, z}
      (%i7) setdifference ({}, S_1);
      (%o7)                          {}

  Categories: Sets

Function: setequalp (a, b)

  Returns `true` if sets a and b have the same number of elements and `is(x = y)` is `true` for `x` in the elements of a and `y` in the elements of b, considered in the order determined by `listify`. Otherwise, `setequalp` returns `false`.

  Examples:

      (%i1) setequalp ({1, 2, 3}, {1, 2, 3});
      (%o1)                         true
      (%i2) setequalp ({a, b, c}, {1, 2, 3});
      (%o2)                         false
      (%i3) setequalp ({x^2 - y^2}, {(x + y) * (x - y)});
      (%o3)                         false

  Categories: Sets · Predicate functions

Function: setify (a)

  Constructs a set from the elements of the list a. Duplicate elements of the list a are deleted and the elements are sorted according to the predicate `orderlessp`.

  `setify` complains if a is not a literal list.

  Examples:

      (%i1) setify ([1, 2, 3, a, b, c]);
      (%o1)                  {1, 2, 3, a, b, c}
      (%i2) setify ([a, b, c, a, b, c]);
      (%o2)                       {a, b, c}
      (%i3) setify ([7, 13, 11, 1, 3, 9, 5]);
      (%o3)                {1, 3, 5, 7, 9, 11, 13}

  Categories: Lists

Function: setp (a)

  Returns `true` if and only if a is a Maxima set.

  `setp` returns `true` for unsimplified sets (that is, sets with redundant members) as well as simplified sets.

  `setp` is equivalent to the Maxima function `setp(a) := not atom(a) and op(a) = 'set`.

  Examples:

      (%i1) simp : false;
      (%o1)                         false
      (%i2) {a, a, a};
      (%o2)                       {a, a, a}
      (%i3) setp (%);
      (%o3)                         true

  Categories: Sets · Predicate functions

Function: set_partitions
`set_partitions` (a)
`set_partitions` (a, n)

  Returns the set of all partitions of a, or a subset of that set.

  `set_partitions(a, n)` returns a set of all decompositions of a into n nonempty disjoint subsets.

  `set_partitions(a)` returns the set of all partitions.

  `stirling2` returns the cardinality of the set of partitions of a set.

  A set of sets P is a partition of a set S when

  1. each member of P is a nonempty set,
  2. distinct members of P are disjoint,
  3. the union of the members of P equals S.

  Examples:

  The empty set is a partition of itself, the conditions 1 and 2 being vacuously true.

      (%i1) set_partitions ({});
      (%o1)                         {{}}

  The cardinality of the set of partitions of a set can be found using `stirling2`.

      (%i1) s: {0, 1, 2, 3, 4, 5}$
      (%i2) p: set_partitions (s, 3)$
      (%i3) cardinality(p) = stirling2 (6, 3);
      (%o3)                        90 = 90

  Each member of `p` should have n = 3 members; let's check.

      (%i1) s: {0, 1, 2, 3, 4, 5}$
      (%i2) p: set_partitions (s, 3)$
      (%i3) map (cardinality, p);
      (%o3)                          {3}

  Finally, for each member of `p`, the union of its members should equal `s`; again let's check.

      (%i1) s: {0, 1, 2, 3, 4, 5}$
      (%i2) p: set_partitions (s, 3)$
      (%i3) map (lambda ([x], apply (union, listify (x))), p);
      (%o3)                 {{0, 1, 2, 3, 4, 5}}

  Categories: Sets

Function: some
`some` (f, a)
`some` (f, L_1, ..., L_n)

  Returns `true` if the predicate f is `true` for one or more given arguments.

  Given one set as the second argument, `some(f, s)` returns `true` if `is(f(a_i))` returns `true` for one or more a_i in s. `some` may or may not evaluate f for all a_i in s. Since sets are unordered, `some` may evaluate `f(a_i)` in any order.

  Given one or more lists as arguments, `some(f, L_1, ..., L_n)` returns `true` if `is(f(x_1, ..., x_n))` returns `true` for one or more x_1, ..., x_n in L_1, ..., L_n, respectively. `some` may or may not evaluate f for some combinations x_1, ..., x_n. `some` evaluates lists in the order of increasing index.

  Given an empty set `{}` or empty lists `[]` as arguments, `some` returns `false`.

  When the global flag `maperror` is `true`, all lists L_1, ..., L_n must have equal lengths. When `maperror` is `false`, list arguments are effectively truncated to the length of the shortest list.

  Return values of the predicate f which evaluate (via `is`) to something other than `true` or `false` are governed by the global flag `prederror`. When `prederror` is `true`, such values are treated as `false`. When `prederror` is `false`, such values are treated as `unknown`.

  Examples:

  `some` applied to a single set. The predicate is a function of one argument.

      (%i1) some (integerp, {1, 2, 3, 4, 5, 6});
      (%o1)                         true
      (%i2) some (atom, {1, 2, sin(3), 4, 5 + y, 6});
      (%o2)                         true

  `some` applied to two lists. The predicate is a function of two arguments.

      (%i1) some ("=", [a, b, c], [a, b, c]);
      (%o1)                         true
      (%i2) some ("#", [a, b, c], [a, b, c]);
      (%o2)                         false

  Return values of the predicate f which evaluate to something other than `true` or `false` are governed by the global flag `prederror`.

      (%i1) prederror : false;
      (%o1)                         false
      (%i2) map (lambda ([a, b], is (a < b)), [x, y, z],
                 [x^2, y^2, z^2]);
      (%o2)              [unknown, unknown, unknown]
      (%i3) some ("<", [x, y, z], [x^2, y^2, z^2]);
      (%o3)                        unknown
      (%i4) some ("<", [x, y, z], [x^2, y^2, z + 1]);
      (%o4)                         true
      (%i5) prederror : true;
      (%o5)                         true
      (%i6) some ("<", [x, y, z], [x^2, y^2, z^2]);
      (%o6)                         false
      (%i7) some ("<", [x, y, z], [x^2, y^2, z + 1]);
      (%o7)                         true

  Categories: Sets · Lists

Function: stirling1 (n, m)

  Represents the Stirling number of the first kind.

  When n and m are nonnegative integers, the magnitude of `stirling1 (n, m)` is the number of permutations of a set with n members that have m cycles.

  `stirling1` is a simplifying function. Maxima knows the following identities:

  1. stirling1(1,k) = kron_delta(1,k), k >= 0,(see http://dlmf.nist.gov/26.8.E2)
  2. stirling1(n,n) = 1, n >= 0 (see http://dlmf.nist.gov/26.8.E1)
  3. stirling1(n,n-1) = -binomial(n,2), n >= 1, (see http://dlmf.nist.gov/26.8.E16)
  4. stirling1(n,0) = kron_delta(n,0), n >=0 (see http://dlmf.nist.gov/26.8.E14 and http://dlmf.nist.gov/26.8.E1)
  5. stirling1(n,1) =(-1)^(n-1) (n-1)!, n >= 1 (see http://dlmf.nist.gov/26.8.E14)
  6. stirling1(n,k) = 0, n >= 0 and k > n.

  These identities are applied when the arguments are literal integers or symbols declared as integers, and the first argument is nonnegative. `stirling1` does not simplify for non-integer arguments.

  Examples:

      (%i1) declare (n, integer)$
      (%i2) assume (n >= 0)$
      (%i3) stirling1 (n, n);
      (%o3)                           1

  Categories: Integers

Function: stirling2 (n, m)

  Represents the Stirling number of the second kind.

  When n and m are nonnegative integers, `stirling2 (n, m)` is the number of ways a set with cardinality n can be partitioned into m disjoint subsets.

  `stirling2` is a simplifying function. Maxima knows the following identities.

  1. stirling2(n,0) = 1, n >= 1 (see http://dlmf.nist.gov/26.8.E17 and stirling2(0,0) = 1)
  2. stirling2(n,n) = 1, n >= 0, (see http://dlmf.nist.gov/26.8.E4)
  3. stirling2(n,1) = 1, n >= 1, (see http://dlmf.nist.gov/26.8.E17 and stirling2(0,1) = 0)
  4. stirling2(n,2) = 2^(n-1) -1, n >= 1, (see http://dlmf.nist.gov/26.8.E17)
  5. stirling2(n,n-1) = binomial(n,2), n>= 1 (see http://dlmf.nist.gov/26.8.E16)
  6. stirling2(n,k) = 0, n >= 0 and k > n.

  These identities are applied when the arguments are literal integers or symbols declared as integers, and the first argument is nonnegative. `stirling2` does not simplify for non-integer arguments.

  Examples:

      (%i1) declare (n, integer)$
      (%i2) assume (n >= 0)$
      (%i3) stirling2 (n, n);
      (%o3)                           1

  `stirling2` does not simplify for non-integer arguments.

      (%i1) stirling2 (%pi, %pi);
      (%o1)                  stirling2(%pi, %pi)

  Categories: Integers

Function: subset (a, f)

  Returns the subset of the set a that satisfies the predicate f.

  `subset` returns a set which comprises the elements of a for which f returns anything other than `false`. `subset` does not apply `is` to the return value of f.

  `subset` complains if a is not a literal set.

  See also `partition_set`.

  Examples:

      (%i1) subset ({1, 2, x, x + y, z, x + y + z}, atom);
      (%o1)                     {1, 2, x, z}
      (%i2) subset ({1, 2, 7, 8, 9, 14}, evenp);
      (%o2)                      {2, 8, 14}

  Categories: Sets

Function: subsetp (a, b)

  Returns `true` if and only if the set a is a subset of b.

  `subsetp` complains if either a or b is not a literal set.

  Examples:

      (%i1) subsetp ({1, 2, 3}, {a, 1, b, 2, c, 3});
      (%o1)                         true
      (%i2) subsetp ({a, 1, b, 2, c, 3}, {1, 2, 3});
      (%o2)                         false

  Categories: Sets · Predicate functions

Function: symmdifference (a_1, …, a_n)

  Returns the symmetric difference of sets a_1, …, a_n.

  Given two arguments, `symmdifference (a, b)` is the same as `union (setdifference (a, b), setdifference (b, a))`.

  `symmdifference` complains if any argument is not a literal set.

  Examples:

      (%i1) S_1 : {a, b, c};
      (%o1)                       {a, b, c}
      (%i2) S_2 : {1, b, c};
      (%o2)                       {1, b, c}
      (%i3) S_3 : {a, b, z};
      (%o3)                       {a, b, z}
      (%i4) symmdifference ();
      (%o4)                          {}
      (%i5) symmdifference (S_1);
      (%o5)                       {a, b, c}
      (%i6) symmdifference (S_1, S_2);
      (%o6)                        {1, a}
      (%i7) symmdifference (S_1, S_2, S_3);
      (%o7)                        {1, b, z}
      (%i8) symmdifference ({}, S_1, S_2, S_3);
      (%o8)                        {1,b, z}

  Categories: Sets

Function: union (a_1, ..., a_n)

  Returns the union of the sets a_1 through a_n.

  `union()` (with no arguments) returns the empty set.

  `union` complains if any argument is not a literal set.

  Examples:

      (%i1) S_1 : {a, b, c + d, %e};
      (%o1)                   {%e, a, b, d + c}
      (%i2) S_2 : {%pi, %i, %e, c + d};
      (%o2)                 {%e, %i, %pi, d + c}
      (%i3) S_3 : {17, 29, 1729, %pi, %i};
      (%o3)                {17, 29, 1729, %i, %pi}
      (%i4) union ();
      (%o4)                          {}
      (%i5) union (S_1);
      (%o5)                   {%e, a, b, d + c}
      (%i6) union (S_1, S_2);
      (%o6)              {%e, %i, %pi, a, b, d + c}
      (%i7) union (S_1, S_2, S_3);
      (%o7)       {17, 29, 1729, %e, %i, %pi, a, b, d + c}
      (%i8) union ({}, S_1, S_2, S_3);
      (%o8)       {17, 29, 1729, %e, %i, %pi, a, b, d + c}

  Categories: Sets

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# 36. Function Definition

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## 36.1 Introduction to Function Definition

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## 36.2 Function

Categories: Function definition · Programming

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### 36.2.1 Ordinary functions

To define a function in Maxima you use the `:=` operator. E.g.

    f(x) := sin(x)

defines a function `f`. Anonymous functions may also be created using `lambda`. For example

    lambda ([i, j], ...)

can be used instead of `f` where

    f(i,j) := block ([], ...);
    map (lambda ([i], i+1), l)

would return a list with 1 added to each term.

You may also define a function with a variable number of arguments, by having a final argument which is assigned to a list of the extra arguments:

    (%i1) f ([u]) := u;
    (%o1)                      f([u]) := u
    (%i2) f (1, 2, 3, 4);
    (%o2)                     [1, 2, 3, 4]
    (%i3) f (a, b, [u]) := [a, b, u];
    (%o3)               f(a, b, [u]) := [a, b, u]
    (%i4) f (1, 2, 3, 4, 5, 6);
    (%o4)                 [1, 2, [3, 4, 5, 6]]

The right hand side of a function is an expression. Thus if you want a sequence of expressions, you do

    f(x) := (expr1, expr2, ...., exprn);

and the value of exprn is what is returned by the function.

If you wish to make a `return` from some expression inside the function then you must use `block` and `return`.

    block ([], expr1, ..., if (a > 10) then return(a), ..., exprn)

is itself an expression, and so could take the place of the right hand side of a function definition. Here it may happen that the return happens earlier than the last expression.

The first `[]` in the block, may contain a list of variables and variable assignments, such as `[a: 3, b, c: []]`, which would cause the three variables `a`,`b`,and `c` to not refer to their global values, but rather have these special values for as long as the code executes inside the `block`, or inside functions called from inside the `block`. This is called dynamic binding, since the variables last from the start of the block to the time it exits. Once you return from the `block`, or throw out of it, the old values (if any) of the variables will be restored. It is certainly a good idea to protect your variables in this way. Note that the assignments in the block variables, are done in parallel. This means, that if you had used `c: a` in the above, the value of `c` would have been the value of `a` at the time you just entered the block, but before `a` was bound. Thus doing something like

    block ([a: a], expr1, ... a: a+3, ..., exprn)

will protect the external value of `a` from being altered, but would let you access what that value was. Thus the right hand side of the assignments, is evaluated in the entering context, before any binding occurs. Using just `block ([x], ...)` would cause the `x` to have itself as value, just as if it would have if you entered a fresh Maxima session.

The actual arguments to a function are treated in exactly same way as the variables in a block. Thus in

    f(x) := (expr1, ..., exprn);

and

    f(1);

we would have a similar context for evaluation of the expressions as if we had done

    block ([x: 1], expr1, ..., exprn)

Inside functions, when the right hand side of a definition, may be computed at runtime, it is useful to use `define` and possibly `buildq`.

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### 36.2.2 Array functions

An array function stores the function value the first time it is called with a given argument, and returns the stored value, without recomputing it, when that same argument is given. Such a function is often called a memoizing function.

Array function names are appended to the global list `arrays` (not the global list `functions`). `arrayinfo` returns the list of arguments for which there are stored values, and `listarray` returns the stored values. `dispfun` and `fundef` return the array function definition.

`arraymake` constructs an array function call, analogous to `funmake` for ordinary functions. `arrayapply` applies an array function to its arguments, analogous to `apply` for ordinary functions. There is nothing exactly analogous to `map` for array functions, although `map(lambda([x], a[x]), L)` or `makelist(a[x], x, L)`, where L is a list, are not too far off the mark.

`remarray` removes an array function definition (including any stored function values), analogous to `remfunction` for ordinary functions.

`kill(a[x])` removes the value of the array function a stored for the argument x; the next time a is called with argument x, the function value is recomputed. However, there is no way to remove all of the stored values at once, except for `kill(a)` or `remarray(a)`, which also remove the function definition.

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## 36.3 Macros

Function: buildq (L, expr)

  Substitutes variables named by the list L into the expression expr, in parallel, without evaluating expr. The resulting expression is simplified, but not evaluated, after `buildq` carries out the substitution.

  The elements of L are symbols or assignment expressions `symbol: value`, evaluated in parallel. That is, the binding of a variable on the right-hand side of an assignment is the binding of that variable in the context from which `buildq` was called, not the binding of that variable in the variable list L. If some variable in L is not given an explicit assignment, its binding in `buildq` is the same as in the context from which `buildq` was called.

  Then the variables named by L are substituted into expr in parallel. That is, the substitution for every variable is determined before any substitution is made, so the substitution for one variable has no effect on any other.

  If any variable x appears as `splice (x)` in expr, then x must be bound to a list, and the list is spliced (interpolated) into expr instead of substituted.

  Any variables in expr not appearing in L are carried into the result verbatim, even if they have bindings in the context from which `buildq` was called.

  Examples

  `a` is explicitly bound to `x`, while `b` has the same binding (namely 29) as in the calling context, and `c` is carried through verbatim. The resulting expression is not evaluated until the explicit evaluation `''%`.

      (%i1) (a: 17, b: 29, c: 1729)$
      (%i2) buildq ([a: x, b], a + b + c);
      (%o2)                      x + c + 29
      (%i3) ''%;
      (%o3)                       x + 1758

  `e` is bound to a list, which appears as such in the arguments of `foo`, and interpolated into the arguments of `bar`.

      (%i1) buildq ([e: [a, b, c]], foo (x, e, y));
      (%o1)                 foo(x, [a, b, c], y)
      (%i2) buildq ([e: [a, b, c]], bar (x, splice (e), y));
      (%o2)                  bar(x, a, b, c, y)

  The result is simplified after substitution. If simplification were applied before substitution, these two results would be the same.

      (%i1) buildq ([e: [a, b, c]], splice (e) + splice (e));
      (%o1)                    2 c + 2 b + 2 a
      (%i2) buildq ([e: [a, b, c]], 2 * splice (e));
      (%o2)                        2 a b c

  The variables in L are bound in parallel; if bound sequentially, the first result would be `foo (b, b)`. Substitutions are carried out in parallel; compare the second result with the result of `subst`, which carries out substitutions sequentially.

      (%i1) buildq ([a: b, b: a], foo (a, b));
      (%o1)                       foo(b, a)
      (%i2) buildq ([u: v, v: w, w: x, x: y, y: z, z: u],
                    bar (u, v, w, x, y, z));
      (%o2)                 bar(v, w, x, y, z, u)
      (%i3) subst ([u=v, v=w, w=x, x=y, y=z, z=u],
                   bar (u, v, w, x, y, z));
      (%o3)                 bar(u, u, u, u, u, u)

  Construct a list of equations with some variables or expressions on the left-hand side and their values on the right-hand side. `macroexpand` shows the expression returned by `show_values`.

      (%i1) show_values ([L]) ::= buildq ([L], map ("=", 'L, L));
      (%o1)   show_values([L]) ::= buildq([L], map("=", 'L, L))
      (%i2) (a: 17, b: 29, c: 1729)$
      (%i3) show_values (a, b, c - a - b);
      (%o3)          [a = 17, b = 29, c - b - a = 1683]
      (%i4) macroexpand (show_values (a, b, c - a - b));
      (%o4)    map(=, '([a, b, c - b - a]), [a, b, c - b - a])

  Given a function of several arguments, create another function for which some of the arguments are fixed.

      (%i1) curry (f, [a]) :=
              buildq ([f, a], lambda ([[x]], apply (f, append (a, x))))$
      (%i2) by3 : curry ("*", 3);
      (%o2)        lambda([[x]], apply(*, append([3], x)))
      (%i3) by3 (a + b);
      (%o3)                       3 (b + a)

  Categories: Function definition

Function: macroexpand (expr)

  Returns the macro expansion of expr without evaluating it, when `expr` is a macro function call. Otherwise, `macroexpand` returns expr.

  If the expansion of expr yields another macro function call, that macro function call is also expanded.

  `macroexpand` quotes its argument. However, if the expansion of a macro function call has side effects, those side effects are executed.

  See also `::=`, `macros`, and `macroexpand1`..

  Examples

      (%i1) g (x) ::= x / 99;
                                          x
      (%o1)                      g(x) ::= --
                                          99
      (%i2) h (x) ::= buildq ([x], g (x - a));
      (%o2)            h(x) ::= buildq([x], g(x - a))
      (%i3) a: 1234;
      (%o3)                         1234
      (%i4) macroexpand (h (y));
                                    y - a
      (%o4)                         -----
                                     99
      (%i5) h (y);
                                  y - 1234
      (%o5)                       --------
                                     99

  Categories: Function application

Function: macroexpand1 (expr)

  Returns the macro expansion of expr without evaluating it, when `expr` is a macro function call. Otherwise, `macroexpand1` returns expr.

  `macroexpand1` quotes its argument. However, if the expansion of a macro function call has side effects, those side effects are executed.

  If the expansion of expr yields another macro function call, that macro function call is not expanded.

  See also `::=`, `macros`, and `macroexpand`.

  Examples

      (%i1) g (x) ::= x / 99;
                                          x
      (%o1)                      g(x) ::= --
                                          99
      (%i2) h (x) ::= buildq ([x], g (x - a));
      (%o2)            h(x) ::= buildq([x], g(x - a))
      (%i3) a: 1234;
      (%o3)                         1234
      (%i4) macroexpand1 (h (y));
      (%o4)                       g(y - a)
      (%i5) h (y);
                                  y - 1234
      (%o5)                       --------
                                     99

  Categories: Function application

Global variable: macros

  Default value: `[]`

  `macros` is the list of user-defined macro functions. The macro function definition operator `::=` puts a new macro function onto this list, and `kill`, `remove`, and `remfunction` remove macro functions from the list.

  See also `infolists`.

  Categories: Function definition · Global variables

Function: splice (a)

  Splices (interpolates) the list named by the atom a into an expression, but only if `splice` appears within `buildq`; otherwise, `splice` is treated as an undefined function. If appearing within `buildq` as a alone (without `splice`), a is substituted (not interpolated) as a list into the result. The argument of `splice` can only be an atom; it cannot be a literal list or an expression which yields a list.

  Typically `splice` supplies the arguments for a function or operator. For a function `f`, the expression `f (splice (a))` within `buildq` expands to `f (a[1], a[2], a[3], ...)`. For an operator `o`, the expression `"o" (splice (a))` within `buildq` expands to `"o" (a[1], a[2], a[3], ...)`, where `o` may be any type of operator (typically one which takes multiple arguments). Note that the operator must be enclosed in double quotes `"`.

  Examples

      (%i1) buildq ([x: [1, %pi, z - y]], foo (splice (x)) / length (x));
                             foo(1, %pi, z - y)
      (%o1)                -----------------------
                           length([1, %pi, z - y])
      (%i2) buildq ([x: [1, %pi]], "/" (splice (x)));
                                      1
      (%o2)                          ---
                                     %pi
      (%i3) matchfix ("<>", "<>");
      (%o3)                          <>
      (%i4) buildq ([x: [1, %pi, z - y]], "<>" (splice (x)));
      (%o4)                   <>1, %pi, z - y<>

  Categories: Function definition

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## 36.4 Functions and Variables for Function Definition

Function: apply (F, [x_1, …, x_n])

  Constructs and evaluates an expression `F(arg_1, ..., arg_n)`.

  `apply` does not attempt to distinguish array functions from ordinary functions; when F is the name of an array function, `apply` evaluates `F(...)` (that is, a function call with parentheses instead of square brackets). `arrayapply` evaluates a function call with square brackets in this case.

  See also `funmake` and `args`.

  Examples:

  `apply` evaluates its arguments. In this example, `min` is applied to the value of `L`.

      (%i1) L : [1, 5, -10.2, 4, 3];
      (%o1)                 [1, 5, - 10.2, 4, 3]
      (%i2) apply (min, L);
      (%o2)                        - 10.2

  `apply` evaluates arguments, even if the function F quotes them.

      (%i1) F (x) := x / 1729;
                                         x
      (%o1)                     F(x) := ----
                                        1729
      (%i2) fname : F;
      (%o2)                           F
      (%i3) dispfun (F);
                                         x
      (%t3)                     F(x) := ----
                                        1729

      (%o3)                         [%t3]
      (%i4) dispfun (fname);
      fundef: no such function: fname
       -- an error. To debug this try: debugmode(true);
      (%i5) apply (dispfun, [fname]);
                                         x
      (%t5)                     F(x) := ----
                                        1729

      (%o5)                         [%t5]

  `apply` evaluates the function name F. Single quote `'` defeats evaluation. `demoivre` is the name of a global variable and also a function.

      (%i1) demoivre;
      (%o1)                         false
      (%i2) demoivre (exp (%i * x));
      (%o2)                  %i sin(x) + cos(x)
      (%i3) apply (demoivre, [exp (%i * x)]);
      apply: found false where a function was expected.
       -- an error. To debug this try: debugmode(true);
      (%i4) apply ('demoivre, [exp (%i * x)]);
      (%o4)                  %i sin(x) + cos(x)

  How to convert a nested list into a matrix:

      (%i1) a:[[1,2],[3,4]];
      (%o1)                   [[1, 2], [3, 4]]
      (%i2) apply(matrix,a);
                                  [ 1  2 ]
      (%o2)                       [      ]
                                  [ 3  4 ]

  Categories: Function application

Function: block
`block` ([v_1, …, v_m], expr_1, …, expr_n)
`block` (expr_1, …, expr_n)

  The function `block` allows to make the variables v_1, …, v_m to be local for a sequence of commands. If these variables are already bound `block` saves the current values of the variables v_1, …, v_m (if any) upon entry to the block, then unbinds the variables so that they evaluate to themselves; The local variables may be bound to arbitrary values within the block but when the block is exited the saved values are restored, and the values assigned within the block are lost.

  If there is no need to define local variables then the list at the beginning of the `block` command may be omitted. In this case if neither `return` nor `go` are used `block` behaves similar to the following construct:

      ( expr_1, expr_2,... , expr_n );

  expr_1, …, expr_n will be evaluated in sequence and the value of the last expression will be returned. The sequence can be modified by the `go`, `throw`, and `return` functions. The last expression is expr_n unless `return` or an expression containing `throw` is evaluated.

  The declaration `local(v_1, ..., v_m)` within `block` saves the properties associated with the symbols v_1, …, v_m, removes any properties before evaluating other expressions, and restores any saved properties on exit from the block. Some declarations are implemented as properties of a symbol, including `:=`, `array`, `dependencies`, `atvalue`, `matchdeclare`, `atomgrad`, `constant`, `nonscalar`, `assume`, and some others. The effect of `local` is to make such declarations effective only within the block; otherwise declarations within a block are actually global declarations.

  `block` may appear within another `block`. Local variables are established each time a new `block` is evaluated. Local variables appear to be global to any enclosed blocks. If a variable is non-local in a block, its value is the value most recently assigned by an enclosing block, if any, otherwise, it is the value of the variable in the global environment. This policy may coincide with the usual understanding of "dynamic scope".

  The value of the block is the value of the last statement or the value of the argument to the function `return` which may be used to exit explicitly from the block. The function `go` may be used to transfer control to the statement of the block that is tagged with the argument to `go`. To tag a statement, precede it by an atomic argument as another statement in the block. For example: `block ([x], x:1, loop, x: x+1, ..., go(loop), ...)`. The argument to `go` must be the name of a tag appearing within the block. One cannot use `go` to transfer to a tag in a block other than the one containing the `go`.

  Blocks typically appear on the right side of a function definition but can be used in other places as well.

  See also `return` and `go`.

  Categories: Expressions · Programming

Function: break (expr_1, …, expr_n)

  Evaluates and prints expr_1, …, expr_n and then causes a Maxima break at which point the user can examine and change his environment. Upon typing `exit;` the computation resumes.

  Categories: Debugging

Function: catch (expr_1, …, expr_n)

  Evaluates expr_1, …, expr_n one by one; if any leads to the evaluation of an expression of the form `throw (arg)`, then the value of the `catch` is the value of `throw (arg)`, and no further expressions are evaluated. This "non-local return" thus goes through any depth of nesting to the nearest enclosing `catch`. If there is no `catch` enclosing a `throw`, an error message is printed.

  If the evaluation of the arguments does not lead to the evaluation of any `throw` then the value of `catch` is the value of expr_n.

      (%i1) lambda ([x], if x < 0 then throw(x) else f(x))$
      (%i2) g(l) := catch (map (''%, l))$
      (%i3) g ([1, 2, 3, 7]);
      (%o3)               [f(1), f(2), f(3), f(7)]
      (%i4) g ([1, 2, -3, 7]);
      (%o4)                          - 3

  The function `g` returns a list of `f` of each element of `l` if `l` consists only of non-negative numbers; otherwise, `g` "catches" the first negative element of `l` and "throws" it up.

  Categories: Programming

Function: compfile
`compfile` (filename, f_1, …, f_n)
`compfile` (filename, functions)
`compfile` (filename, all)

  Translates Maxima functions into Lisp and writes the translated code into the file filename.

  `compfile(filename, f_1, ..., f_n)` translates the specified functions. `compfile (filename, functions)` and `compfile (filename, all)` translate all user-defined functions.

  The Lisp translations are not evaluated, nor is the output file processed by the Lisp compiler. `translate` creates and evaluates Lisp translations. `compile_file` translates Maxima into Lisp, and then executes the Lisp compiler.

  See also `translate`, `translate_file`, and `compile_file`.

  Categories: Translation and compilation

Function: compile
`compile` (f_1, …, f_n)
`compile` (functions)
`compile` (all)

  Translates Maxima functions f_1, …, f_n into Lisp, evaluates the Lisp translations, and calls the Lisp function `COMPILE` on each translated function. `compile` returns a list of the names of the compiled functions.

  `compile (all)` or `compile (functions)` compiles all user-defined functions.

  `compile` quotes its arguments; the quote-quote operator `''` defeats quotation.

  Compiling a function to native code can mean a big increase in speed and might cause the memory footprint to reduce drastically. Code tends to be especially effective when the flexibility it needs to provide is limited. If compilation doesn't provide the speed that is needed a few ways to limit the code's functionality are the following:

  - If the function accesses global variables the complexity of the function can be drastically be reduced by limiting these variables to one data type, for example using `mode_declare` or a statement like the following one: `put(x_1, bigfloat, numerical_type)`
  - The compiler might warn about undeclared variables if text could either be a named option to a command or (if they are assigned a value to) the name of a variable. Prepending the option with a single quote `'` tells the compiler that the text is meant as an option.

  Categories: Translation and compilation

Function: define
`define` (f(x_1, …, x_n), expr)
`define` (f[x_1, …, x_n], expr)
`define` (f[x_1, …, x_n](y_1, …, y_m), expr)
`define` (funmake (f, [x_1, …, x_n]), expr)
`define` (arraymake (f, [x_1, …, x_n]), expr)
`define` (ev (expr_1), expr_2)

  Defines a function named f with arguments x_1, …, x_n and function body expr. `define` always evaluates its second argument (unless explicitly quoted). The function so defined may be an ordinary Maxima function (with arguments enclosed in parentheses) or an array function (with arguments enclosed in square brackets).

  When the last or only function argument x_n is a list of one element, the function defined by `define` accepts a variable number of arguments. Actual arguments are assigned one-to-one to formal arguments x_1, …, x_(n - 1), and any further actual arguments, if present, are assigned to x_n as a list.

  When the first argument of `define` is an expression of the form `f(x_1, ..., x_n)` or `f[x_1, ..., x_n]`, the function arguments are evaluated but f is not evaluated, even if there is already a function or variable by that name.

  When the first argument is an expression with operator `funmake`, `arraymake`, or `ev`, the first argument is evaluated; this allows for the function name to be computed, as well as the body.

  All function definitions appear in the same namespace; defining a function `f` within another function `g` does not automatically limit the scope of `f` to `g`. However, `local(f)` makes the definition of function `f` effective only within the block or other compound expression in which `local` appears.

  If some formal argument x_k is a quoted symbol (after evaluation), the function defined by `define` does not evaluate the corresponding actual argument. Otherwise all actual arguments are evaluated.

  See also `:=` and `::=`.

  Examples:

  `define` always evaluates its second argument (unless explicitly quoted).

      (%i1) expr : cos(y) - sin(x);
      (%o1)                    cos(y) - sin(x)
      (%i2) define (F1 (x, y), expr);
      (%o2)              F1(x, y) := cos(y) - sin(x)
      (%i3) F1 (a, b);
      (%o3)                    cos(b) - sin(a)
      (%i4) F2 (x, y) := expr;
      (%o4)                   F2(x, y) := expr
      (%i5) F2 (a, b);
      (%o5)                    cos(y) - sin(x)

  The function defined by `define` may be an ordinary Maxima function or an array function.

      (%i1) define (G1 (x, y), x.y - y.x);
      (%o1)               G1(x, y) := x . y - y . x
      (%i2) define (G2 [x, y], x.y - y.x);
      (%o2)                G2     := x . y - y . x
                             x, y

  When the last or only function argument x_n is a list of one element, the function defined by `define` accepts a variable number of arguments.

      (%i1) define (H ([L]), '(apply ("+", L)));
      (%o1)                H([L]) := apply("+", L)
      (%i2) H (a, b, c);
      (%o2)                       c + b + a

  When the first argument is an expression with operator `funmake`, `arraymake`, or `ev`, the first argument is evaluated.

      (%i1) [F : I, u : x];
      (%o1)                        [I, x]
      (%i2) funmake (F, [u]);
      (%o2)                         I(x)
      (%i3) define (funmake (F, [u]), cos(u) + 1);
      (%o3)                  I(x) := cos(x) + 1
      (%i4) define (arraymake (F, [u]), cos(u) + 1);
      (%o4)                   I  := cos(x) + 1
                               x
      (%i5) define (foo (x, y), bar (y, x));
      (%o5)                foo(x, y) := bar(y, x)
      (%i6) define (ev (foo (x, y)), sin(x) - cos(y));
      (%o6)             bar(y, x) := sin(x) - cos(y)

  Categories: Function definition

Function: define_variable (name, default_value, mode)

  Introduces a global variable into the Maxima environment. `define_variable` is useful in user-written packages, which are often translated or compiled.

  `define_variable` carries out the following steps:

  1. `mode_declare (name, mode)` declares the mode of name to the translator. See `mode_declare` for a list of the possible modes.
  2. If the variable is unbound, default_value is assigned to name.
  3. Associates name with a test function to ensure that name is only assigned values of the declared mode.

  The `value_check` property can be assigned to any variable which has been defined via `define_variable` with a mode other than `any`. The `value_check` property is a lambda expression or the name of a function of one variable, which is called when an attempt is made to assign a value to the variable. The argument of the `value_check` function is the would-be assigned value.

  `define_variable` evaluates `default_value`, and quotes `name` and `mode`. `define_variable` returns the current value of `name`, which is `default_value` if `name` was unbound before, and otherwise it is the previous value of `name`.

  Examples:

  `foo` is a Boolean variable, with the initial value `true`.

      (%i1) define_variable (foo, true, boolean);
      (%o1)                         true
      (%i2) foo;
      (%o2)                         true
      (%i3) foo: false;
      (%o3)                         false
      (%i4) foo: %pi;
      translator: foo was declared with mode boolean
                                                , but it has value: %pi
       -- an error. To debug this try: debugmode(true);
      (%i5) foo;
      (%o5)                         false

  `bar` is an integer variable, which must be prime.

      (%i1) define_variable (bar, 2, integer);
      (%o1)                           2
      (%i2) qput (bar, prime_test, value_check);
      (%o2)                      prime_test
      (%i3) prime_test (y) := if not primep(y) then
                                 error (y, "is not prime.");
      (%o3) prime_test(y) := if not primep(y)
                                         then error(y, "is not prime.")
      (%i4) bar: 1439;
      (%o4)                         1439
      (%i5) bar: 1440;
      1440 is not prime.
       -- an error. To debug this try: debugmode(true);
      (%i6) bar;
      (%o6)                         1439

  `baz_quux` is a variable which cannot be assigned a value. The mode `any_check` is like `any`, but `any_check` enables the `value_check` mechanism, and `any` does not.

      (%i1) define_variable (baz_quux, 'baz_quux, any_check);
      (%o1)                       baz_quux
      (%i2) F: lambda ([y], if y # 'baz_quux then
                       error ("Cannot assign to `baz_quux'."));
      (%o2) lambda([y], if y # 'baz_quux
                              then error(Cannot assign to `baz_quux'.))
      (%i3) qput (baz_quux, ''F, value_check);
      (%o3) lambda([y], if y # 'baz_quux
                              then error(Cannot assign to `baz_quux'.))
      (%i4) baz_quux: 'baz_quux;
      (%o4)                       baz_quux
      (%i5) baz_quux: sqrt(2);
      Cannot assign to `baz_quux'.
       -- an error. To debug this try: debugmode(true);
      (%i6) baz_quux;
      (%o6)                       baz_quux

  Categories: Translation and compilation

Function: dispfun
`dispfun` (f_1, …, f_n)
`dispfun` (all)

  Displays the definition of the user-defined functions f_1, …, f_n. Each argument may be the name of a macro (defined with `::=`), an ordinary function (defined with `:=` or `define`), an array function (defined with `:=` or `define`, but enclosing arguments in square brackets `[ ]`), a subscripted function (defined with `:=` or `define`, but enclosing some arguments in square brackets and others in parentheses `( )`), one of a family of subscripted functions selected by a particular subscript value, or a subscripted function defined with a constant subscript.

  `dispfun (all)` displays all user-defined functions as given by the `functions`, `arrays`, and `macros` lists, omitting subscripted functions defined with constant subscripts.

  `dispfun` creates an intermediate expression label (`%t1`, `%t2`, etc.) for each displayed function, and assigns the function definition to the label. In contrast, `fundef` returns the function definition.

  `dispfun` quotes its arguments; the quote-quote operator `''` defeats quotation. `dispfun` returns the list of intermediate expression labels corresponding to the displayed functions.

  Examples:

      (%i1) m(x, y) ::= x^(-y);
                                           - y
      (%o1)                   m(x, y) ::= x
      (%i2) f(x, y) :=  x^(-y);
                                           - y
      (%o2)                    f(x, y) := x
      (%i3) g[x, y] :=  x^(-y);
                                          - y
      (%o3)                     g     := x
                                 x, y
      (%i4) h[x](y) :=  x^(-y);
                                          - y
      (%o4)                     h (y) := x
                                 x
      (%i5) i[8](y) :=  8^(-y);
                                          - y
      (%o5)                     i (y) := 8
                                 8
      (%i6) dispfun (m, f, g, h, h[5], h[10], i[8]);
                                           - y
      (%t6)                   m(x, y) ::= x

                                           - y
      (%t7)                    f(x, y) := x

                                          - y
      (%t8)                     g     := x
                                 x, y

                                          - y
      (%t9)                     h (y) := x
                                 x

                                          1
      (%t10)                     h (y) := --
                                  5        y
                                          5

                                           1
      (%t11)                    h  (y) := ---
                                 10         y
                                          10

                                          - y
      (%t12)                    i (y) := 8
                                 8

      (%o12)       [%t6, %t7, %t8, %t9, %t10, %t11, %t12]
      (%i13) ''%;
                           - y              - y            - y
      (%o13) [m(x, y) ::= x   , f(x, y) := x   , g     := x   ,
                                                  x, y
                        - y           1              1             - y
              h (y) := x   , h (y) := --, h  (y) := ---, i (y) := 8   ]
               x              5        y   10         y   8
                                      5             10

  Categories: Function definition · Display functions

Function: fullmap (f, expr_1, …)

  Similar to `map`, but `fullmap` keeps mapping down all subexpressions until the main operators are no longer the same.

  `fullmap` is used by the Maxima simplifier for certain matrix manipulations; thus, Maxima sometimes generates an error message concerning `fullmap` even though `fullmap` was not explicitly called by the user.

  Examples:

      (%i1) a + b * c;
      (%o1)                        b c + a
      (%i2) fullmap (g, %);
      (%o2)                   g(b) g(c) + g(a)
      (%i3) map (g, %th(2));
      (%o3)                     g(b c) + g(a)

  Categories: Function application · Expressions

Function: fullmapl (f, list_1, …)

  Similar to `fullmap`, but `fullmapl` only maps onto lists and matrices.

  Example:

      (%i1) fullmapl ("+", [3, [4, 5]], [[a, 1], [0, -1.5]]);
      (%o1)                [[a + 3, 4], [4, 3.5]]

  Categories: Function application · Expressions

System variable: functions

  Default value: `[]`

  `functions` is the list of ordinary Maxima functions in the current session. An ordinary function is a function constructed by `define` or `:=` and called with parentheses `()`. A function may be defined at the Maxima prompt or in a Maxima file loaded by `load` or `batch`.

  Array functions (called with square brackets, e.g., `F[x]`) and subscripted functions (called with square brackets and parentheses, e.g., `F[x](y)`) are listed by the global variable `arrays`, and not by `functions`.

  Lisp functions are not kept on any list.

  Examples:

      (%i1) F_1 (x) := x - 100;
      (%o1)                   F_1(x) := x - 100
      (%i2) F_2 (x, y) := x / y;
                                            x
      (%o2)                    F_2(x, y) := -
                                            y
      (%i3) define (F_3 (x), sqrt (x));
      (%o3)                   F_3(x) := sqrt(x)
      (%i4) G_1 [x] := x - 100;
      (%o4)                    G_1  := x - 100
                                  x
      (%i5) G_2 [x, y] := x / y;
                                           x
      (%o5)                     G_2     := -
                                   x, y    y
      (%i6) define (G_3 [x], sqrt (x));
      (%o6)                    G_3  := sqrt(x)
                                  x
      (%i7) H_1 [x] (y) := x^y;
                                            y
      (%o7)                     H_1 (y) := x
                                   x
      (%i8) functions;
      (%o8)              [F_1(x), F_2(x, y), F_3(x)]
      (%i9) arrays;
      (%o9)                 [G_1, G_2, G_3, H_1]

  Categories: Function definition · Global variables

Function: fundef (f)

  Returns the definition of the function f.

  The argument may be

  - the name of a macro (defined with `::=`),
  - an ordinary function (defined with `:=` or `define`),
  - an array function (defined with `:=` or `define`, but enclosing arguments in square brackets `[ ]`),
  - a subscripted function (defined with `:=` or `define`, but enclosing some arguments in square brackets and others in parentheses `( )`),
  - one of a family of subscripted functions selected by a particular subscript value,
  - or a subscripted function defined with a constant subscript.

  `fundef` quotes its argument; the quote-quote operator `''` defeats quotation.

  `fundef (f)` returns the definition of f. In contrast, `dispfun (f)` creates an intermediate expression label and assigns the definition to the label.

  Categories: Function definition

Function: funmake (F, [arg_1, …, arg_n])

  Returns an expression `F(arg_1, ..., arg_n)`. The return value is simplified, but not evaluated, so the function F is not called, even if it exists.

  `funmake` does not attempt to distinguish array functions from ordinary functions; when F is the name of an array function, `funmake` returns `F(...)` (that is, a function call with parentheses instead of square brackets). `arraymake` returns a function call with square brackets in this case.

  `funmake` evaluates its arguments.

  See also `apply` and `args`.

  Examples:

  `funmake` applied to an ordinary Maxima function.

      (%i1) F (x, y) := y^2 - x^2;
                                         2    2
      (%o1)                  F(x, y) := y  - x
      (%i2) funmake (F, [a + 1, b + 1]);
      (%o2)                    F(a + 1, b + 1)
      (%i3) ''%;
                                    2          2
      (%o3)                  (b + 1)  - (a + 1)

  `funmake` applied to a macro.

      (%i1) G (x) ::= (x - 1)/2;
                                        x - 1
      (%o1)                    G(x) ::= -----
                                          2
      (%i2) funmake (G, [u]);
      (%o2)                         G(u)
      (%i3) ''%;
                                    u - 1
      (%o3)                         -----
                                      2

  `funmake` applied to a subscripted function.

      (%i1) H [a] (x) := (x - 1)^a;
                                              a
      (%o1)                   H (x) := (x - 1)
                               a
      (%i2) funmake (H [n], [%e]);
                                             n
      (%o2)               lambda([x], (x - 1) )(%e)
      (%i3) ''%;
                                          n
      (%o3)                       (%e - 1)
      (%i4) funmake ('(H [n]), [%e]);
      (%o4)                        H (%e)
                                    n
      (%i5) ''%;
                                          n
      (%o5)                       (%e - 1)

  `funmake` applied to a symbol which is not a defined function of any kind.

      (%i1) funmake (A, [u]);
      (%o1)                         A(u)
      (%i2) ''%;
      (%o2)                         A(u)

  `funmake` evaluates its arguments, but not the return value.

      (%i1) det(a,b,c) := b^2 -4*a*c;
                                          2
      (%o1)              det(a, b, c) := b  - 4 a c
      (%i2) (x : 8, y : 10, z : 12);
      (%o2)                          12
      (%i3) f : det;
      (%o3)                          det
      (%i4) funmake (f, [x, y, z]);
      (%o4)                    det(8, 10, 12)
      (%i5) ''%;
      (%o5)                         - 284

  Maxima simplifies `funmake`'s return value.

      (%i1) funmake (sin, [%pi / 2]);
      (%o1)                           1

  Categories: Function application · Expressions

Function: lambda
`lambda` ([x_1, …, x_m], expr_1, …, expr_n)
`lambda` ([[L]], expr_1, …, expr_n)
`lambda` ([x_1, …, x_m, [L]], expr_1, …, expr_n)

  Defines and returns a lambda expression (that is, an anonymous function). The function may have required arguments x_1, …, x_m and/or optional arguments L, which appear within the function body as a list. The return value of the function is expr_n. A lambda expression can be assigned to a variable and evaluated like an ordinary function. A lambda expression may appear in some contexts in which a function name is expected.

  When the function is evaluated, unbound local variables x_1, …, x_m are created. `lambda` may appear within `block` or another `lambda`; local variables are established each time another `block` or `lambda` is evaluated. Local variables appear to be global to any enclosed `block` or `lambda`. If a variable is not local, its value is the value most recently assigned in an enclosing `block` or `lambda`, if any, otherwise, it is the value of the variable in the global environment. This policy may coincide with the usual understanding of "dynamic scope".

  After local variables are established, expr_1 through expr_n are evaluated in turn. The special variable `%%`, representing the value of the preceding expression, is recognized. `throw` and `catch` may also appear in the list of expressions.

  `return` cannot appear in a lambda expression unless enclosed by `block`, in which case `return` defines the return value of the block and not of the lambda expression, unless the block happens to be expr_n. Likewise, `go` cannot appear in a lambda expression unless enclosed by `block`.

  `lambda` quotes its arguments; the quote-quote operator `''` defeats quotation.

  Examples:

  - A lambda expression can be assigned to a variable and evaluated like an ordinary function.

      (%i1) f: lambda ([x], x^2);
                                            2
      (%o1)                    lambda([x], x )
      (%i2) f(a);
                                      2
      (%o2)                          a

  - A lambda expression may appear in contexts in which a function evaluation is expected.

      (%i1) lambda ([x], x^2) (a);
                                      2
      (%o1)                          a
      (%i2) apply (lambda ([x], x^2), [a]);
                                      2
      (%o2)                          a
      (%i3) map (lambda ([x], x^2), [a, b, c, d, e]);
                              2   2   2   2   2
      (%o3)                 [a , b , c , d , e ]

  - Argument variables are local variables. Other variables appear to be global variables. Global variables are evaluated at the time the lambda expression is evaluated, unless some special evaluation is forced by some means, such as `''`.

      (%i1) a: %pi$
      (%i2) b: %e$
      (%i3) g: lambda ([a], a*b);
      (%o3)                   lambda([a], a b)
      (%i4) b: %gamma$
      (%i5) g(1/2);
                                   %gamma
      (%o5)                        ------
                                     2
      (%i6) g2: lambda ([a], a*''b);
      (%o6)                 lambda([a], a %gamma)
      (%i7) b: %e$
      (%i8) g2(1/2);
                                   %gamma
      (%o8)                        ------
                                     2

  - Lambda expressions may be nested. Local variables within the outer lambda expression appear to be global to the inner expression unless masked by local variables of the same names.

      (%i1) h: lambda ([a, b], h2: lambda ([a], a*b), h2(1/2));
                                                         1
      (%o1)     lambda([a, b], h2 : lambda([a], a b), h2(-))
                                                         2
      (%i2) h(%pi, %gamma);
                                   %gamma
      (%o2)                        ------
                                     2

  - Since `lambda` quotes its arguments, lambda expression `i` below does not define a "multiply by `a`" function. Such a function can be defined via `buildq`, as in lambda expression `i2` below.

      (%i1) i: lambda ([a], lambda ([x], a*x));
      (%o1)             lambda([a], lambda([x], a x))
      (%i2) i(1/2);
      (%o2)                   lambda([x], a x)
      (%i3) i2: lambda([a], buildq([a: a], lambda([x], a*x)));
      (%o3)    lambda([a], buildq([a : a], lambda([x], a x)))
      (%i4) i2(1/2);
                                          1
      (%o4)                  lambda([x], (-) x)
                                          2
      (%i5) i2(1/2)(%pi);
                                     %pi
      (%o5)                          ---
                                      2

  - A lambda expression may take a variable number of arguments, which are indicated by `[L]` as the sole or final argument. The arguments appear within the function body as a list.

      (%i1) f : lambda ([aa, bb, [cc]], aa * cc + bb);
      (%o1)          lambda([aa, bb, [cc]], aa cc + bb)
      (%i2) f (foo, %i, 17, 29, 256);
      (%o2)       [17 foo + %i, 29 foo + %i, 256 foo + %i]
      (%i3) g : lambda ([[aa]], apply ("+", aa));
      (%o3)             lambda([[aa]], apply(+, aa))
      (%i4) g (17, 29, x, y, z, %e);
      (%o4)                  z + y + x + %e + 46

  Categories: Function definition

Function: local (v_1, …, v_n)

  Saves the properties associated with the symbols v_1, …, v_n, removes any properties before evaluating other expressions, and restores any saved properties on exit from the block or other compound expression in which `local` appears.

  Some declarations are implemented as properties of a symbol, including `:=`, `array`, `dependencies`, `atvalue`, `matchdeclare`, `atomgrad`, `constant`, `nonscalar`, `assume`, and some others. The effect of `local` is to make such declarations effective only within the block or other compound expression in which `local` appears; otherwise such declarations are global declarations.

  `local` can only appear in `block` or in the body of a function definition or `lambda` expression, and only one occurrence is permitted in each.

  `local` quotes its arguments. `local` returns `done`.

  Example:

  A local function definition.

      (%i1) foo (x) := 1 - x;
      (%o1)                    foo(x) := 1 - x
      (%i2) foo (100);
      (%o2)                         - 99
      (%i3) block (local (foo), foo (x) := 2 * x, foo (100));
      (%o3)                          200
      (%i4) foo (100);
      (%o4)                         - 99

  Categories: Function definition · Programming

Option variable: macroexpansion

  Default value: `false`

  `macroexpansion` controls whether the expansion (that is, the return value) of a macro function is substituted for the macro function call. A substitution may speed up subsequent expression evaluations, at the cost of storing the expansion.

  `false`

    The expansion of a macro function is not substituted for the macro function call.

  `expand`

    The first time a macro function call is evaluated, the expansion is stored. The expansion is not recomputed on subsequent calls; any side effects (such as `print` or assignment to global variables) happen only when the macro function call is first evaluated. Expansion in an expression does not affect other expressions which have the same macro function call.

  `displace`

    The first time a macro function call is evaluated, the expansion is substituted for the call, thus modifying the expression from which the macro function was called. The expansion is not recomputed on subsequent calls; any side effects happen only when the macro function call is first evaluated. Expansion in an expression does not affect other expressions which have the same macro function call.

  Examples

  When `macroexpansion` is `false`, a macro function is called every time the calling expression is evaluated, and the calling expression is not modified.

      (%i1) f (x) := h (x) / g (x);
                                        h(x)
      (%o1)                     f(x) := ----
                                        g(x)
      (%i2) g (x) ::= block (print ("x + 99 is equal to", x),
                             return (x + 99));
      (%o2) g(x) ::= block(print("x + 99 is equal to", x),
                                                        return(x + 99))
      (%i3) h (x) ::= block (print ("x - 99 is equal to", x),
                             return (x - 99));
      (%o3) h(x) ::= block(print("x - 99 is equal to", x),
                                                        return(x - 99))
      (%i4) macroexpansion: false;
      (%o4)                         false
      (%i5) f (a * b);
      x - 99 is equal to x
      x + 99 is equal to x
                                  a b - 99
      (%o5)                       --------
                                  a b + 99
      (%i6) dispfun (f);
                                        h(x)
      (%t6)                     f(x) := ----
                                        g(x)

      (%o6)                         [%t6]
      (%i7) f (a * b);
      x - 99 is equal to x
      x + 99 is equal to x
                                  a b - 99
      (%o7)                       --------
                                  a b + 99

  When `macroexpansion` is `expand`, a macro function is called once, and the calling expression is not modified.

      (%i1) f (x) := h (x) / g (x);
                                        h(x)
      (%o1)                     f(x) := ----
                                        g(x)
      (%i2) g (x) ::= block (print ("x + 99 is equal to", x),
                             return (x + 99));
      (%o2) g(x) ::= block(print("x + 99 is equal to", x),
                                                        return(x + 99))
      (%i3) h (x) ::= block (print ("x - 99 is equal to", x),
                             return (x - 99));
      (%o3) h(x) ::= block(print("x - 99 is equal to", x),
                                                        return(x - 99))
      (%i4) macroexpansion: expand;
      (%o4)                        expand
      (%i5) f (a * b);
      x - 99 is equal to x
      x + 99 is equal to x
                                  a b - 99
      (%o5)                       --------
                                  a b + 99
      (%i6) dispfun (f);
                            mmacroexpanded(x - 99, h(x))
      (%t6)         f(x) := ----------------------------
                            mmacroexpanded(x + 99, g(x))

      (%o6)                         [%t6]
      (%i7) f (a * b);
                                  a b - 99
      (%o7)                       --------
                                  a b + 99

  When `macroexpansion` is `displace`, a macro function is called once, and the calling expression is modified.

      (%i1) f (x) := h (x) / g (x);
                                        h(x)
      (%o1)                     f(x) := ----
                                        g(x)
      (%i2) g (x) ::= block (print ("x + 99 is equal to", x),
                             return (x + 99));
      (%o2) g(x) ::= block(print("x + 99 is equal to", x),
                                                        return(x + 99))
      (%i3) h (x) ::= block (print ("x - 99 is equal to", x),
                             return (x - 99));
      (%o3) h(x) ::= block(print("x - 99 is equal to", x),
                                                        return(x - 99))
      (%i4) macroexpansion: displace;
      (%o4)                       displace
      (%i5) f (a * b);
      x - 99 is equal to x
      x + 99 is equal to x
                                  a b - 99
      (%o5)                       --------
                                  a b + 99
      (%i6) dispfun (f);
                                       x - 99
      (%t6)                    f(x) := ------
                                       x + 99

      (%o6)                         [%t6]
      (%i7) f (a * b);
                                  a b - 99
      (%o7)                       --------
                                  a b + 99

  Categories: Function application · Global flags

Option variable: mode_checkp

  Default value: `true`

  When `mode_checkp` is `true`, `mode_declare` checks the modes of bound variables.

  Categories: Translation flags and variables

Option variable: mode_check_errorp

  Default value: `false`

  When `mode_check_errorp` is `true`, `mode_declare` calls error.

  Categories: Translation flags and variables

Option variable: mode_check_warnp

  Default value: `true`

  When `mode_check_warnp` is `true`, mode errors are described.

  Categories: Translation flags and variables

Function: mode_declare (y_1, mode_1, …, y_n, mode_n)

  `mode_declare` is used to declare the modes of variables and functions for subsequent translation or compilation of functions. `mode_declare` is typically placed at the beginning of a function definition, at the beginning of a Maxima script, or executed at the interactive prompt.

  The arguments of `mode_declare` are pairs consisting of a variable and a mode which is one of `boolean`, `fixnum`, `number`, `rational`, or `float`. Each variable may also be a list of variables all of which are declared to have the same mode.

  If a variable is an array, and if every element of the array which is referenced has a value then `array (yi, complete, dim1, dim2, ...)` rather than

      array(yi, dim1, dim2, ...)

  should be used when first declaring the bounds of the array. If all the elements of the array are of mode `fixnum` (`float`), use `fixnum` (`float`) instead of `complete`. Also if every element of the array is of the same mode, say `m`, then

      mode_declare (completearray (yi), m))

  should be used for efficient translation.

  Numeric code using arrays might run faster by declaring the expected size of the array, as in:

      mode_declare (completearray (a [10, 10]), float)

  for a floating point number array which is 10 x 10.

  One may declare the mode of the result of a function by using `function (f_1, f_2, ...)` as an argument; here `f_1`, `f_2`, … are the names of functions. For example the expression,

      mode_declare ([function (f_1, f_2, ...)], fixnum)

  declares that the values returned by `f_1`, `f_2`, … are single-word integers.

  `modedeclare` is a synonym for `mode_declare`.

  Categories: Translation and compilation

Function: mode_identity (arg_1, arg_2)

  A special form used with `mode_declare` and `macros` to declare, e.g., a list of lists of flonums, or other compound data object. The first argument to `mode_identity` is a primitive value mode name as given to `mode_declare` (i.e., one of `float`, `fixnum`, `number`, `list`, or `any`), and the second argument is an expression which is evaluated and returned as the value of `mode_identity`. However, if the return value is not allowed by the mode declared in the first argument, an error or warning is signalled. The important thing is that the mode of the expression as determined by the Maxima to Lisp translator, will be that given as the first argument, independent of anything that goes on in the second argument. E.g., `x: 3.3; mode_identity (fixnum, x);` yields an error. `mode_identity (flonum, x)` returns 3.3 . This has a number of uses, e.g., if you knew that `first (l)` returned a number then you might write `mode_identity (number, first (l))`. However, a more efficient way to do it would be to define a new primitive,

      firstnumb (x) ::= buildq ([x], mode_identity (number, first(x)));

  and use `firstnumb` every time you take the first of a list of numbers.

  Categories: Translation and compilation

Function: remfunction
`remfunction` (f_1, …, f_n)
`remfunction` (all)

  Unbinds the function definitions of the symbols f_1, …, f_n. The arguments may be the names of ordinary functions (created by `:=` or `define`) or macro functions (created by `::=`).

  `remfunction (all)` unbinds all function definitions.

  `remfunction` quotes its arguments.

  `remfunction` returns a list of the symbols for which the function definition was unbound. `false` is returned in place of any symbol for which there is no function definition.

  `remfunction` does not apply to array functions or subscripted functions. `remarray` applies to those types of functions.

  Categories: Function definition

Option variable: savedef

  Default value: `true`

  When `savedef` is `true`, the Maxima version of a user function is preserved when the function is translated. This permits the definition to be displayed by `dispfun` and allows the function to be edited.

  When `savedef` is `false`, the names of translated functions are removed from the `functions` list.

  Categories: Translation flags and variables

Option variable: transcompile

  Default value: `true`

  When `transcompile` is `true`, `translate` and `translate_file` generate declarations to make the translated code more suitable for compilation.

  `compfile` sets `transcompile: true` for the duration.

  Categories: Translation flags and variables

Function: translate
`translate` (f_1, …, f_n)
`translate` (functions)
`translate` (all)

  Translates the user-defined functions f_1, …, f_n from the Maxima language into Lisp and evaluates the Lisp translations. Typically the translated functions run faster than the originals.

  `translate (all)` or `translate (functions)` translates all user-defined functions.

  Functions to be translated should include a call to `mode_declare` at the beginning when possible in order to produce more efficient code. For example:

      f (x_1, x_2, ...) := block ([v_1, v_2, ...],
          mode_declare (v_1, mode_1, v_2, mode_2, ...), ...)

  where the x_1, x_2, … are the parameters to the function and the v_1, v_2, … are the local variables.

  The names of translated functions are removed from the `functions` list if `savedef` is `false` (see below) and are added to the `props` lists.

  Functions should not be translated unless they are fully debugged.

  Expressions are assumed simplified; if they are not, correct but non-optimal code gets generated. Thus, the user should not set the `simp` switch to `false` which inhibits simplification of the expressions to be translated.

  The switch `translate`, if `true`, causes automatic translation of a user's function to Lisp.

  Note that translated functions may not run identically to the way they did before translation as certain incompatibilities may exist between the Lisp and Maxima versions. Principally, the `rat` function with more than one argument and the `ratvars` function should not be used if any variables are `mode_declare`'d canonical rational expressions (CRE). Also the `prederror: false` setting will not translate.

  `savedef` - if `true` will cause the Maxima version of a user function to remain when the function is `translate`'d. This permits the definition to be displayed by `dispfun` and allows the function to be edited.

  `transrun` - if `false` will cause the interpreted version of all functions to be run (provided they are still around) rather than the translated version.

  The result returned by `translate` is a list of the names of the functions translated.

  Categories: Translation and compilation

Function: translate_file
`translate_file` (maxima_filename)
`translate_file` (maxima_filename, lisp_filename)

  Translates a file of Maxima code into a file of Lisp code. `translate_file` returns a list of three filenames: the name of the Maxima file, the name of the Lisp file, and the name of file containing additional information about the translation. `translate_file` evaluates its arguments.

  `translate_file ("foo.mac"); load("foo.LISP")` is the same as the command `batch ("foo.mac")` except for certain restrictions, the use of `''` and `%`, for example.

  `translate_file (maxima_filename)` translates a Maxima file maxima_filename into a similarly-named Lisp file. For example, `foo.mac` is translated into `foo.LISP`. The Maxima filename may include a directory name or names, in which case the Lisp output file is written to the same directory from which the Maxima input comes.

  `translate_file (maxima_filename, lisp_filename)` translates a Maxima file maxima_filename into a Lisp file lisp_filename. `translate_file` ignores the filename extension, if any, of `lisp_filename`; the filename extension of the Lisp output file is always `LISP`. The Lisp filename may include a directory name or names, in which case the Lisp output file is written to the specified directory.

  `translate_file` also writes a file of translator warning messages of various degrees of severity. The filename extension of this file is `UNLISP`. This file may contain valuable information, though possibly obscure, for tracking down bugs in translated code. The `UNLISP` file is always written to the same directory from which the Maxima input comes.

  `translate_file` emits Lisp code which causes some declarations and definitions to take effect as soon as the Lisp code is compiled. See `compile_file` for more on this topic.

  See also

  `tr_array_as_ref` `tr_bound_function_applyp`, `tr_exponent` `tr_file_tty_messagesp`, `tr_float_can_branch_complex`, `tr_function_call_default`, `tr_numer`, `tr_optimize_max_loop`, `tr_semicompile`, `tr_state_vars`, `tr_warnings_get`, `tr_warn_bad_function_calls` `tr_warn_fexpr`, `tr_warn_meval`, `tr_warn_mode`, `tr_warn_undeclared`, and `tr_warn_undefined_variable`.

  Categories: Translation and compilation

Option variable: transrun

  Default value: `true`

  When `transrun` is `false` will cause the interpreted version of all functions to be run (provided they are still around) rather than the translated version.

  Categories: Translation flags and variables

Option variable: tr_array_as_ref

  Default value: `true`

  If `translate_fast_arrays` is `false`, array references in Lisp code emitted by `translate_file` are affected by `tr_array_as_ref`. When `tr_array_as_ref` is `true`, array names are evaluated, otherwise array names appear as literal symbols in translated code.

  `tr_array_as_ref` has no effect if `translate_fast_arrays` is `true`.

  Categories: Translation flags and variables

Option variable: tr_bound_function_applyp

  Default value: `true`

  When `tr_bound_function_applyp` is `true`, Maxima gives a warning if a bound variable (such as a function argument) is found being used as a function. `tr_bound_function_applyp` does not affect the code generated in such cases.

  For example, an expression such as `g (f, x) := f (x+1)` will trigger the warning message.

  Categories: Translation flags and variables

Option variable: tr_file_tty_messagesp

  Default value: `false`

  When `tr_file_tty_messagesp` is `true`, messages generated by `translate_file` during translation of a file are displayed on the console and inserted into the UNLISP file. When `false`, messages about translation of the file are only inserted into the UNLISP file.

  Categories: Translation flags and variables

Option variable: tr_float_can_branch_complex

  Default value: `true`

  Tells the Maxima-to-Lisp translator to assume that the functions `acos`, `asin`, `asec`, and `acsc` can return complex results.

  The ostensible effect of `tr_float_can_branch_complex` is the following. However, it appears that this flag has no effect on the translator output.

  When it is `true` then `acos(x)` is of mode `any` even if `x` is of mode `float` (as set by `mode_declare`). When `false` then `acos(x)` is of mode `float` if and only if `x` is of mode `float`.

  Categories: Translation flags and variables

Option variable: tr_function_call_default

  Default value: `general`

  `false` means give up and call `meval`, `expr` means assume Lisp fixed arg function. `general`, the default gives code good for `mexprs` and `mlexprs` but not `macros`. `general` assures variable bindings are correct in compiled code. In `general` mode, when translating F(X), if F is a bound variable, then it assumes that `apply (f, [x])` is meant, and translates a such, with appropriate warning. There is no need to turn this off. With the default settings, no warning messages implies full compatibility of translated and compiled code with the Maxima interpreter.

  Categories: Translation flags and variables

Option variable: tr_numer

  Default value: `false`

  When `tr_numer` is `true`, `numer` properties are used for atoms which have them, e.g. `%pi`.

  Categories: Translation flags and variables

Option variable: tr_optimize_max_loop

  Default value: 100

  `tr_optimize_max_loop` is the maximum number of times the macro-expansion and optimization pass of the translator will loop in considering a form. This is to catch macro expansion errors, and non-terminating optimization properties.

  Categories: Translation flags and variables

Option variable: tr_semicompile

  Default value: `false`

  When `tr_semicompile` is `true`, `translate_file` and `compfile` output forms which will be macroexpanded but not compiled into machine code by the Lisp compiler.

  Categories: Translation flags and variables

System variable: tr_state_vars

  Default value:

      [transcompile, tr_semicompile, tr_warn_undeclared, tr_warn_meval,
      tr_warn_fexpr, tr_warn_mode, tr_warn_undefined_variable,
      tr_function_call_default, tr_array_as_ref,tr_numer]

  The list of the switches that affect the form of the translated output. This information is useful to system people when trying to debug the translator. By comparing the translated product to what should have been produced for a given state, it is possible to track down bugs.

  Categories: Translation flags and variables

Function: tr_warnings_get ()

  Prints a list of warnings which have been given by the translator during the current translation.

  Categories: Translation and compilation

Option variable: tr_warn_bad_function_calls

  Default value: `true`

  - Gives a warning when when function calls are being made which may not be correct due to improper declarations that were made at translate time.

  Categories: Translation flags and variables

Option variable: tr_warn_fexpr

  Default value: `compfile`

  - Gives a warning if any FEXPRs are encountered. FEXPRs should not normally be output in translated code, all legitimate special program forms are translated.

  Categories: Translation flags and variables

Option variable: tr_warn_meval

  Default value: `compfile`

  - Gives a warning if the function `meval` gets called. If `meval` is called that indicates problems in the translation.

  Categories: Translation flags and variables

Option variable: tr_warn_mode

  Default value: `all`

  - Gives a warning when variables are assigned values inappropriate for their mode.

  Categories: Translation flags and variables

Option variable: tr_warn_undeclared

  Default value: `compile`

  - Determines when to send warnings about undeclared variables to the TTY.

  Categories: Translation flags and variables

Option variable: tr_warn_undefined_variable

  Default value: `all`

  - Gives a warning when undefined global variables are seen.

  Categories: Translation flags and variables

Function: compile_file
`compile_file` (filename)
`compile_file` (filename, compiled_filename)
`compile_file` (filename, compiled_filename, lisp_filename)

  Translates the Maxima file filename into Lisp, executes the Lisp compiler, and, if the translation and compilation succeed, loads the compiled code into Maxima.

  `compile_file` returns a list of the names of four files: the original Maxima file, the Lisp translation, notes on translation, and the compiled code. If the compilation fails, the fourth item is `false`.

  Some declarations and definitions take effect as soon as the Lisp code is compiled (without loading the compiled code). These include functions defined with the `:=` operator, macros define with the `::=` operator, `alias`, `declare`, `define_variable`, `mode_declare`, and `infix`, `matchfix`, `nofix`, `postfix`, `prefix`, and `compfile`.

  Assignments and function calls are not evaluated until the compiled code is loaded. In particular, within the Maxima file, assignments to the translation flags (`tr_numer`, etc.) have no effect on the translation.

  filename may not contain `:lisp` statements.

  `compile_file` evaluates its arguments.

  Categories: Translation and compilation

Function: declare_translated (f_1, f_2, …)

  When translating a file of Maxima code to Lisp, it is important for the translator to know which functions it sees in the file are to be called as translated or compiled functions, and which ones are just Maxima functions or undefined. Putting this declaration at the top of the file, lets it know that although a symbol does which does not yet have a Lisp function value, will have one at call time. `(MFUNCTION-CALL fn arg1 arg2 ...)` is generated when the translator does not know `fn` is going to be a Lisp function.

  Categories: Translation and compilation

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# 37. Program Flow

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## 37.1 Lisp and Maxima

Maxima is written in Lisp, and it is easy to access Lisp functions and variables from Maxima and vice versa. Lisp and Maxima symbols are distinguished by a naming convention. A Lisp symbol which begins with a dollar sign `$` corresponds to a Maxima symbol without the dollar sign.

A Maxima symbol which begins with a question mark `?` corresponds to a Lisp symbol without the question mark. For example, the Maxima symbol `foo` corresponds to the Lisp symbol `$FOO`, while the Maxima symbol `?foo` corresponds to the Lisp symbol `FOO`. Note that `?foo` is written without a space between `?` and `foo`; otherwise it might be mistaken for `describe ("foo")`.

Hyphen `-`, asterisk `*`, or other special characters in Lisp symbols must be escaped by backslash `\` where they appear in Maxima code. For example, the Lisp identifier `*foo-bar*` is written `?\*foo\-bar\*` in Maxima.

Lisp code may be executed from within a Maxima session. A single line of Lisp (containing one or more forms) may be executed by the special command `:lisp`. For example,

    (%i1) :lisp (foo $x $y)

calls the Lisp function `foo` with Maxima variables `x` and `y` as arguments. The `:lisp` construct can appear at the interactive prompt or in a file processed by `batch` or `demo`, but not in a file processed by `load`, `batchload`, `translate_file`, or `compile_file`.

The function `to_lisp` opens an interactive Lisp session. Entering `(to-maxima)` closes the Lisp session and returns to Maxima.

Lisp functions and variables which are to be visible in Maxima as functions and variables with ordinary names (no special punctuation) must have Lisp names beginning with the dollar sign `$`.

Maxima is case-sensitive, distinguishing between lowercase and uppercase letters in identifiers. There are some rules governing the translation of names between Lisp and Maxima.

1. A Lisp identifier not enclosed in vertical bars corresponds to a Maxima identifier in lowercase. Whether the Lisp identifier is uppercase, lowercase, or mixed case, is ignored. E.g., Lisp `$foo`, `$FOO`, and `$Foo` all correspond to Maxima `foo`. But this is because `$foo`, `$FOO` and `$Foo` are converted by the Lisp reader by default to the Lisp symbol `$FOO`.
2. A Lisp identifier which is all uppercase or all lowercase and enclosed in vertical bars corresponds to a Maxima identifier with case reversed. That is, uppercase is changed to lowercase and lowercase to uppercase. E.g., Lisp `|$FOO|` and `|$foo|` correspond to Maxima `foo` and `FOO`, respectively.
3. A Lisp identifier which is mixed uppercase and lowercase and enclosed in vertical bars corresponds to a Maxima identifier with the same case. E.g., Lisp `|$Foo|` corresponds to Maxima `Foo`.

The `#$` Lisp macro allows the use of Maxima expressions in Lisp code. `#$expr$` expands to a Lisp expression equivalent to the Maxima expression expr.

    (msetq $foo #$[x, y]$)

This has the same effect as entering

    (%i1) foo: [x, y];

The Lisp function `displa` prints an expression in Maxima format.

    (%i1) :lisp #$[x, y, z]$
    ((MLIST SIMP) $X $Y $Z)
    (%i1) :lisp (displa '((MLIST SIMP) $X $Y $Z))
    [x, y, z]
    NIL

Functions defined in Maxima are not ordinary Lisp functions. The Lisp function `mfuncall` calls a Maxima function. For example:

    (%i1) foo(x,y) := x*y$
    (%i2) :lisp (mfuncall '$foo 'a 'b)
    ((MTIMES SIMP) A B)

Some Lisp functions are shadowed in the Maxima package, namely the following.

       complement     continue      //
       float          functionp     array
       exp            listen        signum
       atan           asin          acos
       asinh          acosh         atanh
       tanh           cosh          sinh
       tan            break         gcd

Categories: Programming

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## 37.2 Garbage Collection

Symbolic computation tends to create a good deal of garbage (temporary or intermediate results that are eventually not used), and effective handling of this can be crucial to successful completion of some programs.

Under GCL, on UNIX systems where the mprotect system call is available (including SUN OS 4.0 and some variants of BSD) a stratified garbage collection is available. This limits the collection to pages which have been recently written to. See the GCL documentation under ALLOCATE and GBC. At the Lisp level doing (setq si::*notify-gbc* t) will help you determine which areas might need more space.

For other Lisps that run Maxima, we refer the reader to the documentation for that Lisp on how to control GC.

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## 37.3 Introduction to Program Flow

Maxima provides a `do` loop for iteration, as well as more primitive constructs such as `go`.

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## 37.4 Functions and Variables for Program Flow

Function: backtrace
`backtrace` ()
`backtrace` (n)

  Prints the call stack, that is, the list of functions which called the currently active function.

  `backtrace()` prints the entire call stack.

  `backtrace (n)` prints the n most recent functions, including the currently active function.

  `backtrace` can be called from a script, a function, or the interactive prompt (not only in a debugging context).

  Examples:

  - `backtrace()` prints the entire call stack.

        (%i1) h(x) := g(x/7)$
        (%i2) g(x) := f(x-11)$
        (%i3) f(x) := e(x^2)$
        (%i4) e(x) := (backtrace(), 2*x + 13)$
        (%i5) h(10);
        #0: e(x=4489/49)
        #1: f(x=-67/7)
        #2: g(x=10/7)
        #3: h(x=10)
                                      9615
        (%o5)                         ----
                                       49

  - `backtrace (n)` prints the n most recent functions, including the currently active function.

        (%i1) h(x) := (backtrace(1), g(x/7))$
        (%i2) g(x) := (backtrace(1), f(x-11))$
        (%i3) f(x) := (backtrace(1), e(x^2))$
        (%i4) e(x) := (backtrace(1), 2*x + 13)$
        (%i5) h(10);
        #0: h(x=10)
        #0: g(x=10/7)
        #0: f(x=-67/7)
        #0: e(x=4489/49)
                                      9615
        (%o5)                         ----
                                       49

  Categories: Debugging

Special operator: do
Special operator: while
Special operator: unless
Special operator: for
Special operator: from
Special operator: thru
Special operator: step
Special operator: next
Special operator: in

  The `do` statement is used for performing iteration. The general form of the `do` statements maxima supports is:

  - `for variable: initial_value step increment thru limit do body`
  - `for variable: initial_value step increment while condition do body`
  - `for variable: initial_value step increment unless condition do body`
  - `for variable in list do body`

  If the loop is expected to generate a list as output the command `makelist` may be the appropriate command to use instead, See section Performance considerations for Lists.

  initial_value, increment, limit, and body can be any expression. list is a list. If the increment is 1 then "`step 1`" may be omitted; As always, if `body` needs to contain more than one command these commands can be specified as a comma-separated list surrounded by parenthesis or as a `block`. Due to its great generality the `do` statement will be described in two parts. The first form of the `do` statement (which is shown in the first three items above) is analogous to that used in several other programming languages (Fortran, Algol, PL/I, etc.); then the other features will be mentioned.

  The execution of the `do` statement proceeds by first assigning the initial_value to the variable (henceforth called the control-variable). Then: (1) If the control-variable has exceeded the limit of a `thru` specification, or if the condition of the `unless` is `true`, or if the condition of the `while` is `false` then the `do` terminates. (2) The body is evaluated. (3) The increment is added to the control-variable. The process from (1) to (3) is performed repeatedly until the termination condition is satisfied. One may also give several termination conditions in which case the `do` terminates when any of them is satisfied.

  In general the `thru` test is satisfied when the control-variable is greater than the limit if the increment was non-negative, or when the control-variable is less than the limit if the increment was negative. The increment and limit may be non-numeric expressions as long as this inequality can be determined. However, unless the increment is syntactically negative (e.g. is a negative number) at the time the `do` statement is input, Maxima assumes it will be positive when the `do` is executed. If it is not positive, then the `do` may not terminate properly.

  Note that the limit, increment, and termination condition are evaluated each time through the loop. Thus if any of these involve much computation, and yield a result that does not change during all the executions of the body, then it is more efficient to set a variable to their value prior to the `do` and use this variable in the `do` form.

  The value normally returned by a `do` statement is the atom `done`. However, the function `return` may be used inside the body to exit the `do` prematurely and give it any desired value. Note however that a `return` within a `do` that occurs in a `block` will exit only the `do` and not the `block`. Note also that the `go` function may not be used to exit from a `do` into a surrounding `block`.

  The control-variable is always local to the `do` and thus any variable may be used without affecting the value of a variable with the same name outside of the `do`. The control-variable is unbound after the `do` terminates.

      (%i1) for a:-3 thru 26 step 7 do display(a)$
                                   a = - 3

                                    a = 4

                                   a = 11

                                   a = 18

                                   a = 25

      (%i1) s: 0$
      (%i2) for i: 1 while i <= 10 do s: s+i;
      (%o2)                         done
      (%i3) s;
      (%o3)                          55

  Note that the condition `while i <= 10` is equivalent to `unless i > 10` and also `thru 10`.

      (%i1) series: 1$
      (%i2) term: exp (sin (x))$
      (%i3) for p: 1 unless p > 7 do
                (term: diff (term, x)/p,
                 series: series + subst (x=0, term)*x^p)$
      (%i4) series;
                        7    6     5    4    2
                       x    x     x    x    x
      (%o4)            -- - --- - -- - -- + -- + x + 1
                       90   240   15   8    2

  which gives 8 terms of the Taylor series for `e^sin(x)`.

      (%i1) poly: 0$
      (%i2) for i: 1 thru 5 do
                for j: i step -1 thru 1 do
                    poly: poly + i*x^j$
      (%i3) poly;
                        5      4       3       2
      (%o3)          5 x  + 9 x  + 12 x  + 14 x  + 15 x
      (%i4) guess: -3.0$
      (%i5) for i: 1 thru 10 do
                (guess: subst (guess, x, 0.5*(x + 10/x)),
                 if abs (guess^2 - 10) < 0.00005 then return (guess));
      (%o5)                  - 3.162280701754386

  This example computes the negative square root of 10 using the Newton- Raphson iteration a maximum of 10 times. Had the convergence criterion not been met the value returned would have been `done`.

  Instead of always adding a quantity to the control-variable one may sometimes wish to change it in some other way for each iteration. In this case one may use `next expression` instead of `step increment`. This will cause the control-variable to be set to the result of evaluating expression each time through the loop.

      (%i6) for count: 2 next 3*count thru 20 do display (count)$
                                  count = 2

                                  count = 6

                                 count = 18

  As an alternative to `for variable: value ...do...` the syntax `for variable from value ...do...` may be used. This permits the `from value` to be placed after the `step` or `next` value or after the termination condition. If `from value` is omitted then 1 is used as the initial value.

  Sometimes one may be interested in performing an iteration where the control-variable is never actually used. It is thus permissible to give only the termination conditions omitting the initialization and updating information as in the following example to compute the square-root of 5 using a poor initial guess.

      (%i1) x: 1000$
      (%i2) thru 20 do x: 0.5*(x + 5.0/x)$
      (%i3) x;
      (%o3)                   2.23606797749979
      (%i4) sqrt(5), numer;
      (%o4)                   2.23606797749979

  If it is desired one may even omit the termination conditions entirely and just give `do body` which will continue to evaluate the body indefinitely. In this case the function `return` should be used to terminate execution of the `do`.

      (%i1) newton (f, x):= ([y, df, dfx], df: diff (f ('x), 'x),
                do (y: ev(df), x: x - f(x)/y,
                    if abs (f (x)) < 5e-6 then return (x)))$
      (%i2) sqr (x) := x^2 - 5.0$
      (%i3) newton (sqr, 1000);
      (%o3)                   2.236068027062195

  (Note that `return`, when executed, causes the current value of `x` to be returned as the value of the `do`. The `block` is exited and this value of the `do` is returned as the value of the `block` because the `do` is the last statement in the block.)

  One other form of the `do` is available in Maxima. The syntax is:

      for variable in list end_tests do body

  The elements of list are any expressions which will successively be assigned to the `variable` on each iteration of the body. The optional termination tests end_tests can be used to terminate execution of the `do`; otherwise it will terminate when the list is exhausted or when a `return` is executed in the body. (In fact, `list` may be any non-atomic expression, and successive parts are taken.)

      (%i1)  for f in [log, rho, atan] do ldisp(f(1))$
      (%t1)                                  0
      (%t2)                                rho(1)
                                           %pi
      (%t3)                                 ---
                                            4
      (%i4) ev(%t3,numer);
      (%o4)                             0.78539816

  Categories: Programming

Function: errcatch (expr_1, …, expr_n)

  Evaluates expr_1, …, expr_n one by one and returns `[expr_n]` (a list) if no error occurs. If an error occurs in the evaluation of any argument, `errcatch` prevents the error from propagating and returns the empty list `[]` without evaluating any more arguments.

  `errcatch` is useful in `batch` files where one suspects an error might occur which would terminate the `batch` if the error weren't caught.

  Categories: Programming

Function: error (expr_1, …, expr_n)
System variable: error

  Evaluates and prints expr_1, …, expr_n, and then causes an error return to top level Maxima or to the nearest enclosing `errcatch`.

  The variable `error` is set to a list describing the error. The first element of `error` is a format string, which merges all the strings among the arguments expr_1, …, expr_n, and the remaining elements are the values of any non-string arguments.

  `errormsg()` formats and prints `error`. This is effectively reprinting the most recent error message.

  Categories: Programming

Option variable: error_size

  Default value: 10

  `error_size` modifies error messages according to the size of expressions which appear in them. If the size of an expression (as determined by the Lisp function `ERROR-SIZE`) is greater than `error_size`, the expression is replaced in the message by a symbol, and the symbol is assigned the expression. The symbols are taken from the list `error_syms`.

  Otherwise, the expression is smaller than `error_size`, and the expression is displayed in the message.

  See also `error` and `error_syms`.

  Example:

  The size of `U`, as determined by `ERROR-SIZE`, is 24.

      (%i1) U: (C^D^E + B + A)/(cos(X-1) + 1)$

      (%i2) error_size: 20$

      (%i3) error ("Example expression is", U);

      Example expression is errexp1
       -- an error.  Quitting.  To debug this try debugmode(true);
      (%i4) errexp1;
                                  E
                                 D
                                C   + B + A
      (%o4)                    --------------
                               cos(X - 1) + 1
      (%i5) error_size: 30$

      (%i6) error ("Example expression is", U);

                               E
                              D
                             C   + B + A
      Example expression is --------------
                            cos(X - 1) + 1
       -- an error.  Quitting.  To debug this try debugmode(true);

  Categories: Debugging · Display flags and variables

Option variable: error_syms

  Default value: `[errexp1, errexp2, errexp3]`

  In error messages, expressions larger than `error_size` are replaced by symbols, and the symbols are set to the expressions. The symbols are taken from the list `error_syms`. The first too-large expression is replaced by `error_syms[1]`, the second by `error_syms[2]`, and so on.

  If there are more too-large expressions than there are elements of `error_syms`, symbols are constructed automatically, with the n-th symbol equivalent to `concat ('errexp, n)`.

  See also `error` and `error_size`.

  Categories: Debugging · Display flags and variables

Function: errormsg ()

  Reprints the most recent error message. The variable `error` holds the message, and `errormsg` formats and prints it.

  Categories: Programming

Option variable: errormsg

  Default value: `true`

  When `false` the output of error messages is suppressed.

  The option variable `errormsg` can not be set in a block to a local value. The global value of `errormsg` is always present.

      (%i1) errormsg;
      (%o1)                         true
      (%i2) sin(a,b);
      sin: wrong number of arguments.
       -- an error. To debug this try: debugmode(true);
      (%i3) errormsg:false;
      (%o3)                         false
      (%i4) sin(a,b);
       -- an error. To debug this try: debugmode(true);

  The option variable `errormsg` can not be set in a block to a local value.

      (%i1) f(bool):=block([errormsg:bool], print ("value of errormsg is",errormsg))$
      (%i2) errormsg:true;
      (%o2)                         true
      (%i3) f(false);
      value of errormsg is true
      (%o3)                         true
      (%i4) errormsg:false;
      (%o4)                         false
      (%i5) f(true);
      value of errormsg is false
      (%o5)                         false

  Categories: Programming

Function: go (tag)

  is used within a `block` to transfer control to the statement of the block which is tagged with the argument to `go`. To tag a statement, precede it by an atomic argument as another statement in the `block`. For example:

      block ([x], x:1, loop, x+1, ..., go(loop), ...)

  The argument to `go` must be the name of a tag appearing in the same `block`. One cannot use `go` to transfer to tag in a `block` other than the one containing the `go`.

  Categories: Programming

Special operator: if

  Represents conditional evaluation. Various forms of `if` expressions are recognized.

  `if cond_1 then expr_1 else expr_0` evaluates to expr_1 if cond_1 evaluates to `true`, otherwise the expression evaluates to expr_0.

  The command `if cond_1 then expr_1 elseif cond_2 then expr_2 elseif ... else expr_0` evaluates to expr_k if cond_k is `true` and all preceding conditions are `false`. If none of the conditions are `true`, the expression evaluates to `expr_0`.

  A trailing `else false` is assumed if `else` is missing. That is, the command `if cond_1 then expr_1` is equivalent to `if cond_1 then expr_1 else false`, and the command `if cond_1 then expr_1 elseif ... elseif cond_n then expr_n` is equivalent to `if cond_1 then expr_1 elseif ... elseif cond_n then expr_n else false`.

  The alternatives expr_0, …, expr_n may be any Maxima expressions, including nested `if` expressions. The alternatives are neither simplified nor evaluated unless the corresponding condition is `true`.

  The conditions cond_1, …, cond_n are expressions which potentially or actually evaluate to `true` or `false`. When a condition does not actually evaluate to `true` or `false`, the behavior of `if` is governed by the global flag `prederror`. When `prederror` is `true`, it is an error if any evaluated condition does not evaluate to `true` or `false`. Otherwise, conditions which do not evaluate to `true` or `false` are accepted, and the result is a conditional expression.

  Among other elements, conditions may comprise relational and logical operators as follows.

      Operation            Symbol      Type

      less than            <           relational infix
      less than            <=
        or equal to                    relational infix
      equality (syntactic) =           relational infix
      negation of =        #           relational infix
      equality (value)     equal       relational function
      negation of equal    notequal    relational function
      greater than         >=
        or equal to                    relational infix
      greater than         >           relational infix
      and                  and         logical infix
      or                   or          logical infix
      not                  not         logical prefix

  Categories: Programming · Predicate functions

Function: map (f, expr_1, …, expr_n)

  Returns an expression whose leading operator is the same as that of the expressions expr_1, …, expr_n but whose subparts are the results of applying f to the corresponding subparts of the expressions. f is either the name of a function of n arguments or is a `lambda` form of n arguments.

  `maperror` - if `false` will cause all of the mapping functions to (1) stop when they finish going down the shortest expr_i if not all of the expr_i are of the same length and (2) apply f to [expr_1, expr_2, …] if the expr_i are not all the same type of object. If `maperror` is `true` then an error message will be given in the above two instances.

  One of the uses of this function is to `map` a function (e.g. `partfrac`) onto each term of a very large expression where it ordinarily wouldn't be possible to use the function on the entire expression due to an exhaustion of list storage space in the course of the computation.

  See also `scanmap`, `maplist`, `outermap`, `matrixmap` and `apply`.

      (%i1) map(f,x+a*y+b*z);
      (%o1)                        f(b z) + f(a y) + f(x)
      (%i2) map(lambda([u],partfrac(u,x)),x+1/(x^3+4*x^2+5*x+2));
                                 1       1        1
      (%o2)                     ----- - ----- + -------- + x
                               x + 2   x + 1          2
                                               (x + 1)
      (%i3) map(ratsimp, x/(x^2+x)+(y^2+y)/y);
                                            1
      (%o3)                            y + ----- + 1
                                          x + 1
      (%i4) map("=",[a,b],[-0.5,3]);
      (%o4)                          [a = - 0.5, b = 3]

  Categories: Function application

Function: mapatom (expr)

  Returns `true` if and only if expr is treated by the mapping routines as an atom. "Mapatoms" are atoms, numbers (including rational numbers), and subscripted variables.

  Categories: Predicate functions

Option variable: maperror

  Default value: `true`

  When `maperror` is `false`, causes all of the mapping functions, for example

      map (f, expr_1, expr_2, ...)

  to (1) stop when they finish going down the shortest expr_i if not all of the expr_i are of the same length and (2) apply f to [expr_1, expr_2, …] if the expr_i are not all the same type of object.

  If `maperror` is `true` then an error message is displayed in the above two instances.

  Categories: Function application

Option variable: mapprint

  Default value: `true`

  When `mapprint` is `true`, various information messages from `map`, `maplist`, and `fullmap` are produced in certain situations. These include situations where `map` would use `apply`, or `map` is truncating on the shortest list.

  If `mapprint` is `false`, these messages are suppressed.

  Categories: Function application

Function: maplist (f, expr_1, …, expr_n)

  Returns a list of the applications of f to the parts of the expressions expr_1, …, expr_n. f is the name of a function, or a lambda expression.

  `maplist` differs from `map(f, expr_1, ..., expr_n)` which returns an expression with the same main operator as expr_i has (except for simplifications and the case where `map` does an `apply`).

  Categories: Function application

Option variable: prederror

  Default value: `false`

  When `prederror` is `true`, an error message is displayed whenever the predicate of an `if` statement or an `is` function fails to evaluate to either `true` or `false`.

  If `false`, `unknown` is returned instead in this case. The `prederror: false` mode is not supported in translated code; however, `maybe` is supported in translated code.

  See also `is` and `maybe`.

  Categories: Programming · Predicate functions

Function: return (value)

  May be used to exit explicitly from the current `block`, `while`, `for` or `do` loop bringing its argument. It therefore can be compared with the `return` statement found in other programming languages but it yields one difference: In maxima only returns from the current block, not from the entire function it was called in. In this aspect it more closely resembles the `break` statement from C.

      (%i1) for i:1 thru 10 do o:i;
      (%o1)                         done
      (%i2) for i:1 thru 10 do if i=3 then return(i);
      (%o2)                           3
      (%i3) for i:1 thru 10 do
          (
              block([i],
                  i:3,
                  return(i)
              ),
              return(8)
          );
      (%o3)                           8
      (%i4) block([i],
          i:4,
          block([o],
              o:5,
              return(o)
          ),
          return(i),
          return(10)
       );
      (%o4)                           4

  See also `for`, `while`, `do` and `block`.

  Categories: Programming

Function: scanmap
`scanmap` (f, expr)
`scanmap` (f, expr, bottomup)

  Recursively applies f to expr, in a top down manner. This is most useful when complete factorization is desired, for example:

      (%i1) exp:(a^2+2*a+1)*y + x^2$
      (%i2) scanmap(factor,exp);
                                          2      2
      (%o2)                         (a + 1)  y + x

  Note the way in which `scanmap` applies the given function `factor` to the constituent subexpressions of expr; if another form of expr is presented to `scanmap` then the result may be different. Thus, `%o2` is not recovered when `scanmap` is applied to the expanded form of `exp`:

      (%i3) scanmap(factor,expand(exp));
                                 2                  2
      (%o3)                      a  y + 2 a y + y + x

  Here is another example of the way in which `scanmap` recursively applies a given function to all subexpressions, including exponents:

      (%i4) expr : u*v^(a*x+b) + c$
      (%i5) scanmap('f, expr);
                          f(f(f(a) f(x)) + f(b))
      (%o5) f(f(f(u) f(f(v)                      )) + f(c))

  `scanmap (f, expr, bottomup)` applies f to expr in a bottom-up manner. E.g., for undefined `f`,

      scanmap(f,a*x+b) ->
         f(a*x+b) -> f(f(a*x)+f(b)) -> f(f(f(a)*f(x))+f(b))
      scanmap(f,a*x+b,bottomup) -> f(a)*f(x)+f(b)
          -> f(f(a)*f(x))+f(b) ->
           f(f(f(a)*f(x))+f(b))

  In this case, you get the same answer both ways.

  Categories: Function application

Function: throw (expr)

  Evaluates expr and throws the value back to the most recent `catch`. `throw` is used with `catch` as a nonlocal return mechanism.

  Categories: Programming

Function: outermap (f, a_1, …, a_n)

  Applies the function f to each one of the elements of the outer product a_1 cross a_2 … cross a_n.

  f is the name of a function of n arguments or a lambda expression of n arguments. Each argument a_k may be a list or nested list, or a matrix, or any other kind of expression.

  The `outermap` return value is a nested structure. Let x be the return value. Then x has the same structure as the first list, nested list, or matrix argument, `x[i_1]...[i_m]` has the same structure as the second list, nested list, or matrix argument, `x[i_1]...[i_m][j_1]...[j_n]` has the same structure as the third list, nested list, or matrix argument, and so on, where m, n, … are the numbers of indices required to access the elements of each argument (one for a list, two for a matrix, one or more for a nested list). Arguments which are not lists or matrices have no effect on the structure of the return value.

  Note that the effect of `outermap` is different from that of applying f to each one of the elements of the outer product returned by `cartesian_product`. `outermap` preserves the structure of the arguments in the return value, while `cartesian_product` does not.

  `outermap` evaluates its arguments.

  See also `map`, `maplist`, and `apply`.

  Examples:

  Elementary examples of `outermap`. To show the argument combinations more clearly, `F` is left undefined.

      (%i1) outermap (F, [a, b, c], [1, 2, 3]);
      (%o1) [[F(a, 1), F(a, 2), F(a, 3)], [F(b, 1), F(b, 2), F(b, 3)],
                                           [F(c, 1), F(c, 2), F(c, 3)]]
      (%i2) outermap (F, matrix ([a, b], [c, d]), matrix ([1, 2], [3, 4]));
               [ [ F(a, 1)  F(a, 2) ]  [ F(b, 1)  F(b, 2) ] ]
               [ [                  ]  [                  ] ]
               [ [ F(a, 3)  F(a, 4) ]  [ F(b, 3)  F(b, 4) ] ]
      (%o2)    [                                            ]
               [ [ F(c, 1)  F(c, 2) ]  [ F(d, 1)  F(d, 2) ] ]
               [ [                  ]  [                  ] ]
               [ [ F(c, 3)  F(c, 4) ]  [ F(d, 3)  F(d, 4) ] ]
      (%i3) outermap (F, [a, b], x, matrix ([1, 2], [3, 4]));
             [ F(a, x, 1)  F(a, x, 2) ]  [ F(b, x, 1)  F(b, x, 2) ]
      (%o3) [[                        ], [                        ]]
             [ F(a, x, 3)  F(a, x, 4) ]  [ F(b, x, 3)  F(b, x, 4) ]
      (%i4) outermap (F, [a, b], matrix ([1, 2]), matrix ([x], [y]));
             [ [ F(a, 1, x) ]  [ F(a, 2, x) ] ]
      (%o4) [[ [            ]  [            ] ],
             [ [ F(a, 1, y) ]  [ F(a, 2, y) ] ]
                                    [ [ F(b, 1, x) ]  [ F(b, 2, x) ] ]
                                    [ [            ]  [            ] ]]
                                    [ [ F(b, 1, y) ]  [ F(b, 2, y) ] ]
      (%i5) outermap ("+", [a, b, c], [1, 2, 3]);
      (%o5) [[a + 1, a + 2, a + 3], [b + 1, b + 2, b + 3],
                                                 [c + 1, c + 2, c + 3]]

  A closer examination of the `outermap` return value. The first, second, and third arguments are a matrix, a list, and a matrix, respectively. The return value is a matrix. Each element of that matrix is a list, and each element of each list is a matrix.

      (%i1) arg_1 :  matrix ([a, b], [c, d]);
                                  [ a  b ]
      (%o1)                       [      ]
                                  [ c  d ]
      (%i2) arg_2 : [11, 22];
      (%o2)                       [11, 22]
      (%i3) arg_3 : matrix ([xx, yy]);
      (%o3)                      [ xx  yy ]
      (%i4) xx_0 : outermap (lambda ([x, y, z], x / y + z), arg_1,
                                                         arg_2, arg_3);
                     [  [      a        a  ]  [      a        a  ]  ]
                     [ [[ xx + --  yy + -- ], [ xx + --  yy + -- ]] ]
                     [  [      11       11 ]  [      22       22 ]  ]
      (%o4)  Col 1 = [                                              ]
                     [  [      c        c  ]  [      c        c  ]  ]
                     [ [[ xx + --  yy + -- ], [ xx + --  yy + -- ]] ]
                     [  [      11       11 ]  [      22       22 ]  ]
                       [  [      b        b  ]  [      b        b  ]  ]
                       [ [[ xx + --  yy + -- ], [ xx + --  yy + -- ]] ]
                       [  [      11       11 ]  [      22       22 ]  ]
               Col 2 = [                                              ]
                       [  [      d        d  ]  [      d        d  ]  ]
                       [ [[ xx + --  yy + -- ], [ xx + --  yy + -- ]] ]
                       [  [      11       11 ]  [      22       22 ]  ]
      (%i5) xx_1 : xx_0 [1][1];
                 [      a        a  ]  [      a        a  ]
      (%o5)     [[ xx + --  yy + -- ], [ xx + --  yy + -- ]]
                 [      11       11 ]  [      22       22 ]
      (%i6) xx_2 : xx_0 [1][1] [1];
                            [      a        a  ]
      (%o6)                 [ xx + --  yy + -- ]
                            [      11       11 ]
      (%i7) xx_3 : xx_0 [1][1] [1] [1][1];
                                        a
      (%o7)                        xx + --
                                        11
      (%i8) [op (arg_1), op (arg_2), op (arg_3)];
      (%o8)                  [matrix, [, matrix]
      (%i9) [op (xx_0), op (xx_1), op (xx_2)];
      (%o9)                  [matrix, [, matrix]

  `outermap` preserves the structure of the arguments in the return value, while `cartesian_product` does not.

      (%i1) outermap (F, [a, b, c], [1, 2, 3]);
      (%o1) [[F(a, 1), F(a, 2), F(a, 3)], [F(b, 1), F(b, 2), F(b, 3)],
                                           [F(c, 1), F(c, 2), F(c, 3)]]
      (%i2) setify (flatten (%));
      (%o2) {F(a, 1), F(a, 2), F(a, 3), F(b, 1), F(b, 2), F(b, 3),
                                             F(c, 1), F(c, 2), F(c, 3)}
      (%i3) map (lambda ([L], apply (F, L)),
                           cartesian_product ({a, b, c}, {1, 2, 3}));
      (%o3) {F(a, 1), F(a, 2), F(a, 3), F(b, 1), F(b, 2), F(b, 3),
                                             F(c, 1), F(c, 2), F(c, 3)}
      (%i4) is (equal (%, %th (2)));
      (%o4)                         true

  Categories: Function application

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# 38. Debugging

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## 38.1 Source Level Debugging

Maxima has a built-in source level debugger. The user can set a breakpoint at a function, and then step line by line from there. The call stack may be examined, together with the variables bound at that level.

The command `:help` or `:h` shows the list of debugger commands. (In general, commands may be abbreviated if the abbreviation is unique. If not unique, the alternatives will be listed.) Within the debugger, the user can also use any ordinary Maxima functions to examine, define, and manipulate variables and expressions.

A breakpoint is set by the `:br` command at the Maxima prompt. Within the debugger, the user can advance one line at a time using the `:n` ("next") command. The `:bt` ("backtrace") command shows a list of stack frames. The `:r` ("resume") command exits the debugger and continues with execution. These commands are demonstrated in the example below.

    (%i1) load ("/tmp/foobar.mac");

    (%o1)                           /tmp/foobar.mac

    (%i2) :br foo
    Turning on debugging debugmode(true)
    Bkpt 0 for foo (in /tmp/foobar.mac line 1)

    (%i2) bar (2,3);
    Bkpt 0:(foobar.mac 1)
    /tmp/foobar.mac:1::

    (dbm:1) :bt                        <-- :bt typed here gives a backtrace
    #0: foo(y=5)(foobar.mac line 1)
    #1: bar(x=2,y=3)(foobar.mac line 9)

    (dbm:1) :n                         <-- Here type :n to advance line
    (foobar.mac 2)
    /tmp/foobar.mac:2::

    (dbm:1) :n                         <-- Here type :n to advance line
    (foobar.mac 3)
    /tmp/foobar.mac:3::

    (dbm:1) u;                         <-- Investigate value of u
    28

    (dbm:1) u: 33;                     <-- Change u to be 33
    33

    (dbm:1) :r                         <-- Type :r to resume the computation

    (%o2)                                1094

The file `/tmp/foobar.mac` is the following:

    foo(y) := block ([u:y^2],
      u: u+3,
      u: u^2,
      u);

    bar(x,y) := (
      x: x+2,
      y: y+2,
      x: foo(y),
      x+y);

USE OF THE DEBUGGER THROUGH EMACS

If the user is running the code under GNU emacs in a shell window (dbl shell), or is running the graphical interface version, Xmaxima, then if he stops at a break point, he will see his current position in the source file which will be displayed in the other half of the window, either highlighted in red, or with a little arrow pointing at the right line. He can advance single lines at a time by typing M-n (Alt-n).

Under Emacs you should run in a `dbl` shell, which requires the `dbl.el` file in the elisp directory. Make sure you install the elisp files or add the Maxima elisp directory to your path: e.g., add the following to your ``.emacs'` file or the ``site-init.el'`

    (setq load-path (cons "/usr/share/maxima/5.9.1/emacs" load-path))
    (autoload 'dbl "dbl")

then in emacs

    M-x dbl

should start a shell window in which you can run programs, for example Maxima, gcl, gdb etc. This shell window also knows about source level debugging, and display of source code in the other window.

The user may set a break point at a certain line of the file by typing `C-x space`. This figures out which function the cursor is in, and then it sees which line of that function the cursor is on. If the cursor is on, say, line 2 of `foo`, then it will insert in the other window the command, "`:br foo 2`", to break `foo` at its second line. To have this enabled, the user must have maxima-mode.el turned on in the window in which the file `foobar.mac` is visiting. There are additional commands available in that file window, such as evaluating the function into the Maxima, by typing `Alt-Control-x`.

Categories: Debugging

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## 38.2 Keyword Commands

Keyword commands are special keywords which are not interpreted as Maxima expressions. A keyword command can be entered at the Maxima prompt or the debugger prompt, although not at the break prompt. Keyword commands start with a colon, '`:`'. For example, to evaluate a Lisp form you may type `:lisp` followed by the form to be evaluated.

    (%i1) :lisp (+ 2 3)
    5

The number of arguments taken depends on the particular command. Also, you need not type the whole command, just enough to be unique among the break keywords. Thus `:br` would suffice for `:break`.

The keyword commands are listed below.

`:break F n`

  Set a breakpoint in function `F` at line offset `n` from the beginning of the function. If `F` is given as a string, then it is assumed to be a file, and `n` is the offset from the beginning of the file. The offset is optional. If not given, it is assumed to be zero (first line of the function or file).

`:bt`

  Print a backtrace of the stack frames

`:continue`

  Continue the computation

`:delete`

  Delete the specified breakpoints, or all if none are specified

`:disable`

  Disable the specified breakpoints, or all if none are specified

`:enable`

  Enable the specified breakpoints, or all if none are specified

`:frame n`

  Print stack frame `n`, or the current frame if none is specified

`:help`

  Print help on a debugger command, or all commands if none is specified

`:info`

  Print information about item

`:lisp some-form`

  Evaluate `some-form` as a Lisp form

`:lisp-quiet some-form`

  Evaluate Lisp form `some-form` without any output

`:next`

  Like `:step`, except `:next` steps over function calls

`:quit`

  Quit the current debugger level without completing the computation

`:resume`

  Continue the computation

`:step`

  Continue the computation until it reaches a new source line

`:top`

  Return to the Maxima prompt (from any debugger level) without completing the computation

Categories: Debugging

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## 38.3 Functions and Variables for Debugging

Option variable: debugmode

  Default value: `false`

  When a Maxima error occurs, Maxima will start the debugger if `debugmode` is `true`. The user may enter commands to examine the call stack, set breakpoints, step through Maxima code, and so on. See `debugging` for a list of debugger commands.

  Enabling `debugmode` will not catch Lisp errors.

  Categories: Debugging · Global flags

Option variable: refcheck

  Default value: `false`

  When `refcheck` is `true`, Maxima prints a message each time a bound variable is used for the first time in a computation.

  Categories: Evaluation · Console interaction · Global flags

Option variable: setcheck

  Default value: `false`

  If `setcheck` is set to a list of variables (which can be subscripted), Maxima prints a message whenever the variables, or subscripted occurrences of them, are bound with the ordinary assignment operator `:`, the `::` assignment operator, or function argument binding, but not the function assignment `:=` nor the macro assignment `::=` operators. The message comprises the name of the variable and the value it is bound to.

  `setcheck` may be set to `all` or `true` thereby including all variables.

  Each new assignment of `setcheck` establishes a new list of variables to check, and any variables previously assigned to `setcheck` are forgotten.

  The names assigned to `setcheck` must be quoted if they would otherwise evaluate to something other than themselves. For example, if `x`, `y`, and `z` are already bound, then enter

      setcheck: ['x, 'y, 'z]$

  to put them on the list of variables to check.

  No printout is generated when a variable on the `setcheck` list is assigned to itself, e.g., `X: 'X`.

  Categories: Console interaction · Global flags

Option variable: setcheckbreak

  Default value: `false`

  When `setcheckbreak` is `true`, Maxima will present a break prompt whenever a variable on the `setcheck` list is assigned a new value. The break occurs before the assignment is carried out. At this point, `setval` holds the value to which the variable is about to be assigned. Hence, one may assign a different value by assigning to `setval`.

  See also `setcheck` and `setval`.

  Categories: Console interaction · Global flags

System variable: setval

  Holds the value to which a variable is about to be set when a `setcheckbreak` occurs. Hence, one may assign a different value by assigning to `setval`.

  See also `setcheck` and `setcheckbreak`.

  Categories: Console interaction · Global variables

Function: timer (f_1, …, f_n)
`timer` (all)
`timer` ()

  Given functions f_1, …, f_n, `timer` puts each one on the list of functions for which timing statistics are collected. `timer(f)$ timer(g)$` puts `f` and then `g` onto the list; the list accumulates from one call to the next.

  `timer(all)` puts all user-defined functions (as named by the global variable `functions`) on the list of timed functions.

  With no arguments, `timer` returns the list of timed functions.

  Maxima records how much time is spent executing each function on the list of timed functions. `timer_info` returns the timing statistics, including the average time elapsed per function call, the number of calls, and the total time elapsed. `untimer` removes functions from the list of timed functions.

  `timer` quotes its arguments. `f(x) := x^2$ g:f$ timer(g)$` does not put `f` on the timer list.

  If `trace(f)` is in effect, then `timer(f)` has no effect; `trace` and `timer` cannot both be in effect at the same time.

  See also `timer_devalue`.

  Categories: Debugging

Function: untimer (f_1, …, f_n)
`untimer` ()

  Given functions f_1, …, f_n, `untimer` removes each function from the timer list.

  With no arguments, `untimer` removes all functions currently on the timer list.

  After `untimer (f)` is executed, `timer_info (f)` still returns previously collected timing statistics, although `timer_info()` (with no arguments) does not return information about any function not currently on the timer list. `timer (f)` resets all timing statistics to zero and puts `f` on the timer list again.

  Categories: Debugging

Option variable: timer_devalue

  Default value: `false`

  When `timer_devalue` is `true`, Maxima subtracts from each timed function the time spent in other timed functions. Otherwise, the time reported for each function includes the time spent in other functions. Note that time spent in untimed functions is not subtracted from the total time.

  See also `timer` and `timer_info`.

  Categories: Debugging · Global flags

Function: timer_info (f_1, ..., f_n)
`timer_info` ()

  Given functions f_1, ..., f_n, `timer_info` returns a matrix containing timing information for each function. With no arguments, `timer_info` returns timing information for all functions currently on the timer list.

  The matrix returned by `timer_info` contains the function name, time per function call, number of function calls, total time, and `gctime`, which meant "garbage collection time" in the original Macsyma but is now always zero.

  The data from which `timer_info` constructs its return value can also be obtained by the `get` function:

      get(f, 'calls);  get(f, 'runtime);  get(f, 'gctime);

  See also `timer`.

  Categories: Debugging

Function: trace (f_1, …, f_n)
`trace` (all)
`trace` ()

  Given functions f_1, …, f_n, `trace` instructs Maxima to print out debugging information whenever those functions are called. `trace(f)$ trace(g)$` puts `f` and then `g` onto the list of functions to be traced; the list accumulates from one call to the next.

  `trace(all)` puts all user-defined functions (as named by the global variable `functions`) on the list of functions to be traced.

  With no arguments, `trace` returns a list of all the functions currently being traced.

  The `untrace` function disables tracing. See also `trace_options`.

  `trace` quotes its arguments. Thus, `f(x) := x^2$ g:f$ trace(g)$` does not put `f` on the trace list.

  When a function is redefined, it is removed from the timer list. Thus after `timer(f)$ f(x) := x^2$`, function `f` is no longer on the timer list.

  If `timer (f)` is in effect, then `trace (f)` has no effect; `trace` and `timer` can't both be in effect for the same function.

  Categories: Debugging

Function: trace_options (f, option_1, …, option_n)
`trace_options` (f)

  Sets the trace options for function f. Any previous options are superseded. `trace_options (f, ...)` has no effect unless `trace (f)` is also called (either before or after `trace_options`).

  `trace_options (f)` resets all options to their default values.

  The option keywords are:

  - `noprint` Do not print a message at function entry and exit.
  - `break` Put a breakpoint before the function is entered, and after the function is exited. See `break`.
  - `lisp_print` Display arguments and return values as Lisp objects.
  - `info` Print `-> true` at function entry and exit.
  - `errorcatch` Catch errors, giving the option to signal an error, retry the function call, or specify a return value.

  Trace options are specified in two forms. The presence of the option keyword alone puts the option into effect unconditionally. (Note that option foo is not put into effect by specifying `foo: true` or a similar form; note also that keywords need not be quoted.) Specifying the option keyword with a predicate function makes the option conditional on the predicate.

  The argument list to the predicate function is always `[level, direction, function, item]` where `level` is the recursion level for the function, `direction` is either `enter` or `exit`, `function` is the name of the function, and `item` is the argument list (on entering) or the return value (on exiting).

  Here is an example of unconditional trace options:

      (%i1) ff(n) := if equal(n, 0) then 1 else n * ff(n - 1)$

      (%i2) trace (ff)$

      (%i3) trace_options (ff, lisp_print, break)$

      (%i4) ff(3);

  Here is the same function, with the `break` option conditional on a predicate:

      (%i5) trace_options (ff, break(pp))$

      (%i6) pp (level, direction, function, item) := block (print (item),
          return (function = 'ff and level = 3 and direction = exit))$

      (%i7) ff(6);

  Categories: Debugging

Function: untrace
`untrace` (f_1, …, f_n)
`untrace` ()

  Given functions f_1, …, f_n, `untrace` disables tracing enabled by the `trace` function. With no arguments, `untrace` disables tracing for all functions.

  `untrace` returns a list of the functions for which it disabled tracing.

  Categories: Debugging

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# 39. alt-display

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## 39.1 Introduction to alt-display

The alt-display package provides a means to change the way that Maxima displays its output. The *alt-display1d* and *alt-display2d* Lisp hooks were introduced to Maxima in 2002, but were not easily accessible from the Maxima REPL until the introduction of this package.

The package provides a general purpose function to define alternative display functions, and a separate function to set the display function. The package also provides customized display functions to produce output in TeX, Texinfo, XML and all three output formats within Texinfo.

Here is a sample session:

    (%i1) load("alt-display.mac")$
    (%i2) set_alt_display(2,tex_display)$

    (%i3) x/(x^2+y^2) = 1;
    \mbox{\tt\red({\it \%o_3}) \black}$${{x}\over{y^2+x^2}}=1$$

    (%i4) set_alt_display(2,mathml_display)$

    (%i5) x/(x^2+y^2) = 1;
    <math xmlns="http://www.w3.org/1998/Math/MathML"> <mi>mlabel</mi>
    <mfenced separators=""><msub><mi>%o</mi> <mn>5</mn></msub>
    <mo>,</mo><mfrac><mrow><mi>x</mi> </mrow> <mrow><msup><mrow>
    <mi>y</mi> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup><mrow>
    <mi>x</mi> </mrow> <mn>2</mn> </msup> </mrow></mfrac> <mo>=</mo>
    <mn>1</mn> </mfenced> </math>

    (%i6) set_alt_display(2,multi_display_for_texinfo)$

    (%i7) x/(x^2+y^2) = 1;

    @iftex
    @tex
    \mbox{\tt\red({\it \%o_7}) \black}$${{x}\over{y^2+x^2}}=1$$
    @end tex
    @end iftex
    @ifhtml
    @html

    <math xmlns="http://www.w3.org/1998/Math/MathML"> <mi>mlabel</mi>
    <mfenced separators=""><msub><mi>%o</mi> <mn>7</mn></msub>
    <mo>,</mo><mfrac><mrow><mi>x</mi> </mrow> <mrow><msup><mrow>
    <mi>y</mi> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup><mrow>
    <mi>x</mi> </mrow> <mn>2</mn> </msup> </mrow></mfrac> <mo>=</mo>
    <mn>1</mn> </mfenced> </math>
    @end html
    @end ifhtml
    @ifinfo
    @example
    (%o7) x/(y^2+x^2) = 1
    @end example
    @end ifinfo

If the alternative display function causes an error, the error is trapped and the display function is reset to the default display. In the following example, the `error` function is set to display the output. This throws an error, which is handled by resetting the 2d-display to the default.

    (%i8) set_alt_display(2,?error)$

    (%i9) x;

    Error in *alt-display2d*.
    Messge: Condition designator ((MLABEL) $%O9 $X) is not of type (OR SYMBOL STRING
                                                                 FUNCTION).
    *alt-display2d* reset to nil.
     -- an error. To debug this try: debugmode(true);

    (%i10) x;
    (%o10)                                 x

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## 39.2 Functions and Variables for alt-display

Function: define_alt_display (function(input), expr)

  This function is similar to `define`: it evaluates its arguments and expands into a function definition. The function is a function of a single input input. For convenience, a substitution is applied to expr after evaluation, to provide easy access to Lisp variable names.

  Set a time-stamp on each prompt:

      (%i1) load("alt-display.mac")$

      (%i2) display2d:false$

      (%i3) define_alt_display(time_stamp(x),
                         block([alt_display1d:false,alt_display2d:false],
                               simp:true,
                               prompt_prefix:printf(false,"~a~%",timedate()),
                               displa(x)));

      (%o3) time_stamp(x):=block([simp:false],
                       block([?\*alt\-display1d\*:false,?\*alt\-display2d\*:false],
                             simp:true,
                             ?\*prompt\-prefix\*:printf(false,"~a~%",timedate()),
                             ?displa(x)))
      (%i4) set_alt_display(1,time_stamp);

      (%o4) done
      2016-10-03 09:49:27-05:00
      (%i5)

  The input line `%i3` defines `time_stamp` using `define_alt_display`. The output line `%o3` shows that the Maxima variable names `alt_display1d`, `alt_display2d` and `prompt_prefix` have been replaced by their Lisp translations, as has `displa` been replaced by `?displa` (the display function).

  The display variables `alt_display1d` and `alt_display2d` are both bound to `false` in the body of `time_stamp` to prevent an infinite recursion in `displa`.

  Categories: Package alt-display

Function: info_display (form)

  This is an alias for the default 1-d display function. It may be used as an alternative 1-d or 2-d display function.

      (%i1) load("alt-display.mac")$

      (%i2) set_alt_display(2,info_display);

      (%o2) done
      (%i3) x/y;

      (%o3) x/y

  Categories: Package alt-display

Function: mathml_display (form)

  Produces MathML output.

      (%i1) load("alt-display.mac")$

      (%i2) set_alt_display(2,mathml_display);
      <math xmlns="http://www.w3.org/1998/Math/MathML"> <mi>mlabel</mi>
       <mfenced separators=""><msub><mi>%o</mi> <mn>2</mn></msub>
       <mo>,</mo><mi>done</mi> </mfenced> </math>

  Categories: Package alt-display

Function: tex_display (form)

  Produces TeX output.

      (%i2) set_alt_display(2,tex_display);
      \mbox{\tt\red({\it \%o_2}) \black}$$\mathbf{done}$$
      (%i3) x/(x^2+y^2);
      \mbox{\tt\red({\it \%o_3}) \black}$${{x}\over{y^2+x^2}}$$

  Categories: Package alt-display

Function: multi_display_for_texinfo (form)

  Produces Texinfo output using all three display functions.

      (%i2) set_alt_display(2,multi_display_for_texinfo)$

      (%i3) x/(x^2+y^2);

      @iftex
      @tex
      \mbox{\tt\red({\it \%o_3}) \black}$${{x}\over{y^2+x^2}}$$
      @end tex
      @end iftex
      @ifhtml
      @html

         <math xmlns="http://www.w3.org/1998/Math/MathML"> <mi>mlabel</mi>
         <mfenced separators=""><msub><mi>%o</mi> <mn>3</mn></msub>
         <mo>,</mo><mfrac><mrow><mi>x</mi> </mrow> <mrow><msup><mrow>
         <mi>y</mi> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup><mrow>
         <mi>x</mi> </mrow> <mn>2</mn> </msup> </mrow></mfrac> </mfenced> </math>
      @end html
      @end ifhtml
      @ifinfo
      @example
      (%o3) x/(y^2+x^2)
      @end example
      @end ifinfo

  Categories: Package alt-display

Functions: reset_displays ()

  Resets the prompt prefix and suffix to the empty string, and sets both 1-d and 2-d display functions to the default.

  Categories: Package alt-display

Function: set_alt_display (num, display-function)

  The input num is the display to set; it may be either 1 or 2. The second input display-function is the display function to use. The display function may be either a Maxima function or a `lambda` expression.

  Here is an example where the display function is a `lambda` expression; it just displays the result as TeX.

      (%i1) load("alt-display.mac")$

      (%i2) set_alt_display(2, lambda([form], tex(?caddr(form))))$

      (%i3) integrate(exp(-t^2),t,0,inf);
      $${{\sqrt{\pi}}\over{2}}$$

  A user-defined display function should take care that it prints its output. A display function that returns a string will appear to display nothing, nor cause any errors.

  Categories: Package alt-display

Function: set_prompt (fix, expr)

  Set the prompt prefix or suffix to expr. The input fix must evaluate to one of `prefix`, `suffix`, `general`, `prolog` or `epilog`. The input expr must evaluate to either a string or `false`; if `false`, the fix is reset to the default value.

      (%i1) load("alt-display.mac")$
      (%i2) set_prompt('prefix,printf(false,"It is now: ~a~%",timedate()))$

      It is now: 2014-01-07 15:23:23-05:00
      (%i3)

  The following example shows the effect of each option, except `prolog`. Note that the `epilog` prompt is printed as Maxima closes down. The `general` is printed between the end of input and the output, unless the input line ends in `$`.

  Here is an example to show where the prompt strings are placed.

      (%i1) load("alt-display.mac")$

      (%i2) set_prompt(prefix,"<<prefix>> ",suffix,"<<suffix>> ",general,
                 printf(false,"<<general>>~%"),epilog,printf(false,"<<epilog>>~%"));

      (%o2)                                done
      <<prefix>> (%i3) <<suffix>> x/y;
      <<general>>
                                             x
      (%o3)                                  -
                                             y
      <<prefix>> (%i4) <<suffix>> quit();
      <<general>>
      <<epilog>>

  Here is an example that shows how to colorize the input and output when Maxima is running in a terminal or terminal emulator like Emacs(8).

  Each prompt string starts with the ASCII escape character (27) followed by an open square bracket (91); each string ends with a lower-case m (109). The webpages http://misc.flogisoft.com/bash/tip_colors_and_formatting and http://www.tldp.org/HOWTO/Bash-Prompt-HOWTO/x329.html provide information on how to use control strings to set the terminal colors.

  Categories: Package alt-display

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# 40. asympa

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## 40.1 Introduction to asympa

Function: asympa

  `asympa` is a package for asymptotic analysis. The package contains simplification functions for asymptotic analysis, including the "big O" and "little o" functions that are widely used in complexity analysis and numerical analysis.

  `load ("asympa")` loads this package.

  Categories: Package asympa

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## 40.2 Functions and variables for asympa

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# 41. augmented_lagrangian

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## 41.1 Functions and Variables for augmented_lagrangian

Function: augmented_lagrangian_method
`augmented_lagrangian_method` (FOM, xx, C, yy)
`augmented_lagrangian_method` (FOM, xx, C, yy, optional_args)
`augmented_lagrangian_method` ([FOM, grad], xx, C, yy)
`augmented_lagrangian_method` ([FOM, grad], xx, C, yy, optional_args)

  Returns an approximate minimum of the expression FOM with respect to the variables xx, holding the constraints C equal to zero. yy is a list of initial guesses for xx. The method employed is the augmented Lagrangian method (see Refs [1] and [2]).

  grad, if present, is the gradient of FOM with respect to xx, represented as a list of expressions, one for each variable in xx. If not present, the gradient is constructed automatically.

  FOM and each element of grad, if present, must be ordinary expressions, not names of functions or lambda expressions.

  `optional_args` represents additional arguments, specified as `symbol = value`. The optional arguments recognized are:

  `niter`

    Number of iterations of the augmented Lagrangian algorithm

  `lbfgs_tolerance`

    Tolerance supplied to LBFGS

  `iprint`

    IPRINT parameter (a list of two integers which controls verbosity) supplied to LBFGS

  `%lambda`

    Initial value of `%lambda` to be used for calculating the augmented Lagrangian

  This implementation minimizes the augmented Lagrangian by applying the limited-memory BFGS (LBFGS) algorithm, which is a quasi-Newton algorithm.

  `load("augmented_lagrangian")` loads this function.

  See also lbfgs

  References:

  [1] http://www-fp.mcs.anl.gov/otc/Guide/OptWeb/continuous/constrained/nonlinearcon/auglag.html

  [2] http://www.cs.ubc.ca/spider/ascher/542/chap10.pdf

  Examples:

      (%i1) load ("lbfgs");
      (%o1)  /home/gunter/src/maxima-code/share/lbfgs/lbfgs.mac
      (%i2) load ("augmented_lagrangian");
      (%o2) /home/gunter/src/maxima-code/share/contrib/augmented_lagra\
      ngian.mac
      (%i3) FOM: x^2 + 2*y^2;
                                     2    2
      (%o3)                       2 y  + x
      (%i4) xx: [x, y];
      (%o4)                        [x, y]
      (%i5) C: [x + y - 1];
      (%o5)                      [y + x - 1]
      (%i6) yy: [1, 1];
      (%o6)                        [1, 1]
      (%i7) augmented_lagrangian_method(FOM, xx, C, yy, iprint=[-1,0]);
      (%o7) [[x = 0.666659841080023, y = 0.333340272455448],
                                       %lambda = [- 1.333337940892518]]

  Same example as before, but this time the gradient is supplied as an argument.

      (%i1) load ("lbfgs")$
      (%i2) load ("augmented_lagrangian")$
      (%i3) FOM: x^2 + 2*y^2;
                                     2    2
      (%o3)                       2 y  + x
      (%i4) xx: [x, y];
      (%o4)                        [x, y]
      (%i5) grad : [2*x, 4*y];
      (%o5)                      [2 x, 4 y]
      (%i6) C: [x + y - 1];
      (%o6)                      [y + x - 1]
      (%i7) yy: [1, 1];
      (%o7)                        [1, 1]
      (%i8) augmented_lagrangian_method ([FOM, grad], xx, C, yy,
                                   iprint = [-1, 0]);
      (%o8) [[x = 0.6666598410800247, y = 0.3333402724554464],
                                       %lambda = [- 1.333337940892525]]

  Categories: Numerical methods · Optimization · Share packages · Package augmented_lagrangian

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# 42. Bernstein

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## 42.1 Functions and Variables for Bernstein

Function: bernstein_poly (k, n, x)

  Provided `k` is not a negative integer, the Bernstein polynomials are defined by `bernstein_poly(k,n,x) = binomial(n,k) x^k (1-x)^(n-k)`; for a negative integer `k`, the Bernstein polynomial `bernstein_poly(k,n,x)` vanishes. When either `k` or `n` are non integers, the option variable `bernstein_explicit` controls the expansion of the Bernstein polynomials into its explicit form; example:

      (%i1) load(bernstein)$

      (%i2) bernstein_poly(k,n,x);
      (%o2)                bernstein_poly(k, n, x)
      (%i3) bernstein_poly(k,n,x), bernstein_explicit : true;
                                             n - k  k
      (%o3)            binomial(n, k) (1 - x)      x

  The Bernstein polynomials have both a gradef property and an integrate property:

      (%i4) diff(bernstein_poly(k,n,x),x);
      (%o4) (bernstein_poly(k - 1, n - 1, x)
                                       - bernstein_poly(k, n - 1, x)) n
      (%i5) integrate(bernstein_poly(k,n,x),x);
      (%o5)
                                                                  k + 1
       hypergeometric([k + 1, k - n], [k + 2], x) binomial(n, k) x
       ----------------------------------------------------------------
                                    k + 1

  For numeric inputs, both real and complex, the Bernstein polynomials evaluate to a numeric result:

      (%i6) bernstein_poly(5,9, 1/2 + %i);
                              39375 %i   39375
      (%o6)                   -------- + -----
                                128       256
      (%i7) bernstein_poly(5,9, 0.5b0 + %i);
      (%o7)           3.076171875b2 %i + 1.5380859375b2

  To use `bernstein_poly`, first `load("bernstein")`.

Variable: bernstein_explicit

  Default value: `false`

  When either `k` or `n` are non integers, the option variable `bernstein_explicit` controls the expansion of `bernstein(k,n,x)` into its explicit form; example:

      (%i1) bernstein_poly(k,n,x);
      (%o1)                bernstein_poly(k, n, x)
      (%i2) bernstein_poly(k,n,x), bernstein_explicit : true;
                                             n - k  k
      (%o2)            binomial(n, k) (1 - x)      x

  When both `k` and `n` are explicitly integers, `bernstein(k,n,x)` always expands to its explicit form.

Function: multibernstein_poly ([k1,k2,…, kp], [n1,n2,…, np], [x1,x2,…, xp])

  The multibernstein polynomial `multibernstein_poly ([k1, k2, ..., kp], [n1, n2, ..., np], [x1, x2, ..., xp])` is the product of bernstein polynomials `bernstein_poly(k1, n1, x1) bernstein_poly(k2, n2, x2) ... bernstein_poly(kp, np, xp)`.

  To use `multibernstein_poly`, first `load("bernstein")`.

Function: bernstein_approx (f, [x1, x1, …, xn], n)

  Return the `n`-th order uniform Bernstein polynomial approximation for the function `(x1, x2, ..., xn) |--> f`. Examples

      (%i1) bernstein_approx(f(x),[x], 2);
                       2       1                          2
      (%o1)      f(1) x  + 2 f(-) (1 - x) x + f(0) (1 - x)
                               2
      (%i2) bernstein_approx(f(x,y),[x,y], 2);
                     2  2       1                2
      (%o2) f(1, 1) x  y  + 2 f(-, 1) (1 - x) x y
                                2
                        2  2          1   2
       + f(0, 1) (1 - x)  y  + 2 f(1, -) x  (1 - y) y
                                      2
             1  1                               1         2
       + 4 f(-, -) (1 - x) x (1 - y) y + 2 f(0, -) (1 - x)  (1 - y) y
             2  2                               2
                  2        2       1                      2
       + f(1, 0) x  (1 - y)  + 2 f(-, 0) (1 - x) x (1 - y)
                                   2
                        2        2
       + f(0, 0) (1 - x)  (1 - y)

  To use `bernstein_approx`, first `load("bernstein")`.

Function: bernstein_expand (e, [x1, x1, …, xn])

  Express the polynomial `e` exactly as a linear combination of multi-variable Bernstein polynomials.

      (%i1) bernstein_expand(x*y+1,[x,y]);
      (%o1)    2 x y + (1 - x) y + x (1 - y) + (1 - x) (1 - y)
      (%i2) expand(%);
      (%o2)                        x y + 1

  Maxima signals an error when the first argument isn't a polynomial.

  To use `bernstein_expand`, first `load("bernstein")`.

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# 43. bitwise

The package `bitwise` provides functions that allow to manipulate bits of integer constants. As always maxima attempts to simplify the result of the operation if the actual value of a constant isn't known considering attributes that might be known for the variables, see the `declare` mechanism.

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## 43.1 Functions and Variables for bitwise

Function: bit_not (int)

  Inverts all bits of a signed integer. The result of this action reads `-int - 1`.

      (%i1) load("bitwise")$
      (%i2) bit_not(i);
      (%o2)                      bit_not(i)
      (%i3) bit_not(bit_not(i));
      (%o3)                           i
      (%i4) bit_not(3);
      (%o4)                          - 4
      (%i5) bit_not(100);
      (%o5)                         - 101
      (%i6) bit_not(-101);
      (%o6)                          100

  Categories: Number theory Binary operations Package bitwise

Function: bit_and (int1, ...)

  This function calculates a bitwise `and` of two or more signed integers.

      (%i1) load("bitwise")$
      (%i2) bit_and(i,i);
      (%o2)                           i
      (%i3) bit_and(i,i,i);
      (%o3)                           i
      (%i4) bit_and(1,3);
      (%o4)                           1
      (%i5) bit_and(-7,7);
      (%o5)                           1

  If it is known if one of the parameters to `bit_and` is even this information is taken into consideration by the function.

      (%i1) load("bitwise")$
      (%i2) declare(e,even,o,odd);
      (%o2)                         done
      (%i3) bit_and(1,e);
      (%o3)                           0
      (%i4) bit_and(1,o);
      (%o4)                           1

  Categories: Number theory Binary operations Package bitwise

Function: bit_or (int1, ...)

  This function calculates a bitwise `or` of two or more signed integers.

      (%i1) load("bitwise")$
      (%i2) bit_or(i,i);
      (%o2)                           i
      (%i3) bit_or(i,i,i);
      (%o3)                           i
      (%i4) bit_or(1,3);
      (%o4)                           3
      (%i5) bit_or(-7,7);
      (%o5)                          - 1

  If it is known if one of the parameters to `bit_or` is even this information is taken into consideration by the function.

      (%i1) load("bitwise")$
      (%i2) declare(e,even,o,odd);
      (%o2)                         done
      (%i3) bit_or(1,e);
      (%o3)                         e + 1
      (%i4) bit_or(1,o);
      (%o4)                           o

  Categories: Number theory Binary operations Package bitwise

Function: bit_xor (int1, ...)

  This function calculates a bitwise `or` of two or more signed integers.

      (%i1) load("bitwise")$
      (%i2) bit_xor(i,i);
      (%o2)                           0
      (%i3) bit_xor(i,i,i);
      (%o3)                           i
      (%i4) bit_xor(1,3);
      (%o4)                           2
      (%i5) bit_xor(-7,7);
      (%o5)                          - 2

  If it is known if one of the parameters to `bit_xor` is even this information is taken into consideration by the function.

      (%i1) load("bitwise")$
      (%i2) declare(e,even,o,odd);
      (%o2)                         done
      (%i3) bit_xor(1,e);
      (%o3)                         e + 1
      (%i4) bit_xor(1,o);
      (%o4)                         o - 1

  Categories: Number theory Binary operations Package bitwise

Function: bit_lsh (int, nBits)

  This function shifts all bits of the signed integer `int` to the left by `nBits` bits. The width of the integer is extended by `nBits` for this process. The result of `bit_lsh` therefore is `int * 2`.

      (%i1) load("bitwise")$
      (%i2) bit_lsh(0,1);
      (%o2)                           0
      (%i3) bit_lsh(1,0);
      (%o3)                           1
      (%i4) bit_lsh(1,1);
      (%o4)                           2
      (%i5) bit_lsh(1,i);
      (%o5)                     bit_lsh(1, i)
      (%i6) bit_lsh(-3,1);
      (%o6)                          - 6
      (%i7) bit_lsh(-2,1);
      (%o7)                          - 4

  Categories: Number theory Binary operations Package bitwise

Function: bit_rsh (int, nBits)

  This function shifts all bits of the signed integer `int` to the right by `nBits` bits. The width of the integer is reduced by `nBits` for this process.

      (%i1) load("bitwise")$
      (%i2) bit_rsh(0,1);
      (%o2)                           0
      (%i3) bit_rsh(2,0);
      (%o3)                           2
      (%i4) bit_rsh(2,1);
      (%o4)                           1
      (%i5) bit_rsh(2,2);
      (%o5)                           0
      (%i6) bit_rsh(-3,1);
      (%o6)                          - 2
      (%i7) bit_rsh(-2,1);
      (%o7)                          - 1
      (%i8) bit_rsh(-2,2);
      (%o8)                          - 1

  Categories: Number theory Binary operations Package bitwise

Function: bit_length (int)

  determines how many bits a variable needs to be long in order to store the number `int`. This function only operates on positive numbers.

      (%i1) load("bitwise")$
      (%i2) bit_length(0);
      (%o2)                           0
      (%i3) bit_length(1);
      (%o3)                           1
      (%i4) bit_length(7);
      (%o4)                           3
      (%i5) bit_length(8);
      (%o5)                           4

  Categories: Number theory Binary operations Package bitwise

Function: bit_onep (int, nBit)

  determines if bits `nBit` is set in the signed integer `int`.

      (%i1) load("bitwise")$
      (%i2) bit_onep(85,0);
      (%o2)                         true
      (%i3) bit_onep(85,1);
      (%o3)                         false
      (%i4) bit_onep(85,2);
      (%o4)                         true
      (%i5) bit_onep(85,3);
      (%o5)                         false
      (%i6) bit_onep(85,100);
      (%o6)                         false
      (%i7) bit_onep(i,100);
      (%o7)                   bit_onep(i, 100)

  For signed numbers the sign bit is interpreted to be more than `nBit` to the left of the leftmost bit of `int` that reads `1`.

      (%i1) load("bitwise")$
      (%i2) bit_onep(-2,0);
      (%o2)                         false
      (%i3) bit_onep(-2,1);
      (%o3)                         true
      (%i4) bit_onep(-2,2);
      (%o4)                         true
      (%i5) bit_onep(-2,3);
      (%o5)                         true
      (%i6) bit_onep(-2,4);
      (%o6)                         true

  If it is known if the number to be tested is even this information is taken into consideration by the function.

      (%i1) load("bitwise")$
      (%i2) declare(e,even,o,odd);
      (%o2)                         done
      (%i3) bit_onep(e,0);
      (%o3)                         false
      (%i4) bit_onep(o,0);
      (%o4)                         true

  Categories: Number theory Binary operations Package bitwise

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# 44. bode

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## 44.1 Functions and Variables for bode

Function: bode_gain (H, range, ...plot_opts...)

  Function to draw Bode gain plots.

  Examples (1 through 7 from

      http://www.swarthmore.edu/NatSci/echeeve1/Ref/Bode/BodeHow.html,

  8 from Ron Crummett):

      (%i1) load("bode")$

      (%i2) H1 (s) := 100 * (1 + s) / ((s + 10) * (s + 100))$

      (%i3) bode_gain (H1 (s), [w, 1/1000, 1000])$

      (%i4) H2 (s) := 1 / (1 + s/omega0)$

      (%i5) bode_gain (H2 (s), [w, 1/1000, 1000]), omega0 = 10$

      (%i6) H3 (s) := 1 / (1 + s/omega0)^2$

      (%i7) bode_gain (H3 (s), [w, 1/1000, 1000]), omega0 = 10$

      (%i8) H4 (s) := 1 + s/omega0$

      (%i9) bode_gain (H4 (s), [w, 1/1000, 1000]), omega0 = 10$

      (%i10) H5 (s) := 1/s$

      (%i11) bode_gain (H5 (s), [w, 1/1000, 1000])$

      (%i12) H6 (s) := 1/((s/omega0)^2 + 2 * zeta * (s/omega0) + 1)$

      (%i13) bode_gain (H6 (s), [w, 1/1000, 1000]),
                        omega0 = 10, zeta = 1/10$

      (%i14) H7 (s) := (s/omega0)^2 + 2 * zeta * (s/omega0) + 1$

      (%i15) bode_gain (H7 (s), [w, 1/1000, 1000]),
                        omega0 = 10, zeta = 1/10$

      (%i16) H8 (s) := 0.5 / (0.0001 * s^3 + 0.002 * s^2 + 0.01 * s)$

      (%i17) bode_gain (H8 (s), [w, 1/1000, 1000])$

  To use this function write first `load("bode")`. See also `bode_phase`.

  Categories: Plotting · Package bode

Function: bode_phase (H, range, ...plot_opts...)

  Function to draw Bode phase plots.

  Examples (1 through 7 from

      http://www.swarthmore.edu/NatSci/echeeve1/Ref/Bode/BodeHow.html,

  8 from Ron Crummett):

      (%i1) load("bode")$

      (%i2) H1 (s) := 100 * (1 + s) / ((s + 10) * (s + 100))$

      (%i3) bode_phase (H1 (s), [w, 1/1000, 1000])$

      (%i4) H2 (s) := 1 / (1 + s/omega0)$

      (%i5) bode_phase (H2 (s), [w, 1/1000, 1000]), omega0 = 10$

      (%i6) H3 (s) := 1 / (1 + s/omega0)^2$

      (%i7) bode_phase (H3 (s), [w, 1/1000, 1000]), omega0 = 10$

      (%i8) H4 (s) := 1 + s/omega0$

      (%i9) bode_phase (H4 (s), [w, 1/1000, 1000]), omega0 = 10$

      (%i10) H5 (s) := 1/s$

      (%i11) bode_phase (H5 (s), [w, 1/1000, 1000])$

      (%i12) H6 (s) := 1/((s/omega0)^2 + 2 * zeta * (s/omega0) + 1)$

      (%i13) bode_phase (H6 (s), [w, 1/1000, 1000]),
                         omega0 = 10, zeta = 1/10$

      (%i14) H7 (s) := (s/omega0)^2 + 2 * zeta * (s/omega0) + 1$

      (%i15) bode_phase (H7 (s), [w, 1/1000, 1000]),
                         omega0 = 10, zeta = 1/10$

      (%i16) H8 (s) := 0.5 / (0.0001 * s^3 + 0.002 * s^2 + 0.01 * s)$

      (%i17) bode_phase (H8 (s), [w, 1/1000, 1000])$

      (%i18) block ([bode_phase_unwrap : false],
                    bode_phase (H8 (s), [w, 1/1000, 1000]));

      (%i19) block ([bode_phase_unwrap : true],
                    bode_phase (H8 (s), [w, 1/1000, 1000]));

  To use this function write first `load("bode")`. See also `bode_gain`.

  Categories: Plotting · Package bode

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# 45. celine

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## 45.1 Introduction to celine

Maxima implementation of Sister Celine's method. Barton Willis wrote this code. It is released under the Creative Commons CC0 license.

Celine's method is described in Sections 4.1-4.4 of the book "A=B", by Marko Petkovsek, Herbert S. Wilf, and Doron Zeilberger. This book is available at http://www.math.rutgers.edu/~zeilberg/AeqB.pdf

Let f = F(n,k). The function celine returns a set of recursion relations for F of the form

p_0(n) * fff(n,k) + p_1(n) * fff(n+1,k) + ... + p_p(n) * fff(n+p,k+q),

where p_0 through p_p are polynomials. If Maxima is unable to determine that sum(sum(a(i,j) * F(n+i,k+j),i,0,p),j,0,q) / F(n,k) is a rational function of n and k, celine returns the empty set. When f involves parameters (variables other than n or k), celine might make assumptions about these parameters. Using 'put' with a key of 'proviso,' Maxima saves these assumptions on the input label.

To use this function, first load the package integer_sequence, opsubst, and to_poly_solve.

Examples:

    (%i1) load("integer_sequence")$
    (%i2) load("opsubst")$
    (%i3) load("to_poly_solve")$
    (%i4) load("celine")$
    (%i5) celine(n!,n,k,1,0);
    (%o5)       {fff(n + 1, k) - n fff(n, k) - fff(n, k)}

Verification that this result is correct:

    (%i1) load("integer_sequence")$
    (%i2) load("opsubst")$
    (%i3) load("to_poly_solve")$
    (%i4) load("celine")$
    (%i5) g1:{fff(n+1,k)-n*fff(n,k)-fff(n,k)};
    (%o5)       {fff(n + 1, k) - n fff(n, k) - fff(n, k)}
    (%i6) ratsimp(minfactorial(first(g1))),fff(n,k) := n!;
    (%o6)                           0

An example with parameters including the test that the result of the example is correct:

    (%i1) load("integer_sequence")$
    (%i2) load("opsubst")$
    (%i3) load("to_poly_solve")$
    (%i4) load("celine")$
    (%i5) e : pochhammer(a,k) * pochhammer(-k,n) / (pochhammer(b,k));
                               (a)  (- k)
                                  k      n
    (%o5)                      -----------
                                  (b)
                                     k
    (%i6) recur : celine(e,n,k,2,1);
    (%o6) {fff(n + 2, k + 1) - fff(n + 2, k) - b fff(n + 1, k + 1)
     + n ((- fff(n + 1, k + 1)) + 2 fff(n + 1, k) - a fff(n, k)
     - fff(n, k)) + a (fff(n + 1, k) - fff(n, k)) + 2 fff(n + 1, k)
        2
     - n  fff(n, k)}
    (%i7) /* Test this result for correctness */
    (%i8) first(%), fff(n,k) := ''(e)$
    (%i9) makefact(makegamma(%))$
    (%o9)                           0
    (%i10) minfactorial(factor(minfactorial(factor(%))));

The proviso data suggests that setting a = b may result in a lower order recursion which is shown by the following example:

    (%i1) load("integer_sequence")$
    (%i2) load("opsubst")$
    (%i3) load("to_poly_solve")$
    (%i4) load("celine")$
    (%i5) e : pochhammer(a,k) * pochhammer(-k,n) / (pochhammer(b,k));
                               (a)  (- k)
                                  k      n
    (%o5)                      -----------
                                  (b)
                                     k
    (%i6) recur : celine(e,n,k,2,1);
    (%o6) {fff(n + 2, k + 1) - fff(n + 2, k) - b fff(n + 1, k + 1)
     + n ((- fff(n + 1, k + 1)) + 2 fff(n + 1, k) - a fff(n, k)
     - fff(n, k)) + a (fff(n + 1, k) - fff(n, k)) + 2 fff(n + 1, k)
        2
     - n  fff(n, k)}
    (%i7) get('%,'proviso);
    (%o7)                         false
    (%i8) celine(subst(b=a,e),n,k,1,1);
    (%o8) {fff(n + 1, k + 1) - fff(n + 1, k) + n fff(n, k)
                                                         + fff(n, k)}

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# 46. clebsch_gordan

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## 46.1 Functions and Variables for clebsch_gordan

Function: clebsch_gordan (j1, j2, m1, m2, j, m)

  Compute the Clebsch-Gordan coefficient `<j1, j2, m1, m2 | j, m>`.

  Categories: clebsch_gordan

Function: racah_v (a, b, c, a1, b1, c1)

  Compute Racah's V coefficient (computed in terms of a related Clebsch-Gordan coefficient).

  Categories: clebsch_gordan

Function: racah_w (j1, j2, j5, j4, j3, j6)

  Compute Racah's W coefficient (computed in terms of a Wigner `6j` symbol)

  Categories: clebsch_gordan

Function: wigner_3j (j1, j2, j3, m1, m2, m3)

  Compute Wigner's `3j` symbol (computed in terms of a related Clebsch-Gordan coefficient).

  Categories: clebsch_gordan

Function: wigner_6j (j1, j2, j3, j4, j5, j6)

  Compute Wigner's `6j` symbol.

  Categories: clebsch_gordan

Function: wigner_9j (a, b, c, d, e, f, g, h, i, j,)

  Compute Wigner's `9j` symbol.

  Categories: clebsch_gordan

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# 47. cobyla

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## 47.1 Introduction to cobyla

`fmin_cobyla` is a Common Lisp translation (via `f2cl`) of the Fortran constrained optimization routine COBYLA by Powell[1][2][3].

COBYLA minimizes an objective function F(X) subject to M inequality constraints of the form g(X) >= 0 on X, where X is a vector of variables that has N components.

Equality constraints g(X)=0 can often be implemented by a pair of inequality constraints g(X)>=0 and -g(X)>= 0. Maxima's interface to COBYLA allows equality constraints and internally converts the equality constraints to a pair of inequality constraints.

The algorithm employs linear approximations to the objective and constraint functions, the approximations being formed by linear interpolation at N+1 points in the space of the variables. The interpolation points are regarded as vertices of a simplex. The parameter RHO controls the size of the simplex and it is reduced automatically from RHOBEG to RHOEND. For each RHO the subroutine tries to achieve a good vector of variables for the current size, and then RHO is reduced until the value RHOEND is reached. Therefore RHOBEG and RHOEND should be set to reasonable initial changes to and the required accuracy in the variables respectively, but this accuracy should be viewed as a subject for experimentation because it is not guaranteed. The routine treats each constraint individually when calculating a change to the variables, rather than lumping the constraints together into a single penalty function. The name of the subroutine is derived from the phrase Constrained Optimization BY Linear Approximations.

References:

[1] Fortran Code is from http://plato.asu.edu/sub/nlores.html#general

[2] M. J. D. Powell, "A direct search optimization method that models the objective and constraint functions by linear interpolation," in Advances in Optimization and Numerical Analysis, eds. S. Gomez and J.-P. Hennart (Kluwer Academic: Dordrecht, 1994), p. 51-67.

[3] M. J. D. Powell, "Direct search algorithms for optimization calculations," Acta Numerica 7, 287-336 (1998). Also available as University of Cambridge, Department of Applied Mathematics and Theoretical Physics, Numerical Analysis Group, Report NA1998/04 from http://www.damtp.cam.ac.uk/user/na/reports.html

Categories: Numerical methods · Optimization · Share packages · Package cobyla

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## 47.2 Functions and Variables for cobyla

Function: fmin_cobyla
`fmin_cobyla` (F, X, Y)
`fmin_cobyla` (F, X, Y, optional_args)

  Returns an approximate minimum of the expression F with respect to the variables X, subject to an optional set of constraints. Y is a list of initial guesses for X.

  F must be an ordinary expressions, not names of functions or lambda expressions.

  `optional_args` represents additional arguments, specified as `symbol = value`. The optional arguments recognized are:

  `constraints`

    List of inequality and equality constraints that must be satisfied by X. The inequality constraints must be actual inequalities of the form `g(X) >= h(X)` or `g(X) <= h(X)`. The equality constraints must be of the form `g(X) = h(X)`.

  `rhobeg`

    Initial value of the internal RHO variable which controls the size of simplex. (Defaults to 1.0)

  `rhoend`

    The desired final value rho parameter. It is approximately the accuracy in the variables. (Defaults to 1d-6.)

  `iprint`

    Verbose output level. (Defaults to 0)

  `maxfun`

    The maximum number of function evaluations. (Defaults to 1000).

  On return, a vector is given:

  1. The value of the variables giving the minimum. This is a list of elements of the form `var = value` for each of the variables listed in X.
  2. The minimized function value
  3. The number of function evaluations.
  4. Return code with the following meanings
     1. 0 - No errors.
     2. 1 - Limit on maximum number of function evaluations reached.
     3. 2 - Rounding errors inhibiting progress.

  `load(fmin_cobyla)` loads this function.

Function: bf_fmin_cobyla
`bf_fmin_cobyla` (F, X, Y)
`bf_fmin_cobyla` (F, X, Y, optional_args)

  This function is identical to `fmin_cobyla`, except that bigfloat operations are used, and the default value for rhoend is `10^(fpprec/2)`.

  See `fmin_cobyla` for more information.

  `load(bf_fmin_cobyla)` loads this function.

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## 47.3 Examples for cobyla

Minimize x1*x2 with 1-x1^2-x2^2 >= 0. The theoretical solution is x1 = 1/sqrt(2), x2 = -1/sqrt(2).

    (%i1) load(fmin_cobyla)$
    (%i2) fmin_cobyla(x1*x2, [x1, x2], [1,1],
                      constraints = [x1^2+x2^2<=1], iprint=1);
       Normal return from subroutine COBYLA

       NFVALS =   66   F =-5.000000E-01    MAXCV = 1.999845E-12
       X = 7.071058E-01  -7.071077E-01
    (%o2) [[x1 = 0.70710584934848, x2 = - 0.7071077130248],
           - 0.49999999999926, [[-1.999955756559757e-12],[]], 66]

There are additional examples in the share/cobyla/ex directory.

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# 48. contrib_ode

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## 48.1 Introduction to contrib_ode

Maxima's ordinary differential equation (ODE) solver `ode2` solves elementary linear ODEs of first and second order. The function `contrib_ode` extends `ode2` with additional methods for linear and non-linear first order ODEs and linear homogeneous second order ODEs. The code is still under development and the calling sequence may change in future releases. Once the code has stabilized it may be moved from the contrib directory and integrated into Maxima.

This package must be loaded with the command `load('contrib_ode)` before use.

The calling convention for `contrib_ode` is identical to `ode2`. It takes three arguments: an ODE (only the left hand side need be given if the right hand side is 0), the dependent variable, and the independent variable. When successful, it returns a list of solutions.

The form of the solution differs from `ode2`. As non-linear equations can have multiple solutions, `contrib_ode` returns a list of solutions. Each solution can have a number of forms:

- an explicit solution for the dependent variable,
- an implicit solution for the dependent variable,
- a parametric solution in terms of variable `%t`, or
- a tranformation into another ODE in variable `%u`.

`%c` is used to represent the constant of integration for first order equations. `%k1` and `%k2` are the constants for second order equations. If `contrib_ode` cannot obtain a solution for whatever reason, it returns `false`, after perhaps printing out an error message.

It is necessary to return a list of solutions, as even first order non-linear ODEs can have multiple solutions. For example:

    (%i1) load('contrib_ode)$
    (%i2) eqn:x*'diff(y,x)^2-(1+x*y)*'diff(y,x)+y=0;
                        dy 2             dy
    (%o2)            x (--)  - (1 + x y) -- + y = 0
                        dx               dx
    (%i3) contrib_ode(eqn,y,x);
                        dy 2             dy
    (%t3)            x (--)  - (1 + x y) -- + y = 0
                        dx               dx

                  first order equation not linear in y'

                                                 x
    (%o3)             [y = log(x) + %c, y = %c %e ]
    (%i4) method;
    (%o4)                        factor

Nonlinear ODEs can have singular solutions without constants of integration, as in the second solution of the following example:

    (%i1) load('contrib_ode)$
    (%i2) eqn:'diff(y,x)^2+x*'diff(y,x)-y=0;
                           dy 2     dy
    (%o2)                 (--)  + x -- - y = 0
                           dx       dx
    (%i3) contrib_ode(eqn,y,x);
                           dy 2     dy
    (%t3)                 (--)  + x -- - y = 0
                           dx       dx

                  first order equation not linear in y'

                                               2
                                     2        x
    (%o3)              [y = %c x + %c , y = - --]
                                              4
    (%i4) method;
    (%o4)                       clairault

The following ODE has two parametric solutions in terms of the dummy variable `%t`. In this case the parametric solutions can be manipulated to give explicit solutions.

    (%i1) load('contrib_ode)$
    (%i2) eqn:'diff(y,x)=(x+y)^2;
                              dy          2
    (%o2)                     -- = (x + y)
                              dx
    (%i3) contrib_ode(eqn,y,x);
    (%o3) [[x = %c - atan(sqrt(%t)), y = (- x) - sqrt(%t)],
                         [x = atan(sqrt(%t)) + %c, y = sqrt(%t) - x]]
    (%i4) method;
    (%o4)                       lagrange

The following example (Kamke 1.112) demonstrates an implicit solution.

    (%i1) load('contrib_ode)$
    (%i2) assume(x>0,y>0);
    (%o2)                    [x > 0, y > 0]
    (%i3) eqn:x*'diff(y,x)-x*sqrt(y^2+x^2)-y;
                         dy           2    2
    (%o3)              x -- - x sqrt(y  + x ) - y
                         dx
    (%i4) contrib_ode(eqn,y,x);
                                      y
    (%o4)                  [x - asinh(-) = %c]
                                      x
    (%i5) method;
    (%o5)                          lie

The following Riccati equation is transformed into a linear second order ODE in the variable `%u`. Maxima is unable to solve the new ODE, so it is returned unevaluated.

    (%i1) load('contrib_ode)$
    (%i2) eqn:x^2*'diff(y,x)=a+b*x^n+c*x^2*y^2;
                        2 dy      2  2      n
    (%o2)              x  -- = c x  y  + b x  + a
                          dx
    (%i3) contrib_ode(eqn,y,x);
                   d%u
                   ---                            2
                   dx        2  a       n - 2    d %u
    (%o3)  [[y = - ----, %u c  (-- + b x     ) + ---- c = 0]]
                   %u c          2                 2
                                x                dx
    (%i4) method;
    (%o4)                        riccati

For first order ODEs `contrib_ode` calls `ode2`. It then tries the following methods: factorization, Clairault, Lagrange, Riccati, Abel and Lie symmetry methods. The Lie method is not attempted on Abel equations if the Abel method fails, but it is tried if the Riccati method returns an unsolved second order ODE.

For second order ODEs `contrib_ode` calls `ode2` then `odelin`.

Extensive debugging traces and messages are displayed if the command `put('contrib_ode,true,'verbose)` is executed.

Categories: Differential equations · Share packages · Package contrib_ode

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## 48.2 Functions and Variables for contrib_ode

Function: contrib_ode (eqn, y, x)

  Returns a list of solutions of the ODE eqn with independent variable x and dependent variable y.

  Categories: Package contrib_ode

Function: odelin (eqn, y, x)

  `odelin` solves linear homogeneous ODEs of first and second order with independent variable x and dependent variable y. It returns a fundamental solution set of the ODE.

  For second order ODEs, `odelin` uses a method, due to Bronstein and Lafaille, that searches for solutions in terms of given special functions.

      (%i1) load('contrib_ode)$
      (%i2) odelin(x*(x+1)*'diff(y,x,2)+(x+5)*'diff(y,x,1)+(-4)*y,y,x);
             gauss_a(- 6, - 2, - 3, - x)  gauss_b(- 6, - 2, - 3, - x)
      (%o2) {---------------------------, ---------------------------}
                          4                            4
                         x                            x

  Categories: Package contrib_ode

Function: ode_check (eqn, soln)

  Returns the value of ODE eqn after substituting a possible solution soln. The value is equivalent to zero if soln is a solution of eqn.

      (%i1) load('contrib_ode)$
      (%i2) eqn:'diff(y,x,2)+(a*x+b)*y;
                               2
                              d y
      (%o2)                   --- + (b + a x) y
                                2
                              dx
      (%i3) ans:[y = bessel_y(1/3,2*(a*x+b)^(3/2)/(3*a))*%k2*sqrt(a*x+b)
               +bessel_j(1/3,2*(a*x+b)^(3/2)/(3*a))*%k1*sqrt(a*x+b)];
                                        3/2
                          1  2 (b + a x)
      (%o3) [y = bessel_y(-, --------------) %k2 sqrt(a x + b)
                          3       3 a
                                                3/2
                                  1  2 (b + a x)
                       + bessel_j(-, --------------) %k1 sqrt(a x + b)]
                                  3       3 a
      (%i4) ode_check(eqn,ans[1]);
      (%o4)                           0

  Categories: Package contrib_ode

System variable: method

  The variable `method` is set to the successful solution method.

  Categories: Package contrib_ode

Variable: %c

  `%c` is the integration constant for first order ODEs.

  Categories: Package contrib_ode

Variable: %k1

  `%k1` is the first integration constant for second order ODEs.

  Categories: Package contrib_ode

Variable: %k2

  `%k2` is the second integration constant for second order ODEs.

  Categories: Package contrib_ode

Function: gauss_a (a, b, c, x)

  `gauss_a(a,b,c,x)` and `gauss_b(a,b,c,x)` are 2F1 geometric functions. They represent any two independent solutions of the hypergeometric differential equation `x(1-x) diff(y,x,2) + [c-(a+b+1)x] diff(y,x) - aby = 0` (A&S 15.5.1).

  The only use of these functions is in solutions of ODEs returned by `odelin` and `contrib_ode`. The definition and use of these functions may change in future releases of Maxima.

  See also `gauss_b`, `dgauss_a` and `gauss_b`.

  Categories: Package contrib_ode

Function: gauss_b (a, b, c, x)

  See `gauss_a`.

  Categories: Package contrib_ode

Function: dgauss_a (a, b, c, x)

  The derivative with respect to x of `gauss_a(a, b, c, x)`.

  Categories: Package contrib_ode

Function: dgauss_b (a, b, c, x)

  The derivative with respect to x of `gauss_b(a, b, c, x)`.

  Categories: Package contrib_ode

Function: kummer_m (a, b, x)

  Kummer's M function, as defined in Abramowitz and Stegun, Handbook of Mathematical Functions, Section 13.1.2.

  The only use of this function is in solutions of ODEs returned by `odelin` and `contrib_ode`. The definition and use of this function may change in future releases of Maxima.

  See also `kummer_u`, `dkummer_m`, and `dkummer_u`.

  Categories: Package contrib_ode

Function: kummer_u (a, b, x)

  Kummer's U function, as defined in Abramowitz and Stegun, Handbook of Mathematical Functions, Section 13.1.3.

  See `kummer_m`.

  Categories: Package contrib_ode

Function: dkummer_m (a, b, x)

  The derivative with respect to x of `kummer_m(a, b, x)`.

  Categories: Package contrib_ode

Function: dkummer_u (a, b, x)

  The derivative with respect to x of `kummer_u(a, b, x)`.

  Categories: Package contrib_ode

Function: bessel_simplify (expr)

  Simplifies expressions containing Bessel functions bessel_j, bessel_y, bessel_i, bessel_k, hankel_1, hankel_2, strauve_h and strauve_l. Recurrence relations (given in Abramowitz and Stegun, Handbook of Mathematical Functions, Section 9.1.27) are used to replace functions of highest order n by functions of order n-1 and n-2.

  This process repeated until all the orders differ by less than 2.

      (%i1) load('contrib_ode)$
      (%i2) bessel_simplify(4*bessel_j(n,x^2)*(x^2-n^2/x^2)
        +x*((bessel_j(n-2,x^2)-bessel_j(n,x^2))*x
        -(bessel_j(n,x^2)-bessel_j(n+2,x^2))*x)
        -2*bessel_j(n+1,x^2)+2*bessel_j(n-1,x^2));
      (%o2)                           0
      (%i3) bessel_simplify(-2*bessel_j(1,z)*z^3-10*bessel_j(2,z)*z^2
       +15*%pi*bessel_j(1,z)*struve_h(3,z)*z-15*%pi*struve_h(1,z)*bessel_j(3,z)*z
       -15*%pi*bessel_j(0,z)*struve_h(2,z)*z+15*%pi*struve_h(0,z)*bessel_j(2,z)*z
       -30*%pi*bessel_j(1,z)*struve_h(2,z)+30*%pi*struve_h(1,z)*bessel_j(2,z));
      (%o3)                           0

  Categories: Package contrib_ode Bessel functions Special functions

Function: expintegral_e_simplify (expr)

  Simplify expressions containing exponential integral expintegral_e using the recurrence (A&S 5.1.14).

  expintegral_e(n+1,z) = (1/n) * (exp(-z)-z*expintegral_e(n,z)) n = 1,2,3 ....

  Categories: Package contrib_ode Exponential Integrals Special functions

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## 48.3 Possible improvements to contrib_ode

These routines are work in progress. I still need to:

- Extend the FACTOR method `ode1_factor` to work for multiple roots.
- Extend the FACTOR method `ode1_factor` to attempt to solve higher order factors. At present it only attemps to solve linear factors.
- Fix the LAGRANGE routine `ode1_lagrange` to prefer real roots over complex roots.
- Add additional methods for Riccati equations.
- Improve the detection of Abel equations of second kind. The exisiting pattern matching is weak.
- Work on the Lie symmetry group routine `ode1_lie`. There are quite a few problems with it: some parts are unimplemented; some test cases seem to run forever; other test cases crash; yet others return very complex "solutions". I wonder if it really ready for release yet.
- Add more test cases.

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## 48.4 Test cases for contrib_ode

The routines have been tested on a approximately one thousand test cases from Murphy, Kamke, Zwillinger and elsewhere. These are included in the tests subdirectory.

- The Clairault routine `ode1_clairault` finds all known solutions, including singular solutions, of the Clairault equations in Murphy and Kamke.
- The other routines often return a single solution when multiple solutions exist.
- Some of the "solutions" from `ode1_lie` are overly complex and impossible to check.
- There are some crashes.

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## 48.5 References for contrib_ode

1. E. Kamke, Differentialgleichungen Losungsmethoden und Losungen, Vol 1, Geest & Portig, Leipzig, 1961
2. G. M. Murphy, Ordinary Differential Equations and Their Solutions, Van Nostrand, New York, 1960
3. D. Zwillinger, Handbook of Differential Equations, 3rd edition, Academic Press, 1998
4. F. Schwarz, Symmetry Analysis of Abel's Equation, Studies in Applied Mathematics, 100:269-294 (1998)
5. F. Schwarz, Algorithmic Solution of Abel's Equation, Computing 61, 39-49 (1998)
6. E. S. Cheb-Terrab, A. D. Roche, Symmetries and First Order ODE Patterns, Computer Physics Communications 113 (1998), p 239. (http://lie.uwaterloo.ca/papers/ode_vii.pdf)
7. E. S. Cheb-Terrab, T. Kolokolnikov, First Order ODEs, Symmetries and Linear Transformations, European Journal of Applied Mathematics, Vol. 14, No. 2, pp. 231-246 (2003). (http://arxiv.org/abs/math-ph/0007023,
   http://lie.uwaterloo.ca/papers/ode_iv.pdf)
8. G. W. Bluman, S. C. Anco, Symmetry and Integration Methods for Differential Equations, Springer, (2002)
9. M. Bronstein, S. Lafaille, Solutions of linear ordinary differential equations in terms of special functions, Proceedings of ISSAC 2002, Lille, ACM Press, 23-28. (http://www-sop.inria.fr/cafe/Manuel.Bronstein/publications/issac2002.pdf)

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# 49. descriptive

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## 49.1 Introduction to descriptive

Package `descriptive` contains a set of functions for making descriptive statistical computations and graphing. Together with the source code there are three data sets in your Maxima tree: `pidigits.data`, `wind.data` and `biomed.data`.

Any statistics manual can be used as a reference to the functions in package `descriptive`.

For comments, bugs or suggestions, please contact me at 'riotorto AT yahoo DOT com'.

Here is a simple example on how the descriptive functions in `descriptive` do they work, depending on the nature of their arguments, lists or matrices,

    (%i1) load (descriptive)$
    (%i2) /* univariate sample */   mean ([a, b, c]);
                                c + b + a
    (%o2)                       ---------
                                    3
    (%i3) matrix ([a, b], [c, d], [e, f]);
                                [ a  b ]
                                [      ]
    (%o3)                       [ c  d ]
                                [      ]
                                [ e  f ]
    (%i4) /* multivariate sample */ mean (%);
                          e + c + a  f + d + b
    (%o4)                [---------, ---------]
                              3          3

Note that in multivariate samples the mean is calculated for each column.

In case of several samples with possible different sizes, the Maxima function `map` can be used to get the desired results for each sample,

    (%i1) load (descriptive)$
    (%i2) map (mean, [[a, b, c], [d, e]]);
                            c + b + a  e + d
    (%o2)                  [---------, -----]
                                3        2

In this case, two samples of sizes 3 and 2 were stored into a list.

Univariate samples must be stored in lists like

    (%i1) s1 : [3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5];
    (%o1)           [3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5]

and multivariate samples in matrices as in

    (%i1) s2 : matrix ([13.17, 9.29], [14.71, 16.88], [18.50, 16.88],
                 [10.58, 6.63], [13.33, 13.25], [13.21,  8.12]);
                            [ 13.17  9.29  ]
                            [              ]
                            [ 14.71  16.88 ]
                            [              ]
                            [ 18.5   16.88 ]
    (%o1)                   [              ]
                            [ 10.58  6.63  ]
                            [              ]
                            [ 13.33  13.25 ]
                            [              ]
                            [ 13.21  8.12  ]

In this case, the number of columns equals the random variable dimension and the number of rows is the sample size.

Data can be introduced by hand, but big samples are usually stored in plain text files. For example, file `pidigits.data` contains the first 100 digits of number `%pi`:

          3
          1
          4
          1
          5
          9
          2
          6
          5
          3 ...

In order to load these digits in Maxima,

    (%i1) s1 : read_list (file_search ("pidigits.data"))$
    (%i2) length (s1);
    (%o2)                          100

On the other hand, file `wind.data` contains daily average wind speeds at 5 meteorological stations in the Republic of Ireland (This is part of a data set taken at 12 meteorological stations. The original file is freely downloadable from the StatLib Data Repository and its analysis is discused in Haslett, J., Raftery, A. E. (1989) Space-time Modelling with Long-memory Dependence: Assessing Ireland's Wind Power Resource, with Discussion. Applied Statistics 38, 1-50). This loads the data:

    (%i1) s2 : read_matrix (file_search ("wind.data"))$
    (%i2) length (s2);
    (%o2)                          100
    (%i3) s2 [%]; /* last record */
    (%o3)            [3.58, 6.0, 4.58, 7.62, 11.25]

Some samples contain non numeric data. As an example, file `biomed.data` (which is part of another bigger one downloaded from the StatLib Data Repository) contains four blood measures taken from two groups of patients, `A` and `B`, of different ages,

    (%i1) s3 : read_matrix (file_search ("biomed.data"))$
    (%i2) length (s3);
    (%o2)                          100
    (%i3) s3 [1]; /* first record */
    (%o3)            [A, 30, 167.0, 89.0, 25.6, 364]

The first individual belongs to group `A`, is 30 years old and his/her blood measures were 167.0, 89.0, 25.6 and 364.

One must take care when working with categorical data. In the next example, symbol `a` is assigned a value in some previous moment and then a sample with categorical value `a` is taken,

    (%i1) a : 1$
    (%i2) matrix ([a, 3], [b, 5]);
                                [ 1  3 ]
    (%o2)                       [      ]
                                [ b  5 ]

Categories: Descriptive statistics · Share packages · Package descriptive

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## 49.2 Functions and Variables for data manipulation

Function: build_sample
`build_sample` (list)
`build_sample` (matrix)

  Builds a sample from a table of absolute frequencies. The input table can be a matrix or a list of lists, all of them of equal size. The number of columns or the length of the lists must be greater than 1. The last element of each row or list is interpreted as the absolute frequency. The output is always a sample in matrix form.

  Examples:

  Univariate frequency table.

      (%i1) load (descriptive)$
      (%i2) sam1: build_sample([[6,1], [j,2], [2,1]]);
                             [ 6 ]
                             [   ]
                             [ j ]
      (%o2)                  [   ]
                             [ j ]
                             [   ]
                             [ 2 ]
      (%i3) mean(sam1);
                            2 j + 8
      (%o3)                [-------]
                               4
      (%i4) barsplot(sam1) $

  Multivariate frequency table.

      (%i1) load (descriptive)$
      (%i2) sam2: build_sample([[6,3,1], [5,6,2], [u,2,1],[6,8,2]]) ;
                                 [ 6  3 ]
                                 [      ]
                                 [ 5  6 ]
                                 [      ]
                                 [ 5  6 ]
      (%o2)                      [      ]
                                 [ u  2 ]
                                 [      ]
                                 [ 6  8 ]
                                 [      ]
                                 [ 6  8 ]
      (%i3) cov(sam2);
             [   2                 2                            ]
             [  u  + 158   (u + 28)     2 u + 174   11 (u + 28) ]
             [  -------- - ---------    --------- - ----------- ]
      (%o3)  [     6          36            6           12      ]
             [                                                  ]
             [ 2 u + 174   11 (u + 28)            21            ]
             [ --------- - -----------            --            ]
             [     6           12                 4             ]
      (%i4) barsplot(sam2, grouping=stacked) $

  Categories: Package descriptive

Function: continuous_freq
`continuous_freq` (data)
`continuous_freq` (data, m)

  The first argument of `continuous_freq` must be a list or 1-dimensional array (as created by `make_array`) of numbers. Divides the range in intervals and counts how many values are inside them. The second argument is optional and either equals the number of classes we want, 10 by default, or equals a list containing the class limits and the number of classes we want, or a list containing only the limits.

  If sample values are all equal, this function returns only one class of amplitude 2.

  Examples:

  Optional argument indicates the number of classes we want. The first list in the output contains the interval limits, and the second the corresponding counts: there are 16 digits inside the interval `[0, 1.8]`, 24 digits in `(1.8, 3.6]`, and so on.

      (%i1) load (descriptive)$
      (%i2) s1 : read_list (file_search ("pidigits.data"))$
      (%i3) continuous_freq (s1, 5);
      (%o3) [[0, 1.8, 3.6, 5.4, 7.2, 9.0], [16, 24, 18, 17, 25]]

  Optional argument indicates we want 7 classes with limits -2 and 12:

      (%i1) load (descriptive)$
      (%i2) s1 : read_list (file_search ("pidigits.data"))$
      (%i3) continuous_freq (s1, [-2,12,7]);
      (%o3) [[- 2, 0, 2, 4, 6, 8, 10, 12], [8, 20, 22, 17, 20, 13, 0]]

  Optional argument indicates we want the default number of classes with limits -2 and 12:

      (%i1) load (descriptive)$
      (%i2) s1 : read_list (file_search ("pidigits.data"))$
      (%i3) continuous_freq (s1, [-2,12]);
                      3  4  11  18     32  39  46  53
      (%o3)  [[- 2, - -, -, --, --, 5, --, --, --, --, 12],
                      5  5  5   5      5   5   5   5
                     [0, 8, 20, 12, 18, 9, 8, 25, 0, 0]]

  The first argument may be an array.

      (%i1) load (descriptive)$
      (%i2) s1 : read_list (file_search ("pidigits.data"))$
      (%i3) a1 : make_array (fixnum, length (s1)) $
      (%i4) fillarray (a1, s1);
      (%o4) {Lisp Array:
      #(3 1 4 1 5 9 2 6 5 3 5 8 9 7 9 3 2 3 8 4 6 2 6 4 3 3 8 3 2 7 9 \
      5 0 2 8 8 4 1 9 7 1 6 9 3 9 9 3 7 5 1 0 5 8 2 0 9 7 4 9 4 4 5 9
        2 3 0 7 8 1 6 4 0 6 2 8 6 2 0 8 9 9 8 6 2 8 0 3 4 8 2 5 3 4 2 \
      1 1 7 0 6 7)}
      (%i5) continuous_freq (a1);
                 9   9  27  18  9  27  63  36  81
      (%o5) [[0, --, -, --, --, -, --, --, --, --, 9],
                 10  5  10  5   2  5   10  5   10
                                   [8, 8, 12, 12, 10, 8, 9, 8, 12, 13]]

  Categories: Package descriptive

Function: discrete_freq (data)

  Counts absolute frequencies in discrete samples, both numeric and categorical. Its unique argument is a list, or 1-dimensional array (as created by `make_array`).

      (%i1) load (descriptive)$
      (%i2) s1 : read_list (file_search ("pidigits.data"))$
      (%i3) discrete_freq (s1);
      (%o3) [[0, 1, 2, 3, 4, 5, 6, 7, 8, 9],
                                   [8, 8, 12, 12, 10, 8, 9, 8, 12, 13]]

  The first list gives the sample values and the second their absolute frequencies. Commands `? col` and `? transpose` should help you to understand the last input.

  The argument may be an array.

      (%i1) load (descriptive)$
      (%i2) s1 : read_list (file_search ("pidigits.data"))$
      (%i3) a1 : make_array (fixnum, length (s1)) $
      (%i4) fillarray (a1, s1);
      (%o4) {Lisp Array:
      #(3 1 4 1 5 9 2 6 5 3 5 8 9 7 9 3 2 3 8 4 6 2 6 4 3 3 8 3 2 7 9 \
      5 0 2 8 8 4 1 9 7 1 6 9 3 9 9 3 7 5 1 0 5 8 2 0 9 7 4 9 4 4 5 9
        2 3 0 7 8 1 6 4 0 6 2 8 6 2 0 8 9 9 8 6 2 8 0 3 4 8 2 5 3 4 2 \
      1 1 7 0 6 7)}
      (%i5) discrete_freq (a1);
      (%o5) [[0, 1, 2, 3, 4, 5, 6, 7, 8, 9],
                                   [8, 8, 12, 12, 10, 8, 9, 8, 12, 13]]

  Categories: Package descriptive

Function: standardize
`standardize` (list)
`standardize` (matrix)

  Subtracts to each element of the list the sample mean and divides the result by the standard deviation. When the input is a matrix, `standardize` subtracts to each row the multivariate mean, and then divides each component by the corresponding standard deviation.

  Categories: Package descriptive

Function: subsample
`subsample` (data_matrix, predicate_function)
`subsample` (data_matrix, predicate_function, col_num1, col_num2, ...)

  This is a sort of variant of the Maxima `submatrix` function. The first argument is the data matrix, the second is a predicate function and optional additional arguments are the numbers of the columns to be taken. Its behaviour is better understood with examples.

  These are multivariate records in which the wind speed in the first meteorological station were greater than 18. See that in the lambda expression the i-th component is referred to as `v[i]`.

      (%i1) load (descriptive)$
      (%i2) s2 : read_matrix (file_search ("wind.data"))$
      (%i3) subsample (s2, lambda([v], v[1] > 18));
                    [ 19.38  15.37  15.12  23.09  25.25 ]
                    [                                   ]
                    [ 18.29  18.66  19.08  26.08  27.63 ]
      (%o3)         [                                   ]
                    [ 20.25  21.46  19.95  27.71  23.38 ]
                    [                                   ]
                    [ 18.79  18.96  14.46  26.38  21.84 ]

  In the following example, we request only the first, second and fifth components of those records with wind speeds greater or equal than 16 in station number 1 and less than 25 knots in station number 4. The sample contains only data from stations 1, 2 and 5. In this case, the predicate function is defined as an ordinary Maxima function.

      (%i1) load (descriptive)$
      (%i2) s2 : read_matrix (file_search ("wind.data"))$
      (%i3) g(x):= x[1] >= 16 and x[4] < 25$
      (%i4) subsample (s2, g, 1, 2, 5);
                           [ 19.38  15.37  25.25 ]
                           [                     ]
                           [ 17.33  14.67  19.58 ]
      (%o4)                [                     ]
                           [ 16.92  13.21  21.21 ]
                           [                     ]
                           [ 17.25  18.46  23.87 ]

  Here is an example with the categorical variables of `biomed.data`. We want the records corresponding to those patients in group `B` who are older than 38 years.

      (%i1) load (descriptive)$
      (%i2) s3 : read_matrix (file_search ("biomed.data"))$
      (%i3) h(u):= u[1] = B and u[2] > 38 $
      (%i4) subsample (s3, h);
                      [ B  39  28.0  102.3  17.1  146 ]
                      [                               ]
                      [ B  39  21.0  92.4   10.3  197 ]
                      [                               ]
                      [ B  39  23.0  111.5  10.0  133 ]
                      [                               ]
                      [ B  39  26.0  92.6   12.3  196 ]
      (%o4)           [                               ]
                      [ B  39  25.0  98.7   10.0  174 ]
                      [                               ]
                      [ B  39  21.0  93.2   5.9   181 ]
                      [                               ]
                      [ B  39  18.0  95.0   11.3  66  ]
                      [                               ]
                      [ B  39  39.0  88.5   7.6   168 ]

  Probably, the statistical analysis will involve only the blood measures,

      (%i1) load (descriptive)$
      (%i2) s3 : read_matrix (file_search ("biomed.data"))$
      (%i3) subsample (s3, lambda([v], v[1] = B and v[2] > 38),
                       3, 4, 5, 6);
                         [ 28.0  102.3  17.1  146 ]
                         [                        ]
                         [ 21.0  92.4   10.3  197 ]
                         [                        ]
                         [ 23.0  111.5  10.0  133 ]
                         [                        ]
                         [ 26.0  92.6   12.3  196 ]
      (%o3)              [                        ]
                         [ 25.0  98.7   10.0  174 ]
                         [                        ]
                         [ 21.0  93.2   5.9   181 ]
                         [                        ]
                         [ 18.0  95.0   11.3  66  ]
                         [                        ]
                         [ 39.0  88.5   7.6   168 ]

  This is the multivariate mean of `s3`,

      (%i1) load (descriptive)$
      (%i2) s3 : read_matrix (file_search ("biomed.data"))$
      (%i3) mean (s3);
             65 B + 35 A  317          6 NA + 8144.999999999999
      (%o3) [-----------, ---, 87.178, ------------------------,
                 100      10                     100
                                                          3 NA + 19587
                                                  18.123, ------------]
                                                              100

  Here, the first component is meaningless, since `A` and `B` are categorical, the second component is the mean age of individuals in rational form, and the fourth and last values exhibit some strange behaviour. This is because symbol `NA` is used here to indicate non available data, and the two means are nonsense. A possible solution would be to take out from the matrix those rows with `NA` symbols, although this deserves some loss of information.

      (%i1) load (descriptive)$
      (%i2) s3 : read_matrix (file_search ("biomed.data"))$
      (%i3) g(v):= v[4] # NA and v[6] # NA $
      (%i4) mean (subsample (s3, g, 3, 4, 5, 6));
      (%o4) [79.4923076923077, 86.2032967032967, 16.93186813186813,
                                                                  2514
                                                                  ----]
                                                                   13

  Categories: Package descriptive

Function: transform_sample (matrix, varlist, exprlist)

  Transforms the sample matrix, where each column is called according to varlist, following expressions in exprlist.

  Examples:

  The second argument assigns names to the three columns. With these names, a list of expressions define the transformation of the sample.

      (%i1) load (descriptive)$
      (%i2) data: matrix([3,2,7],[3,7,2],[8,2,4],[5,2,4]) $
      (%i3) transform_sample(data, [a,b,c], [c, a*b, log(a)]);
                                     [ 7  6   log(3) ]
                                     [               ]
                                     [ 2  21  log(3) ]
      (%o3)                          [               ]
                                     [ 4  16  log(8) ]
                                     [               ]
                                     [ 4  10  log(5) ]

  Add a constant column and remove the third variable.

      (%i1) load (descriptive)$
      (%i2) data: matrix([3,2,7],[3,7,2],[8,2,4],[5,2,4]) $
      (%i3) transform_sample(data, [a,b,c], [makelist(1,k,length(data)),a,b]);
                                        [ 1  3  2 ]
                                        [         ]
                                        [ 1  3  7 ]
      (%o3)                             [         ]
                                        [ 1  8  2 ]
                                        [         ]
                                        [ 1  5  2 ]

  Categories: Package descriptive

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## 49.3 Functions and Variables for descriptive statistics

Function: mean
`mean` (list)
`mean` (matrix)

  This is the sample mean, defined as

                             n
                           ====
                   _   1   \
                   x = -    >    x
                       n   /      i
                           ====
                           i = 1

  Example:

      (%i1) load (descriptive)$
      (%i2) s1 : read_list (file_search ("pidigits.data"))$
      (%i3) mean (s1);
                                     471
      (%o3)                          ---
                                     100
      (%i4) %, numer;
      (%o4)                         4.71
      (%i5) s2 : read_matrix (file_search ("wind.data"))$
      (%i6) mean (s2);
      (%o6)     [9.9485, 10.1607, 10.8685, 15.7166, 14.8441]

  Categories: Package descriptive

Function: var
`var` (list)
`var` (matrix)

  This is the sample variance, defined as

                           n
                         ====
                 2   1   \          _ 2
                s  = -    >    (x - x)
                     n   /       i
                         ====
                         i = 1

  Example:

      (%i1) load (descriptive)$
      (%i2) s1 : read_list (file_search ("pidigits.data"))$
      (%i3) var (s1), numer;
      (%o3)                   8.425899999999999

  See also function `var1`.

  Categories: Package descriptive

Function: var1
`var1` (list)
`var1` (matrix)

  This is the sample variance, defined as

                           n
                         ====
                     1   \          _ 2
                    ---   >    (x - x)
                    n-1  /       i
                         ====
                         i = 1

  Example:

      (%i1) load (descriptive)$
      (%i2) s1 : read_list (file_search ("pidigits.data"))$
      (%i3) var1 (s1), numer;
      (%o3)                    8.5110101010101
      (%i4) s2 : read_matrix (file_search ("wind.data"))$
      (%i5) var1 (s2);
      (%o5) [17.39586540404041, 15.13912778787879, 15.63204924242424,
                                  32.50152569696971, 24.66977392929294]

  See also function `var`.

  Categories: Package descriptive

Function: std
`std` (list)
`std` (matrix)

  This is the square root of the function `var`, the variance with denominator n.

  Example:

      (%i1) load (descriptive)$
      (%i2) s1 : read_list (file_search ("pidigits.data"))$
      (%i3) std (s1), numer;
      (%o3)                   2.902740084816414
      (%i4) s2 : read_matrix (file_search ("wind.data"))$
      (%i5) std (s2);
      (%o5) [4.149928523480858, 3.871399812729241, 3.933920277534866,
                                  5.672434260526957, 4.941970881136392]

  See also functions `var` and `std1`.

  Categories: Package descriptive

Function: std1
`std1` (list)
`std1` (matrix)

  This is the square root of the function `var1`, the variance with denominator n-1.

  Example:

      (%i1) load (descriptive)$
      (%i2) s1 : read_list (file_search ("pidigits.data"))$
      (%i3) std1 (s1), numer;
      (%o3)                   2.917363553109228
      (%i4) s2 : read_matrix (file_search ("wind.data"))$
      (%i5) std1 (s2);
      (%o5) [4.170835096721089, 3.89090320978032, 3.953738641137555,
                                  5.701010936401517, 4.966867617451963]

  See also functions `var1` and `std`.

  Categories: Package descriptive

Function: noncentral_moment
`noncentral_moment` (list, k)
`noncentral_moment` (matrix, k)

  The non central moment of order k, defined as

                             n
                           ====
                       1   \      k
                       -    >    x
                       n   /      i
                           ====
                           i = 1

  Example:

      (%i1) load (descriptive)$
      (%i2) s1 : read_list (file_search ("pidigits.data"))$
      (%i3) noncentral_moment (s1, 1), numer; /* the mean */
      (%o3)                         4.71
      (%i5) s2 : read_matrix (file_search ("wind.data"))$
      (%i6) noncentral_moment (s2, 5);
      (%o6) [319793.8724761505, 320532.1923892463,
            391249.5621381556, 2502278.205988911, 1691881.797742255]

  See also function `central_moment`.

  Categories: Package descriptive

Function: central_moment
`central_moment` (list, k)
`central_moment` (matrix, k)

  The central moment of order k, defined as

                          n
                        ====
                    1   \          _ k
                    -    >    (x - x)
                    n   /       i
                        ====
                        i = 1

  Example:

      (%i1) load (descriptive)$
      (%i2) s1 : read_list (file_search ("pidigits.data"))$
      (%i3) central_moment (s1, 2), numer; /* the variance */
      (%o3)                   8.425899999999999
      (%i5) s2 : read_matrix (file_search ("wind.data"))$
      (%i6) central_moment (s2, 3);
      (%o6) [11.29584771375004, 16.97988248298583, 5.626661952750102,
                                   37.5986572057918, 25.85981904394192]

  See also functions `central_moment` and `mean`.

  Categories: Package descriptive

Function: cv
`cv` (list)
`cv` (matrix)

  The variation coefficient is the quotient between the sample standard deviation (`std`) and the `mean`,

      (%i1) load (descriptive)$
      (%i2) s1 : read_list (file_search ("pidigits.data"))$
      (%i3) cv (s1), numer;
      (%o3)                   .6193977819764815
      (%i4) s2 : read_matrix (file_search ("wind.data"))$
      (%i5) cv (s2);
      (%o5) [.4192426091090204, .3829365309260502, 0.363779605385983,
                                  .3627381836021478, .3346021393989506]

  See also functions `std` and `mean`.

  Categories: Package descriptive

Function: smin
`smin` (list)
`smin` (matrix)

  This is the minimum value of the sample list. When the argument is a matrix, `smin` returns a list containing the minimum values of the columns, which are associated to statistical variables.

      (%i1) load (descriptive)$
      (%i2) s1 : read_list (file_search ("pidigits.data"))$
      (%i3) smin (s1);
      (%o3)                           0
      (%i4) s2 : read_matrix (file_search ("wind.data"))$
      (%i5) smin (s2);
      (%o5)             [0.58, 0.5, 2.67, 5.25, 5.17]

  See also function `smax`.

  Categories: Package descriptive

Function: smax
`smax` (list)
`smax` (matrix)

  This is the maximum value of the sample list. When the argument is a matrix, `smax` returns a list containing the maximum values of the columns, which are associated to statistical variables.

      (%i1) load (descriptive)$
      (%i2) s1 : read_list (file_search ("pidigits.data"))$
      (%i3) smax (s1);
      (%o3)                           9
      (%i4) s2 : read_matrix (file_search ("wind.data"))$
      (%i5) smax (s2);
      (%o5)          [20.25, 21.46, 20.04, 29.63, 27.63]

  See also function `smin`.

  Categories: Package descriptive

Function: range
`range` (list)
`range` (matrix)

  The range is the difference between the extreme values.

  Example:

      (%i1) load (descriptive)$
      (%i2) s1 : read_list (file_search ("pidigits.data"))$
      (%i3) range (s1);
      (%o3)                           9
      (%i4) s2 : read_matrix (file_search ("wind.data"))$
      (%i5) range (s2);
      (%o5)          [19.67, 20.96, 17.37, 24.38, 22.46]

  Categories: Package descriptive

Function: quantile
`quantile` (list, p)
`quantile` (matrix, p)

  This is the p-quantile, with p a number in [0, 1], of the sample list. Although there are several definitions for the sample quantile (Hyndman, R. J., Fan, Y. (1996) Sample quantiles in statistical packages. American Statistician, 50, 361-365), the one based on linear interpolation is implemented in package descriptive

  Example:

      (%i1) load (descriptive)$
      (%i2) s1 : read_list (file_search ("pidigits.data"))$
      (%i3) /* 1st and 3rd quartiles */
               [quantile (s1, 1/4), quantile (s1, 3/4)], numer;
      (%o3)                      [2.0, 7.25]
      (%i4) s2 : read_matrix (file_search ("wind.data"))$
      (%i5) quantile (s2, 1/4);
      (%o5)    [7.2575, 7.477500000000001, 7.82, 11.28, 11.48]

  Categories: Package descriptive

Function: median
`median` (list)
`median` (matrix)

  Once the sample is ordered, if the sample size is odd the median is the central value, otherwise it is the mean of the two central values.

  Example:

      (%i1) load (descriptive)$
      (%i2) s1 : read_list (file_search ("pidigits.data"))$
      (%i3) median (s1);
                                      9
      (%o3)                           -
                                      2
      (%i4) s2 : read_matrix (file_search ("wind.data"))$
      (%i5) median (s2);
      (%o5)         [10.06, 9.855, 10.73, 15.48, 14.105]

  The median is the 1/2-quantile.

  See also function `quantile`.

  Categories: Package descriptive

Function: qrange
`qrange` (list)
`qrange` (matrix)

  The interquartilic range is the difference between the third and first quartiles, `quantile(list,3/4) - quantile(list,1/4)`,

      (%i1) load (descriptive)$
      (%i2) s1 : read_list (file_search ("pidigits.data"))$
      (%i3) qrange (s1);
                                     21
      (%o3)                          --
                                     4
      (%i4) s2 : read_matrix (file_search ("wind.data"))$
      (%i5) qrange (s2);
      (%o5) [5.385, 5.572499999999998, 6.022500000000001,
                                  8.729999999999999, 6.649999999999999]

  See also function `quantile`.

  Categories: Package descriptive

Function: mean_deviation
`mean_deviation` (list)
`mean_deviation` (matrix)

  The mean deviation, defined as

                           n
                         ====
                     1   \          _
                     -    >    |x - x|
                     n   /       i
                         ====
                         i = 1

  Example:

      (%i1) load (descriptive)$
      (%i2) s1 : read_list (file_search ("pidigits.data"))$
      (%i3) mean_deviation (s1);
                                     51
      (%o3)                          --
                                     20
      (%i4) s2 : read_matrix (file_search ("wind.data"))$
      (%i5) mean_deviation (s2);
      (%o5) [3.287959999999999, 3.075342, 3.23907, 4.715664000000001,
                                                     4.028546000000002]

  See also function `mean`.

  Categories: Package descriptive

Function: median_deviation
`median_deviation` (list)
`median_deviation` (matrix)

  The median deviation, defined as

                       n
                     ====
                 1   \
                 -    >    |x - med|
                 n   /       i
                     ====
                     i = 1

  where `med` is the median of list.

  Example:

      (%i1) load (descriptive)$
      (%i2) s1 : read_list (file_search ("pidigits.data"))$
      (%i3) median_deviation (s1);
                                      5
      (%o3)                           -
                                      2
      (%i4) s2 : read_matrix (file_search ("wind.data"))$
      (%i5) median_deviation (s2);
      (%o5)           [2.75, 2.755, 3.08, 4.315, 3.31]

  See also function `mean`.

  Categories: Package descriptive

Function: harmonic_mean
`harmonic_mean` (list)
`harmonic_mean` (matrix)

  The harmonic mean, defined as

                        n
                     --------
                      n
                     ====
                     \     1
                      >    --
                     /     x
                     ====   i
                     i = 1

  Example:

      (%i1) load (descriptive)$
      (%i2) y : [5, 7, 2, 5, 9, 5, 6, 4, 9, 2, 4, 2, 5]$
      (%i3) harmonic_mean (y), numer;
      (%o3)                   3.901858027632205
      (%i4) s2 : read_matrix (file_search ("wind.data"))$
      (%i5) harmonic_mean (s2);
      (%o5) [6.948015590052786, 7.391967752360356, 9.055658197151745,
                                  13.44199028193692, 13.01439145898509]

  See also functions `mean` and `geometric_mean`.

  Categories: Package descriptive

Function: geometric_mean
`geometric_mean` (list)
`geometric_mean` (matrix)

  The geometric mean, defined as

                       /  n      \ 1/n
                       | /===\   |
                       |  ! !    |
                       |  ! !  x |
                       |  ! !   i|
                       | i = 1   |
                       \         /

  Example:

      (%i1) load (descriptive)$
      (%i2) y : [5, 7, 2, 5, 9, 5, 6, 4, 9, 2, 4, 2, 5]$
      (%i3) geometric_mean (y), numer;
      (%o3)                   4.454845412337012
      (%i4) s2 : read_matrix (file_search ("wind.data"))$
      (%i5) geometric_mean (s2);
      (%o5) [8.82476274347979, 9.22652604739361, 10.0442675714889,
                                  14.61274126349021, 13.96184163444275]

  See also functions `mean` and `harmonic_mean`.

  Categories: Package descriptive

Function: kurtosis
`kurtosis` (list)
`kurtosis` (matrix)

  The kurtosis coefficient, defined as

                          n
                        ====
                  1     \          _ 4
                 ----    >    (x - x)  - 3
                    4   /       i
                 n s    ====
                        i = 1

  Example:

      (%i1) load (descriptive)$
      (%i2) s1 : read_list (file_search ("pidigits.data"))$
      (%i3) kurtosis (s1), numer;
      (%o3)                  - 1.273247946514421
      (%i4) s2 : read_matrix (file_search ("wind.data"))$
      (%i5) kurtosis (s2);
      (%o5) [- .2715445622195385, 0.119998784429451,
           - .4275233490482861, - .6405361979019522, - .4952382132352935]

  See also functions `mean`, `var` and `skewness`.

  Categories: Package descriptive

Function: skewness
`skewness` (list)
`skewness` (matrix)

  The skewness coefficient, defined as

                          n
                        ====
                  1     \          _ 3
                 ----    >    (x - x)
                    3   /       i
                 n s    ====
                        i = 1

  Example:

      (%i1) load (descriptive)$
      (%i2) s1 : read_list (file_search ("pidigits.data"))$
      (%i3) skewness (s1), numer;
      (%o3)                  .009196180476450424
      (%i4) s2 : read_matrix (file_search ("wind.data"))$
      (%i5) skewness (s2);
      (%o5) [.1580509020000978, .2926379232061854, .09242174416107717,
                                  .2059984348148687, .2142520248890831]

  See also functions `mean`,, `var` and `kurtosis`.

  Categories: Package descriptive

Function: pearson_skewness
`pearson_skewness` (list)
`pearson_skewness` (matrix)

  Pearson's skewness coefficient, defined as

                      _
                   3 (x - med)
                   -----------
                        s

  where med is the median of list.

  Example:

      (%i1) load (descriptive)$
      (%i2) s1 : read_list (file_search ("pidigits.data"))$
      (%i3) pearson_skewness (s1), numer;
      (%o3)                   .2159484029093895
      (%i4) s2 : read_matrix (file_search ("wind.data"))$
      (%i5) pearson_skewness (s2);
      (%o5) [- .08019976629211892, .2357036272952649,
               .1050904062491204, .1245042340592368, .4464181795804519]

  See also functions `mean`, `var` and `median`.

  Categories: Package descriptive

Function: quartile_skewness
`quartile_skewness` (list)
`quartile_skewness` (matrix)

  The quartile skewness coefficient, defined as

                     c    - 2 c    + c
                      3/4      1/2    1/4
                     --------------------
                         c    - c
                          3/4    1/4

  where c_p is the p-quantile of sample list.

  Example:

      (%i1) load (descriptive)$
      (%i2) s1 : read_list (file_search ("pidigits.data"))$
      (%i3) quartile_skewness (s1), numer;
      (%o3)                  .04761904761904762
      (%i4) s2 : read_matrix (file_search ("wind.data"))$
      (%i5) quartile_skewness (s2);
      (%o5) [- 0.0408542246982353, .1467025572005382,
             0.0336239103362392, .03780068728522298, .2105263157894735]

  See also function `quantile`.

  Categories: Package descriptive

Function: km
`km` (list, option ...)
`km` (matrix, option ...)

  Kaplan Meier estimator of the survival, or reliability, function S(x)=1-F(x).

  Data can be introduced as a list of pairs, or as a two column matrix. The first component is the observed time, and the second component a censoring index (1 = non censored, 0 = right censored).

  The optional argument is the name of the variable in the returned expression, which is x by default.

  Examples:

  Sample as a list of pairs.

      (%i1) load ("descriptive")$
      (%i2) S: km([[2,1], [3,1], [5,0], [8,1]]);
                             charfun((3 <= x) and (x < 8))
      (%o2) charfun(x < 0) + -----------------------------
                                           2
                      3 charfun((2 <= x) and (x < 3))
                    + -------------------------------
                                     4
                    + charfun((0 <= x) and (x < 2))
      (%i3) load ("draw")$
      (%i4) draw2d(
              line_width = 3, grid = true,
              explicit(S, x, -0.1, 10))$

  Estimate survival probabilities.

      (%i1) load ("descriptive")$
      (%i2) S(t):= ''(km([[2,1], [3,1], [5,0], [8,1]], t)) $
      (%i3) S(6);
                                  1
      (%o3)                       -
                                  2

  Categories: Package descriptive

Function: cdf_empirical
`cdf_empirical` (list, option ...)
`cdf_empirical` (matrix, option ...)

  Empirical distribution function F(x).

  Data can be introduced as a list of numbers, or as a one column matrix.

  The optional argument is the name of the variable in the returned expression, which is x by default.

  Example:

  Empirical distribution function.

      (%i1) load ("descriptive")$
      (%i2) F(x):= ''(cdf_empirical([1,3,3,5,7,7,7,8,9]));
      (%o2) F(x) := (charfun(x >= 9) + charfun(x >= 8)
                     + 3 charfun(x >= 7) + charfun(x >= 5)
                     + 2 charfun(x >= 3) + charfun(x >= 1))/9
      (%i3) F(6);
                                 4
      (%o3)                      -
                                 9
      (%i4) load(draw)$
      (%i5) draw2d(
              line_width = 3,
              grid       = true,
              explicit(F(z), z, -2, 12)) $

  Categories: Package descriptive

Function: cov (matrix)

  The covariance matrix of the multivariate sample, defined as

                    n
                   ====
                1  \           _        _
            S = -   >    (X  - X) (X  - X)'
                n  /       j        j
                   ====
                   j = 1

  where X_j is the j-th row of the sample matrix.

  Example:

      (%i1) load ("descriptive")$
      (%i2) s2 : read_matrix (file_search ("wind.data"))$
      (%i3) fpprintprec : 7$  /* change precision for pretty output */
      (%i4) cov (s2);
            [ 17.22191  13.61811  14.37217  19.39624  15.42162 ]
            [                                                  ]
            [ 13.61811  14.98774  13.30448  15.15834  14.9711  ]
            [                                                  ]
      (%o4) [ 14.37217  13.30448  15.47573  17.32544  16.18171 ]
            [                                                  ]
            [ 19.39624  15.15834  17.32544  32.17651  20.44685 ]
            [                                                  ]
            [ 15.42162  14.9711   16.18171  20.44685  24.42308 ]

  See also function `cov1`.

  Categories: Package descriptive

Function: cov1 (matrix)

  The covariance matrix of the multivariate sample, defined as

                    n
                   ====
               1   \           _        _
         S  = ---   >    (X  - X) (X  - X)'
          1   n-1  /       j        j
                   ====
                   j = 1

  where X_j is the j-th row of the sample matrix.

  Example:

      (%i1) load ("descriptive")$
      (%i2) s2 : read_matrix (file_search ("wind.data"))$
      (%i3) fpprintprec : 7$ /* change precision for pretty output */
      (%i4) cov1 (s2);
            [ 17.39587  13.75567  14.51734  19.59216  15.5774  ]
            [                                                  ]
            [ 13.75567  15.13913  13.43887  15.31145  15.12232 ]
            [                                                  ]
      (%o4) [ 14.51734  13.43887  15.63205  17.50044  16.34516 ]
            [                                                  ]
            [ 19.59216  15.31145  17.50044  32.50153  20.65338 ]
            [                                                  ]
            [ 15.5774   15.12232  16.34516  20.65338  24.66977 ]

  See also function `cov`.

  Categories: Package descriptive

Function: global_variances
`global_variances` (matrix)
`global_variances` (matrix, options ...)

  Function `global_variances` returns a list of global variance measures:

  - total variance: `trace(S_1)`,
  - mean variance: `trace(S_1)/p`,
  - generalized variance: `determinant(S_1)`,
  - generalized standard deviation: `sqrt(determinant(S_1))`,
  - efective variance `determinant(S_1)^(1/p)`, (defined in: Peña, D. (2002) Análisis de datos multivariantes; McGraw-Hill, Madrid.)
  - efective standard deviation: `determinant(S_1)^(1/(2*p))`.

  where p is the dimension of the multivariate random variable and S_1 the covariance matrix returned by `cov1`.

  Option:

  - `'data`, default `'true`, indicates whether the input matrix contains the sample data, in which case the covariance matrix `cov1` must be calculated, or not, and then the covariance matrix (symmetric) must be given, instead of the data.

  Example:

      (%i1) load ("descriptive")$
      (%i2) s2 : read_matrix (file_search ("wind.data"))$
      (%i3) global_variances (s2);
      (%o3) [105.338342060606, 21.06766841212119, 12874.34690469686,
               113.4651792608501, 6.636590811800795, 2.576158149609762]

  Calculate the `global_variances` from the covariance matrix.

      (%i1) load ("descriptive")$
      (%i2) s2 : read_matrix (file_search ("wind.data"))$
      (%i3) s : cov1 (s2)$
      (%i4) global_variances (s, data=false);
      (%o4) [105.338342060606, 21.06766841212119, 12874.34690469686,
               113.4651792608501, 6.636590811800795, 2.576158149609762]

  See also `cov` and `cov1`.

  Categories: Package descriptive

Function: cor
`cor` (matrix)
`cor` (matrix, logical_value)

  The correlation matrix of the multivariate sample.

  Option:

  - `'data`, default `'true`, indicates whether the input matrix contains the sample data, in which case the covariance matrix `cov1` must be calculated, or not, and then the covariance matrix (symmetric) must be given, instead of the data.

  Example:

      (%i1) load ("descriptive")$
      (%i2) fpprintprec : 7 $
      (%i3) s2 : read_matrix (file_search ("wind.data"))$
      (%i4) cor (s2);
            [   1.0     .8476339  .8803515  .8239624  .7519506 ]
            [                                                  ]
            [ .8476339    1.0     .8735834  .6902622  0.782502 ]
            [                                                  ]
      (%o4) [ .8803515  .8735834    1.0     .7764065  .8323358 ]
            [                                                  ]
            [ .8239624  .6902622  .7764065    1.0     .7293848 ]
            [                                                  ]
            [ .7519506  0.782502  .8323358  .7293848    1.0    ]

  Calculate de correlation matrix from the covariance matrix.

      (%i1) load ("descriptive")$
      (%i2) fpprintprec : 7 $
      (%i3) s2 : read_matrix (file_search ("wind.data"))$
      (%i4) s : cov1 (s2)$
      (%i5) cor (s, data=false); /* this is faster */
            [   1.0     .8476339  .8803515  .8239624  .7519506 ]
            [                                                  ]
            [ .8476339    1.0     .8735834  .6902622  0.782502 ]
            [                                                  ]
      (%o5) [ .8803515  .8735834    1.0     .7764065  .8323358 ]
            [                                                  ]
            [ .8239624  .6902622  .7764065    1.0     .7293848 ]
            [                                                  ]
            [ .7519506  0.782502  .8323358  .7293848    1.0    ]

  See also `cov` and `cov1`.

  Categories: Package descriptive

Function: list_correlations
`list_correlations` (matrix)
`list_correlations` (matrix, options ...)

  Function `list_correlations` returns a list of correlation measures:

  - precision matrix: the inverse of the covariance matrix S_1,

               -1     ij
              S   = (s  )
               1         i,j = 1,2,...,p

  - multiple correlation vector: (R_1^2, R_2^2, ..., R_p^2), with

               2          1
              R  = 1 - -------
               i        ii
                       s   s
                            ii

    being an indicator of the goodness of fit of the linear multivariate regression model on X_i when the rest of variables are used as regressors.

  - partial correlation matrix: with element (i, j) being

                                 ij
                                s
              r        = - ------------
               ij.rest     / ii  jj\ 1/2
                           |s   s  |
                           \       /

  Option:

  - `'data`, default `'true`, indicates whether the input matrix contains the sample data, in which case the covariance matrix `cov1` must be calculated, or not, and then the covariance matrix (symmetric) must be given, instead of the data.

  Example:

      (%i1) load ("descriptive")$
      (%i2) s2 : read_matrix (file_search ("wind.data"))$
      (%i3) z : list_correlations (s2)$
      (%i4) fpprintprec : 5$ /* for pretty output */
      (%i5) z[1];  /* precision matrix */
            [  .38486   - .13856   - .15626   - .10239    .031179  ]
            [                                                      ]
            [ - .13856   .34107    - .15233    .038447   - .052842 ]
            [                                                      ]
      (%o5) [ - .15626  - .15233    .47296    - .024816  - .10054  ]
            [                                                      ]
            [ - .10239   .038447   - .024816   .10937    - .034033 ]
            [                                                      ]
            [ .031179   - .052842  - .10054   - .034033   .14834   ]
      (%i6) z[2];  /* multiple correlation vector */
      (%o6)      [.85063, .80634, .86474, .71867, .72675]
      (%i7) z[3];  /* partial correlation matrix */
            [  - 1.0     .38244   .36627   .49908   - .13049 ]
            [                                                ]
            [  .38244    - 1.0    .37927  - .19907   .23492  ]
            [                                                ]
      (%o7) [  .36627    .37927   - 1.0    .10911    .37956  ]
            [                                                ]
            [  .49908   - .19907  .10911   - 1.0     .26719  ]
            [                                                ]
            [ - .13049   .23492   .37956   .26719    - 1.0   ]

  See also `cov` and `cov1`.

  Categories: Package descriptive

Function: principal_components
`principal_components` (matrix)
`principal_components` (matrix, options ...)

  Calculates the principal componentes of a multivariate sample. Principal components are used in multivariate statistical analysis to reduce the dimensionality of the sample.

  Option:

  - `'data`, default `'true`, indicates whether the input matrix contains the sample data, in which case the covariance matrix `cov1` must be calculated, or not, and then the covariance matrix (symmetric) must be given, instead of the data.

  The output of function `principal_components` is a list with the following results:

  - variances of the principal components,
  - percentage of total variance explained by each principal component,
  - rotation matrix.

  Examples:

  In this sample, the first component explains 83.13 per cent of total variance.

      (%i1) load ("descriptive")$
      (%i2) s2 : read_matrix (file_search ("wind.data"))$
      (%i3) fpprintprec:4 $
      (%i4) res: principal_components(s2);
      0 errors, 0 warnings
      (%o4) [[87.57, 8.753, 5.515, 1.889, 1.613],
      [83.13, 8.31, 5.235, 1.793, 1.531],
      [ .4149  .03379   - .4757  - 0.581   - .5126 ]
      [                                            ]
      [ 0.369  - .3657  - .4298   .7237    - .1469 ]
      [                                            ]
      [ .3959  - .2178  - .2181  - .2749    .8201  ]]
      [                                            ]
      [ .5548   .7744    .1857    .2319    .06498  ]
      [                                            ]
      [ .4765  - .4669   0.712   - .09605  - .1969 ]
      (%i5) /* accumulated percentages  */
          block([ap: copy(res[2])],
            for k:2 thru length(ap) do ap[k]: ap[k]+ap[k-1],
            ap);
      (%o5)                 [83.13, 91.44, 96.68, 98.47, 100.0]
      (%i6) /* sample dimension */
            p: length(first(res));
      (%o6)                                  5
      (%i7) /* plot percentages to select number of
               principal components for further work */
           draw2d(
              fill_density = 0.2,
              apply(bars, makelist([k, res[2][k], 1/2], k, p)),
              points_joined = true,
              point_type    = filled_circle,
              point_size    = 3,
              points(makelist([k, res[2][k]], k, p)),
              xlabel = "Variances",
              ylabel = "Percentages",
              xtics  = setify(makelist([concat("PC",k),k], k, p))) $

  In case de covariance matrix is known, it can be passed to the function, but option `data=false` must be used.

      (%i1) load ("descriptive")$
      (%i2) S: matrix([1,-2,0],[-2,5,0],[0,0,2]);
                                      [  1   - 2  0 ]
                                      [             ]
      (%o2)                           [ - 2   5   0 ]
                                      [             ]
                                      [  0    0   2 ]
      (%i3) fpprintprec:4 $
      (%i4) /* the argumment is a covariance matrix */
            res: principal_components(S, data=false);
      0 errors, 0 warnings
                                                        [ - .3827  0.0  .9239 ]
                                                        [                     ]
      (%o4) [[5.828, 2.0, .1716], [72.86, 25.0, 2.145], [  .9239   0.0  .3827 ]]
                                                        [                     ]
                                                        [   0.0    1.0   0.0  ]
      (%i5) /* transformation to get the principal components
               from original records */
            matrix([a1,b2,c3],[a2,b2,c2]).last(res);
                   [ .9239 b2 - .3827 a1  1.0 c3  .3827 b2 + .9239 a1 ]
      (%o5)        [                                                  ]
                   [ .9239 b2 - .3827 a2  1.0 c2  .3827 b2 + .9239 a2 ]

  Categories: Package descriptive

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## 49.4 Functions and Variables for statistical graphs

Function: barsplot (data1, data2, …, option_1, option_2, …)

  Plots bars diagrams for discrete statistical variables, both for one or multiple samples.

  data can be a list of outcomes representing one sample, or a matrix of m rows and n columns, representing n samples of size m each.

  Available options are:

  - box_width (default, `3/4`): relative width of rectangles. This value must be in the range `[0,1]`.
  - grouping (default, `clustered`): indicates how multiple samples are shown. Valid values are: `clustered` and `stacked`.
  - groups_gap (default, `1`): a positive integer number representing the gap between two consecutive groups of bars.
  - bars_colors (default, `[]`): a list of colors for multiple samples. When there are more samples than specified colors, the extra necessary colors are chosen at random. See `color` to learn more about them.
  - frequency (default, `absolute`): indicates the scale of the ordinates. Possible values are: `absolute`, `relative`, and `percent`.
  - ordering (default, `orderlessp`): possible values are `orderlessp` or `ordergreatp`, indicating how statistical outcomes should be ordered on the x-axis.
  - sample_keys (default, `[]`): a list with the strings to be used in the legend. When the list length is other than 0 or the number of samples, an error message is returned.
  - start_at (default, `0`): indicates where the plot begins to be plotted on the x axis.
  - All global `draw` options, except `xtics`, which is internally assigned by `barsplot`. If you want to set your own values for this option or want to build complex scenes, make use of `barsplot_description`. See example below.
  - The following local draw options: `key`, `color_draw`, `fill_color`, `fill_density` and `line_width`. See also `barsplot`.

  There is also a function `wxbarsplot` for creating embedded histograms in interfaces wxMaxima and iMaxima. `barsplot` in a multiplot context.

  Examples:

  Univariate sample in matrix form. Absolute frequencies.

      (%i1) load ("descriptive")$
      (%i2) m : read_matrix (file_search ("biomed.data"))$
      (%i3) barsplot(
              col(m,2),
              title        = "Ages",
              xlabel       = "years",
              box_width    = 1/2,
              fill_density = 3/4)$

  Two samples of different sizes, with relative frequencies and user declared colors.

      (%i1) load ("descriptive")$
      (%i2) l1:makelist(random(10),k,1,50)$
      (%i3) l2:makelist(random(10),k,1,100)$
      (%i4) barsplot(
              l1,l2,
              box_width    = 1,
              fill_density = 1,
              bars_colors  = [black, grey],
              frequency = relative,
              sample_keys = ["A", "B"])$

  Four non numeric samples of equal size.

      (%i1) load ("descriptive")$
      (%i2) barsplot(
              makelist([Yes, No, Maybe][random(3)+1],k,1,50),
              makelist([Yes, No, Maybe][random(3)+1],k,1,50),
              makelist([Yes, No, Maybe][random(3)+1],k,1,50),
              makelist([Yes, No, Maybe][random(3)+1],k,1,50),
              title  = "Asking for something to four groups",
              ylabel = "# of individuals",
              groups_gap   = 3,
              fill_density = 0.5,
              ordering     = ordergreatp)$

  Stacked bars.

      (%i1) load ("descriptive")$
      (%i2) barsplot(
              makelist([Yes, No, Maybe][random(3)+1],k,1,50),
              makelist([Yes, No, Maybe][random(3)+1],k,1,50),
              makelist([Yes, No, Maybe][random(3)+1],k,1,50),
              makelist([Yes, No, Maybe][random(3)+1],k,1,50),
              title  = "Asking for something to four groups",
              ylabel = "# of individuals",
              grouping     = stacked,
              fill_density = 0.5,
              ordering     = ordergreatp)$

  For bars diagrams related options, see `barsplot` of package draw See also functions `histogram` and `piechart`.

  Categories: Package descriptive · Plotting

Function: barsplot_description (…)

  Function `barsplot_description` creates a graphic object suitable for creating complex scenes, together with other graphic objects.

  Example: `barsplot` in a multiplot context.

      (%i1) load ("descriptive")$
      (%i2) l1:makelist(random(10),k,1,50)$
      (%i3) l2:makelist(random(10),k,1,100)$
      (%i4) bp1 :
              barsplot_description(
               l1,
               box_width = 1,
               fill_density = 0.5,
               bars_colors = [blue],
               frequency = relative)$
      (%i5) bp2 :
              barsplot_description(
               l2,
               box_width = 1,
               fill_density = 0.5,
               bars_colors = [red],
               frequency = relative)$
      (%i6) draw(gr2d(bp1), gr2d(bp2))$

  Categories: Package descriptive · Plotting

Function: boxplot (data)
`boxplot` (data, option_1, option_2, …)

  This function plots box-and-whisker diagrams. Argument data can be a list, which is not of great interest, since these diagrams are mainly used for comparing different samples, or a matrix, so it is possible to compare two or more components of a multivariate statistical variable. But it is also allowed data to be a list of samples with possible different sample sizes, in fact this is the only function in package `descriptive` that admits this type of data structure.

  The box is plotted from the first quartile to the third, with an horizontal segment situated at the second quartile or median. By default, lower and upper whiskers are plotted at the minimum and maximum values, respectively. Option range can be used to indicate that values greater than `quantile(x,3/4)+range*(quantile(x,3/4)-quantile(x,1/4))` or less than `quantile(x,1/4)-range*(quantile(x,3/4)-quantile(x,1/4))` must be considered as outliers, in which case they are plotted as isolated points, and the whiskers are located at the extremes of the rest of the sample.

  Available options are:

  - box_width (default, `3/4`): relative width of boxes. This value must be in the range `[0,1]`.
  - box_orientation (default, `vertical`): possible values: `vertical` and `horizontal`.
  - range (default, `inf`): positive coefficient of the interquartilic range to set outliers boundaries.
  - outliers_size (default, `1`): circle size for isolated outliers.
  - All `draw` options, except `points_joined`, `point_size`, `point_type`, `xtics`, `ytics`, `xrange`, and `yrange`, which are internally assigned by `boxplot`. If you want to set your own values for this options or want to build complex scenes, make use of `boxplot_description`.
  - The following local `draw` options: `key`, `color`, and `line_width`.

  There is also a function `wxboxplot` for creating embedded histograms in interfaces wxMaxima and iMaxima.

  Examples:

  Box-and-whisker diagram from a multivariate sample.

      (%i1) load ("descriptive")$
      (%i2) s2 : read_matrix(file_search("wind.data"))$
      (%i3) boxplot(s2,
              box_width  = 0.2,
              title      = "Windspeed in knots",
              xlabel     = "Stations",
              color      = red,
              line_width = 2)$

  Box-and-whisker diagram from three samples of different sizes.

      (%i1) load ("descriptive")$
      (%i2) A :
             [[6, 4, 6, 2, 4, 8, 6, 4, 6, 4, 3, 2],
              [8, 10, 7, 9, 12, 8, 10],
              [16, 13, 17, 12, 11, 18, 13, 18, 14, 12]]$
      (%i3) boxplot (A, box_orientation = horizontal)$

  Option range can be used to handle outliers.

      (%i1) load ("descriptive")$
      (%i2) B: [[7, 15, 5, 8, 6, 5, 7, 3, 1],
                [10, 8, 12, 8, 11, 9, 20],
                [23, 17, 19, 7, 22, 19]] $
      (%i3) boxplot (B, range=1)$
      (%i4) boxplot (B, range=1.5, box_orientation = horizontal)$
      (%i5) draw2d(
              boxplot_description(
                B,
                range            = 1.5,
                line_width       = 3,
                outliers_size    = 2,
                color            = red,
                background_color = light_gray),
              xtics = {["Low",1],["Medium",2],["High",3]}) $

  Categories: Package descriptive · Plotting

Function: boxplot_description (…)

  Function `boxplot_description` creates a graphic object suitable for creating complex scenes, together with other graphic objects.

  Categories: Package descriptive · Plotting

Function: histogram
`histogram` (list)
`histogram` (list, option_1, option_2, …)
`histogram` (one_column_matrix)
`histogram` (one_column_matrix, option_1, option_2, …)
`histogram` (one_row_matrix)
`histogram` (one_row_matrix, option_1, option_2, …)

  This function plots an histogram from a continuous sample. Sample data must be stored in a list of numbers or a one dimensional matrix.

  Available options are:

  - nclasses (default, `10`): number of classes of the histogram, or a list indicating the limits of the classes and the number of them, or only the limits. This option also accepts bounds for varying bin widths, or a symbol with the name of one of the three optimal algorithms available for the number of classes: `'fd` (Freedman, D. and Diaconis, P. (1981) On the histogram as a density estimator: L_2 theory. Zeitschrift fuer Wahrscheinlichkeitstheorie und verwandte Gebiete 57, 453-476.), `'scott` (Scott, D. W. (1979) On optimal and data-based histograms. Biometrika 66, 605-610.), and `'sturges` (Sturges, H. A. (1926) The choice of a class interval. Journal of the American Statistical Association 21, 65-66).
  - frequency (default, `absolute`): indicates the scale of the ordinates. Possible values are: `absolute`, `relative`, `percent`, and `density`. With `density`, the histogram area has a total area of one.
  - htics (default, `auto`): format of the histogram tics. Possible values are: `auto`, `endpoints`, `intervals`, or a list of labels.
  - All global `draw` options, except `xrange`, `yrange`, and `xtics`, which are internally assigned by `histogram`. If you want to set your own values for these options, make use of `histogram_description`. See examples bellow.
  - The following local draw options: `key`, `color`, `fill_color`, `fill_density` and `line_width`. See also `barsplot`.

  There is also a function `wxhistogram` for creating embedded histograms in interfaces wxMaxima and iMaxima.

  Examples:

  A simple with eight classes:

      (%i1) load ("descriptive")$
      (%i2) s1 : read_list (file_search ("pidigits.data"))$
      (%i3) histogram (
                 s1,
                 nclasses     = 8,
                 title        = "pi digits",
                 xlabel       = "digits",
                 ylabel       = "Absolute frequency",
                 fill_color   = grey,
                 fill_density = 0.6)$

  Setting the limits of the histogram to -2 and 12, with 3 classes. Also, we introduce predefined tics:

      (%i1) load ("descriptive")$
      (%i2) s1 : read_list (file_search ("pidigits.data"))$
      (%i3) histogram (
                 s1,
                 nclasses     = [-2,12,3],
                 htics        = ["A", "B", "C"],
                 terminal     = png,
                 fill_color   = "#23afa0",
                 fill_density = 0.6)$

  Bounds for varying bin widths.

      (%i1) load ("descriptive")$
      (%i2) s1 : read_list (file_search ("pidigits.data"))$
      (%i3) histogram (s1, nclasses = {0,3,6,7,11})$

  Freedmann - Diakonis robust method for optimal search of the number of classes.

      (%i1) load ("descriptive")$
      (%i2) s1 : read_list (file_search ("pidigits.data"))$
      (%i3) histogram(s1, nclasses=fd) $

  Categories: Package descriptive · Plotting

Function: histogram_description (…)

  Function `histogram_description` creates a graphic object suitable for creating complex scenes, together with other graphic objects. We make use of `histogram_description` for setting the `xrange` and adding an explicit curve into the scene:

      (%i1) load ("descriptive")$
      (%i2) ( load("distrib"),
              m: 14, s: 2,
              s2: random_normal(m, s, 1000) ) $
      (%i3) draw2d(
              grid   = true,
              xrange = [5, 25],
              histogram_description(
                s2,
                nclasses     = 9,
                frequency    = density,
                fill_density = 0.5),
              explicit(pdf_normal(x,m,s), x, m - 3*s, m + 3* s))$

  Categories: Package descriptive · Plotting

Function: piechart
`piechart` (list)
`piechart` (list, option_1, option_2, …)
`piechart` (one_column_matrix)
`piechart` (one_column_matrix, option_1, option_2, …)
`piechart` (one_row_matrix)
`piechart` (one_row_matrix, option_1, option_2, …)

  Similar to `barsplot`, but plots sectors instead of rectangles.

  Available options are:

  - sector_colors (default, `[]`): a list of colors for sectors. When there are more sectors than specified colors, the extra necessary colors are chosen at random. See `color` to learn more about them.
  - pie_center (default, `[0,0]`): diagram's center.
  - pie_radius (default, `1`): diagram's radius.
  - All global `draw` options, except `key`, which is internally assigned by `piechart`. If you want to set your own values for this option or want to build complex scenes, make use of `piechart_description`.
  - The following local `draw` options: `key`, `color`, `fill_density` and `line_width`. See also `ellipse`

  There is also a function `wxpiechart` for creating embedded histograms in interfaces wxMaxima and iMaxima.

  Example:

      (%i1) load ("descriptive")$
      (%i2) s1 : read_list (file_search ("pidigits.data"))$
      (%i3) piechart(
              s1,
              xrange  = [-1.1, 1.3],
              yrange  = [-1.1, 1.1],
              title   = "Digit frequencies in pi")$

  See also function `barsplot`.

  Categories: Package descriptive · Plotting

Function: piechart_description (…)

  Function `piechart_description` creates a graphic object suitable for creating complex scenes, together with other graphic objects.

  Categories: Package descriptive · Plotting

Function: scatterplot
`scatterplot` (list)
`scatterplot` (list, option_1, option_2, …)
`scatterplot` (matrix)
`scatterplot` (matrix, option_1, option_2, …)

  Plots scatter diagrams both for univariate (list) and multivariate (matrix) samples.

  Available options are the same admitted by `histogram`.

  There is also a function `wxscatterplot` for creating embedded histograms in interfaces wxMaxima and iMaxima.

  Examples:

  Univariate scatter diagram from a simulated Gaussian sample.

      (%i1) load ("descriptive")$
      (%i2) load ("distrib")$
      (%i3) scatterplot(
              random_normal(0,1,200),
              xaxis      = true,
              point_size = 2,
              dimensions = [600,150])$

  Two dimensional scatter plot.

      (%i1) load ("descriptive")$
      (%i2) s2 : read_matrix (file_search ("wind.data"))$
      (%i3) scatterplot(
             submatrix(s2, 1,2,3),
             title      = "Data from stations #4 and #5",
             point_type = diamant,
             point_size = 2,
             color      = blue)$

  Three dimensional scatter plot.

      (%i1) load ("descriptive")$
      (%i2) s2 : read_matrix (file_search ("wind.data"))$
      (%i3) scatterplot(submatrix (s2, 1,2), nclasses=4)$

  Five dimensional scatter plot, with five classes histograms.

      (%i1) load ("descriptive")$
      (%i2) s2 : read_matrix (file_search ("wind.data"))$
      (%i3) scatterplot(
              s2,
              nclasses     = 5,
              frequency    = relative,
              fill_color   = blue,
              fill_density = 0.3,
              xtics        = 5)$

  For plotting isolated or line-joined points in two and three dimensions, see `points`. See also `histogram`.

  Categories: Package descriptive · Plotting

Function: scatterplot_description (…)

  Function `scatterplot_description` creates a graphic object suitable for creating complex scenes, together with other graphic objects.

  Categories: Package descriptive · Plotting

Function: starplot (data1, data2, …, option_1, option_2, …)

  Plots star diagrams for discrete statistical variables, both for one or multiple samples.

  data can be a list of outcomes representing one sample, or a matrix of m rows and n columns, representing n samples of size m each.

  Available options are:

  - stars_colors (default, `[]`): a list of colors for multiple samples. When there are more samples than specified colors, the extra necessary colors are chosen at random. See `color` to learn more about them.
  - frequency (default, `absolute`): indicates the scale of the radii. Possible values are: `absolute` and `relative`.
  - ordering (default, `orderlessp`): possible values are `orderlessp` or `ordergreatp`, indicating how statistical outcomes should be ordered.
  - sample_keys (default, `[]`): a list with the strings to be used in the legend. When the list length is other than 0 or the number of samples, an error message is returned.
  - star_center (default, `[0,0]`): diagram's center.
  - star_radius (default, `1`): diagram's radius.
  - All global `draw` options, except `points_joined`, `point_type`, and `key`, which are internally assigned by `starplot`. If you want to set your own values for this options or want to build complex scenes, make use of `starplot_description`.
  - The following local `draw` option: `line_width`.

  There is also a function `wxstarplot` for creating embedded histograms in interfaces wxMaxima and iMaxima.

  Example:

  Plot based on absolute frequencies. Location and radius defined by the user.

      (%i1) load ("descriptive")$
      (%i2) l1: makelist(random(10),k,1,50)$
      (%i3) l2: makelist(random(10),k,1,200)$
      (%i4) starplot(
              l1, l2,
              stars_colors = [blue,red],
              sample_keys = ["1st sample", "2nd sample"],
              star_center = [1,2],
              star_radius = 4,
              proportional_axes = xy,
              line_width = 2 ) $

  Categories: Package descriptive · Plotting

Function: starplot_description (…)

  Function `starplot_description` creates a graphic object suitable for creating complex scenes, together with other graphic objects.

  Categories: Package descriptive · Plotting

Function: stemplot
`stemplot` (data)
`stemplot` (data, option)

  Plots stem and leaf diagrams.

  Unique available option is:

  - leaf_unit (default, `1`): indicates the unit of the leaves; must be a power of 10.

  Example:

      (%i1) load ("descriptive")$
      (%i2) load(distrib)$
      (%i3) stemplot(
              random_normal(15, 6, 100),
              leaf_unit = 0.1);
      -5|4
       0|37
       1|7
       3|6
       4|4
       5|4
       6|57
       7|0149
       8|3
       9|1334588
      10|07888
      11|01144467789
      12|12566889
      13|24778
      14|047
      15|223458
      16|4
      17|11557
      18|000247
      19|4467799
      20|00
      21|1
      22|2335
      23|01457
      24|12356
      25|455
      27|79
      key: 6|3 =  6.3
      (%o3)                  done

  Categories: Package descriptive · Plotting

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# 50. diag

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## 50.1 Functions and Variables for diag

Function: diag (lm)

  Constructs a matrix that is the block sum of the elements of lm. The elements of lm are assumed to be matrices; if an element is scalar, it treated as a 1 by 1 matrix.

  The resulting matrix will be square if each of the elements of lm is square.

  Example:

      (%i1) load("diag")$

      (%i2) a1:matrix([1,2,3],[0,4,5],[0,0,6])$

      (%i3) a2:matrix([1,1],[1,0])$

      (%i4) diag([a1,x,a2]);
                         [ 1  2  3  0  0  0 ]
                         [                  ]
                         [ 0  4  5  0  0  0 ]
                         [                  ]
                         [ 0  0  6  0  0  0 ]
      (%o4)              [                  ]
                         [ 0  0  0  x  0  0 ]
                         [                  ]
                         [ 0  0  0  0  1  1 ]
                         [                  ]
                         [ 0  0  0  0  1  0 ]
      (%i5) diag ([matrix([1,2]), 3]);
                              [ 1  2  0 ]
      (%o5)                   [         ]
                              [ 0  0  3 ]

  To use this function write first `load("diag")`.

  Categories: Matrices · Share packages · Package diag

Function: JF (lambda,n)

  Returns the Jordan cell of order n with eigenvalue lambda.

  Example:

      (%i1) load("diag")$

      (%i2) JF(2,5);
                          [ 2  1  0  0  0 ]
                          [               ]
                          [ 0  2  1  0  0 ]
                          [               ]
      (%o2)               [ 0  0  2  1  0 ]
                          [               ]
                          [ 0  0  0  2  1 ]
                          [               ]
                          [ 0  0  0  0  2 ]
      (%i3) JF(3,2);
                               [ 3  1 ]
      (%o3)                    [      ]
                               [ 0  3 ]

  To use this function write first `load("diag")`.

  Categories: Package diag

Function: jordan (mat)

  Returns the Jordan form of matrix mat, encoded as a list in a particular format. To get the corresponding matrix, call the function `dispJordan` using the output of `jordan` as the argument.

  The elements of the returned list are themselves lists. The first element of each is an eigenvalue of mat. The remaining elements are positive integers which are the lengths of the Jordan blocks for this eigenvalue. These integers are listed in decreasing order. Eigenvalues are not repeated.

  The functions `dispJordan`, `minimalPoly` and `ModeMatrix` expect the output of a call to `jordan` as an argument. If you construct this argument by hand, rather than by calling `jordan`, you must ensure that each eigenvalue only appears once and that the block sizes are listed in decreasing order, otherwise the functions might give incorrect answers.

  Example:

      (%i1) load("diag")$
      (%i2) A: matrix([2,0,0,0,0,0,0,0],
                      [1,2,0,0,0,0,0,0],
                      [-4,1,2,0,0,0,0,0],
                      [2,0,0,2,0,0,0,0],
                      [-7,2,0,0,2,0,0,0],
                      [9,0,-2,0,1,2,0,0],
                      [-34,7,1,-2,-1,1,2,0],
                      [145,-17,-16,3,9,-2,0,3])$
      (%i3) jordan (A);
      (%o3)                [[2, 3, 3, 1], [3, 1]]
      (%i4) dispJordan (%);
                         [ 2  1  0  0  0  0  0  0 ]
                         [                        ]
                         [ 0  2  1  0  0  0  0  0 ]
                         [                        ]
                         [ 0  0  2  0  0  0  0  0 ]
                         [                        ]
                         [ 0  0  0  2  1  0  0  0 ]
      (%o4)              [                        ]
                         [ 0  0  0  0  2  1  0  0 ]
                         [                        ]
                         [ 0  0  0  0  0  2  0  0 ]
                         [                        ]
                         [ 0  0  0  0  0  0  2  0 ]
                         [                        ]
                         [ 0  0  0  0  0  0  0  3 ]

  To use this function write first `load("diag")`. See also `dispJordan` and `minimalPoly`.

  Categories: Package diag

Function: dispJordan (l)

  Returns a matrix in Jordan canonical form (JCF) corresponding to the list of eigenvalues and multiplicities given by l. This list should be in the format given by the `jordan` function. See `jordan` for details of this format.

  Example:

      (%i1) load("diag")$

      (%i2) b1:matrix([0,0,1,1,1],
                      [0,0,0,1,1],
                      [0,0,0,0,1],
                      [0,0,0,0,0],
                      [0,0,0,0,0])$

      (%i3) jordan(b1);
      (%o3)                  [[0, 3, 2]]
      (%i4) dispJordan(%);
                          [ 0  1  0  0  0 ]
                          [               ]
                          [ 0  0  1  0  0 ]
                          [               ]
      (%o4)               [ 0  0  0  0  0 ]
                          [               ]
                          [ 0  0  0  0  1 ]
                          [               ]
                          [ 0  0  0  0  0 ]

  To use this function write first `load("diag")`. See also `jordan` and `minimalPoly`.

  Categories: Package diag

Function: minimalPoly (l)

  Returns the minimal polynomial of the matrix whose Jordan form is described by the list l. This list should be in the format given by the `jordan` function. See `jordan` for details of this format.

  Example:

      (%i1) load("diag")$

      (%i2) a:matrix([2,1,2,0],
                     [-2,2,1,2],
                     [-2,-1,-1,1],
                     [3,1,2,-1])$

      (%i3) jordan(a);
      (%o3)               [[- 1, 1], [1, 3]]
      (%i4) minimalPoly(%);
                                  3
      (%o4)                (x - 1)  (x + 1)

  To use this function write first `load("diag")`. See also `jordan` and `dispJordan`.

  Categories: Package diag

Function: ModeMatrix (A, [jordan_info])

  Returns an invertible matrix M such that (M^^-1).A.M is the Jordan form of A.

  To calculate this, Maxima must find the Jordan form of A, which might be quite computationally expensive. If that has already been calculated by a previous call to `jordan`, pass it as a second argument, jordan_info. See `jordan` for details of the required format.

  Example:

      (%i1) load("diag")$
      (%i2) A: matrix([2,1,2,0], [-2,2,1,2], [-2,-1,-1,1], [3,1,2,-1])$
      (%i3) M: ModeMatrix (A);
                            [  1    - 1   1   1 ]
                            [                   ]
                            [   1               ]
                            [ - -   - 1   0   0 ]
                            [   9               ]
                            [                   ]
      (%o3)                 [   13              ]
                            [ - --   1   - 1  0 ]
                            [   9               ]
                            [                   ]
                            [  17               ]
                            [  --   - 1   1   1 ]
                            [  9                ]
      (%i4) is ((M^^-1) . A . M = dispJordan (jordan (A)));
      (%o4)                         true

  Note that, in this example, the Jordan form of `A` is computed twice. To avoid this, we could have stored the output of `jordan(A)` in a variable and passed that to both `ModeMatrix` and `dispJordan`.

  To use this function write first `load("diag")`. See also `jordan` and `dispJordan`.

  Categories: Package diag

Function: mat_function (f,A)

  Returns f(A), where f is an analytic function and A a matrix. This computation is based on the Taylor expansion of f. It is not efficient for numerical evaluation, but can give symbolic answers for small matrices.

  Example 1:

  The exponential of a matrix. We only give the first row of the answer, since the output is rather large.

      (%i1) load("diag")$
      (%i2) A: matrix ([0,1,0], [0,0,1], [-1,-3,-3])$
      (%i3) ratsimp (mat_function (exp, t*A)[1]);
                 2              - t                   2   - t
               (t  + 2 t + 2) %e       2        - t  t  %e
      (%o3)   [--------------------, (t  + t) %e   , --------]
                        2                               2

  Example 2:

  Comparison with the Taylor series for the exponential and also comparing `exp(%i*A)` with sine and cosine.

      (%i1) load("diag")$
      (%i2) A: matrix ([0,1,1,1],
                       [0,0,0,1],
                       [0,0,0,1],
                       [0,0,0,0])$
      (%i3) ratsimp (mat_function (exp, t*A));
                             [           2     ]
                             [ 1  t  t  t  + t ]
                             [                 ]
      (%o3)                  [ 0  1  0    t    ]
                             [                 ]
                             [ 0  0  1    t    ]
                             [                 ]
                             [ 0  0  0    1    ]
      (%i4) minimalPoly (jordan (A));
                                      3
      (%o4)                          x
      (%i5) ratsimp (ident(4) + t*A + 1/2*(t^2)*A^^2);
                             [           2     ]
                             [ 1  t  t  t  + t ]
                             [                 ]
      (%o5)                  [ 0  1  0    t    ]
                             [                 ]
                             [ 0  0  1    t    ]
                             [                 ]
                             [ 0  0  0    1    ]
      (%i6) ratsimp (mat_function (exp, %i*t*A));
                        [                        2 ]
                        [ 1  %i t  %i t  %i t - t  ]
                        [                          ]
      (%o6)             [ 0   1     0      %i t    ]
                        [                          ]
                        [ 0   0     1      %i t    ]
                        [                          ]
                        [ 0   0     0        1     ]
      (%i7) ratsimp (mat_function (cos, t*A) + %i*mat_function (sin, t*A));
                        [                        2 ]
                        [ 1  %i t  %i t  %i t - t  ]
                        [                          ]
      (%o7)             [ 0   1     0      %i t    ]
                        [                          ]
                        [ 0   0     1      %i t    ]
                        [                          ]
                        [ 0   0     0        1     ]

  Example 3:

  Power operations.

      (%i1) load("diag")$
      (%i2) A: matrix([1,2,0], [0,1,0], [1,0,1])$
      (%i3) integer_pow(x) := block ([k], declare (k, integer), x^k)$
      (%i4) mat_function (integer_pow, A);
                             [ 1     2 k     0 ]
                             [                 ]
      (%o4)                  [ 0      1      0 ]
                             [                 ]
                             [ k  (k - 1) k  1 ]
      (%i5) A^^20;
                               [ 1   40   0 ]
                               [            ]
      (%o5)                    [ 0    1   0 ]
                               [            ]
                               [ 20  380  1 ]

  To use this function write first `load("diag")`.

  Categories: Package diag

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# 51. distrib

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## 51.1 Introduction to distrib

Package `distrib` contains a set of functions for making probability computations on both discrete and continuous univariate models.

What follows is a short reminder of basic probabilistic related definitions.

Let f(x) be the density function of an absolute continuous random variable X. The distribution function is defined as

                           x
                          /
                          [
                   F(x) = I     f(u) du
                          ]
                          /
                           minf

which equals the probability Pr(X <= x).

The mean value is a localization parameter and is defined as

                         inf
                        /
                        [
               E[X]  =  I   x f(x) dx
                        ]
                        /
                         minf

The variance is a measure of variation,

                     inf
                    /
                    [                    2
             V[X] = I     f(x) (x - E[X])  dx
                    ]
                    /
                     minf

which is a positive real number. The square root of the variance is the standard deviation, D[X]=sqrt(V[X]), and it is another measure of variation.

The skewness coefficient is a measure of non-symmetry,

                     inf
                    /
                1   [                    3
      SK[X] = ----- I     f(x) (x - E[X])  dx
                  3 ]
              D[X]  /
                     minf

And the kurtosis coefficient measures the peakedness of the distribution,

                     inf
                    /
                1   [                    4
      KU[X] = ----- I     f(x) (x - E[X])  dx - 3
                  4 ]
              D[X]  /
                     minf

If X is gaussian, KU[X]=0. In fact, both skewness and kurtosis are shape parameters used to measure the non-gaussianity of a distribution.

If the random variable X is discrete, the density, or probability, function f(x) takes positive values within certain countable set of numbers x_i, and zero elsewhere. In this case, the distribution function is

                           ====
                           \
                    F(x) =  >    f(x )
                           /        i
                           ====
                          x <= x
                           i

The mean, variance, standard deviation, skewness coefficient and kurtosis coefficient take the form

                           ====
                           \
                    E[X] =  >  x  f(x ) ,
                           /    i    i
                           ====
                            x
                             i

                    ====
                    \                     2
            V[X] =   >    f(x ) (x - E[X])  ,
                    /        i    i
                    ====
                     x
                      i

                   D[X] = sqrt(V[X]),

                         ====
                  1      \                     3
      SK[X] =  -------    >    f(x ) (x - E[X])
               D[X]^3    /        i    i
                         ====
                          x
                           i

and

                         ====
                  1      \                     4
      KU[X] =  -------    >    f(x ) (x - E[X])   - 3 ,
               D[X]^4    /        i    i
                         ====
                          x
                           i

respectively.

There is a naming convention in package `distrib`. Every function name has two parts, the first one makes reference to the function or parameter we want to calculate,

    Functions:
       Density function            (pdf_*)
       Distribution function       (cdf_*)
       Quantile                    (quantile_*)
       Mean                        (mean_*)
       Variance                    (var_*)
       Standard deviation          (std_*)
       Skewness coefficient        (skewness_*)
       Kurtosis coefficient        (kurtosis_*)
       Random variate              (random_*)

The second part is an explicit reference to the probabilistic model,

    Continuous distributions:
       Normal              (*normal)
       Student             (*student_t)
       Chi^2               (*chi2)
       Noncentral Chi^2    (*noncentral_chi2)
       F                   (*f)
       Exponential         (*exp)
       Lognormal           (*lognormal)
       Gamma               (*gamma)
       Beta                (*beta)
       Continuous uniform  (*continuous_uniform)
       Logistic            (*logistic)
       Pareto              (*pareto)
       Weibull             (*weibull)
       Rayleigh            (*rayleigh)
       Laplace             (*laplace)
       Cauchy              (*cauchy)
       Gumbel              (*gumbel)

    Discrete distributions:
       Binomial             (*binomial)
       Poisson              (*poisson)
       Bernoulli            (*bernoulli)
       Geometric            (*geometric)
       Discrete uniform     (*discrete_uniform)
       hypergeometric       (*hypergeometric)
       Negative binomial    (*negative_binomial)
       Finite discrete      (*general_finite_discrete)

For example, `pdf_student_t(x,n)` is the density function of the Student distribution with n degrees of freedom, `std_pareto(a,b)` is the standard deviation of the Pareto distribution with parameters a and b and `kurtosis_poisson(m)` is the kurtosis coefficient of the Poisson distribution with mean m.

In order to make use of package `distrib` you need first to load it by typing

    (%i1) load(distrib)$

For comments, bugs or suggestions, please contact the author at 'riotorto AT yahoo DOT com'.

Categories: Statistical functions · Share packages · Package distrib

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## 51.2 Functions and Variables for continuous distributions

Function: pdf_normal (x,m,s)

  Returns the value at x of the density function of a Normal(m,s) random variable, with s>0. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: cdf_normal (x,m,s)

  Returns the value at x of the distribution function of a Normal(m,s) random variable, with s>0. This function is defined in terms of Maxima's built-in error function `erf`.

      (%i1) load (distrib)$
      (%i2) cdf_normal(x,m,s);
                                          x - m
                                    erf(---------)
                                        sqrt(2) s    1
      (%o2)                         -------------- + -
                                          2          2

  See also `erf`.

  Categories: Package distrib

Function: quantile_normal (q,m,s)

  Returns the q-quantile of a Normal(m,s) random variable, with s>0; in other words, this is the inverse of `cdf_normal`. Argument q must be an element of [0,1]. To make use of this function, write first `load(distrib)`.

      (%i1) load (distrib)$
      (%i2) quantile_normal(95/100,0,1);
                                            9
      (%o2)             sqrt(2) inverse_erf(--)
                                            10
      (%i3) float(%);
      (%o3)               1.644853626951472

  Categories: Package distrib

Function: mean_normal (m,s)

  Returns the mean of a Normal(m,s) random variable, with s>0, namely m. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: var_normal (m,s)

  Returns the variance of a Normal(m,s) random variable, with s>0, namely s^2. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: std_normal (m,s)

  Returns the standard deviation of a Normal(m,s) random variable, with s>0, namely s. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: skewness_normal (m,s)

  Returns the skewness coefficient of a Normal(m,s) random variable, with s>0, which is always equal to 0. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: kurtosis_normal (m,s)

  Returns the kurtosis coefficient of a Normal(m,s) random variable, with s>0, which is always equal to 0. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: random_normal (m,s)
`random_normal` (m,s,n)

  Returns a Normal(m,s) random variate, with s>0. Calling `random_normal` with a third argument n, a random sample of size n will be simulated.

  This is an implementation of the Box-Mueller algorithm, as described in Knuth, D.E. (1981) Seminumerical Algorithms. The Art of Computer Programming. Addison-Wesley.

  To make use of this function, write first `load(distrib)`.

  Categories: Package distrib · Random numbers

Function: pdf_student_t (x,n)

  Returns the value at x of the density function of a Student random variable t(n), with n>0 degrees of freedom. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: cdf_student_t (x,n)

  Returns the value at x of the distribution function of a Student random variable t(n), with n>0 degrees of freedom.

      (%i1) load (distrib)$
      (%i2) cdf_student_t(1/2, 7/3);
                                               7  1  28
                   beta_incomplete_regularized(-, -, --)
                                               6  2  31
      (%o2)    1 - -------------------------------------
                                     2
      (%i3) float(%);
      (%o3)                .6698450596140415

  Categories: Package distrib

Function: quantile_student_t (q,n)

  Returns the q-quantile of a Student random variable t(n), with n>0; in other words, this is the inverse of `cdf_student_t`. Argument q must be an element of [0,1]. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: mean_student_t (n)

  Returns the mean of a Student random variable t(n), with n>0, which is always equal to 0. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: var_student_t (n)

  Returns the variance of a Student random variable t(n), with n>2.

      (%i1) load (distrib)$
      (%i2) var_student_t(n);
                                      n
      (%o2)                         -----
                                    n - 2

  Categories: Package distrib

Function: std_student_t (n)

  Returns the standard deviation of a Student random variable t(n), with n>2. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: skewness_student_t (n)

  Returns the skewness coefficient of a Student random variable t(n), with n>3, which is always equal to 0. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: kurtosis_student_t (n)

  Returns the kurtosis coefficient of a Student random variable t(n), with n>4. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: random_student_t (n)
`random_student_t` (n,m)

  Returns a Student random variate t(n), with n>0. Calling `random_student_t` with a second argument m, a random sample of size m will be simulated.

  The implemented algorithm is based on the fact that if Z is a normal random variable N(0,1) and S^2 is a chi square random variable with n degrees of freedom, Chi^2(n), then

                                 Z
                       X = -------------
                           /   2  \ 1/2
                           |  S   |
                           | ---  |
                           \  n   /

  is a Student random variable with n degrees of freedom, t(n).

  To make use of this function, write first `load(distrib)`.

  Categories: Package distrib · Random numbers

Function: pdf_noncentral_student_t (x,n,ncp)

  Returns the value at x of the density function of a noncentral Student random variable nc_t(n,ncp), with n>0 degrees of freedom and noncentrality parameter ncp. To make use of this function, write first `load(distrib)`.

  Sometimes an extra work is necessary to get the final result.

      (%i1) load (distrib)$
      (%i2) expand(pdf_noncentral_student_t(3,5,0.1));
                                 7/2                         7/2
            0.04296414417400905 5      1.323650307289301e-6 5
      (%o2) ------------------------ + -------------------------
               3/2   5/2                       sqrt(%pi)
              2    14    sqrt(%pi)
                                                              7/2
                                         1.94793720435093e-4 5
                                       + ------------------------
                                                   %pi
      (%i3) float(%);
      (%o3)          .02080593159405669

  Categories: Package distrib

Function: cdf_noncentral_student_t (x,n,ncp)

  Returns the value at x of the distribution function of a noncentral Student random variable nc_t(n,ncp), with n>0 degrees of freedom and noncentrality parameter ncp. This function has no closed form and it is numerically computed.

      (%i1) load (distrib)$
      (%i2) cdf_noncentral_student_t(-2,5,-5);
      (%o2)          .9952030093319743

  Categories: Package distrib

Function: quantile_noncentral_student_t (q,n,ncp)

  Returns the q-quantile of a noncentral Student random variable nc_t(n,ncp), with n>0 degrees of freedom and noncentrality parameter ncp; in other words, this is the inverse of `cdf_noncentral_student_t`. Argument q must be an element of [0,1]. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: mean_noncentral_student_t (n,ncp)

  Returns the mean of a noncentral Student random variable nc_t(n,ncp), with n>1 degrees of freedom and noncentrality parameter ncp. To make use of this function, write first `load(distrib)`.

      (%i1) load (distrib)$
      (%i2) mean_noncentral_student_t(df,k);
                         df - 1
                   gamma(------) sqrt(df) k
                           2
      (%o2)        ------------------------
                                    df
                      sqrt(2) gamma(--)
                                    2

  Categories: Package distrib

Function: var_noncentral_student_t (n,ncp)

  Returns the variance of a noncentral Student random variable nc_t(n,ncp), with n>2 degrees of freedom and noncentrality parameter ncp. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: std_noncentral_student_t (n,ncp)

  Returns the standard deviation of a noncentral Student random variable nc_t(n,ncp), with n>2 degrees of freedom and noncentrality parameter ncp. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: skewness_noncentral_student_t (n,ncp)

  Returns the skewness coefficient of a noncentral Student random variable nc_t(n,ncp), with n>3 degrees of freedom and noncentrality parameter ncp. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: kurtosis_noncentral_student_t (n,ncp)

  Returns the kurtosis coefficient of a noncentral Student random variable nc_t(n,ncp), with n>4 degrees of freedom and noncentrality parameter ncp. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: random_noncentral_student_t (n,ncp)
`random_noncentral_student_t` (n,ncp,m)

  Returns a noncentral Student random variate nc_t(n,ncp), with n>0. Calling `random_noncentral_student_t` with a third argument m, a random sample of size m will be simulated.

  The implemented algorithm is based on the fact that if X is a normal random variable N(ncp,1) and S^2 is a chi square random variable with n degrees of freedom, Chi^2(n), then

                                 X
                       U = -------------
                           /   2  \ 1/2
                           |  S   |
                           | ---  |
                           \  n   /

  is a noncentral Student random variable with n degrees of freedom and noncentrality parameter ncp, nc_t(n,ncp).

  To make use of this function, write first `load(distrib)`.

  Categories: Package distrib · Random numbers

Function: pdf_chi2 (x,n)

  Returns the value at x of the density function of a Chi-square random variable Chi^2(n), with n>0. The Chi^2(n) random variable is equivalent to the Gamma(n/2,2).

      (%i1) load (distrib)$
      (%i2) pdf_chi2(x,n);
                               n/2 - 1   - x/2
                              x        %e
      (%o2)                   ----------------
                                n/2       n
                               2    gamma(-)
                                          2

  Categories: Package distrib

Function: cdf_chi2 (x,n)

  Returns the value at x of the distribution function of a Chi-square random variable Chi^2(n), with n>0.

      (%i1) load (distrib)$
      (%i2) cdf_chi2(3,4);
                                                     3
      (%o2)      1 - gamma_incomplete_regularized(2, -)
                                                     2
      (%i3) float(%);
      (%o3)               .4421745996289256

  Categories: Package distrib

Function: quantile_chi2 (q,n)

  Returns the q-quantile of a Chi-square random variable Chi^2(n), with n>0; in other words, this is the inverse of `cdf_chi2`. Argument q must be an element of [0,1].

  This function has no closed form and it is numerically computed.

      (%i1) load (distrib)$
      (%i2) quantile_chi2(0.99,9);
      (%o2)                   21.66599433346194

  Categories: Package distrib

Function: mean_chi2 (n)

  Returns the mean of a Chi-square random variable Chi^2(n), with n>0.

  The Chi^2(n) random variable is equivalent to the Gamma(n/2,2).

      (%i1) load (distrib)$
      (%i2) mean_chi2(n);
      (%o2)                           n

  Categories: Package distrib

Function: var_chi2 (n)

  Returns the variance of a Chi-square random variable Chi^2(n), with n>0.

  The Chi^2(n) random variable is equivalent to the Gamma(n/2,2).

      (%i1) load (distrib)$
      (%i2) var_chi2(n);
      (%o2)                          2 n

  Categories: Package distrib

Function: std_chi2 (n)

  Returns the standard deviation of a Chi-square random variable Chi^2(n), with n>0.

  The Chi^2(n) random variable is equivalent to the Gamma(n/2,2).

      (%i1) load (distrib)$
      (%i2) std_chi2(n);
      (%o2)                    sqrt(2) sqrt(n)

  Categories: Package distrib

Function: skewness_chi2 (n)

  Returns the skewness coefficient of a Chi-square random variable Chi^2(n), with n>0.

  The Chi^2(n) random variable is equivalent to the Gamma(n/2,2).

      (%i1) load (distrib)$
      (%i2) skewness_chi2(n);
                                           3/2
                                          2
      (%o2)                              -------
                                         sqrt(n)

  Categories: Package distrib

Function: kurtosis_chi2 (n)

  Returns the kurtosis coefficient of a Chi-square random variable Chi^2(n), with n>0.

  The Chi^2(n) random variable is equivalent to the Gamma(n/2,2).

      (%i1) load (distrib)$
      (%i2) kurtosis_chi2(n);
                                     12
      (%o2)                          --
                                     n

  Categories: Package distrib

Function: random_chi2 (n)
`random_chi2` (n,m)

  Returns a Chi-square random variate Chi^2(n), with n>0. Calling `random_chi2` with a second argument m, a random sample of size m will be simulated.

  The simulation is based on the Ahrens-Cheng algorithm. See `random_gamma` for details.

  To make use of this function, write first `load(distrib)`.

  Categories: Package distrib · Random numbers

Function: pdf_noncentral_chi2 (x,n,ncp)

  Returns the value at x of the density function of a noncentral Chi-square random variable nc_Chi^2(n,ncp), with n>0 and noncentrality parameter ncp>=0. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: cdf_noncentral_chi2 (x,n,ncp)

  Returns the value at x of the distribution function of a noncentral Chi-square random variable nc_Chi^2(n,ncp), with n>0 and noncentrality parameter ncp>=0. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: quantile_noncentral_chi2 (q,n,ncp)

  Returns the q-quantile of a noncentral Chi-square random variable nc_Chi^2(n,ncp), with n>0 and noncentrality parameter ncp>=0; in other words, this is the inverse of `cdf_noncentral_chi2`. Argument q must be an element of [0,1].

  This function has no closed form and it is numerically computed.

  Categories: Package distrib

Function: mean_noncentral_chi2 (n,ncp)

  Returns the mean of a noncentral Chi-square random variable nc_Chi^2(n,ncp), with n>0 and noncentrality parameter ncp>=0.

  Categories: Package distrib

Function: var_noncentral_chi2 (n,ncp)

  Returns the variance of a noncentral Chi-square random variable nc_Chi^2(n,ncp), with n>0 and noncentrality parameter ncp>=0.

  Categories: Package distrib

Function: std_noncentral_chi2 (n,ncp)

  Returns the standard deviation of a noncentral Chi-square random variable nc_Chi^2(n,ncp), with n>0 and noncentrality parameter ncp>=0.

  Categories: Package distrib

Function: skewness_noncentral_chi2 (n,ncp)

  Returns the skewness coefficient of a noncentral Chi-square random variable nc_Chi^2(n,ncp), with n>0 and noncentrality parameter ncp>=0.

  Categories: Package distrib

Function: kurtosis_noncentral_chi2 (n,ncp)

  Returns the kurtosis coefficient of a noncentral Chi-square random variable nc_Chi^2(n,ncp), with n>0 and noncentrality parameter ncp>=0.

  Categories: Package distrib

Function: random_noncentral_chi2 (n,ncp)
`random_noncentral_chi2` (n,ncp,m)

  Returns a noncentral Chi-square random variate nc_Chi^2(n,ncp), with n>0 and noncentrality parameter ncp>=0. Calling `random_noncentral_chi2` with a third argument m, a random sample of size m will be simulated.

  To make use of this function, write first `load(distrib)`.

  Categories: Package distrib · Random numbers

Function: pdf_f (x,m,n)

  Returns the value at x of the density function of a F random variable F(m,n), with m,n>0. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: cdf_f (x,m,n)

  Returns the value at x of the distribution function of a F random variable F(m,n), with m,n>0.

      (%i1) load (distrib)$
      (%i2) cdf_f(2,3,9/4);
                                               9  3  3
      (%o2)    1 - beta_incomplete_regularized(-, -, --)
                                               8  2  11
      (%i3) float(%);
      (%o3)                 0.66756728179008

  Categories: Package distrib

Function: quantile_f (q,m,n)

  Returns the q-quantile of a F random variable F(m,n), with m,n>0; in other words, this is the inverse of `cdf_f`. Argument q must be an element of [0,1].

      (%i1) load (distrib)$
      (%i2) quantile_f(2/5,sqrt(3),5);
      (%o2)                   0.518947838573693

  Categories: Package distrib

Function: mean_f (m,n)

  Returns the mean of a F random variable F(m,n), with m>0, n>2. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: var_f (m,n)

  Returns the variance of a F random variable F(m,n), with m>0, n>4. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: std_f (m,n)

  Returns the standard deviation of a F random variable F(m,n), with m>0, n>4. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: skewness_f (m,n)

  Returns the skewness coefficient of a F random variable F(m,n), with m>0, n>6. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: kurtosis_f (m,n)

  Returns the kurtosis coefficient of a F random variable F(m,n), with m>0, n>8. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: random_f (m,n)
`random_f` (m,n,k)

  Returns a F random variate F(m,n), with m,n>0. Calling `random_f` with a third argument k, a random sample of size k will be simulated.

  The simulation algorithm is based on the fact that if X is a Chi^2(m) random variable and Y is a Chi^2(n) random variable, then

                              n X
                          F = ---
                              m Y

  is a F random variable with m and n degrees of freedom, F(m,n).

  To make use of this function, write first `load(distrib)`.

  Categories: Package distrib · Random numbers

Function: pdf_exp (x,m)

  Returns the value at x of the density function of an Exponential(m) random variable, with m>0.

  The Exponential(m) random variable is equivalent to the Weibull(1,1/m).

      (%i1) load (distrib)$
      (%i2) pdf_exp(x,m);
                                      - m x
      (%o2)                       m %e

  Categories: Package distrib

Function: cdf_exp (x,m)

  Returns the value at x of the distribution function of an Exponential(m) random variable, with m>0.

  The Exponential(m) random variable is equivalent to the Weibull(1,1/m).

      (%i1) load (distrib)$
      (%i2) cdf_exp(x,m);
                                       - m x
      (%o2)                      1 - %e

  Categories: Package distrib

Function: quantile_exp (q,m)

  Returns the q-quantile of an Exponential(m) random variable, with m>0; in other words, this is the inverse of `cdf_exp`. Argument q must be an element of [0,1].

  The Exponential(m) random variable is equivalent to the Weibull(1,1/m).

      (%i1) load (distrib)$
      (%i2) quantile_exp(0.56,5);
      (%o2)                   .1641961104139661
      (%i3) quantile_exp(0.56,m);
                                   0.8209805520698303
      (%o3)                        ------------------
                                           m

  Categories: Package distrib

Function: mean_exp (m)

  Returns the mean of an Exponential(m) random variable, with m>0.

  The Exponential(m) random variable is equivalent to the Weibull(1,1/m).

      (%i1) load (distrib)$
      (%i2) mean_exp(m);
                                      1
      (%o2)                           -
                                      m

  Categories: Package distrib

Function: var_exp (m)

  Returns the variance of an Exponential(m) random variable, with m>0.

  The Exponential(m) random variable is equivalent to the Weibull(1,1/m).

      (%i1) load (distrib)$
      (%i2) var_exp(m);
                                     1
      (%o2)                          --
                                      2
                                     m

  Categories: Package distrib

Function: std_exp (m)

  Returns the standard deviation of an Exponential(m) random variable, with m>0.

  The Exponential(m) random variable is equivalent to the Weibull(1,1/m).

      (%i1) load (distrib)$
      (%i2) std_exp(m);
                                      1
      (%o2)                           -
                                      m

  Categories: Package distrib

Function: skewness_exp (m)

  Returns the skewness coefficient of an Exponential(m) random variable, with m>0.

  The Exponential(m) random variable is equivalent to the Weibull(1,1/m).

      (%i1) load (distrib)$
      (%i2) skewness_exp(m);
      (%o2)                           2

  Categories: Package distrib

Function: kurtosis_exp (m)

  Returns the kurtosis coefficient of an Exponential(m) random variable, with m>0.

  The Exponential(m) random variable is equivalent to the Weibull(1,1/m).

      (%i1) load (distrib)$
      (%i2) kurtosis_exp(m);
      (%o3)                           6

  Categories: Package distrib

Function: random_exp (m)
`random_exp` (m,k)

  Returns an Exponential(m) random variate, with m>0. Calling `random_exp` with a second argument k, a random sample of size k will be simulated.

  The simulation algorithm is based on the general inverse method.

  To make use of this function, write first `load(distrib)`.

  Categories: Package distrib · Random numbers

Function: pdf_lognormal (x,m,s)

  Returns the value at x of the density function of a Lognormal(m,s) random variable, with s>0. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: cdf_lognormal (x,m,s)

  Returns the value at x of the distribution function of a Lognormal(m,s) random variable, with s>0. This function is defined in terms of Maxima's built-in error function `erf`.

      (%i1) load (distrib)$
      (%i2) cdf_lognormal(x,m,s);
                                 log(x) - m
                             erf(----------)
                                 sqrt(2) s     1
      (%o2)                  --------------- + -
                                    2          2

  See also `erf`.

  Categories: Package distrib

Function: quantile_lognormal (q,m,s)

  Returns the q-quantile of a Lognormal(m,s) random variable, with s>0; in other words, this is the inverse of `cdf_lognormal`. Argument q must be an element of [0,1]. To make use of this function, write first `load(distrib)`.

      (%i1) load (distrib)$
      (%i2) quantile_lognormal(95/100,0,1);
                        sqrt(2) inverse_erf(9/10)
      (%o2)           %e
      (%i3) float(%);
      (%o3)               5.180251602233015

  Categories: Package distrib

Function: mean_lognormal (m,s)

  Returns the mean of a Lognormal(m,s) random variable, with s>0. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: var_lognormal (m,s)

  Returns the variance of a Lognormal(m,s) random variable, with s>0. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: std_lognormal (m,s)

  Returns the standard deviation of a Lognormal(m,s) random variable, with s>0. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: skewness_lognormal (m,s)

  Returns the skewness coefficient of a Lognormal(m,s) random variable, with s>0. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: kurtosis_lognormal (m,s)

  Returns the kurtosis coefficient of a Lognormal(m,s) random variable, with s>0. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: random_lognormal (m,s)
`random_lognormal` (m,s,n)

  Returns a Lognormal(m,s) random variate, with s>0. Calling `random_lognormal` with a third argument n, a random sample of size n will be simulated.

  Log-normal variates are simulated by means of random normal variates. See `random_normal` for details.

  To make use of this function, write first `load(distrib)`.

  Categories: Package distrib · Random numbers

Function: pdf_gamma (x,a,b)

  Returns the value at x of the density function of a Gamma(a,b) random variable, with a,b>0. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: cdf_gamma (x,a,b)

  Returns the value at x of the distribution function of a Gamma(a,b) random variable, with a,b>0.

      (%i1) load (distrib)$
      (%i2) cdf_gamma(3,5,21);
                                                    1
      (%o2)     1 - gamma_incomplete_regularized(5, -)
                                                    7
      (%i3) float(%);
      (%o3)              4.402663157376807E-7

  Categories: Package distrib

Function: quantile_gamma (q,a,b)

  Returns the q-quantile of a Gamma(a,b) random variable, with a,b>0; in other words, this is the inverse of `cdf_gamma`. Argument q must be an element of [0,1]. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: mean_gamma (a,b)

  Returns the mean of a Gamma(a,b) random variable, with a,b>0. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: var_gamma (a,b)

  Returns the variance of a Gamma(a,b) random variable, with a,b>0. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: std_gamma (a,b)

  Returns the standard deviation of a Gamma(a,b) random variable, with a,b>0. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: skewness_gamma (a,b)

  Returns the skewness coefficient of a Gamma(a,b) random variable, with a,b>0. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: kurtosis_gamma (a,b)

  Returns the kurtosis coefficient of a Gamma(a,b) random variable, with a,b>0. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: random_gamma (a,b)
`random_gamma` (a,b,n)

  Returns a Gamma(a,b) random variate, with a,b>0. Calling `random_gamma` with a third argument n, a random sample of size n will be simulated.

  The implemented algorithm is a combinantion of two procedures, depending on the value of parameter a:

  For a>=1, Cheng, R.C.H. and Feast, G.M. (1979). Some simple gamma variate generators. Appl. Stat., 28, 3, 290-295.

  For 0<a<1, Ahrens, J.H. and Dieter, U. (1974). Computer methods for sampling from gamma, beta, poisson and binomial cdf_tributions. Computing, 12, 223-246.

  To make use of this function, write first `load(distrib)`.

  Categories: Package distrib · Random numbers

Function: pdf_beta (x,a,b)

  Returns the value at x of the density function of a Beta(a,b) random variable, with a,b>0. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: cdf_beta (x,a,b)

  Returns the value at x of the distribution function of a Beta(a,b) random variable, with a,b>0.

      (%i1) load (distrib)$
      (%i2) cdf_beta(1/3,15,2);
                                   11
      (%o2)                     --------
                                14348907
      (%i3) float(%);
      (%o3)              7.666089131388195E-7

  Categories: Package distrib

Function: quantile_beta (q,a,b)

  Returns the q-quantile of a Beta(a,b) random variable, with a,b>0; in other words, this is the inverse of `cdf_beta`. Argument q must be an element of [0,1]. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: mean_beta (a,b)

  Returns the mean of a Beta(a,b) random variable, with a,b>0. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: var_beta (a,b)

  Returns the variance of a Beta(a,b) random variable, with a,b>0. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: std_beta (a,b)

  Returns the standard deviation of a Beta(a,b) random variable, with a,b>0. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: skewness_beta (a,b)

  Returns the skewness coefficient of a Beta(a,b) random variable, with a,b>0. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: kurtosis_beta (a,b)

  Returns the kurtosis coefficient of a Beta(a,b) random variable, with a,b>0. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: random_beta (a,b)
`random_beta` (a,b,n)

  Returns a Beta(a,b) random variate, with a,b>0. Calling `random_beta` with a third argument n, a random sample of size n will be simulated.

  The implemented algorithm is defined in Cheng, R.C.H. (1978). Generating Beta Variates with Nonintegral Shape Parameters. Communications of the ACM, 21:317-322

  To make use of this function, write first `load(distrib)`.

  Categories: Package distrib · Random numbers

Function: pdf_continuous_uniform (x,a,b)

  Returns the value at x of the density function of a Continuous Uniform(a,b) random variable, with a<b. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: cdf_continuous_uniform (x,a,b)

  Returns the value at x of the distribution function of a Continuous Uniform(a,b) random variable, with a<b. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: quantile_continuous_uniform (q,a,b)

  Returns the q-quantile of a Continuous Uniform(a,b) random variable, with a<b; in other words, this is the inverse of `cdf_continuous_uniform`. Argument q must be an element of [0,1]. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: mean_continuous_uniform (a,b)

  Returns the mean of a Continuous Uniform(a,b) random variable, with a<b. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: var_continuous_uniform (a,b)

  Returns the variance of a Continuous Uniform(a,b) random variable, with a<b. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: std_continuous_uniform (a,b)

  Returns the standard deviation of a Continuous Uniform(a,b) random variable, with a<b. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: skewness_continuous_uniform (a,b)

  Returns the skewness coefficient of a Continuous Uniform(a,b) random variable, with a<b. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: kurtosis_continuous_uniform (a,b)

  Returns the kurtosis coefficient of a Continuous Uniform(a,b) random variable, with a<b. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: random_continuous_uniform (a,b)
`random_continuous_uniform` (a,b,n)

  Returns a Continuous Uniform(a,b) random variate, with a<b. Calling `random_continuous_uniform` with a third argument n, a random sample of size n will be simulated.

  This is a direct application of the `random` built-in Maxima function.

  See also `random`. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib · Random numbers

Function: pdf_logistic (x,a,b)

  Returns the value at x of the density function of a Logistic(a,b) random variable , with b>0. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: cdf_logistic (x,a,b)

  Returns the value at x of the distribution function of a Logistic(a,b) random variable , with b>0. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: quantile_logistic (q,a,b)

  Returns the q-quantile of a Logistic(a,b) random variable , with b>0; in other words, this is the inverse of `cdf_logistic`. Argument q must be an element of [0,1]. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: mean_logistic (a,b)

  Returns the mean of a Logistic(a,b) random variable , with b>0. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: var_logistic (a,b)

  Returns the variance of a Logistic(a,b) random variable , with b>0. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: std_logistic (a,b)

  Returns the standard deviation of a Logistic(a,b) random variable , with b>0. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: skewness_logistic (a,b)

  Returns the skewness coefficient of a Logistic(a,b) random variable , with b>0. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: kurtosis_logistic (a,b)

  Returns the kurtosis coefficient of a Logistic(a,b) random variable , with b>0. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: random_logistic (a,b)
`random_logistic` (a,b,n)

  Returns a Logistic(a,b) random variate, with b>0. Calling `random_logistic` with a third argument n, a random sample of size n will be simulated.

  The implemented algorithm is based on the general inverse method.

  To make use of this function, write first `load(distrib)`.

  Categories: Package distrib · Random numbers

Function: pdf_pareto (x,a,b)

  Returns the value at x of the density function of a Pareto(a,b) random variable, with a,b>0. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: cdf_pareto (x,a,b)

  Returns the value at x of the distribution function of a Pareto(a,b) random variable, with a,b>0. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: quantile_pareto (q,a,b)

  Returns the q-quantile of a Pareto(a,b) random variable, with a,b>0; in other words, this is the inverse of `cdf_pareto`. Argument q must be an element of [0,1]. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: mean_pareto (a,b)

  Returns the mean of a Pareto(a,b) random variable, with a>1,b>0. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: var_pareto (a,b)

  Returns the variance of a Pareto(a,b) random variable, with a>2,b>0. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: std_pareto (a,b)

  Returns the standard deviation of a Pareto(a,b) random variable, with a>2,b>0. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: skewness_pareto (a,b)

  Returns the skewness coefficient of a Pareto(a,b) random variable, with a>3,b>0. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: kurtosis_pareto (a,b)

  Returns the kurtosis coefficient of a Pareto(a,b) random variable, with a>4,b>0. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: random_pareto (a,b)
`random_pareto` (a,b,n)

  Returns a Pareto(a,b) random variate, with a>0,b>0. Calling `random_pareto` with a third argument n, a random sample of size n will be simulated.

  The implemented algorithm is based on the general inverse method.

  To make use of this function, write first `load(distrib)`.

  Categories: Package distrib · Random numbers

Function: pdf_weibull (x,a,b)

  Returns the value at x of the density function of a Weibull(a,b) random variable, with a,b>0. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: cdf_weibull (x,a,b)

  Returns the value at x of the distribution function of a Weibull(a,b) random variable, with a,b>0. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: quantile_weibull (q,a,b)

  Returns the q-quantile of a Weibull(a,b) random variable, with a,b>0; in other words, this is the inverse of `cdf_weibull`. Argument q must be an element of [0,1]. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: mean_weibull (a,b)

  Returns the mean of a Weibull(a,b) random variable, with a,b>0. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: var_weibull (a,b)

  Returns the variance of a Weibull(a,b) random variable, with a,b>0. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: std_weibull (a,b)

  Returns the standard deviation of a Weibull(a,b) random variable, with a,b>0. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: skewness_weibull (a,b)

  Returns the skewness coefficient of a Weibull(a,b) random variable, with a,b>0. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: kurtosis_weibull (a,b)

  Returns the kurtosis coefficient of a Weibull(a,b) random variable, with a,b>0. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: random_weibull (a,b)
`random_weibull` (a,b,n)

  Returns a Weibull(a,b) random variate, with a,b>0. Calling `random_weibull` with a third argument n, a random sample of size n will be simulated.

  The implemented algorithm is based on the general inverse method.

  To make use of this function, write first `load(distrib)`.

  Categories: Package distrib · Random numbers

Function: pdf_rayleigh (x,b)

  Returns the value at x of the density function of a Rayleigh(b) random variable, with b>0.

  The Rayleigh(b) random variable is equivalent to the Weibull(2,1/b).

      (%i1) load (distrib)$
      (%i2) pdf_rayleigh(x,b);
                                          2  2
                                 2     - b  x
      (%o2)                   2 b  x %e

  Categories: Package distrib

Function: cdf_rayleigh (x,b)

  Returns the value at x of the distribution function of a Rayleigh(b) random variable, with b>0.

  The Rayleigh(b) random variable is equivalent to the Weibull(2,1/b).

      (%i1) load (distrib)$
      (%i2) cdf_rayleigh(x,b);
                                         2  2
                                      - b  x
      (%o2)                     1 - %e

  Categories: Package distrib

Function: quantile_rayleigh (q,b)

  Returns the q-quantile of a Rayleigh(b) random variable, with b>0; in other words, this is the inverse of `cdf_rayleigh`. Argument q must be an element of [0,1].

  The Rayleigh(b) random variable is equivalent to the Weibull(2,1/b).

      (%i1) load (distrib)$
      (%i2) quantile_rayleigh(0.99,b);
                              2.145966026289347
      (%o2)                   -----------------
                                      b

  Categories: Package distrib

Function: mean_rayleigh (b)

  Returns the mean of a Rayleigh(b) random variable, with b>0.

  The Rayleigh(b) random variable is equivalent to the Weibull(2,1/b).

      (%i1) load (distrib)$
      (%i2) mean_rayleigh(b);
                                  sqrt(%pi)
      (%o2)                       ---------
                                     2 b

  Categories: Package distrib

Function: var_rayleigh (b)

  Returns the variance of a Rayleigh(b) random variable, with b>0.

  The Rayleigh(b) random variable is equivalent to the Weibull(2,1/b).

      (%i1) load (distrib)$
      (%i2) var_rayleigh(b);
                                       %pi
                                   1 - ---
                                        4
      (%o2)                        -------
                                      2
                                     b

  Categories: Package distrib

Function: std_rayleigh (b)

  Returns the standard deviation of a Rayleigh(b) random variable, with b>0.

  The Rayleigh(b) random variable is equivalent to the Weibull(2,1/b).

      (%i1) load (distrib)$
      (%i2) std_rayleigh(b);
                                         %pi
                                sqrt(1 - ---)
                                          4
      (%o2)                     -------------
                                      b

  Categories: Package distrib

Function: skewness_rayleigh (b)

  Returns the skewness coefficient of a Rayleigh(b) random variable, with b>0.

  The Rayleigh(b) random variable is equivalent to the Weibull(2,1/b).

      (%i1) load (distrib)$
      (%i2) skewness_rayleigh(b);
                               3/2
                            %pi      3 sqrt(%pi)
                            ------ - -----------
                              4           4
      (%o2)                 --------------------
                                     %pi 3/2
                                (1 - ---)
                                      4

  Categories: Package distrib

Function: kurtosis_rayleigh (b)

  Returns the kurtosis coefficient of a Rayleigh(b) random variable, with b>0.

  The Rayleigh(b) random variable is equivalent to the Weibull(2,1/b).

      (%i1) load (distrib)$
      (%i2) kurtosis_rayleigh(b);
                                        2
                                   3 %pi
                               2 - ------
                                     16
      (%o2)                    ---------- - 3
                                    %pi 2
                               (1 - ---)
                                     4

  Categories: Package distrib

Function: random_rayleigh (b)
`random_rayleigh` (b,n)

  Returns a Rayleigh(b) random variate, with b>0. Calling `random_rayleigh` with a second argument n, a random sample of size n will be simulated.

  The implemented algorithm is based on the general inverse method.

  To make use of this function, write first `load(distrib)`.

  Categories: Package distrib · Random numbers

Function: pdf_laplace (x,a,b)

  Returns the value at x of the density function of a Laplace(a,b) random variable, with b>0. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: cdf_laplace (x,a,b)

  Returns the value at x of the distribution function of a Laplace(a,b) random variable, with b>0. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: quantile_laplace (q,a,b)

  Returns the q-quantile of a Laplace(a,b) random variable, with b>0; in other words, this is the inverse of `cdf_laplace`. Argument q must be an element of [0,1]. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: mean_laplace (a,b)

  Returns the mean of a Laplace(a,b) random variable, with b>0. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: var_laplace (a,b)

  Returns the variance of a Laplace(a,b) random variable, with b>0. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: std_laplace (a,b)

  Returns the standard deviation of a Laplace(a,b) random variable, with b>0. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: skewness_laplace (a,b)

  Returns the skewness coefficient of a Laplace(a,b) random variable, with b>0. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: kurtosis_laplace (a,b)

  Returns the kurtosis coefficient of a Laplace(a,b) random variable, with b>0. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: random_laplace (a,b)
`random_laplace` (a,b,n)

  Returns a Laplace(a,b) random variate, with b>0. Calling `random_laplace` with a third argument n, a random sample of size n will be simulated.

  The implemented algorithm is based on the general inverse method.

  To make use of this function, write first `load(distrib)`.

  Categories: Package distrib · Random numbers

Function: pdf_cauchy (x,a,b)

  Returns the value at x of the density function of a Cauchy(a,b) random variable, with b>0. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: cdf_cauchy (x,a,b)

  Returns the value at x of the distribution function of a Cauchy(a,b) random variable, with b>0. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: quantile_cauchy (q,a,b)

  Returns the q-quantile of a Cauchy(a,b) random variable, with b>0; in other words, this is the inverse of `cdf_cauchy`. Argument q must be an element of [0,1]. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: random_cauchy (a,b)
`random_cauchy` (a,b,n)

  Returns a Cauchy(a,b) random variate, with b>0. Calling `random_cauchy` with a third argument n, a random sample of size n will be simulated.

  The implemented algorithm is based on the general inverse method.

  To make use of this function, write first `load(distrib)`.

  Categories: Package distrib · Random numbers

Function: pdf_gumbel (x,a,b)

  Returns the value at x of the density function of a Gumbel(a,b) random variable, with b>0. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: cdf_gumbel (x,a,b)

  Returns the value at x of the distribution function of a Gumbel(a,b) random variable, with b>0. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: quantile_gumbel (q,a,b)

  Returns the q-quantile of a Gumbel(a,b) random variable, with b>0; in other words, this is the inverse of `cdf_gumbel`. Argument q must be an element of [0,1]. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: mean_gumbel (a,b)

  Returns the mean of a Gumbel(a,b) random variable, with b>0.

      (%i1) load (distrib)$
      (%i2) mean_gumbel(a,b);
      (%o2)                     %gamma b + a

  where symbol `%gamma` stands for the Euler-Mascheroni constant. See also `%gamma`.

  Categories: Package distrib

Function: var_gumbel (a,b)

  Returns the variance of a Gumbel(a,b) random variable, with b>0. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: std_gumbel (a,b)

  Returns the standard deviation of a Gumbel(a,b) random variable, with b>0. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: skewness_gumbel (a,b)

  Returns the skewness coefficient of a Gumbel(a,b) random variable, with b>0.

      (%i1) load (distrib)$
      (%i2) skewness_gumbel(a,b);
                                        3/2
                                     2 6    zeta(3)
      (%o2)                          --------------
                                             3
                                          %pi

  where `zeta` stands for the Riemann's zeta function.

  Categories: Package distrib

Function: kurtosis_gumbel (a,b)

  Returns the kurtosis coefficient of a Gumbel(a,b) random variable, with b>0. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib · Package distrib

Function: random_gumbel (a,b)
`random_gumbel` (a,b,n)

  Returns a Gumbel(a,b) random variate, with b>0. Calling `random_gumbel` with a third argument n, a random sample of size n will be simulated.

  The implemented algorithm is based on the general inverse method.

  To make use of this function, write first `load(distrib)`.

  Categories: Package distrib · Random numbers

[ < ] | [ > ] |  | [ << ] | [ Up ] | [ >> ] |  |  |  |  | [Top] | [Contents] | [Index] | [ ? ]

## 51.3 Functions and Variables for discrete distributions

Function: pdf_general_finite_discrete (x,v)

  Returns the value at x of the probability function of a general finite discrete random variable, with vector probabilities v, such that `Pr(X=i) = v_i`. Vector v can be a list of nonnegative expressions, whose components will be normalized to get a vector of probabilities. To make use of this function, write first `load(distrib)`.

      (%i1) load (distrib)$
      (%i2) pdf_general_finite_discrete(2, [1/7, 4/7, 2/7]);
                                      4
      (%o2)                           -
                                      7
      (%i3) pdf_general_finite_discrete(2, [1, 4, 2]);
                                      4
      (%o3)                           -
                                      7

  Categories: Package distrib

Function: cdf_general_finite_discrete (x,v)

  Returns the value at x of the distribution function of a general finite discrete random variable, with vector probabilities v.

  See `pdf_general_finite_discrete` for more details.

      (%i1) load (distrib)$
      (%i2) cdf_general_finite_discrete(2, [1/7, 4/7, 2/7]);
                                      5
      (%o2)                           -
                                      7
      (%i3) cdf_general_finite_discrete(2, [1, 4, 2]);
                                      5
      (%o3)                           -
                                      7
      (%i4) cdf_general_finite_discrete(2+1/2, [1, 4, 2]);
                                      5
      (%o4)                           -
                                      7

  Categories: Package distrib

Function: quantile_general_finite_discrete (q,v)

  Returns the q-quantile of a general finite discrete random variable, with vector probabilities v.

  See `pdf_general_finite_discrete` for more details.

  Categories: Package distrib

Function: mean_general_finite_discrete (v)

  Returns the mean of a general finite discrete random variable, with vector probabilities v.

  See `pdf_general_finite_discrete` for more details.

  Categories: Package distrib

Function: var_general_finite_discrete (v)

  Returns the variance of a general finite discrete random variable, with vector probabilities v.

  See `pdf_general_finite_discrete` for more details.

  Categories: Package distrib

Function: std_general_finite_discrete (v)

  Returns the standard deviation of a general finite discrete random variable, with vector probabilities v.

  See `pdf_general_finite_discrete` for more details.

  Categories: Package distrib

Function: skewness_general_finite_discrete (v)

  Returns the skewness coefficient of a general finite discrete random variable, with vector probabilities v.

  See `pdf_general_finite_discrete` for more details.

  Categories: Package distrib

Function: kurtosis_general_finite_discrete (v)

  Returns the kurtosis coefficient of a general finite discrete random variable, with vector probabilities v.

  See `pdf_general_finite_discrete` for more details.

  Categories: Package distrib

Function: random_general_finite_discrete (v)
`random_general_finite_discrete` (v,m)

  Returns a general finite discrete random variate, with vector probabilities v. Calling `random_general_finite_discrete` with a second argument m, a random sample of size m will be simulated.

  See `pdf_general_finite_discrete` for more details.

      (%i1) load (distrib)$
      (%i2) random_general_finite_discrete([1,3,1,5]);
      (%o2)                          4
      (%i3) random_general_finite_discrete([1,3,1,5], 10);
      (%o3)           [4, 2, 2, 3, 2, 4, 4, 1, 2, 2]

  Categories: Package distrib · Random numbers

Function: pdf_binomial (x,n,p)

  Returns the value at x of the probability function of a Binomial(n,p) random variable, with 0 \leq p \leq 1 and n a positive integer. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: cdf_binomial (x,n,p)

  Returns the value at x of the distribution function of a Binomial(n,p) random variable, with 0 \leq p \leq 1 and n a positive integer.

      (%i1) load (distrib)$
      (%i2) cdf_binomial(5,7,1/6);
                                  7775
      (%o2)                       ----
                                  7776
      (%i3) float(%);
      (%o3)               .9998713991769548

  Categories: Package distrib

Function: quantile_binomial (q,n,p)

  Returns the q-quantile of a Binomial(n,p) random variable, with 0 \leq p \leq 1 and n a positive integer; in other words, this is the inverse of `cdf_binomial`. Argument q must be an element of [0,1]. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: mean_binomial (n,p)

  Returns the mean of a Binomial(n,p) random variable, with 0 \leq p \leq 1 and n a positive integer. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: var_binomial (n,p)

  Returns the variance of a Binomial(n,p) random variable, with 0 \leq p \leq 1 and n a positive integer. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: std_binomial (n,p)

  Returns the standard deviation of a Binomial(n,p) random variable, with 0 \leq p \leq 1 and n a positive integer. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: skewness_binomial (n,p)

  Returns the skewness coefficient of a Binomial(n,p) random variable, with 0 \leq p \leq 1 and n a positive integer. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: kurtosis_binomial (n,p)

  Returns the kurtosis coefficient of a Binomial(n,p) random variable, with 0 \leq p \leq 1 and n a positive integer. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: random_binomial (n,p)
`random_binomial` (n,p,m)

  Returns a Binomial(n,p) random variate, with 0 \leq p \leq 1 and n a positive integer. Calling `random_binomial` with a third argument m, a random sample of size m will be simulated.

  The implemented algorithm is based on the one described in Kachitvichyanukul, V. and Schmeiser, B.W. (1988) Binomial Random Variate Generation. Communications of the ACM, 31, Feb., 216.

  To make use of this function, write first `load(distrib)`.

  Categories: Package distrib · Random numbers

Function: pdf_poisson (x,m)

  Returns the value at x of the probability function of a Poisson(m) random variable, with m>0. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: cdf_poisson (x,m)

  Returns the value at x of the distribution function of a Poisson(m) random variable, with m>0.

      (%i1) load (distrib)$
      (%i2) cdf_poisson(3,5);
      (%o2)       gamma_incomplete_regularized(4, 5)
      (%i3) float(%);
      (%o3)               .2650259152973623

  Categories: Package distrib

Function: quantile_poisson (q,m)

  Returns the q-quantile of a Poisson(m) random variable, with m>0; in other words, this is the inverse of `cdf_poisson`. Argument q must be an element of [0,1]. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: mean_poisson (m)

  Returns the mean of a Poisson(m) random variable, with m>0. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: var_poisson (m)

  Returns the variance of a Poisson(m) random variable, with m>0. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: std_poisson (m)

  Returns the standard deviation of a Poisson(m) random variable, with m>0. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: skewness_poisson (m)

  Returns the skewness coefficient of a Poisson(m) random variable, with m>0. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: kurtosis_poisson (m)

  Returns the kurtosis coefficient of a Poisson random variable Poi(m), with m>0. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: random_poisson (m)
`random_poisson` (m,n)

  Returns a Poisson(m) random variate, with m>0. Calling `random_poisson` with a second argument n, a random sample of size n will be simulated.

  The implemented algorithm is the one described in Ahrens, J.H. and Dieter, U. (1982) Computer Generation of Poisson Deviates From Modified Normal Distributions. ACM Trans. Math. Software, 8, 2, June,163-179.

  To make use of this function, write first `load(distrib)`.

  Categories: Package distrib · Random numbers

Function: pdf_bernoulli (x,p)

  Returns the value at x of the probability function of a Bernoulli(p) random variable, with 0 \leq p \leq 1.

  The Bernoulli(p) random variable is equivalent to the Binomial(1,p).

      (%i1) load (distrib)$
      (%i2) pdf_bernoulli(1,p);
      (%o2)                           p

  Categories: Package distrib

Function: cdf_bernoulli (x,p)

  Returns the value at x of the distribution function of a Bernoulli(p) random variable, with 0 \leq p \leq 1. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: quantile_bernoulli (q,p)

  Returns the q-quantile of a Bernoulli(p) random variable, with 0 \leq p \leq 1; in other words, this is the inverse of `cdf_bernoulli`. Argument q must be an element of [0,1]. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: mean_bernoulli (p)

  Returns the mean of a Bernoulli(p) random variable, with 0 \leq p \leq 1.

  The Bernoulli(p) random variable is equivalent to the Binomial(1,p).

      (%i1) load (distrib)$
      (%i2) mean_bernoulli(p);
      (%o2)                           p

  Categories: Package distrib

Function: var_bernoulli (p)

  Returns the variance of a Bernoulli(p) random variable, with 0 \leq p \leq 1.

  The Bernoulli(p) random variable is equivalent to the Binomial(1,p).

      (%i1) load (distrib)$
      (%i2) var_bernoulli(p);
      (%o2)                       (1 - p) p

  Categories: Package distrib

Function: std_bernoulli (p)

  Returns the standard deviation of a Bernoulli(p) random variable, with 0 \leq p \leq 1.

  The Bernoulli(p) random variable is equivalent to the Binomial(1,p).

      (%i1) load (distrib)$
      (%i2) std_bernoulli(p);
      (%o2)                           sqrt((1 - p) p)

  Categories: Package distrib

Function: skewness_bernoulli (p)

  Returns the skewness coefficient of a Bernoulli(p) random variable, with 0 \leq p \leq 1.

  The Bernoulli(p) random variable is equivalent to the Binomial(1,p).

      (%i1) load (distrib)$
      (%i2) skewness_bernoulli(p);
                                          1 - 2 p
      (%o2)                           ---------------
                                      sqrt((1 - p) p)

  Categories: Package distrib

Function: kurtosis_bernoulli (p)

  Returns the kurtosis coefficient of a Bernoulli(p) random variable, with 0 \leq p \leq 1.

  The Bernoulli(p) random variable is equivalent to the Binomial(1,p).

      (%i1) load (distrib)$
      (%i2) kurtosis_bernoulli(p);
                               1 - 6 (1 - p) p
      (%o2)                    ---------------
                                  (1 - p) p

  Categories: Package distrib

Function: random_bernoulli (p)
`random_bernoulli` (p,n)

  Returns a Bernoulli(p) random variate, with 0 \leq p \leq 1. Calling `random_bernoulli` with a second argument n, a random sample of size n will be simulated.

  This is a direct application of the `random` built-in Maxima function.

  See also `random`. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib · Random numbers

Function: pdf_geometric (x,p)

  Returns the value at x of the probability function of a Geometric(p) random variable, with 0 < p \leq 1. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: cdf_geometric (x,p)

  Returns the value at x of the distribution function of a Geometric(p) random variable, with 0 < p \leq 1. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: quantile_geometric (q,p)

  Returns the q-quantile of a Geometric(p) random variable, with 0 < p \leq 1; in other words, this is the inverse of `cdf_geometric`. Argument q must be an element of [0,1]. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: mean_geometric (p)

  Returns the mean of a Geometric(p) random variable, with 0 < p \leq 1. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: var_geometric (p)

  Returns the variance of a Geometric(p) random variable, with 0 < p \leq 1. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: std_geometric (p)

  Returns the standard deviation of a Geometric(p) random variable, with 0 < p \leq 1. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: skewness_geometric (p)

  Returns the skewness coefficient of a Geometric(p) random variable, with 0 < p \leq 1. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: kurtosis_geometric (p)

  Returns the kurtosis coefficient of a geometric random variable Geo(p), with 0 < p \leq 1. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: random_geometric (p)
`random_geometric` (p,n)

  Returns a Geometric(p) random variate, with 0 < p \leq 1. Calling `random_geometric` with a second argument n, a random sample of size n will be simulated.

  The algorithm is based on simulation of Bernoulli trials.

  To make use of this function, write first `load(distrib)`.

  Categories: Package distrib · Random numbers

Function: pdf_discrete_uniform (x,n)

  Returns the value at x of the probability function of a Discrete Uniform(n) random variable, with n a strictly positive integer. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: cdf_discrete_uniform (x,n)

  Returns the value at x of the distribution function of a Discrete Uniform(n) random variable, with n a strictly positive integer. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: quantile_discrete_uniform (q,n)

  Returns the q-quantile of a Discrete Uniform(n) random variable, with n a strictly positive integer; in other words, this is the inverse of `cdf_discrete_uniform`. Argument q must be an element of [0,1]. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: mean_discrete_uniform (n)

  Returns the mean of a Discrete Uniform(n) random variable, with n a strictly positive integer. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: var_discrete_uniform (n)

  Returns the variance of a Discrete Uniform(n) random variable, with n a strictly positive integer. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: std_discrete_uniform (n)

  Returns the standard deviation of a Discrete Uniform(n) random variable, with n a strictly positive integer. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: skewness_discrete_uniform (n)

  Returns the skewness coefficient of a Discrete Uniform(n) random variable, with n a strictly positive integer. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: kurtosis_discrete_uniform (n)

  Returns the kurtosis coefficient of a Discrete Uniform(n) random variable, with n a strictly positive integer. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: random_discrete_uniform (n)
`random_discrete_uniform` (n,m)

  Returns a Discrete Uniform(n) random variate, with n a strictly positive integer. Calling `random_discrete_uniform` with a second argument m, a random sample of size m will be simulated.

  This is a direct application of the `random` built-in Maxima function.

  See also `random`. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib · Random numbers

Function: pdf_hypergeometric (x,n1,n2,n)

  Returns the value at x of the probability function of a Hypergeometric(n1,n2,n) random variable, with n1, n2 and n non negative integers and n<=n1+n2. Being n1 the number of objects of class A, n2 the number of objects of class B, and n the size of the sample without replacement, this function returns the probability of event "exactly x objects are of class A".

  To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: cdf_hypergeometric (x,n1,n2,n)

  Returns the value at x of the distribution function of a Hypergeometric(n1,n2,n) random variable, with n1, n2 and n non negative integers and n<=n1+n2. See `pdf_hypergeometric` for a more complete description.

  To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: quantile_hypergeometric (q,n1,n2,n)

  Returns the q-quantile of a Hypergeometric(n1,n2,n) random variable, with n1, n2 and n non negative integers and n<=n1+n2; in other words, this is the inverse of `cdf_hypergeometric`. Argument q must be an element of [0,1]. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: mean_hypergeometric (n1,n2,n)

  Returns the mean of a discrete uniform random variable Hyp(n1,n2,n), with n1, n2 and n non negative integers and n<=n1+n2. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: var_hypergeometric (n1,n2,n)

  Returns the variance of a hypergeometric random variable Hyp(n1,n2,n), with n1, n2 and n non negative integers and n<=n1+n2. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: std_hypergeometric (n1,n2,n)

  Returns the standard deviation of a Hypergeometric(n1,n2,n) random variable, with n1, n2 and n non negative integers and n<=n1+n2. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: skewness_hypergeometric (n1,n2,n)

  Returns the skewness coefficient of a Hypergeometric(n1,n2,n) random variable, with n1, n2 and n non negative integers and n<=n1+n2. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: kurtosis_hypergeometric (n1,n2,n)

  Returns the kurtosis coefficient of a Hypergeometric(n1,n2,n) random variable, with n1, n2 and n non negative integers and n<=n1+n2. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: random_hypergeometric (n1,n2,n)
`random_hypergeometric` (n1,n2,n,m)

  Returns a Hypergeometric(n1,n2,n) random variate, with n1, n2 and n non negative integers and n<=n1+n2. Calling `random_hypergeometric` with a fourth argument m, a random sample of size m will be simulated.

  Algorithm described in Kachitvichyanukul, V., Schmeiser, B.W. (1985) Computer generation of hypergeometric random variates. Journal of Statistical Computation and Simulation 22, 127-145.

  To make use of this function, write first `load(distrib)`.

  Categories: Package distrib · Random numbers

Function: pdf_negative_binomial (x,n,p)

  Returns the value at x of the probability function of a Negative Binomial(n,p) random variable, with 0 < p \leq 1 and n a positive number. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: cdf_negative_binomial (x,n,p)

  Returns the value at x of the distribution function of a Negative Binomial(n,p) random variable, with 0 < p \leq 1 and n a positive number.

      (%i1) load (distrib)$
      (%i2) cdf_negative_binomial(3,4,1/8);
                                  3271
      (%o2)                      ------
                                 524288

  Categories: Package distrib

Function: quantile_negative_binomial (q,n,p)

  Returns the q-quantile of a Negative Binomial(n,p) random variable, with 0 < p \leq 1 and n a positive number; in other words, this is the inverse of `cdf_negative_binomial`. Argument q must be an element of [0,1]. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: mean_negative_binomial (n,p)

  Returns the mean of a Negative Binomial(n,p) random variable, with 0 < p \leq 1 and n a positive number. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: var_negative_binomial (n,p)

  Returns the variance of a Negative Binomial(n,p) random variable, with 0 < p \leq 1 and n a positive number. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: std_negative_binomial (n,p)

  Returns the standard deviation of a Negative Binomial(n,p) random variable, with 0 < p \leq 1 and n a positive number. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: skewness_negative_binomial (n,p)

  Returns the skewness coefficient of a Negative Binomial(n,p) random variable, with 0 < p \leq 1 and n a positive number. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: kurtosis_negative_binomial (n,p)

  Returns the kurtosis coefficient of a Negative Binomial(n,p) random variable, with 0 < p \leq 1 and n a positive number. To make use of this function, write first `load(distrib)`.

  Categories: Package distrib

Function: random_negative_binomial (n,p)
`random_negative_binomial` (n,p,m)

  Returns a Negative Binomial(n,p) random variate, with 0 < p \leq 1 and n a positive number. Calling `random_negative_binomial` with a third argument m, a random sample of size m will be simulated.

  Algorithm described in Devroye, L. (1986) Non-Uniform Random Variate Generation. Springer Verlag, p. 480.

  To make use of this function, write first `load(distrib)`.

  Categories: Package distrib · Random numbers

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# 52. draw

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## 52.1 Introduction to draw

`draw` is a Maxima-Gnuplot and a Maxima-vtk interface.

There are three main functions to be used at Maxima level: `draw2d`, `draw3d` and `draw`.

Follow these links for more elaborated examples of this package:

http://tecnostats.net/Maxima/gnuplot
http://tecnostats.net/Maxima/vtk

You need Gnuplot 4.2 or newer to run this program.

Categories: Plotting · Share packages · Package draw

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## 52.2 Functions and Variables for draw

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### 52.2.1 Scenes

Scene constructor: gr2d (graphic option, ..., graphic object, ...)

  Function `gr2d` builds an object describing a 2D scene. Arguments are graphic options, graphic objects, or lists containing both graphic options and objects. This scene is interpreted sequentially: graphic options affect those graphic objects placed on its right. Some graphic options affect the global appearance of the scene.

  This is the list of graphic objects available for scenes in two dimensions: `bars`, `ellipse`, `explicit`, `image`, `implicit`, `label`, `parametric`, `points`, `polar`, `polygon`, `quadrilateral`, `rectangle`, `triangle`, `vector` and `geomap` (this one defined in package `worldmap`).

  See also `draw` and `draw2d`.

      (%i1) load("draw");
      (%o1)     /maxima/share/draw/draw.lisp
      (%i2) draw(
          gr2d(
              key="sin (x)",grid=[2,2],
              explicit(
                  sin(x),
                  x,0,2*%pi
              )
          ),
          gr2d(
              key="cos (x)",grid=[2,2],
              explicit(
                  cos(x),
                  x,0,2*%pi
              )
          )
       );
      (%o2)           [gr2d(explicit), gr2d(explicit)]

  Categories: Package draw

Scene constructor: gr3d (graphic option, ..., graphic object, ...)

  Function `gr3d` builds an object describing a 3d scene. Arguments are graphic options, graphic objects, or lists containing both graphic options and objects. This scene is interpreted sequentially: graphic options affect those graphic objects placed on its right. Some graphic options affect the global appearance of the scene.

  This is the list of graphic objects available for scenes in three dimensions:
  `cylindrical`, `elevation_grid`, `explicit`, `implicit`, `label`, `mesh`, `parametric`,
  `parametric_surface`, `points`, `quadrilateral`, `spherical`, `triangle`, `tube`,
  `vector`, and `geomap` (this one defined in package `worldmap`).

  See also `draw` and `draw3d`.

  Categories: Package draw

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### 52.2.2 Functions

Function: draw (gr2d, ..., gr3d, ..., options, ...)

  Plots a series of scenes; its arguments are `gr2d` and/or `gr3d` objects, together with some options, or lists of scenes and options. By default, the scenes are put together in one column.

  Function `draw` accepts the following global options: `terminal`, `columns`, `dimensions`, `file_name` and `delay`.

  Functions `draw2d` and `draw3d` are short cuts to be used when only one scene is required, in two or three dimensions, respectively.

  See also `gr2d` and `gr3d`.

  Example:

      (%i1) scene1: gr2d(title="Ellipse",
                         nticks=30,
                         parametric(2*cos(t),5*sin(t),t,0,2*%pi))$
      (%i2) scene2: gr2d(title="Triangle",
                         polygon([4,5,7],[6,4,2]))$
      (%i3) draw(scene1, scene2, columns = 2)$

  The two draw sentences are equivalent:

      (%i1) draw(gr3d(explicit(x^2+y^2,x,-1,1,y,-1,1)));
      (%o1)                          [gr3d(explicit)]
      (%i2) draw3d(explicit(x^2+y^2,x,-1,1,y,-1,1));
      (%o2)                          [gr3d(explicit)]

  An animated gif file:

      (%i1) draw(
              delay     = 100,
              file_name = "zzz",
              terminal  = 'animated_gif,
              gr2d(explicit(x^2,x,-1,1)),
              gr2d(explicit(x^3,x,-1,1)),
              gr2d(explicit(x^4,x,-1,1)));
      End of animation sequence
      (%o1)          [gr2d(explicit), gr2d(explicit), gr2d(explicit)]

  See also `gr2d`, `gr3d`, `draw2d` and `draw3d`.

  Categories: Package draw · File output

Function: draw2d (option, graphic_object, ...)

  This function is a short cut for `draw(gr2d(options, ..., graphic_object, ...))`.

  It can be used to plot a unique scene in 2d.

  See also `draw` and `gr2d`.

  Categories: Package draw · File output

Function: draw3d (option, graphic_object, ...)

  This function is a short cut for `draw(gr3d(options, ..., graphic_object, ...))`.

  It can be used to plot a unique scene in 3d.

  See also `draw` and `gr3d`.

  Categories: Package draw · File output

Function: draw_file (graphic option, ..., graphic object, ...)

  Saves the current plot into a file. Accepted graphics options are: `terminal`, `dimensions` and `file_name`.

  Example:

      (%i1) /* screen plot */
            draw(gr3d(explicit(x^2+y^2,x,-1,1,y,-1,1)))$
      (%i2) /* same plot in eps format */
            draw_file(terminal  = eps,
                      dimensions = [5,5]) $

  Categories: Package draw · File output

Function: multiplot_mode (term)

  This function enables Maxima to work in one-window multiplot mode with terminal term; accepted arguments for this function are `screen`, `wxt`, `aquaterm` and `none`.

  When multiplot mode is enabled, each call to `draw` sends a new plot to the same window, without erasing the previous ones. To disable the multiplot mode, write `multiplot_mode(none)`.

  When multiplot mode is enabled, global option `terminal` is blocked and you have to disable this working mode before changing to another terminal.

  This feature does not work in Windows platforms.

  Example:

      (%i1) set_draw_defaults(
               xrange = [-1,1],
               yrange = [-1,1],
               grid   = true,
               title  = "Step by step plot" )$
      (%i2) multiplot_mode(screen)$
      (%i3) draw2d(color=blue,  explicit(x^2,x,-1,1))$
      (%i4) draw2d(color=red,   explicit(x^3,x,-1,1))$
      (%i5) draw2d(color=brown, explicit(x^4,x,-1,1))$
      (%i6) multiplot_mode(none)$

  Categories: Package draw · File output

Function: set_draw_defaults (graphic option, ..., graphic object, ...)

  Sets user graphics options. This function is useful for plotting a sequence of graphics with common graphics options. Calling this function without arguments removes user defaults.

  Example:

      (%i1) set_draw_defaults(
               xrange = [-10,10],
               yrange = [-2, 2],
               color  = blue,
               grid   = true)$
      (%i2) /* plot with user defaults */
            draw2d(explicit(((1+x)**2/(1+x*x))-1,x,-10,10))$
      (%i3) set_draw_defaults()$
      (%i4) /* plot with standard defaults */
            draw2d(explicit(((1+x)**2/(1+x*x))-1,x,-10,10))$

  Categories: Package draw

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### 52.2.3 Graphics options

Graphic option: adapt_depth

  Default value: 10

  `adapt_depth` is the maximum number of splittings used by the adaptive plotting routine.

  This option is relevant only for 2d `explicit` functions.

  See also `nticks`

  Categories: Package draw

Graphic option: allocation

  Default value: `false`

  With option `allocation` it is possible to place a scene in the output window at will; this is of interest in multiplots. When `false`, the scene is placed automatically, depending on the value assigned to option `columns`. In any other case, `allocation` must be set to a list of two pairs of numbers; the first corresponds to the position of the lower left corner of the scene, and the second pair gives the width and height of the plot. All quantities must be given in relative coordinates, between 0 and 1.

  Examples:

  In site graphics.

      (%i1) draw(
              gr2d(
                explicit(x^2,x,-1,1)),
              gr2d(
                allocation = [[1/4, 1/4],[1/2, 1/2]],
                explicit(x^3,x,-1,1),
                grid = true) ) $

  Multiplot with selected dimensions.

      (%i1) draw(
              terminal = wxt,
              gr2d(
                grid=[5,5],
                allocation = [[0, 0],[1, 1/4]],
                explicit(x^2,x,-1,1)),
              gr3d(
                allocation = [[0, 1/4],[1, 3/4]],
                explicit(x^2+y^2,x,-1,1,y,-1,1) ))$

  See also option `columns`.

  Categories: Package draw

Graphic option: axis_3d

  Default value: `true`

  If `axis_3d` is `true`, the x, y and z axis are shown in 3d scenes.

  Since this is a global graphics option, its position in the scene description does not matter.

  Example:

      (%i1) draw3d(axis_3d = false,
                   explicit(sin(x^2+y^2),x,-2,2,y,-2,2) )$

  See also `axis_bottom`, `axis_left`, `axis_top`, and `axis_right` for axis in 2d.

  Categories: Package draw

Graphic option: axis_bottom

  Default value: `true`

  If `axis_bottom` is `true`, the bottom axis is shown in 2d scenes.

  Since this is a global graphics option, its position in the scene description does not matter.

  Example:

      (%i1) draw2d(axis_bottom = false,
                   explicit(x^3,x,-1,1))$

  See also `axis_left`, `axis_top`, `axis_right` and `axis_3d`.

  Categories: Package draw

Graphic option: axis_left

  Default value: `true`

  If `axis_left` is `true`, the left axis is shown in 2d scenes.

  Since this is a global graphics option, its position in the scene description does not matter.

  Example:

      (%i1) draw2d(axis_left = false,
                   explicit(x^3,x,-1,1))$

  See also `axis_bottom`, `axis_top`, `axis_right` and `axis_3d`.

  Categories: Package draw

Graphic option: axis_right

  Default value: `true`

  If `axis_right` is `true`, the right axis is shown in 2d scenes.

  Since this is a global graphics option, its position in the scene description does not matter.

  Example:

      (%i1) draw2d(axis_right = false,
                   explicit(x^3,x,-1,1))$

  See also `axis_bottom`, `axis_left`, `axis_top` and `axis_3d`.

  Categories: Package draw

Graphic option: axis_top

  Default value: `true`

  If `axis_top` is `true`, the top axis is shown in 2d scenes.

  Since this is a global graphics option, its position in the scene description does not matter.

  Example:

      (%i1) draw2d(axis_top = false,
                   explicit(x^3,x,-1,1))$

  See also `axis_bottom`, `axis_left`, `axis_right`, and `axis_3d`.

  Categories: Package draw

Graphic option: background_color

  Default value: `white`

  Sets the background color for terminals. Default background color is white.

  Since this is a global graphics option, its position in the scene description does not matter.

  This option das not work with terminals `epslatex` and `epslatex_standalone`.

  See also `color`

  Categories: Package draw

Graphic option: border

  Default value: `true`

  If `border` is `true`, borders of polygons are painted according to `line_type` and `line_width`.

  This option affects the following graphic objects:

  - `gr2d`: `polygon`, `rectangle` and `ellipse`.

  Example:

      (%i1) draw2d(color       = brown,
                   line_width  = 8,
                   polygon([[3,2],[7,2],[5,5]]),
                   border      = false,
                   fill_color  = blue,
                   polygon([[5,2],[9,2],[7,5]]) )$

  Categories: Package draw

Graphic option: capping

  Default value: `[false, false]`

  A list with two possible elements, `true` and `false`, indicating whether the extremes of a graphic object `tube` remain closed or open. By default, both extremes are left open.

  Setting `capping = false` is equivalent to `capping = [false, false]`, and `capping = true` is equivalent to `capping = [true, true]`.

  Example:

      (%i1) draw3d(
              capping = [false, true],
              tube(0, 0, a, 1,
                   a, 0, 8) )$

  Categories: Package draw

Graphic option: cbrange

  Default value: `auto`

  If `cbrange` is `auto`, the range for the values which are colored when `enhanced3d` is not `false` is computed automatically. Values outside of the color range use color of the nearest extreme.

  When `enhanced3d` or `colorbox` is `false`, option `cbrange` has no effect.

  If the user wants a specific interval for the colored values, it must be given as a Maxima list, as in `cbrange=[-2, 3]`.

  Since this is a global graphics option, its position in the scene description does not matter.

  Example:

      (%i1) draw3d (
              enhanced3d     = true,
              color          = green,
              cbrange = [-3,10],
              explicit(x^2+y^2, x,-2,2,y,-2,2)) $

  See also `enhanced3d`, `colorbox` and `cbtics`.

  Categories: Package draw

Graphic option: cbtics

  Default value: `auto`

  This graphic option controls the way tic marks are drawn on the colorbox when option `enhanced3d` is not `false`.

  When `enhanced3d` or `colorbox` is `false`, option `cbtics` has no effect.

  See `xtics` for a complete description.

  Example :

      (%i1) draw3d (
              enhanced3d = true,
              color      = green,
              cbtics  = {["High",10],["Medium",05],["Low",0]},
              cbrange = [0, 10],
              explicit(x^2+y^2, x,-2,2,y,-2,2)) $

  See also `enhanced3d`, `colorbox` and `cbrange`.

  Categories: Package draw

Graphic option: color

  Default value: `blue`

  `color` specifies the color for plotting lines, points, borders of polygons and labels.

  Colors can be given as names or in hexadecimal rgb code. If a gnuplot version >= 5.0 is used and the terminal that is in use supports this rgba colors with transparency information are also supported.

  Available color names are:

      white            black            gray0            grey0
      gray10           grey10           gray20           grey20
      gray30           grey30           gray40           grey40
      gray50           grey50           gray60           grey60
      gray70           grey70           gray80           grey80
      gray90           grey90           gray100          grey100
      gray             grey             light_gray       light_grey
      dark_gray        dark_grey        red              light_red
      dark_red         yellow           light_yellow     dark_yellow
      green            light_green      dark_green       spring_green
      forest_green     sea_green        blue             light_blue
      dark_blue        midnight_blue    navy             medium_blue
      royalblue        skyblue          cyan             light_cyan
      dark_cyan        magenta          light_magenta    dark_magenta
      turquoise        light_turquoise  dark_turquoise   pink
      light_pink       dark_pink        coral            light_coral
      orange_red       salmon           light_salmon     dark_salmon
      aquamarine       khaki            dark_khaki       goldenrod
      light_goldenrod  dark_goldenrod   gold             beige
      brown            orange           dark_orange      violet
      dark_violet      plum             purple

  Cromatic componentes in hexadecimal code are introduced in the form `"#rrggbb"`.

  Example:

      (%i1) draw2d(explicit(x^2,x,-1,1), /* default is black */
                   color = red,
                   explicit(0.5 + x^2,x,-1,1),
                   color = blue,
                   explicit(1 + x^2,x,-1,1),
                   color = light_blue,
                   explicit(1.5 + x^2,x,-1,1),
                   color = "#23ab0f",
                   label(["This is a label",0,1.2])  )$

      (%i1) draw2d(
                   line_width=50,
                   color="#FF0000",
                   explicit(sin(x),x,0,10),
                   color="#0000FF80",
                   explicit(cos(x),x,0,10)
            );

  See also `fill_color`.

  Categories: Package draw

Graphic option: colorbox

  Default value: `true`

  If `colorbox` is `true`, a color scale without label is drawn together with `image` 2D objects, or coloured 3d objects. If `colorbox` is `false`, no color scale is shown. If `colorbox` is a string, a color scale with label is drawn.

  Since this is a global graphics option, its position in the scene description does not matter.

  Example:

  Color scale and images.

      (%i1) im: apply('matrix,
                       makelist(makelist(random(200),i,1,30),i,1,30))$
      (%i2) draw(
                gr2d(image(im,0,0,30,30)),
                gr2d(colorbox = false, image(im,0,0,30,30))
            )$

  Color scale and 3D coloured object.

      (%i1) draw3d(
              colorbox   = "Magnitude",
              enhanced3d = true,
              explicit(x^2+y^2,x,-1,1,y,-1,1))$

  See also `palette_draw`.

  Categories: Package draw

Graphic option: columns

  Default value: 1

  `columns` is the number of columns in multiple plots.

  Since this is a global graphics option, its position in the scene description does not matter. It can be also used as an argument of function `draw`.

  Example:

      (%i1) scene1: gr2d(title="Ellipse",
                         nticks=30,
                         parametric(2*cos(t),5*sin(t),t,0,2*%pi))$
      (%i2) scene2: gr2d(title="Triangle",
                         polygon([4,5,7],[6,4,2]))$
      (%i3) draw(scene1, scene2, columns = 2)$

  Categories: Package draw

Graphic option: contour

  Default value: `none`

  Option `contour` enables the user to select where to plot contour lines. Possible values are:

  - `none`: no contour lines are plotted.
  - `base`: contour lines are projected on the xy plane.
  - `surface`: contour lines are plotted on the surface.
  - `both`: two contour lines are plotted: on the xy plane and on the surface.
  - `map`: contour lines are projected on the xy plane, and the view point is set just in the vertical.

  Since this is a global graphics option, its position in the scene description does not matter.

  Example:

      (%i1) draw3d(explicit(20*exp(-x^2-y^2)-10,x,0,2,y,-3,3),
                   contour_levels = 15,
                   contour        = both,
                   surface_hide   = true) $

      (%i1) draw3d(explicit(20*exp(-x^2-y^2)-10,x,0,2,y,-3,3),
                   contour_levels = 15,
                   contour        = map
            ) $

  Categories: Package draw

Graphic option: contour_levels

  Default value: 5

  This graphic option controls the way contours are drawn. `contour_levels` can be set to a positive integer number, a list of three numbers or an arbitrary set of numbers:

  - When option `contour_levels` is bounded to positive integer n, n contour lines will be drawn at equal intervals. By default, five equally spaced contours are plotted.
  - When option `contour_levels` is bounded to a list of length three of the form `[lowest,s,highest]`, contour lines are plotted from `lowest` to `highest` in steps of `s`.
  - When option `contour_levels` is bounded to a set of numbers of the form `{n1, n2, ...}`, contour lines are plotted at values `n1`, `n2`, ...

  Since this is a global graphics option, its position in the scene description does not matter.

  Examples:

  Ten equally spaced contour lines. The actual number of levels can be adjusted to give simple labels.

      (%i1) draw3d(color = green,
                   explicit(20*exp(-x^2-y^2)-10,x,0,2,y,-3,3),
                   contour_levels = 10,
                   contour        = both,
                   surface_hide   = true) $

  From -8 to 8 in steps of 4.

      (%i1) draw3d(color = green,
                   explicit(20*exp(-x^2-y^2)-10,x,0,2,y,-3,3),
                   contour_levels = [-8,4,8],
                   contour        = both,
                   surface_hide   = true) $

  Isolines at levels -7, -6, 0.8 and 5.

      (%i1) draw3d(color = green,
                   explicit(20*exp(-x^2-y^2)-10,x,0,2,y,-3,3),
                   contour_levels = {-7, -6, 0.8, 5},
                   contour        = both,
                   surface_hide   = true) $

  See also `contour`.

  Categories: Package draw

Graphic option: data_file_name

  Default value: `"data.gnuplot"`

  This is the name of the file with the numeric data needed by Gnuplot to build the requested plot.

  Since this is a global graphics option, its position in the scene description does not matter. It can be also used as an argument of function `draw`.

  See example in `gnuplot_file_name`.

  Categories: Package draw

Graphic option: delay

  Default value: 5

  This is the delay in 1/100 seconds of frames in animated gif files.

  Since this is a global graphics option, its position in the scene description does not matter. It can be also used as an argument of function `draw`.

  Example:

      (%i1) draw(
              delay     = 100,
              file_name = "zzz",
              terminal  = 'animated_gif,
              gr2d(explicit(x^2,x,-1,1)),
              gr2d(explicit(x^3,x,-1,1)),
              gr2d(explicit(x^4,x,-1,1)));
      End of animation sequence
      (%o2)          [gr2d(explicit), gr2d(explicit), gr2d(explicit)]

  Option `delay` is only active in animated gif's; it is ignored in any other case.

  See also `terminal`, and `dimensions`.

  Categories: Package draw

Graphic option: dimensions

  Default value: `[600,500]`

  Dimensions of the output terminal. Its value is a list formed by the width and the height. The meaning of the two numbers depends on the terminal you are working with.

  With terminals `gif`, `animated_gif`, `png`, `jpg`, `svg`, `screen`, `wxt`, and `aquaterm`, the integers represent the number of points in each direction. If they are not intergers, they are rounded.

  With terminals `eps`, `eps_color`, `pdf`, and `pdfcairo`, both numbers represent hundredths of cm, which means that, by default, pictures in these formats are 6 cm in width and 5 cm in height.

  Since this is a global graphics option, its position in the scene description does not matter. It can be also used as an argument of function `draw`.

  Examples:

  Option `dimensions` applied to file output and to wxt canvas.

      (%i1) draw2d(
              dimensions = [300,300],
              terminal   = 'png,
              explicit(x^4,x,-1,1)) $
      (%i2) draw2d(
              dimensions = [300,300],
              terminal   = 'wxt,
              explicit(x^4,x,-1,1)) $

  Option `dimensions` applied to eps output. We want an eps file with A4 portrait dimensions.

      (%i1) A4portrait: 100*[21, 29.7]$
      (%i2) draw3d(
              dimensions = A4portrait,
              terminal   = 'eps,
              explicit(x^2-y^2,x,-2,2,y,-2,2)) $

  Categories: Package draw

Graphic option: draw_realpart

  Default value: `true`

  When `true`, functions to be drawn are considered as complex functions whose real part value should be plotted; when `false`, nothing will be plotted when the function does not give a real value.

  This option affects objects `explicit` and `parametric` in 2D and 3D, and `parametric_surface`.

  Example:

  Option `draw_realpart` affects objects `explicit` and `parametric`.

      (%i1) draw2d(
              draw_realpart = false,
              explicit(sqrt(x^2  - 4*x) - x, x, -1, 5),
              color         = red,
              draw_realpart = true,
              parametric(x,sqrt(x^2  - 4*x) - x + 1, x, -1, 5) );

  Categories: Package draw

Graphic option: enhanced3d

  Default value: `none`

  If `enhanced3d` is `none`, surfaces are not colored in 3D plots. In order to get a colored surface, a list must be assigned to option `enhanced3d`, where the first element is an expression and the rest are the names of the variables or parameters used in that expression. A list such `[f(x,y,z), x, y, z]` means that point `[x,y,z]` of the surface is assigned number `f(x,y,z)`, which will be colored according to the actual `palette`. For those 3D graphic objects defined in terms of parameters, it is possible to define the color number in terms of the parameters, as in `[f(u), u]`, as in objects `parametric` and `tube`, or `[f(u,v), u, v]`, as in object `parametric_surface`. While all 3D objects admit the model based on absolute coordinates, `[f(x,y,z), x, y, z]`, only two of them, namely `explicit` and `elevation_grid`, accept also models defined on the `[x,y]` coordinates, `[f(x,y), x, y]`. 3D graphic object `implicit` accepts only the `[f(x,y,z), x, y, z]` model. Object `points` accepts also the `[f(x,y,z), x, y, z]` model, but when points have a chronological nature, model `[f(k), k]` is also valid, being `k` an ordering parameter.

  When `enhanced3d` is assigned something different to `none`, options `color` and `surface_hide` are ignored.

  The names of the variables defined in the lists may be different to those used in the definitions of the graphic objects.

  In order to maintain back compatibility, `enhanced3d = false` is equivalent to `enhanced3d = none`, and `enhanced3d = true` is equivalent to `enhanced3d = [z, x, y, z]`. If an expression is given to `enhanced3d`, its variables must be the same used in the surface definition. This is not necessary when using lists.

  See option `palette` to learn how palettes are specified.

  Examples:

  `explicit` object with coloring defined by the `[f(x,y,z), x, y, z]` model.

      (%i1) draw3d(
               enhanced3d = [x-z/10,x,y,z],
               palette    = gray,
               explicit(20*exp(-x^2-y^2)-10,x,-3,3,y,-3,3))$

  `explicit` object with coloring defined by the `[f(x,y), x, y]` model. The names of the variables defined in the lists may be different to those used in the definitions of the graphic objects; in this case, `r` corresponds to `x`, and `s` to `y`.

      (%i1) draw3d(
               enhanced3d = [sin(r*s),r,s],
               explicit(20*exp(-x^2-y^2)-10,x,-3,3,y,-3,3))$

  `parametric` object with coloring defined by the `[f(x,y,z), x, y, z]` model.

      (%i1) draw3d(
               nticks = 100,
               line_width = 2,
               enhanced3d = [if y>= 0 then 1 else 0, x, y, z],
               parametric(sin(u)^2,cos(u),u,u,0,4*%pi)) $

  `parametric` object with coloring defined by the `[f(u), u]` model. In this case, `(u-1)^2` is a shortcut for `[(u-1)^2,u]`.

      (%i1) draw3d(
               nticks = 60,
               line_width = 3,
               enhanced3d = (u-1)^2,
               parametric(cos(5*u)^2,sin(7*u),u-2,u,0,2))$

  `elevation_grid` object with coloring defined by the `[f(x,y), x, y]` model.

      (%i1) m: apply(
                 matrix,
                 makelist(makelist(cos(i^2/80-k/30),k,1,30),i,1,20)) $
      (%i2) draw3d(
               enhanced3d = [cos(x*y*10),x,y],
               elevation_grid(m,-1,-1,2,2),
               xlabel = "x",
               ylabel = "y");

  `tube` object with coloring defined by the `[f(x,y,z), x, y, z]` model.

      (%i1) draw3d(
               enhanced3d = [cos(x-y),x,y,z],
               palette = gray,
               xu_grid = 50,
               tube(cos(a), a, 0, 1, a, 0, 4*%pi) )$

  `tube` object with coloring defined by the `[f(u), u]` model. Here, `enhanced3d = -a` would be the shortcut for `enhanced3d = [-foo,foo]`.

      (%i1) draw3d(
               capping = [true, false],
               palette = [26,15,-2],
               enhanced3d = [-foo, foo],
               tube(a, a, a^2, 1, a, -2, 2) )$

  `implicit` and `points` objects with coloring defined by the `[f(x,y,z), x, y, z]` model.

      (%i1) draw3d(
               enhanced3d = [x-y,x,y,z],
               implicit((x^2+y^2+z^2-1)*(x^2+(y-1.5)^2+z^2-0.5)=0.015,
                        x,-1,1,y,-1.2,2.3,z,-1,1)) $
      (%i2) m: makelist([random(1.0),random(1.0),random(1.0)],k,1,2000)$

      (%i3) draw3d(
               point_type = filled_circle,
               point_size = 2,
               enhanced3d = [u+v-w,u,v,w],
               points(m) ) $

  When points have a chronological nature, model `[f(k), k]` is also valid, being `k` an ordering parameter.

      (%i1) m:makelist([random(1.0), random(1.0), random(1.0)],k,1,5)$
      (%i2) draw3d(
               enhanced3d = [sin(j), j],
               point_size = 3,
               point_type = filled_circle,
               points_joined = true,
               points(m)) $

  Categories: Package draw

Graphic option: error_type

  Default value: `y`

  Depending on its value, which can be `x`, `y`, or `xy`, graphic object `errors` will draw points with horizontal, vertical, or both, error bars. When `error_type=boxes`, boxes will be drawn instead of crosses.

  See also `errors`.

Graphic option: file_name

  Default value: `"maxima_out"`

  This is the name of the file where terminals `png`, `jpg`, `gif`, `eps`, `eps_color`, `pdf`, `pdfcairo` and `svg` will save the graphic.

  Since this is a global graphics option, its position in the scene description does not matter. It can be also used as an argument of function `draw`.

  Example:

      (%i1) draw2d(file_name = "myfile",
                   explicit(x^2,x,-1,1),
                   terminal  = 'png)$

  See also `terminal`, `dimensions_draw`.

  Categories: Package draw

Graphic option: fill_color

  Default value: `"red"`

  `fill_color` specifies the color for filling polygons and 2d `explicit` functions.

  See `color` to learn how colors are specified.

  Categories: Package draw

Graphic option: fill_density

  Default value: 0

  `fill_density` is a number between 0 and 1 that specifies the intensity of the `fill_color` in `bars` objects.

  See `bars` for examples.

Graphic option: filled_func

  Default value: `false`

  Option `filled_func` controls how regions limited by functions should be filled. When `filled_func` is `true`, the region bounded by the function defined with object `explicit` and the bottom of the graphic window is filled with `fill_color`. When `filled_func` contains a function expression, then the region bounded by this function and the function defined with object `explicit` will be filled. By default, explicit functions are not filled.

  A useful special case is `filled_func=0`, which generates the region bond by the horizontal axis and the explicit function.

  This option affects only the 2d graphic object `explicit`.

  Example:

  Region bounded by an `explicit` object and the bottom of the graphic window.

      (%i1) draw2d(fill_color  = red,
                   filled_func = true,
                   explicit(sin(x),x,0,10) )$

  Region bounded by an `explicit` object and the function defined by option `filled_func`. Note that the variable in `filled_func` must be the same as that used in `explicit`.

      (%i1) draw2d(fill_color  = grey,
                   filled_func = sin(x),
                   explicit(-sin(x),x,0,%pi));

  See also `fill_color` and `explicit`.

  Categories: Package draw

Graphic option: font

  Default value: `""` (empty string)

  This option can be used to set the font face to be used by the terminal. Only one font face and size can be used throughout the plot.

  Since this is a global graphics option, its position in the scene description does not matter.

  See also `font_size`.

  Gnuplot doesn't handle fonts by itself, it leaves this task to the support libraries of the different terminals, each one with its own philosophy about it. A brief summary follows:

  - x11: Uses the normal x11 font server mechanism.

    Example:

        (%i1) draw2d(font      = "Arial",
                     font_size = 20,
                     label(["Arial font, size 20",1,1]))$

  - windows: The windows terminal doesn't support changing of fonts from inside the plot. Once the plot has been generated, the font can be changed right-clicking on the menu of the graph window.
  - png, jpeg, gif: The libgd library uses the font path stored in the environment variable `GDFONTPATH`; in this case, it is only necessary to set option `font` to the font's name. It is also possible to give the complete path to the font file.

    Examples:

    Option `font` can be given the complete path to the font file:

        (%i1) path: "/usr/share/fonts/truetype/freefont/" $
        (%i2) file: "FreeSerifBoldItalic.ttf" $
        (%i3) draw2d(
                font      = concat(path, file),
                font_size = 20,
                color     = red,
                label(["FreeSerifBoldItalic font, size 20",1,1]),
                terminal  = png)$

    If environment variable `GDFONTPATH` is set to the path where font files are allocated, it is possible to set graphic option `font` to the name of the font.

        (%i1) draw2d(
                font      = "FreeSerifBoldItalic",
                font_size = 20,
                color     = red,
                label(["FreeSerifBoldItalic font, size 20",1,1]),
                terminal  = png)$

  - Postscript: Standard Postscript fonts are:
    `"Times-Roman"`, `"Times-Italic"`, `"Times-Bold"`, `"Times-BoldItalic"`,
    `"Helvetica"`, `"Helvetica-Oblique"`, `"Helvetica-Bold"`,
    `"Helvetic-BoldOblique"`, `"Courier"`, `"Courier-Oblique"`, `"Courier-Bold"`,
    and `"Courier-BoldOblique"`.

    Example:

        (%i1) draw2d(
                font      = "Courier-Oblique",
                font_size = 15,
                label(["Courier-Oblique font, size 15",1,1]),
                terminal = eps)$

  - pdf: Uses same fonts as Postscript.
  - pdfcairo: Uses same fonts as wxt.
  - wxt: The pango library finds fonts via the `fontconfig` utility.
  - aqua: Default is `"Times-Roman"`.

  The gnuplot documentation is an important source of information about terminals and fonts.

  Categories: Package draw

Graphic option: font_size

  Default value: 10

  This option can be used to set the font size to be used by the terminal. Only one font face and size can be used throughout the plot. `font_size` is active only when option `font` is not equal to the empty string.

  Since this is a global graphics option, its position in the scene description does not matter.

  See also `font`.

  Categories: Package draw

Graphic option: gnuplot_file_name

  Default value: `"maxout_xxx.gnuplot"` with `"xxx"` being a number that is unique to each concurrently-running maxima process.

  This is the name of the file with the necessary commands to be processed by Gnuplot.

  Since this is a global graphics option, its position in the scene description does not matter. It can be also used as an argument of function `draw`.

  Example:

      (%i1) draw2d(
             file_name = "my_file",
             gnuplot_file_name = "my_commands_for_gnuplot",
             data_file_name    = "my_data_for_gnuplot",
             terminal          = png,
             explicit(x^2,x,-1,1)) $

  See also `data_file_name`.

  Categories: Package draw

Graphic option: grid

  Default value: `false`

  If `grid` is `not false`, a grid will be drawn on the xy plane. If `grid` is assigned true, one grid line per tick of each axis is drawn. If `grid` is assigned a list `nx,ny` with `[nx,ny] > [0,0]` instead `nx` lines per tick of the x axis and `ny` lines per tick of the y axis are drawn.

  Since this is a global graphics option, its position in the scene description does not matter.

  Example:

      (%i1) draw2d(grid = true,
                   explicit(exp(u),u,-2,2))$

      (%i1) draw2d(grid = [2,2],
                   explicit(sin(x),x,0,2*%pi))$

  Categories: Package draw

Graphic option: head_angle

  Default value: 45

  `head_angle` indicates the angle, in degrees, between the arrow heads and the segment.

  This option is relevant only for `vector` objects.

  Example:

      (%i1) draw2d(xrange      = [0,10],
                   yrange      = [0,9],
                   head_length = 0.7,
                   head_angle  = 10,
                   vector([1,1],[0,6]),
                   head_angle  = 20,
                   vector([2,1],[0,6]),
                   head_angle  = 30,
                   vector([3,1],[0,6]),
                   head_angle  = 40,
                   vector([4,1],[0,6]),
                   head_angle  = 60,
                   vector([5,1],[0,6]),
                   head_angle  = 90,
                   vector([6,1],[0,6]),
                   head_angle  = 120,
                   vector([7,1],[0,6]),
                   head_angle  = 160,
                   vector([8,1],[0,6]),
                   head_angle  = 180,
                   vector([9,1],[0,6]) )$

  See also `head_both`, `head_length`, and `head_type`.

  Categories: Package draw

Graphic option: head_both

  Default value: `false`

  If `head_both` is `true`, vectors are plotted with two arrow heads. If `false`, only one arrow is plotted.

  This option is relevant only for `vector` objects.

  Example:

      (%i1) draw2d(xrange      = [0,8],
                   yrange      = [0,8],
                   head_length = 0.7,
                   vector([1,1],[6,0]),
                   head_both   = true,
                   vector([1,7],[6,0]) )$

  See also `head_length`, `head_angle`, and `head_type`.

  Categories: Package draw

Graphic option: head_length

  Default value: 2

  `head_length` indicates, in x-axis units, the length of arrow heads.

  This option is relevant only for `vector` objects.

  Example:

      (%i1) draw2d(xrange      = [0,12],
                   yrange      = [0,8],
                   vector([0,1],[5,5]),
                   head_length = 1,
                   vector([2,1],[5,5]),
                   head_length = 0.5,
                   vector([4,1],[5,5]),
                   head_length = 0.25,
                   vector([6,1],[5,5]))$

  See also `head_both`, `head_angle`, and `head_type`.

  Categories: Package draw

Graphic option: head_type

  Default value: `filled`

  `head_type` is used to specify how arrow heads are plotted. Possible values are: `filled` (closed and filled arrow heads), `empty` (closed but not filled arrow heads), and `nofilled` (open arrow heads).

  This option is relevant only for `vector` objects.

  Example:

      (%i1) draw2d(xrange      = [0,12],
                   yrange      = [0,10],
                   head_length = 1,
                   vector([0,1],[5,5]), /* default type */
                   head_type = 'empty,
                   vector([3,1],[5,5]),
                   head_type = 'nofilled,
                   vector([6,1],[5,5]))$

  See also `head_both`, `head_angle`, and `head_length`.

  Categories: Package draw

Graphic option: interpolate_color

  Default value: `false`

  This option is relevant only when `enhanced3d` is not `false`.

  When `interpolate_color` is `false`, surfaces are colored with homogeneous quadrangles. When `true`, color transitions are smoothed by interpolation.

  `interpolate_color` also accepts a list of two numbers, `[m,n]`. For positive m and n, each quadrangle or triangle is interpolated m times and n times in the respective direction. For negative m and n, the interpolation frequency is chosen so that there will be at least |m| and |n| points drawn; you can consider this as a special gridding function. Zeros, i.e. `interpolate_color=[0,0]`, will automatically choose an optimal number of interpolated surface points.

  Also, `interpolate_color=true` is equivalent to `interpolate_color=[0,0]`.

  Examples:

  Color interpolation with explicit functions.

      (%i1) draw3d(
              enhanced3d   = sin(x*y),
              explicit(20*exp(-x^2-y^2)-10, x ,-3, 3, y, -3, 3)) $

      (%i2) draw3d(
              interpolate_color = true,
              enhanced3d   = sin(x*y),
              explicit(20*exp(-x^2-y^2)-10, x ,-3, 3, y, -3, 3)) $

      (%i3) draw3d(
              interpolate_color = [-10,0],
              enhanced3d   = sin(x*y),
              explicit(20*exp(-x^2-y^2)-10, x ,-3, 3, y, -3, 3)) $

  Color interpolation with the `mesh` graphic object.

  Interpolating colors in parametric surfaces can give unexpected results.

      (%i1) draw3d(
              enhanced3d = true,
              mesh([[1,1,3],   [7,3,1],[12,-2,4],[15,0,5]],
                   [[2,7,8],   [4,3,1],[10,5,8], [12,7,1]],
                   [[-2,11,10],[6,9,5],[6,15,1], [20,15,2]])) $

      (%i2) draw3d(
              enhanced3d        = true,
              interpolate_color = true,
              mesh([[1,1,3],   [7,3,1],[12,-2,4],[15,0,5]],
                   [[2,7,8],   [4,3,1],[10,5,8], [12,7,1]],
                   [[-2,11,10],[6,9,5],[6,15,1], [20,15,2]])) $

      (%i3) draw3d(
              enhanced3d        = true,
              interpolate_color = true,
              view=map,
              mesh([[1,1,3],   [7,3,1],[12,-2,4],[15,0,5]],
                   [[2,7,8],   [4,3,1],[10,5,8], [12,7,1]],
                   [[-2,11,10],[6,9,5],[6,15,1], [20,15,2]])) $

  See also `enhanced3d`.

  Categories: Package draw

Graphic option: ip_grid

  Default value: `[50, 50]`

  `ip_grid` sets the grid for the first sampling in implicit plots.

  This option is relevant only for `implicit` objects.

  Categories: Package draw

Graphic option: ip_grid_in

  Default value: `[5, 5]`

  `ip_grid_in` sets the grid for the second sampling in implicit plots.

  This option is relevant only for `implicit` objects.

  Categories: Package draw

Graphic option: key

  Default value: `""` (empty string)

  `key` is the name of a function in the legend. If `key` is an empty string, no key is assigned to the function.

  This option affects the following graphic objects:

  - `gr2d`: `points`, `polygon`, `rectangle`, `ellipse`, `vector`, `explicit`, `implicit`, `parametric` and `polar`.
  - `gr3d`: `points`, `explicit`, `parametric` and `parametric_surface`.

  Example:

      (%i1) draw2d(key   = "Sinus",
                   explicit(sin(x),x,0,10),
                   key   = "Cosinus",
                   color = red,
                   explicit(cos(x),x,0,10) )$

  Package draw

Graphic option: key_pos

  Default value: `""` (empty string)

  `key_pos` defines at which position the legend will be drawn. If `key` is an empty string, `"top_right"` is used. Available position specifiers are: `top_left`, `top_center`, `top_right`, `center_left`, `center`, `center_right`, `bottom_left`, `bottom_center`, and `bottom_right`.

  Since this is a global graphics option, its position in the scene description does not matter.

  Example:

      (%i1) draw2d(
              key_pos = top_left,
              key   = "x",
              explicit(x,  x,0,10),
              color= red,
              key   = "x squared",
              explicit(x^2,x,0,10))$
      (%i3) draw3d(
              key_pos = center,
              key   = "x",
              explicit(x+y,x,0,10,y,0,10),
              color= red,
              key   = "x squared",
              explicit(x^2+y^2,x,0,10,y,0,10))$

  Categories: Package draw

Graphic option: label_alignment

  Default value: `center`

  `label_alignment` is used to specify where to write labels with respect to the given coordinates. Possible values are: `center`, `left`, and `right`.

  This option is relevant only for `label` objects.

  Example:

      (%i1) draw2d(xrange          = [0,10],
                   yrange          = [0,10],
                   points_joined   = true,
                   points([[5,0],[5,10]]),
                   color           = blue,
                   label(["Centered alignment (default)",5,2]),
                   label_alignment = 'left,
                   label(["Left alignment",5,5]),
                   label_alignment = 'right,
                   label(["Right alignment",5,8]))$

  See also `label_orientation`, and `color`

  Categories: Package draw

Graphic option: label_orientation

  Default value: `horizontal`

  `label_orientation` is used to specify orientation of labels. Possible values are: `horizontal`, and `vertical`.

  This option is relevant only for `label` objects.

  Example:

  In this example, a dummy point is added to get an image. Package `draw` needs always data to draw an scene.

      (%i1) draw2d(xrange     = [0,10],
                   yrange     = [0,10],
                   point_size = 0,
                   points([[5,5]]),
                   color      = navy,
                   label(["Horizontal orientation (default)",5,2]),
                   label_orientation = 'vertical,
                   color             = "#654321",
                   label(["Vertical orientation",1,5]))$

  See also `label_alignment` and `color`

  Categories: Package draw

Graphic option: line_type

  Default value: `solid`

  `line_type` indicates how lines are displayed; possible values are `solid` and `dots`, both available in all terminals, and `dashes`, `short_dashes`, `short_long_dashes`, `short_short_long_dashes`, and `dot_dash`, which are not available in `png`, `jpg`, and `gif` terminals.

  This option affects the following graphic objects:

  - `gr2d`: `points`, `polygon`, `rectangle`, `ellipse`, `vector`, `explicit`, `implicit`, `parametric` and `polar`.
  - `gr3d`: `points`, `explicit`, `parametric` and `parametric_surface`.

  Example:

      (%i1) draw2d(line_type = dots,
                   explicit(1 + x^2,x,-1,1),
                   line_type = solid, /* default */
                   explicit(2 + x^2,x,-1,1))$

  See also `line_width`.

  Categories: Package draw

Graphic option: line_width

  Default value: 1

  `line_width` is the width of plotted lines. Its value must be a positive number.

  This option affects the following graphic objects:

  - `gr2d`: `points`, `polygon`, `rectangle`, `ellipse`, `vector`, `explicit`, `implicit`, `parametric` and `polar`.
  - `gr3d`: `points` and `parametric`.

  Example:

      (%i1) draw2d(explicit(x^2,x,-1,1), /* default width */
                   line_width = 5.5,
                   explicit(1 + x^2,x,-1,1),
                   line_width = 10,
                   explicit(2 + x^2,x,-1,1))$

  See also `line_type`.

  Categories: Package draw

Graphic option: logcb

  Default value: `false`

  If `logcb` is `true`, the tics in the colorbox will be drawn in the logarithmic scale.

  When `enhanced3d` or `colorbox` is `false`, option `logcb` has no effect.

  Since this is a global graphics option, its position in the scene description does not matter.

  Example:

      (%i1) draw3d (
              enhanced3d = true,
              color      = green,
              logcb = true,
              logz  = true,
              palette = [-15,24,-9],
              explicit(exp(x^2-y^2), x,-2,2,y,-2,2)) $

  See also `enhanced3d`, `colorbox` and `cbrange`.

  Categories: Package draw

Graphic option: logx

  Default value: `false`

  If `logx` is `true`, the x axis will be drawn in the logarithmic scale.

  Since this is a global graphics option, its position in the scene description does not matter, with the exception that it should be written before any 2D `explicit` object, so that `draw` can produce a better plot.

  Example:

      (%i1) draw2d(logx = true,
                   explicit(log(x),x,0.01,5))$

  See also `logy`, `logx_secondary`, `logy_secondary`, and `logz`.

  Categories: Package draw

Graphic option: logx_secondary

  Default value: `false`

  If `logx_secondary` is `true`, the secondary x axis will be drawn in the logarithmic scale.

  This option is relevant only for 2d scenes.

  Since this is a global graphics option, its position in the scene description does not matter.

  Example:

      (%i1) draw2d(
              grid = true,
              key="x^2, linear scale",
              color=red,
              explicit(x^2,x,1,100),
              xaxis_secondary = true,
              xtics_secondary = true,
              logx_secondary  = true,
              key = "x^2, logarithmic x scale",
              color = blue,
              explicit(x^2,x,1,100) )$

  See also `logx_draw`, `logy_draw`, `logy_secondary`, and `logz`.

  Categories: Package draw

Graphic option: logy

  Default value: `false`

  If `logy` is `true`, the y axis will be drawn in the logarithmic scale.

  Since this is a global graphics option, its position in the scene description does not matter.

  Example:

      (%i1) draw2d(logy = true,
                   explicit(exp(x),x,0,5))$

  See also `logx_draw`, `logx_secondary`, `logy_secondary`, and `logz`.

  Categories: Package draw

Graphic option: logy_secondary

  Default value: `false`

  If `logy_secondary` is `true`, the secondary y axis will be drawn in the logarithmic scale.

  This option is relevant only for 2d scenes.

  Since this is a global graphics option, its position in the scene description does not matter.

  Example:

      (%i1) draw2d(
              grid = true,
              key="x^2, linear scale",
              color=red,
              explicit(x^2,x,1,100),
              yaxis_secondary = true,
              ytics_secondary = true,
              logy_secondary  = true,
              key = "x^2, logarithmic y scale",
              color = blue,
              explicit(x^2,x,1,100) )$

  See also `logx_draw`, `logy_draw`, `logx_secondary`, and `logz`.

  Categories: Package draw

Graphic option: logz

  Default value: `false`

  If `logz` is `true`, the z axis will be drawn in the logarithmic scale.

  Since this is a global graphics option, its position in the scene description does not matter.

  Example:

      (%i1) draw3d(logz = true,
                   explicit(exp(u^2+v^2),u,-2,2,v,-2,2))$

  See also `logx_draw` and `logy_draw`.

  Categories: Package draw

Graphic option: nticks

  Default value: 29

  In 2d, `nticks` gives the initial number of points used by the adaptive plotting routine for explicit objects. It is also the number of points that will be shown in parametric and polar curves.

  This option affects the following graphic objects:

  - `gr2d`: `ellipse`, `explicit`, `parametric` and `polar`.
  - `gr3d`: `parametric`.

  See also `adapt_depth`

  Example:

      (%i1) draw2d(transparent = true,
                   ellipse(0,0,4,2,0,180),
                   nticks = 5,
                   ellipse(0,0,4,2,180,180) )$

  Categories: Package draw

Graphic option: palette

  Default value: `color`

  `palette` indicates how to map gray levels onto color components. It works together with option `enhanced3d` in 3D graphics, who associates every point of a surfaces to a real number or gray level. It also works with gray images. With `palette`, levels are transformed into colors.

  There are two ways for defining these transformations.

  First, `palette` can be a vector of length three with components ranging from -36 to +36; each value is an index for a formula mapping the levels onto red, green and blue colors, respectively:

       0: 0               1: 0.5           2: 1
       3: x               4: x^2           5: x^3
       6: x^4             7: sqrt(x)       8: sqrt(sqrt(x))
       9: sin(90x)       10: cos(90x)     11: |x-0.5|
      12: (2x-1)^2       13: sin(180x)    14: |cos(180x)|
      15: sin(360x)      16: cos(360x)    17: |sin(360x)|
      18: |cos(360x)|    19: |sin(720x)|  20: |cos(720x)|
      21: 3x             22: 3x-1         23: 3x-2
      24: |3x-1|         25: |3x-2|       26: (3x-1)/2
      27: (3x-2)/2       28: |(3x-1)/2|   29: |(3x-2)/2|
      30: x/0.32-0.78125 31: 2*x-0.84     32: 4x;1;-2x+1.84;x/0.08-11.5
      33: |2*x - 0.5|    34: 2*x          35: 2*x - 0.5
      36: 2*x - 1

  negative numbers mean negative colour component. `palette = gray` and `palette = color` are short cuts for `palette = [3,3,3]` and `palette = [7,5,15]`, respectively.

  Second, `palette` can be a user defined lookup table. In this case, the format for building a lookup table of length `n` is `palette=[color_1, color_2, ..., color_n]`, where `color_i` is a well formed color (see option `color`) such that `color_1` is assigned to the lowest gray level and `color_n` to the highest. The rest of colors are interpolated.

  Since this is a global graphics option, its position in the scene description does not matter.

  Examples:

  It works together with option `enhanced3d` in 3D graphics.

      (%i1) draw3d(
              enhanced3d = [z-x+2*y,x,y,z],
              palette = [32, -8, 17],
              explicit(20*exp(-x^2-y^2)-10,x,-3,3,y,-3,3))$

  It also works with gray images.

      (%i1) im: apply(
                 'matrix,
                  makelist(makelist(random(200),i,1,30),i,1,30))$
      (%i2) /* palette = color, default */
            draw2d(image(im,0,0,30,30))$
      (%i3) draw2d(palette = gray, image(im,0,0,30,30))$
      (%i4) draw2d(palette = [15,20,-4],
                   colorbox=false,
                   image(im,0,0,30,30))$

  `palette` can be a user defined lookup table. In this example, low values of `x` are colored in red, and higher values in yellow.

      (%i1) draw3d(
               palette = [red, blue, yellow],
               enhanced3d = x,
               explicit(x^2+y^2,x,-1,1,y,-1,1)) $

  See also `colorbox` and `enhanced3d`.

  Categories: Package draw

Graphic option: point_size

  Default value: 1

  `point_size` sets the size for plotted points. It must be a non negative number.

  This option has no effect when graphic option `point_type` is set to `dot`.

  This option affects the following graphic objects:

  - `gr2d`: `points`.
  - `gr3d`: `points`.

  Example:

      (%i1) draw2d(points(makelist([random(20),random(50)],k,1,10)),
              point_size = 5,
              points(makelist(k,k,1,20),makelist(random(30),k,1,20)))$

  Categories: Package draw

Graphic option: point_type

  Default value: 1

  `point_type` indicates how isolated points are displayed; the value of this option can be any integer index greater or equal than -1, or the name of a point style: `$none` (-1), `dot` (0), `plus` (1), `multiply` (2), `asterisk` (3), `square` (4), `filled_square` (5), `circle` (6), `filled_circle` (7), `up_triangle` (8), `filled_up_triangle` (9), `down_triangle` (10), `filled_down_triangle` (11), `diamant` (12) and `filled_diamant` (13).

  This option affects the following graphic objects:

  - `gr2d`: `points`.
  - `gr3d`: `points`.

  Example:

      (%i1) draw2d(xrange = [0,10],
                   yrange = [0,10],
                   point_size = 3,
                   point_type = diamant,
                   points([[1,1],[5,1],[9,1]]),
                   point_type = filled_down_triangle,
                   points([[1,2],[5,2],[9,2]]),
                   point_type = asterisk,
                   points([[1,3],[5,3],[9,3]]),
                   point_type = filled_diamant,
                   points([[1,4],[5,4],[9,4]]),
                   point_type = 5,
                   points([[1,5],[5,5],[9,5]]),
                   point_type = 6,
                   points([[1,6],[5,6],[9,6]]),
                   point_type = filled_circle,
                   points([[1,7],[5,7],[9,7]]),
                   point_type = 8,
                   points([[1,8],[5,8],[9,8]]),
                   point_type = filled_diamant,
                   points([[1,9],[5,9],[9,9]]) )$

  Categories: Package draw

Graphic option: points_joined

  Default value: `false`

  When `points_joined` is `true`, points are joined by lines; when `false`, isolated points are drawn. A third possible value for this graphic option is `impulses`; in such case, vertical segments are drawn from points to the x-axis (2D) or to the xy-plane (3D).

  This option affects the following graphic objects:

  - `gr2d`: `points`.
  - `gr3d`: `points`.

  Example:

      (%i1) draw2d(xrange        = [0,10],
                   yrange        = [0,4],
                   point_size    = 3,
                   point_type    = up_triangle,
                   color         = blue,
                   points([[1,1],[5,1],[9,1]]),
                   points_joined = true,
                   point_type    = square,
                   line_type     = dots,
                   points([[1,2],[5,2],[9,2]]),
                   point_type    = circle,
                   color         = red,
                   line_width    = 7,
                   points([[1,3],[5,3],[9,3]]) )$

  Categories: Package draw

Graphic option: proportional_axes

  Default value: `none`

  When `proportional_axes` is equal to `xy` or `xyz`, the aspect ratio of the axis units will be set to 1:1 resulting in a 2D or 3D scene that will be drawn with axes proportional to their relative lengths.

  Since this is a global graphics option, its position in the scene description does not matter.

  This option works with Gnuplot version 4.2.6 or greater.

  Examples:

  Single 2D plot.

      (%i1) draw2d(
              ellipse(0,0,1,1,0,360),
              transparent=true,
              color = blue,
              line_width = 4,
              ellipse(0,0,2,1/2,0,360),
              proportional_axes = 'xy) $

  Multiplot.

      (%i1) draw(
              terminal = wxt,
              gr2d(proportional_axes = 'xy,
                   explicit(x^2,x,0,1)),
              gr2d(explicit(x^2,x,0,1),
                   xrange = [0,1],
                   yrange = [0,2],
                   proportional_axes='xy),
              gr2d(explicit(x^2,x,0,1)))$

  Categories: Package draw

Graphic option: surface_hide

  Default value: `false`

  If `surface_hide` is `true`, hidden parts are not plotted in 3d surfaces.

  Since this is a global graphics option, its position in the scene description does not matter.

  Example:

      (%i1) draw(columns=2,
                 gr3d(explicit(exp(sin(x)+cos(x^2)),x,-3,3,y,-3,3)),
                 gr3d(surface_hide = true,
                      explicit(exp(sin(x)+cos(x^2)),x,-3,3,y,-3,3)) )$

  Categories: Package draw

Graphic option: terminal

  Default value: `screen`

  Selects the terminal to be used by Gnuplot; possible values are: `screen` (default), `png`, `pngcairo`, `jpg`, `gif`, `eps`, `eps_color`, `epslatex`, `epslatex_standalone`, `svg`, `canvas`, `dumb`, `dumb_file`, `pdf`, `pdfcairo`, `wxt`, `animated_gif`, `multipage_pdfcairo`, `multipage_pdf`, `multipage_eps`, `multipage_eps_color`, and `aquaterm`.

  Terminals `screen`, `wxt` and `aquaterm` can be also defined as a list with two elements: the name of the terminal itself and a non negative integer number. In this form, multiple windows can be opened at the same time, each with its corresponding number. This feature does not work in Windows platforms.

  Since this is a global graphics option, its position in the scene description does not matter. It can be also used as an argument of function `draw`.

  N.B. pdfcairo requires Gnuplot 4.3 or newer. `pdf` requires Gnuplot to be compiled with the option `--enable-pdf` and libpdf must be installed. The pdf library is available from: http://www.pdflib.com/en/download/pdflib-family/pdflib-lite/

  Examples:

      (%i1) /* screen terminal (default) */
            draw2d(explicit(x^2,x,-1,1))$
      (%i2) /* png file */
            draw2d(terminal  = 'png,
                   explicit(x^2,x,-1,1))$
      (%i3) /* jpg file */
            draw2d(terminal   = 'jpg,
                   dimensions = [300,300],
                   explicit(x^2,x,-1,1))$
      (%i4) /* eps file */
            draw2d(file_name = "myfile",
                   explicit(x^2,x,-1,1),
                   terminal  = 'eps)$
      (%i5) /* pdf file */
            draw2d(file_name = "mypdf",
                   dimensions = 100*[12.0,8.0],
                   explicit(x^2,x,-1,1),
                   terminal  = 'pdf)$
      (%i6) /* wxwidgets window */
            draw2d(explicit(x^2,x,-1,1),
                   terminal  = 'wxt)$

  Multiple windows.

      (%i1) draw2d(explicit(x^5,x,-2,2), terminal=[screen, 3])$
      (%i2) draw2d(explicit(x^2,x,-2,2), terminal=[screen, 0])$

  An animated gif file.

      (%i1) draw(
              delay     = 100,
              file_name = "zzz",
              terminal  = 'animated_gif,
              gr2d(explicit(x^2,x,-1,1)),
              gr2d(explicit(x^3,x,-1,1)),
              gr2d(explicit(x^4,x,-1,1)));
      End of animation sequence
      (%o1)          [gr2d(explicit), gr2d(explicit), gr2d(explicit)]

  Option `delay` is only active in animated gif's; it is ignored in any other case.

  Multipage output in eps format.

      (%i1) draw(
              file_name = "parabol",
              terminal  = multipage_eps,
              dimensions = 100*[10,10],
              gr2d(explicit(x^2,x,-1,1)),
              gr3d(explicit(x^2+y^2,x,-1,1,y,-1,1))) $

  See also `file_name`, `dimensions_draw` and `delay`.

  Categories: Package draw

Graphic option: title

  Default value: `""` (empty string)

  Option `title`, a string, is the main title for the scene. By default, no title is written.

  Since this is a global graphics option, its position in the scene description does not matter.

  Example:

      (%i1) draw2d(explicit(exp(u),u,-2,2),
                   title = "Exponential function")$

  Categories: Package draw

Graphic option: transform

  Default value: `none`

  If `transform` is `none`, the space is not transformed and graphic objects are drawn as defined. When a space transformation is desired, a list must be assigned to option `transform`. In case of a 2D scene, the list takes the form `[f1(x,y), f2(x,y), x, y]`. In case of a 3D scene, the list is of the form `[f1(x,y,z), f2(x,y,z), f3(x,y,z), x, y, z]`.

  The names of the variables defined in the lists may be different to those used in the definitions of the graphic objects.

  Examples:

  Rotation in 2D.

      (%i1) th : %pi / 4$
      (%i2) draw2d(
              color = "#e245f0",
              proportional_axes = 'xy,
              line_width = 8,
              triangle([3,2],[7,2],[5,5]),
              border     = false,
              fill_color = yellow,
              transform  = [cos(th)*x - sin(th)*y,
                            sin(th)*x + cos(th)*y, x, y],
              triangle([3,2],[7,2],[5,5]) )$

  Translation in 3D.

      (%i1) draw3d(
              color     = "#a02c00",
              explicit(20*exp(-x^2-y^2)-10,x,-3,3,y,-3,3),
              transform = [x+10,y+10,z+10,x,y,z],
              color     = blue,
              explicit(20*exp(-x^2-y^2)-10,x,-3,3,y,-3,3) )$

  Categories: Package draw

Graphic option: transparent

  Default value: `false`

  If `transparent` is `false`, interior regions of polygons are filled according to `fill_color`.

  This option affects the following graphic objects:

  - `gr2d`: `polygon`, `rectangle` and `ellipse`.

  Example:

      (%i1) draw2d(polygon([[3,2],[7,2],[5,5]]),
                   transparent = true,
                   color       = blue,
                   polygon([[5,2],[9,2],[7,5]]) )$

  Categories: Package draw

Graphic option: unit_vectors

  Default value: `false`

  If `unit_vectors` is `true`, vectors are plotted with module 1. This is useful for plotting vector fields. If `unit_vectors` is `false`, vectors are plotted with its original length.

  This option is relevant only for `vector` objects.

  Example:

      (%i1) draw2d(xrange      = [-1,6],
                   yrange      = [-1,6],
                   head_length = 0.1,
                   vector([0,0],[5,2]),
                   unit_vectors = true,
                   color        = red,
                   vector([0,3],[5,2]))$

  Categories: Package draw

Graphic option: user_preamble

  Default value: `""` (empty string)

  Expert Gnuplot users can make use of this option to fine tune Gnuplot's behaviour by writing settings to be sent before the `plot` or `splot` command.

  The value of this option must be a string or a list of strings (one per line).

  Since this is a global graphics option, its position in the scene description does not matter.

  Example:

  Tell Gnuplot to draw axes and grid on top of graphics objects,

      (%i1) draw2d(
              xaxis =true, xaxis_type=solid,
              yaxis =true, yaxis_type=solid,
              user_preamble="set grid front",
              region(x^2+y^2<1 ,x,-1.5,1.5,y,-1.5,1.5))$

  Tell gnuplot to draw all contour lines in black

      (%i1) wxdraw3d(
                contour=both,
                surface_hide=true,enhanced3d=true,wired_surface=true,
                contour_levels=10,
                user_preamble="set for [i=1:8] linetype i dashtype i linecolor 0",
                explicit(sin(x)*cos(y),x,1,10,y,1,10)
            );

  Categories: Package draw

Graphic option: view

  Default value: `[60,30]`

  A pair of angles, measured in degrees, indicating the view direction in a 3D scene. The first angle is the vertical rotation around the x axis, in the range [0, 360]. The second one is the horizontal rotation around the z axis, in the range [0, 360].

  If option `view` is given the value `map`, the view direction is set to be perpendicular to the xy-plane.

  Since this is a global graphics option, its position in the scene description does not matter.

  Example:

      (%i1) draw3d(view = [170, 50],
                   enhanced3d = true,
                   explicit(sin(x^2+y^2),x,-2,2,y,-2,2) )$

      (%i2) draw3d(view = map,
                   enhanced3d = true,
                   explicit(sin(x^2+y^2),x,-2,2,y,-2,2) )$

  Categories: Package draw

Graphic option: wired_surface

  Default value: `false`

  Indicates whether 3D surfaces in `enhanced3d` mode show the grid joinning the points or not.

  Since this is a global graphics option, its position in the scene description does not matter.

  Example:

      (%i1) draw3d(
              enhanced3d    = [sin(x),x,y],
              wired_surface = true,
              explicit(x^2+y^2,x,-1,1,y,-1,1)) $

  Categories: Package draw

Graphic option: x_voxel

  Default value: 10

  `x_voxel` is the number of voxels in the x direction to be used by the marching cubes algorithm implemented by the 3d `implicit` object. It is also used by graphic object `region`.

  Categories: Package draw

Graphic option: xaxis

  Default value: `false`

  If `xaxis` is `true`, the x axis is drawn.

  Since this is a global graphics option, its position in the scene description does not matter.

  Example:

      (%i1) draw2d(explicit(x^3,x,-1,1),
                   xaxis       = true,
                   xaxis_color = blue)$

  See also `xaxis_width`, `xaxis_type` and `xaxis_color`.

  Categories: Package draw

Graphic option: xaxis_color

  Default value: `"black"`

  `xaxis_color` specifies the color for the x axis. See `color` to know how colors are defined.

  Since this is a global graphics option, its position in the scene description does not matter.

  Example:

      (%i1) draw2d(explicit(x^3,x,-1,1),
                   xaxis       = true,
                   xaxis_color = red)$

  See also `xaxis`, `xaxis_width` and `xaxis_type`.

  Categories: Package draw

Graphic option: xaxis_secondary

  Default value: `false`

  If `xaxis_secondary` is `true`, function values can be plotted with respect to the second x axis, which will be drawn on top of the scene.

  Note that this is a local graphics option which only affects to 2d plots.

  Example:

      (%i1) draw2d(
               key   = "Bottom x-axis",
               explicit(x+1,x,1,2),
               color = red,
               key   = "Above x-axis",
               xtics_secondary = true,
               xaxis_secondary = true,
               explicit(x^2,x,-1,1)) $

  See also `xrange_secondary`, `xtics_secondary`, `xtics_rotate_secondary`, `xtics_axis_secondary` and `xaxis_secondary`.

  Package draw

Graphic option: xaxis_type

  Default value: `dots`

  `xaxis_type` indicates how the x axis is displayed; possible values are `solid` and `dots`

  Since this is a global graphics option, its position in the scene description does not matter.

  Example:

      (%i1) draw2d(explicit(x^3,x,-1,1),
                   xaxis       = true,
                   xaxis_type  = solid)$

  See also `xaxis`, `xaxis_width` and `xaxis_color`.

  Categories: Package draw

Graphic option: xaxis_width

  Default value: 1

  `xaxis_width` is the width of the x axis. Its value must be a positive number.

  Since this is a global graphics option, its position in the scene description does not matter.

  Example:

      (%i1) draw2d(explicit(x^3,x,-1,1),
                   xaxis       = true,
                   xaxis_width = 3)$

  See also `xaxis`, `xaxis_type` and `xaxis_color`.

  Categories: Package draw

Graphic option: xlabel

  Default value: `""`

  Option `xlabel`, a string, is the label for the x axis. By default, the axis is labeled with string `"x"`.

  Since this is a global graphics option, its position in the scene description does not matter.

  Example:

      (%i1) draw2d(xlabel = "Time",
                   explicit(exp(u),u,-2,2),
                   ylabel = "Population")$

  See also `xlabel_secondary`, `ylabel`, `ylabel_secondary` and `zlabel_draw`.

  Categories: Package draw

Graphic option: xlabel_secondary

  Default value: `""` (empty string)

  Option `xlabel_secondary`, a string, is the label for the secondary x axis. By default, no label is written.

  Since this is a global graphics option, its position in the scene description does not matter.

  Example:

      (%i1) draw2d(
               xaxis_secondary=true,yaxis_secondary=true,
               xtics_secondary=true,ytics_secondary=true,
               xlabel_secondary="t[s]",
               ylabel_secondary="U[V]",
               explicit(sin(t),t,0,10) )$

  See also `xlabel_draw`, `ylabel_draw`, `ylabel_secondary` and `zlabel_draw`.

  Categories: Package draw

Graphic option: xrange

  Default value: `auto`

  If `xrange` is `auto`, the range for the x coordinate is computed automatically.

  If the user wants a specific interval for x, it must be given as a Maxima list, as in `xrange=[-2, 3]`.

  Since this is a global graphics option, its position in the scene description does not matter.

  Example:

      (%i1) draw2d(xrange = [-3,5],
                   explicit(x^2,x,-1,1))$

  See also `yrange` and `zrange`.

  Categories: Package draw

Graphic option: xrange_secondary

  Default value: `auto`

  If `xrange_secondary` is `auto`, the range for the second x axis is computed automatically.

  If the user wants a specific interval for the second x axis, it must be given as a Maxima list, as in `xrange_secondary=[-2, 3]`.

  Since this is a global graphics option, its position in the scene description does not matter.

  See also `xrange`, `yrange`, `zrange` and `yrange_secondary`.

  Categories: Package draw

Graphic option: xtics

  Default value: `true`

  This graphic option controls the way tic marks are drawn on the x axis.

  - When option `xtics` is bounded to symbol true, tic marks are drawn automatically.
  - When option `xtics` is bounded to symbol false, tic marks are not drawn.
  - When option `xtics` is bounded to a positive number, this is the distance between two consecutive tic marks.
  - When option `xtics` is bounded to a list of length three of the form `[start,incr,end]`, tic marks are plotted from `start` to `end` at intervals of length `incr`.
  - When option `xtics` is bounded to a set of numbers of the form `{n1, n2, ...}`, tic marks are plotted at values `n1`, `n2`, ...
  - When option `xtics` is bounded to a set of pairs of the form `{["label1", n1], ["label2", n2], ...}`, tic marks corresponding to values `n1`, `n2`, ... are labeled with `"label1"`, `"label2"`, ..., respectively.

  Since this is a global graphics option, its position in the scene description does not matter.

  Examples:

  Disable tics.

      (%i1) draw2d(xtics = 'false,
                   explicit(x^3,x,-1,1)  )$

  Tics every 1/4 units.

      (%i1) draw2d(xtics = 1/4,
                   explicit(x^3,x,-1,1)  )$

  Tics from -3/4 to 3/4 in steps of 1/8.

      (%i1) draw2d(xtics = [-3/4,1/8,3/4],
                   explicit(x^3,x,-1,1)  )$

  Tics at points -1/2, -1/4 and 3/4.

      (%i1) draw2d(xtics = {-1/2,-1/4,3/4},
                   explicit(x^3,x,-1,1)  )$

  Labeled tics.

      (%i1) draw2d(xtics = {["High",0.75],["Medium",0],["Low",-0.75]},
                   explicit(x^3,x,-1,1)  )$

  See also `ytics`, and `ztics`.

  Categories: Package draw

Graphic option: xtics_axis

  Default value: `false`

  If `xtics_axis` is `true`, tic marks and their labels are plotted just along the x axis, if it is `false` tics are plotted on the border.

  Since this is a global graphics option, its position in the scene description does not matter.

  Categories: Package draw

Graphic option: xtics_rotate

  Default value: `false`

  If `xtics_rotate` is `true`, tic marks on the x axis are rotated 90 degrees.

  Since this is a global graphics option, its position in the scene description does not matter.

  Categories: Package draw

Graphic option: xtics_rotate_secondary

  Default value: `false`

  If `xtics_rotate_secondary` is `true`, tic marks on the secondary x axis are rotated 90 degrees.

  Since this is a global graphics option, its position in the scene description does not matter.

  Categories: Package draw

Graphic option: xtics_secondary

  Default value: `auto`

  This graphic option controls the way tic marks are drawn on the second x axis.

  See `xtics` for a complete description.

  Categories: Package draw

Graphic option: xtics_secondary_axis

  Default value: `false`

  If `xtics_secondary_axis` is `true`, tic marks and their labels are plotted just along the secondary x axis, if it is `false` tics are plotted on the border.

  Since this is a global graphics option, its position in the scene description does not matter.

  Categories: Package draw

Graphic option: xu_grid

  Default value: 30

  `xu_grid` is the number of coordinates of the first variable (`x` in explicit and `u` in parametric 3d surfaces) to build the grid of sample points.

  This option affects the following graphic objects:

  - `gr3d`: `explicit` and `parametric_surface`.

  Example:

      (%i1) draw3d(xu_grid = 10,
                   yv_grid = 50,
                   explicit(x^2+y^2,x,-3,3,y,-3,3) )$

  See also `yv_grid`.

  Categories: Package draw

Graphic option: xy_file

  Default value: `""` (empty string)

  `xy_file` is the name of the file where the coordinates will be saved after clicking with the mouse button and hitting the 'x' key. By default, no coordinates are saved.

  Since this is a global graphics option, its position in the scene description does not matter.

  Categories: Package draw

Graphic option: xyplane

  Default value: `false`

  Allocates the xy-plane in 3D scenes. When `xyplane` is `false`, the xy-plane is placed automatically; when it is a real number, the xy-plane intersects the z-axis at this level. This option has no effect in 2D scenes.

  Since this is a global graphics option, its position in the scene description does not matter.

  Example:

      (%i1) draw3d(xyplane = %e-2,
                   explicit(x^2+y^2,x,-1,1,y,-1,1))$

  Categories: Package draw

Graphic option: y_voxel

  Default value: 10

  `y_voxel` is the number of voxels in the y direction to be used by the marching cubes algorithm implemented by the 3d `implicit` object. It is also used by graphic object `region`.

  Categories: Package draw

Graphic option: yaxis

  Default value: `false`

  If `yaxis` is `true`, the y axis is drawn.

  Since this is a global graphics option, its position in the scene description does not matter.

  Example:

      (%i1) draw2d(explicit(x^3,x,-1,1),
                   yaxis       = true,
                   yaxis_color = blue)$

  See also `yaxis_width`, `yaxis_type` and `yaxis_color`.

  Categories: Package draw

Graphic option: yaxis_color

  Default value: `"black"`

  `yaxis_color` specifies the color for the y axis. See `color` to know how colors are defined.

  Since this is a global graphics option, its position in the scene description does not matter.

  Example:

      (%i1) draw2d(explicit(x^3,x,-1,1),
                   yaxis       = true,
                   yaxis_color = red)$

  See also `yaxis`, `yaxis_width` and `yaxis_type`.

  Categories: Package draw

Graphic option: yaxis_secondary

  Default value: `false`

  If `yaxis_secondary` is `true`, function values can be plotted with respect to the second y axis, which will be drawn on the right side of the scene.

  Note that this is a local graphics option which only affects to 2d plots.

  Example:

      (%i1) draw2d(
               explicit(sin(x),x,0,10),
               yaxis_secondary = true,
               ytics_secondary = true,
               color = blue,
               explicit(100*sin(x+0.1)+2,x,0,10));

  See also `yrange_secondary`, `ytics_secondary`, `ytics_rotate_secondary` and `ytics_axis_secondary`

  Categories: Package draw

Graphic option: yaxis_type

  Default value: `dots`

  `yaxis_type` indicates how the y axis is displayed; possible values are `solid` and `dots`.

  Since this is a global graphics option, its position in the scene description does not matter.

  Example:

      (%i1) draw2d(explicit(x^3,x,-1,1),
                   yaxis       = true,
                   yaxis_type  = solid)$

  See also `yaxis`, `yaxis_width` and `yaxis_color`.

  Categories: Package draw

Graphic option: yaxis_width

  Default value: 1

  `yaxis_width` is the width of the y axis. Its value must be a positive number.

  Since this is a global graphics option, its position in the scene description does not matter.

  Example:

      (%i1) draw2d(explicit(x^3,x,-1,1),
                   yaxis       = true,
                   yaxis_width = 3)$

  See also `yaxis`, `yaxis_type` and `yaxis_color`.

  Categories: Package draw

Graphic option: ylabel

  Default value: `""`

  Option `ylabel`, a string, is the label for the y axis. By default, the axis is labeled with string `"y"`.

  Since this is a global graphics option, its position in the scene description does not matter.

  Example:

      (%i1) draw2d(xlabel = "Time",
                   ylabel = "Population",
                   explicit(exp(u),u,-2,2) )$

  See also `xlabel_draw`, `xlabel_secondary`, `ylabel_secondary`, and `zlabel_draw`.

  Categories: Package draw

Graphic option: ylabel_secondary

  Default value: `""` (empty string)

  Option `ylabel_secondary`, a string, is the label for the secondary y axis. By default, no label is written.

  Since this is a global graphics option, its position in the scene description does not matter.

  Example:

      (%i1) draw2d(
              key_pos=bottom_right,
              key="current",
              xlabel="t[s]",
              ylabel="I[A]",ylabel_secondary="P[W]",
              explicit(sin(t),t,0,10),
              yaxis_secondary=true,
              ytics_secondary=true,
              color=red,key="Power",
              explicit((sin(t))^2,t,0,10)
          )$

  See also `xlabel_draw`, `xlabel_secondary`, `ylabel_draw` and `zlabel_draw`.

  Categories: Package draw

Graphic option: yrange

  Default value: `auto`

  If `yrange` is `auto`, the range for the y coordinate is computed automatically.

  If the user wants a specific interval for y, it must be given as a Maxima list, as in `yrange=[-2, 3]`.

  Since this is a global graphics option, its position in the scene description does not matter.

  Example:

      (%i1) draw2d(yrange = [-2,3],
                   explicit(x^2,x,-1,1),
                   xrange = [-3,3])$

  See also `xrange`, `yrange_secondary` and `zrange`.

  Categories: Package draw

Graphic option: yrange_secondary

  Default value: `auto`

  If `yrange_secondary` is `auto`, the range for the second y axis is computed automatically.

  If the user wants a specific interval for the second y axis, it must be given as a Maxima list, as in `yrange_secondary=[-2, 3]`.

  Since this is a global graphics option, its position in the scene description does not matter.

  Example:

      (%i1) draw2d(
               explicit(sin(x),x,0,10),
               yaxis_secondary = true,
               ytics_secondary = true,
               yrange = [-3, 3],
               yrange_secondary = [-20, 20],
               color = blue,
               explicit(100*sin(x+0.1)+2,x,0,10)) $

  See also `xrange`, `yrange` and `zrange`.

  Categories: Package draw

Graphic option: ytics

  Default value: `true`

  This graphic option controls the way tic marks are drawn on the y axis.

  See `xtics` for a complete description.

  Categories: Package draw

Graphic option: ytics_axis

  Default value: `false`

  If `ytics_axis` is `true`, tic marks and their labels are plotted just along the y axis, if it is `false` tics are plotted on the border.

  Since this is a global graphics option, its position in the scene description does not matter.

  Categories: Package draw

Graphic option: ytics_rotate

  Default value: `false`

  If `ytics_rotate` is `true`, tic marks on the y axis are rotated 90 degrees.

  Since this is a global graphics option, its position in the scene description does not matter.

  Categories: Package draw

Graphic option: ytics_rotate_secondary

  Default value: `false`

  If `ytics_rotate_secondary` is `true`, tic marks on the secondary y axis are rotated 90 degrees.

  Since this is a global graphics option, its position in the scene description does not matter.

  Categories: Package draw

Graphic option: ytics_secondary

  Default value: `auto`

  This graphic option controls the way tic marks are drawn on the second y axis.

  See `xtics` for a complete description.

  Categories: Package draw

Graphic option: ytics_secondary_axis

  Default value: `false`

  If `ytics_secondary_axis` is `true`, tic marks and their labels are plotted just along the secondary y axis, if it is `false` tics are plotted on the border.

  Since this is a global graphics option, its position in the scene description does not matter.

  Categories: Package draw

Graphic option: yv_grid

  Default value: 30

  `yv_grid` is the number of coordinates of the second variable (`y` in explicit and `v` in parametric 3d surfaces) to build the grid of sample points.

  This option affects the following graphic objects:

  - `gr3d`: `explicit` and `parametric_surface`.

  Example:

      (%i1) draw3d(xu_grid = 10,
                   yv_grid = 50,
                   explicit(x^2+y^2,x,-3,3,y,-3,3) )$

  See also `xu_grid`.

  Categories: Package draw

Graphic option: z_voxel

  Default value: 10

  `z_voxel` is the number of voxels in the z direction to be used by the marching cubes algorithm implemented by the 3d `implicit` object.

  Categories: Package draw

Graphic option: zaxis

  Default value: `false`

  If `zaxis` is `true`, the z axis is drawn in 3D plots. This option has no effect in 2D scenes.

  Since this is a global graphics option, its position in the scene description does not matter.

  Example:

      (%i1) draw3d(explicit(x^2+y^2,x,-1,1,y,-1,1),
                   zaxis       = true,
                   zaxis_type  = solid,
                   zaxis_color = blue)$

  See also `zaxis_width`, `zaxis_type` and `zaxis_color`.

  Categories: Package draw

Graphic option: zaxis_color

  Default value: `"black"`

  `zaxis_color` specifies the color for the z axis. See `color` to know how colors are defined. This option has no effect in 2D scenes.

  Since this is a global graphics option, its position in the scene description does not matter.

  Example:

      (%i1) draw3d(explicit(x^2+y^2,x,-1,1,y,-1,1),
                   zaxis       = true,
                   zaxis_type  = solid,
                   zaxis_color = red)$

  See also `zaxis`, `zaxis_width` and `zaxis_type`.

  Categories: Package draw

Graphic option: zaxis_type

  Default value: `dots`

  `zaxis_type` indicates how the z axis is displayed; possible values are `solid` and `dots`. This option has no effect in 2D scenes.

  Since this is a global graphics option, its position in the scene description does not matter.

  Example:

      (%i1) draw3d(explicit(x^2+y^2,x,-1,1,y,-1,1),
                   zaxis       = true,
                   zaxis_type  = solid)$

  See also `zaxis`, `zaxis_width` and `zaxis_color`.

  Categories: Package draw

Graphic option: zaxis_width

  Default value: 1

  `zaxis_width` is the width of the z axis. Its value must be a positive number. This option has no effect in 2D scenes.

  Since this is a global graphics option, its position in the scene description does not matter.

  Example:

      (%i1) draw3d(explicit(x^2+y^2,x,-1,1,y,-1,1),
                   zaxis       = true,
                   zaxis_type  = solid,
                   zaxis_width = 3)$

  See also `zaxis`, `zaxis_type` and `zaxis_color`.

  Categories: Package draw

Graphic option: zlabel

  Default value: `""`

  Option `zlabel`, a string, is the label for the z axis. By default, the axis is labeled with string `"z"`.

  Since this is a global graphics option, its position in the scene description does not matter.

  Example:

      (%i1) draw3d(zlabel = "Z variable",
                   ylabel = "Y variable",
                   explicit(sin(x^2+y^2),x,-2,2,y,-2,2),
                   xlabel = "X variable" )$

  See also `xlabel_draw`, `ylabel_draw`, and `zlabel_rotate`.

  Categories: Package draw

Graphic option: zlabel_rotate

  Default value: `"auto"`

  This graphics option allows to choose if the z axis label of 3d plots is drawn horizontal (`false`), vertical (`true`) or if maxima automatically chooses an orientation based on the length of the label (`auto`).

  Since this is a global graphics option, its position in the scene description does not matter.

  Example:

      (%i1) draw3d(
                explicit(sin(x)*sin(y),x,0,10,y,0,10),
                zlabel_rotate=false
            )$

  See also `zlabel_draw`.

  Categories: Package draw

Graphic option: zrange

  Default value: `auto`

  If `zrange` is `auto`, the range for the z coordinate is computed automatically.

  If the user wants a specific interval for z, it must be given as a Maxima list, as in `zrange=[-2, 3]`.

  Since this is a global graphics option, its position in the scene description does not matter.

  Example:

      (%i1) draw3d(yrange = [-3,3],
                   zrange = [-2,5],
                   explicit(x^2+y^2,x,-1,1,y,-1,1),
                   xrange = [-3,3])$

  See also `xrange` and `yrange`.

  Categories: Package draw

Graphic option: ztics

  Default value: `true`

  This graphic option controls the way tic marks are drawn on the z axis.

  See `xtics` for a complete description.

  Categories: Package draw

Graphic option: ztics_axis

  Default value: `false`

  If `ztics_axis` is `true`, tic marks and their labels are plotted just along the z axis, if it is `false` tics are plotted on the border.

  Since this is a global graphics option, its position in the scene description does not matter.

  Categories: Package draw

Graphic option: ztics_rotate

  Default value: `false`

  If `ztics_rotate` is `true`, tic marks on the z axis are rotated 90 degrees.

  Since this is a global graphics option, its position in the scene description does not matter.

  Categories: Package draw

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### 52.2.4 Graphics objects

Graphic object: bars ([x1,h1,w1], [x2,h2,w2, ...])

  Draws vertical bars in 2D.

  2D

  `bars ([x1,h1,w1], [x2,h2,w2, ...])` draws bars centered at values x1, x2, ... with heights h1, h2, ... and widths w1, w2, ...

  This object is affected by the following graphic options: `key`, `fill_color`, `fill_density` and `line_width`.

  Example:

      (%i1) draw2d(
             key          = "Group A",
             fill_color   = blue,
             fill_density = 0.2,
             bars([0.8,5,0.4],[1.8,7,0.4],[2.8,-4,0.4]),
             key          = "Group B",
             fill_color   = red,
             fill_density = 0.6,
             line_width   = 4,
             bars([1.2,4,0.4],[2.2,-2,0.4],[3.2,5,0.4]),
             xaxis = true);

  Categories: Package draw

Graphic object: cylindrical (radius, z, minz, maxz, azi, minazi, maxazi)

  Draws 3D functions defined in cylindrical coordinates.

  3D

  `cylindrical(radius, z, minz, maxz, azi, minazi, maxazi)` plots the function `radius(z, azi)` defined in cylindrical coordinates, with variable z taking values from minz to maxz and azimuth azi taking values from minazi to maxazi.

  This object is affected by the following graphic options: `xu_grid`, `yv_grid`, `line_type`, `key`, `wired_surface`, `enhanced3d` and `color`

  Example:

      (%i1) draw3d(cylindrical(1,z,-2,2,az,0,2*%pi))$

  Categories: Package draw

Graphic object: elevation_grid (mat,x0,y0,width,height)

  Draws matrix mat in 3D space. z values are taken from mat, the abscissas range from x0 to x0 + width and ordinates from y0 to y0 + height. Element a(1,1) is projected on point (x0,y0+height), a(1,n) on (x0+width,y0+height), a(m,1) on (x0,y0), and a(m,n) on (x0+width,y0).

  This object is affected by the following graphic options: `line_type`,, `line_width` `key`, `wired_surface`, `enhanced3d` and `color`

  In older versions of Maxima, `elevation_grid` was called `mesh`. See also `mesh`.

  Example:

      (%i1) m: apply(
                  matrix,
                  makelist(makelist(random(10.0),k,1,30),i,1,20)) $
      (%i2) draw3d(
               color = blue,
               elevation_grid(m,0,0,3,2),
               xlabel = "x",
               ylabel = "y",
               surface_hide = true);

  Categories: Package draw

Graphic object: ellipse (xc, yc, a, b, ang1, ang2)

  Draws ellipses and circles in 2D.

  2D

  `ellipse (xc, yc, a, b, ang1, ang2)` plots an ellipse centered at `[xc, yc]` with horizontal and vertical semi axis a and b, respectively, starting at angle ang1 with an amplitude equal to angle ang2.

  This object is affected by the following graphic options: `nticks`, `transparent`, `fill_color`, `border`, `line_width`, `line_type`, `key` and `color`

  Example:

      (%i1) draw2d(transparent = false,
                   fill_color  = red,
                   color       = gray30,
                   transparent = false,
                   line_width  = 5,
                   ellipse(0,6,3,2,270,-270),
                   /* center (x,y), a, b, start & end in degrees */
                   transparent = true,
                   color       = blue,
                   line_width  = 3,
                   ellipse(2.5,6,2,3,30,-90),
                   xrange      = [-3,6],
                   yrange      = [2,9] )$

  Categories: Package draw

Graphic object: errors ([x1, x2, …], [y1, y2, …])

  Draws points with error bars, horizontally, vertically or both, depending on the value of option `error_type`.

  2D

  If `error_type = x`, arguments to `errors` must be of the form `[x, y, xdelta]` or `[x, y, xlow, xhigh]`. If `error_type = y`, arguments must be of the form `[x, y, ydelta]` or `[x, y, ylow, yhigh]`. If `error_type = xy` or `error_type = boxes`, arguments to `errors` must be of the form `[x, y, xdelta, ydelta]` or `[x, y, xlow, xhigh, ylow, yhigh]`.

  See also `error_type`.

  This object is affected by the following graphic options: `error_type`, `points_joined`, `line_width`, `key`, `line_type`, `color` `fill_density`, `xaxis_secondary` and `yaxis_secondary`.

  Option `fill_density` is only relevant when `error_type=boxes`.

  Examples:

  Horizontal error bars.

      (%i1) draw2d(
              error_type = y,
              errors([[1,2,1], [3,5,3], [10,3,1], [17,6,2]]))$

  Vertical and horizontal error bars.

      (%i1) draw2d(
              error_type = 'xy,
              points_joined = true,
              color = blue,
              errors([[1,2,1,2], [3,5,2,1], [10,3,1,1], [17,6,1/2,2]]));

  Categories: Package draw

Graphic object: explicit
`explicit` (fcn,var,minval,maxval)
`explicit` (fcn,var1,minval1,maxval1,var2,minval2,maxval2)

  Draws explicit functions in 2D and 3D.

  2D

  `explicit(fcn,var,minval,maxval)` plots explicit function fcn, with variable var taking values from minval to maxval.

  This object is affected by the following graphic options: `nticks`, `adapt_depth`, `draw_realpart`, `line_width`, `line_type`, `key`, `filled_func`, `fill_color` and `color`

  Example:

      (%i1) draw2d(line_width = 3,
                   color      = blue,
                   explicit(x^2,x,-3,3) )$

      (%i2) draw2d(fill_color  = brown,
                   filled_func = true,
                   explicit(x^2,x,-3,3) )$

  3D

  `explicit(fcn, var1, minval1, maxval1, var2, minval2, maxval2)` plots the explicit function fcn, with variable var1 taking values from minval1 to maxval1 and variable var2 taking values from minval2 to maxval2.

  This object is affected by the following graphic options: `draw_realpart`, `xu_grid`, `yv_grid`, `line_type`, `line_width`, `key`, `wired_surface`, `enhanced3d` and `color`.

  Example:

      (%i1) draw3d(key   = "Gauss",
                   color = "#a02c00",
                   explicit(20*exp(-x^2-y^2)-10,x,-3,3,y,-3,3),
                   yv_grid     = 10,
                   color = blue,
                   key   = "Plane",
                   explicit(x+y,x,-5,5,y,-5,5),
                   surface_hide = true)$

  See also `filled_func` for filled functions.

  Categories: Package draw

Graphic object: image (im,x0,y0,width,height)

  Renders images in 2D.

  2D

  `image (im,x0,y0,width,height)` plots image im in the rectangular region from vertex `(x0,y0)` to `(x0+width,y0+height)` on the real plane. Argument im must be a matrix of real numbers, a matrix of vectors of length three or a picture object.

  If im is a matrix of real numbers or a levels picture object, pixel values are interpreted according to graphic option `palette`, which is a vector of length three with components ranging from -36 to +36; each value is an index for a formula mapping the levels onto red, green and blue colors, respectively:

       0: 0               1: 0.5           2: 1
       3: x               4: x^2           5: x^3
       6: x^4             7: sqrt(x)       8: sqrt(sqrt(x))
       9: sin(90x)       10: cos(90x)     11: |x-0.5|
      12: (2x-1)^2       13: sin(180x)    14: |cos(180x)|
      15: sin(360x)      16: cos(360x)    17: |sin(360x)|
      18: |cos(360x)|    19: |sin(720x)|  20: |cos(720x)|
      21: 3x             22: 3x-1         23: 3x-2
      24: |3x-1|         25: |3x-2|       26: (3x-1)/2
      27: (3x-2)/2       28: |(3x-1)/2|   29: |(3x-2)/2|
      30: x/0.32-0.78125                  31: 2*x-0.84
      32: 4x;1;-2x+1.84;x/0.08-11.5
      33: |2*x - 0.5|    34: 2*x          35: 2*x - 0.5
      36: 2*x - 1

  negative numbers mean negative colour component.

  `palette = gray` and `palette = color` are short cuts for `palette = [3,3,3]` and `palette = [7,5,15]`, respectively.

  If im is a matrix of vectors of length three or an rgb picture object, they are interpreted as red, green and blue color components.

  Examples:

  If im is a matrix of real numbers, pixel values are interpreted according to graphic option `palette`.

      (%i1) im: apply(
                 'matrix,
                  makelist(makelist(random(200),i,1,30),i,1,30))$
      (%i2) /* palette = color, default */
            draw2d(image(im,0,0,30,30))$

      (%i3) draw2d(palette = gray, image(im,0,0,30,30))$

      (%i4) draw2d(palette = [15,20,-4],
                   colorbox=false,
                   image(im,0,0,30,30))$

  See also `colorbox`.

  If im is a matrix of vectors of length three, they are interpreted as red, green and blue color components.

      (%i1) im: apply(
                  'matrix,
                   makelist(
                     makelist([random(300),
                               random(300),
                               random(300)],i,1,30),i,1,30))$
      (%i2) draw2d(image(im,0,0,30,30))$

  Package `draw` automatically loads package `picture`. In this example, a level picture object is built by hand and then rendered.

      (%i1) im: make_level_picture([45,87,2,134,204,16],3,2);
      (%o1)       picture(level, 3, 2, {Array:  #(45 87 2 134 204 16)})
      (%i2) /* default color palette */
            draw2d(image(im,0,0,30,30))$

      (%i3) /* gray palette */
            draw2d(palette = gray,
                   image(im,0,0,30,30))$

  An xpm file is read and then rendered.

      (%i1) load(draw)$
      (%i2) im: read_xpm("myfile.xpm")$
      (%i3) draw2d(image(im,0,0,10,7))$

  See also `make_level_picture`, `make_rgb_picture` and `read_xpm`.

  http://www.telefonica.net/web2/biomates/maxima/gpdraw/image
  contains more elaborated examples.

  Categories: Package draw

Graphic object: implicit
`implicit` (fcn,x,xmin,xmax,y,ymin,ymax)
`implicit` (fcn,x,xmin,xmax,y,ymin,ymax,z,zmin,zmax)

  Draws implicit functions in 2D and 3D.

  2D

  `implicit(fcn,x,xmin,xmax,y,ymin,ymax)` plots the implicit function defined by fcn, with variable x taking values from xmin to xmax, and variable y taking values from ymin to ymax.

  This object is affected by the following graphic options: `ip_grid`, `ip_grid_in`, `line_width`, `line_type`, `key` and `color`.

  Example:

      (%i1) draw2d(grid      = true,
                   line_type = solid,
                   key       = "y^2=x^3-2*x+1",
                   implicit(y^2=x^3-2*x+1, x, -4,4, y, -4,4),
                   line_type = dots,
                   key       = "x^3+y^3 = 3*x*y^2-x-1",
                   implicit(x^3+y^3 = 3*x*y^2-x-1, x,-4,4, y,-4,4),
                   title     = "Two implicit functions" )$

  3D

  `implicit (fcn,x,xmin,xmax, y,ymin,ymax, z,zmin,zmax)` plots the implicit surface defined by fcn, with variable x taking values from xmin to xmax, variable y taking values from ymin to ymax and variable z taking values from zmin to zmax. This object implements the marching cubes algorithm.

  This object is affected by the following graphic options: `x_voxel`, `y_voxel`, `z_voxel`, `line_width`, `line_type`, `key`, `wired_surface`, `enhanced3d` and `color`.

  Example:

      (%i1) draw3d(
              color=blue,
              implicit((x^2+y^2+z^2-1)*(x^2+(y-1.5)^2+z^2-0.5)=0.015,
                       x,-1,1,y,-1.2,2.3,z,-1,1),
              surface_hide=true);

  Categories: Package draw

Graphic object: label
`label` ([string,x,y],...)
`label` ([string,x,y,z],...)

  Writes labels in 2D and 3D.

  Colored labels work only with Gnuplot 4.3 and up.

  This object is affected by the following graphic options: `label_alignment`, `label_orientation` and `color`.

  2D

  `label([string,x,y])` writes the string at point `[x,y]`.

  Example:

      (%i1) draw2d(yrange = [0.1,1.4],
                   color = red,
                   label(["Label in red",0,0.3]),
                   color = "#0000ff",
                   label(["Label in blue",0,0.6]),
                   color = light_blue,
                   label(["Label in light-blue",0,0.9],
                         ["Another light-blue",0,1.2])  )$

  3D

  `label([string,x,y,z])` writes the string at point `[x,y,z]`.

  Example:

      (%i1) draw3d(explicit(exp(sin(x)+cos(x^2)),x,-3,3,y,-3,3),
                   color = red,
                   label(["UP 1",-2,0,3], ["UP 2",1.5,0,4]),
                   color = blue,
                   label(["DOWN 1",2,0,-3]) )$

  Categories: Package draw

Graphic object: mesh (row_1,row_2,...)

  Draws a quadrangular mesh in 3D.

  3D

  Argument row_i is a list of n 3D points of the form `[[x_i1,y_i1,z_i1], ...,[x_in,y_in,z_in]]`, and all rows are of equal length. All these points define an arbitrary surface in 3D and in some sense it's a generalization of the `elevation_grid` object.

  This object is affected by the following graphic options: `line_type`, `line_width`, `color`, `key`, `wired_surface`, `enhanced3d` and `transform`.

  Examples:

  A simple example.

      (%i1) draw3d(
               mesh([[1,1,3],   [7,3,1],[12,-2,4],[15,0,5]],
                    [[2,7,8],   [4,3,1],[10,5,8], [12,7,1]],
                    [[-2,11,10],[6,9,5],[6,15,1], [20,15,2]])) $

  Plotting a triangle in 3D.

      (%i1) draw3d(
              line_width = 2,
              mesh([[1,0,0],[0,1,0]],
                   [[0,0,1],[0,0,1]])) $

  Two quadrilaterals.

      (%i1) draw3d(
              surface_hide = true,
              line_width   = 3,
              color = red,
              mesh([[0,0,0], [0,1,0]],
                   [[2,0,2], [2,2,2]]),
              color = blue,
              mesh([[0,0,2], [0,1,2]],
                   [[2,0,4], [2,2,4]])) $

  Categories: Package draw

Graphic object: parametric
`parametric` (xfun,yfun,par,parmin,parmax)
`parametric` (xfun,yfun,zfun,par,parmin,parmax)

  Draws parametric functions in 2D and 3D.

  This object is affected by the following graphic options: `nticks`, `line_width`, `line_type`, `key`, `color` and `enhanced3d`.

  2D

  The command `parametric(xfun, yfun, par, parmin, parmax)` plots the parametric function `[xfun, yfun]`, with parameter par taking values from parmin to parmax.

  Example:

      (%i1) draw2d(explicit(exp(x),x,-1,3),
                   color = red,
                   key   = "This is the parametric one!!",
                   parametric(2*cos(rrr),rrr^2,rrr,0,2*%pi))$

  3D

  `parametric(xfun, yfun, zfun, par, parmin, parmax)` plots the parametric curve `[xfun, yfun, zfun]`, with parameter par taking values from parmin to parmax.

  Example:

      (%i1) draw3d(explicit(exp(sin(x)+cos(x^2)),x,-3,3,y,-3,3),
                   color = royalblue,
                   parametric(cos(5*u)^2,sin(7*u),u-2,u,0,2),
                   color      = turquoise,
                   line_width = 2,
                   parametric(t^2,sin(t),2+t,t,0,2),
                   surface_hide = true,
                   title = "Surface & curves" )$

  Categories: Package draw

Graphic object: parametric_surface (xfun, yfun, zfun, par1, par1min, par1max, par2, par2min, par2max)

  Draws parametric surfaces in 3D.

  3D

  The command `parametric_surface(xfun, yfun, zfun, par1, par1min, par1max, par2, par2min, par2max)` plots the parametric surface `[xfun, yfun, zfun]`, with parameter par1 taking values from par1min to par1max and parameter par2 taking values from par2min to par2max.

  This object is affected by the following graphic options: `draw_realpart`, `xu_grid`, `yv_grid`, `line_type`, `line_width`, `key`, `wired_surface`, `enhanced3d` and `color`.

  Example:

      (%i1) draw3d(title          = "Sea shell",
                   xu_grid        = 100,
                   yv_grid        = 25,
                   view           = [100,20],
                   surface_hide   = true,
                   parametric_surface(0.5*u*cos(u)*(cos(v)+1),
                                 0.5*u*sin(u)*(cos(v)+1),
                                 u*sin(v) - ((u+3)/8*%pi)^2 - 20,
                                 u, 0, 13*%pi, v, -%pi, %pi) )$

  Categories: Package draw

Graphic object: points
`points` ([[x1,y1], [x2,y2],...])
`points` ([x1,x2,...], [y1,y2,...])
`points` ([y1,y2,...])
`points` ([[x1,y1,z1], [x2,y2,z2],...])
`points` ([x1,x2,...], [y1,y2,...], [z1,z2,...])
`points` (matrix)
`points` (1d_y_array)
`points` (1d_x_array, 1d_y_array)
`points` (1d_x_array, 1d_y_array, 1d_z_array)
`points` (2d_xy_array)
`points` (2d_xyz_array)

  Draws points in 2D and 3D.

  This object is affected by the following graphic options: `point_size`, `point_type`, `points_joined`, `line_width`, `key`, `line_type` and `color`. In 3D mode, it is also affected by `enhanced3d`

  2D

  `points ([[x1,y1], [x2,y2],...])` or `points ([x1,x2,...], [y1,y2,...])` plots points `[x1,y1]`, `[x2,y2]`, etc. If abscissas are not given, they are set to consecutive positive integers, so that `points ([y1,y2,...])` draws points `[1,y1]`, `[2,y2]`, etc. If matrix is a two-column or two-row matrix, `points (matrix)` draws the associated points. If matrix is a one-column or one-row matrix, abscissas are assigned automatically.

  If 1d_y_array is a 1D lisp array of numbers, `points (1d_y_array)` plots them setting abscissas to consecutive positive integers. `points (1d_x_array, 1d_y_array)` plots points with their coordinates taken from the two arrays passed as arguments. If 2d_xy_array is a 2D array with two columns, or with two rows, `points (2d_xy_array)` plots the corresponding points on the plane.

  Examples:

  Two types of arguments for `points`, a list of pairs and two lists of separate coordinates.

      (%i1) draw2d(
              key = "Small points",
              points(makelist([random(20),random(50)],k,1,10)),
              point_type    = circle,
              point_size    = 3,
              points_joined = true,
              key           = "Great points",
              points(makelist(k,k,1,20),makelist(random(30),k,1,20)),
              point_type    = filled_down_triangle,
              key           = "Automatic abscissas",
              color         = red,
              points([2,12,8]))$

  Drawing impulses.

      (%i1) draw2d(
              points_joined = impulses,
              line_width    = 2,
              color         = red,
              points(makelist([random(20),random(50)],k,1,10)))$

  Array with ordinates.

      (%i1) a: make_array (flonum, 100) $
      (%i2) for i:0 thru 99 do a[i]: random(1.0) $
      (%i3) draw2d(points(a)) $

  Two arrays with separate coordinates.

      (%i1) x: make_array (flonum, 100) $
      (%i2) y: make_array (fixnum, 100) $
      (%i3) for i:0 thru 99 do (
              x[i]: float(i/100),
              y[i]: random(10) ) $
      (%i4) draw2d(points(x, y)) $

  A two-column 2D array.

      (%i1) xy: make_array(flonum, 100, 2) $
      (%i2) for i:0 thru 99 do (
              xy[i, 0]: float(i/100),
              xy[i, 1]: random(10) ) $
      (%i3) draw2d(points(xy)) $

  Drawing an array filled with function `read_array`.

      (%i1) a: make_array(flonum,100) $
      (%i2) read_array (file_search ("pidigits.data"), a) $
      (%i3) draw2d(points(a)) $

  3D

  `points([[x1, y1, z1], [x2, y2, z2], ...])` or `points([x1, x2, ...], [y1, y2, ...], [z1, z2,...])` plots points `[x1, y1, z1]`, `[x2, y2, z2]`, etc. If matrix is a three-column or three-row matrix, `points (matrix)` draws the associated points.

  When arguments are lisp arrays, `points (1d_x_array, 1d_y_array, 1d_z_array)` takes coordinates from the three 1D arrays. If 2d_xyz_array is a 2D array with three columns, or with three rows, `points (2d_xyz_array)` plots the corresponding points.

  Examples:

  One tridimensional sample,

      (%i1) load ("numericalio")$
      (%i2) s2 : read_matrix (file_search ("wind.data"))$
      (%i3) draw3d(title = "Daily average wind speeds",
                   point_size = 2,
                   points(args(submatrix (s2, 4, 5))) )$

  Two tridimensional samples,

      (%i1) load ("numericalio")$
      (%i2) s2 : read_matrix (file_search ("wind.data"))$
      (%i3) draw3d(
               title = "Daily average wind speeds. Two data sets",
               point_size = 2,
               key        = "Sample from stations 1, 2 and 3",
               points(args(submatrix (s2, 4, 5))),
               point_type = 4,
               key        = "Sample from stations 1, 4 and 5",
               points(args(submatrix (s2, 2, 3))) )$

  Unidimensional arrays,

      (%i1) x: make_array (fixnum, 10) $
      (%i2) y: make_array (fixnum, 10) $
      (%i3) z: make_array (fixnum, 10) $
      (%i4) for i:0 thru 9 do (
              x[i]: random(10),
              y[i]: random(10),
              z[i]: random(10) ) $
      (%i5) draw3d(points(x,y,z)) $

  Bidimensional colored array,

      (%i1) xyz: make_array(fixnum, 10, 3) $
      (%i2) for i:0 thru 9 do (
              xyz[i, 0]: random(10),
              xyz[i, 1]: random(10),
              xyz[i, 2]: random(10) ) $
      (%i3) draw3d(
               enhanced3d = true,
               points_joined = true,
               points(xyz)) $

  Color numbers explicitly specified by the user.

      (%i1) pts: makelist([t,t^2,cos(t)], t, 0, 15)$
      (%i2) col_num: makelist(k, k, 1, length(pts))$
      (%i3) draw3d(
              enhanced3d = ['part(col_num,k),k],
              point_size = 3,
              point_type = filled_circle,
              points(pts))$

  Categories: Package draw

Graphic object: polar (radius,ang,minang,maxang)

  Draws 2D functions defined in polar coordinates.

  2D

  `polar (radius,ang,minang,maxang)` plots function `radius(ang)` defined in polar coordinates, with variable ang taking values from minang to maxang.

  This object is affected by the following graphic options: `nticks`, `line_width`, `line_type`, `key` and `color`.

  Example:

      (%i1) draw2d(user_preamble = "set grid polar",
                   nticks        = 200,
                   xrange        = [-5,5],
                   yrange        = [-5,5],
                   color         = blue,
                   line_width    = 3,
                   title         = "Hyperbolic Spiral",
                   polar(10/theta,theta,1,10*%pi) )$

  Categories: Package draw

Graphic object: polygon
`polygon` ([[x1, y1], [x2, y2], …])
`polygon` ([x1, x2, …], [y1, y2, …])

  Draws polygons in 2D.

  2D

  The commands `polygon([[x1, y1], [x2, y2], ...])` or `polygon([x1, x2, ...], [y1, y2, ...])` plot on the plane a polygon with vertices `[x1, y1]`, `[x2, y2]`, etc.

  This object is affected by the following graphic options: `transparent`, `fill_color`, `border`, `line_width`, `key`, `line_type` and `color`.

  Example:

      (%i1) draw2d(color      = "#e245f0",
                   line_width = 8,
                   polygon([[3,2],[7,2],[5,5]]),
                   border      = false,
                   fill_color  = yellow,
                   polygon([[5,2],[9,2],[7,5]]) )$

  Categories: Package draw

Graphic object: quadrilateral (point_1, point_2, point_3, point_4)

  Draws a quadrilateral.

  2D

  `quadrilateral([x1, y1], [x2, y2], [x3, y3], [x4, y4])` draws a quadrilateral with vertices `[x1, y1]`, `[x2, y2]`, `[x3, y3]`, and `[x4, y4]`.

  This object is affected by the following graphic options:
  `transparent`, `fill_color`, `border`, `line_width`, `key`, `xaxis_secondary`, `yaxis_secondary`, `line_type`, `transform` and `color`.

  Example:

      (%i1) draw2d(
              quadrilateral([1,1],[2,2],[3,-1],[2,-2]))$

  3D

  `quadrilateral([x1, y1, z1], [x2, y2, z2], [x3, y3, z3], [x4, y4, z4])` draws a quadrilateral with vertices `[x1, y1, z1]`, `[x2, y2, z2]`, `[x3, y3, z3]`, and `[x4, y4, z4]`.

  This object is affected by the following graphic options: `line_type`, `line_width`, `color`, `key`, `enhanced3d` and `transform`.

  Categories: Package draw

Graphic object: rectangle ([x1,y1], [x2,y2])

  Draws rectangles in 2D.

  2D

  `rectangle ([x1,y1], [x2,y2])` draws a rectangle with opposite vertices `[x1,y1]` and `[x2,y2]`.

  This object is affected by the following graphic options: `transparent`, `fill_color`, `border`, `line_width`, `key`, `line_type` and `color`.

  Example:

      (%i1) draw2d(fill_color  = red,
                   line_width  = 6,
                   line_type   = dots,
                   transparent = false,
                   fill_color  = blue,
                   rectangle([-2,-2],[8,-1]), /* opposite vertices */
                   transparent = true,
                   line_type   = solid,
                   line_width  = 1,
                   rectangle([9,4],[2,-1.5]),
                   xrange      = [-3,10],
                   yrange      = [-3,4.5] )$

  Categories: Package draw

Graphic object: region (expr,var1,minval1,maxval1,var2,minval2,maxval2)

  Plots a region on the plane defined by inequalities.

  2D expr is an expression formed by inequalities and boolean operators `and`, `or`, and `not`. The region is bounded by the rectangle defined by [minval1, maxval1] and [minval2, maxval2].

  This object is affected by the following graphic options: `fill_color`, `key`, `x_voxel` and `y_voxel`.

  Example:

      (%i1) draw2d(
              x_voxel = 30,
              y_voxel = 30,
              region(x^2+y^2<1 and x^2+y^2 > 1/2,
                     x, -1.5, 1.5, y, -1.5, 1.5));

Graphic object: spherical (radius, azi, minazi, maxazi, zen, minzen, maxzen)

  Draws 3D functions defined in spherical coordinates.

  3D

  `spherical(radius, azi, minazi, maxazi, zen, minzen, maxzen)` plots the function `radius(azi, zen)` defined in spherical coordinates, with azimuth azi taking values from minazi to maxazi and zenith zen taking values from minzen to maxzen.

  This object is affected by the following graphic options: `xu_grid`, `yv_grid`, `line_type`, `key`, `wired_surface`, `enhanced3d` and `color`.

  Example:

      (%i1) draw3d(spherical(1,a,0,2*%pi,z,0,%pi))$

  Categories: Package draw

Graphic object: triangle (point_1, point_2, point_3)

  Draws a triangle.

  2D

  `triangle ([x1,y1], [x2,y2], [x3,y3])` draws a triangle with vertices `[x1,y1]`, `[x2,y2]`, and `[x3,y3]`.

  This object is affected by the following graphic options:
  `transparent`, `fill_color`, `border`, `line_width`, `key`, `xaxis_secondary`, `yaxis_secondary`, `line_type`, `transform` and `color`.

  Example:

      (%i1) draw2d(
              triangle([1,1],[2,2],[3,-1]))$

  3D

  `triangle ([x1,y1,z1], [x2,y2,z2], [x3,y3,z3])` draws a triangle with vertices `[x1,y1,z1]`, `[x2,y2,z2]`, and `[x3,y3,z3]`.

  This object is affected by the following graphic options: `line_type`, `line_width`, `color`, `key`, `enhanced3d` and `transform`.

  Categories: Package draw

Graphic object: tube (xfun,yfun,zfun,rfun,p,pmin,pmax)

  Draws a tube in 3D with varying diameter.

  3D

  `[xfun,yfun,zfun]` is the parametric curve with parameter p taking values from pmin to pmax. Circles of radius rfun are placed with their centers on the parametric curve and perpendicular to it.

  This object is affected by the following graphic options: `xu_grid`, `yv_grid`, `line_type`, `line_width`, `key`, `wired_surface`, `enhanced3d`, `color` and `capping`.

  Example:

      (%i1) draw3d(
              enhanced3d = true,
              xu_grid = 50,
              tube(cos(a), a, 0, cos(a/10)^2,
                   a, 0, 4*%pi) )$

  Categories: Package draw

Graphic object: vector
`vector` ([x,y], [dx,dy])
`vector` ([x,y,z], [dx,dy,dz])

  Draws vectors in 2D and 3D.

  This object is affected by the following graphic options: `head_both`, `head_length`, `head_angle`, `head_type`, `line_width`, `line_type`, `key` and `color`.

  2D

  `vector([x,y], [dx,dy])` plots vector `[dx,dy]` with origin in `[x,y]`.

  Example:

      (%i1) draw2d(xrange      = [0,12],
                   yrange      = [0,10],
                   head_length = 1,
                   vector([0,1],[5,5]), /* default type */
                   head_type = 'empty,
                   vector([3,1],[5,5]),
                   head_both = true,
                   head_type = 'nofilled,
                   line_type = dots,
                   vector([6,1],[5,5]))$

  3D

  `vector([x,y,z], [dx,dy,dz])` plots vector `[dx,dy,dz]` with origin in `[x,y,z]`.

  Example:

      (%i1) draw3d(color = cyan,
                   vector([0,0,0],[1,1,1]/sqrt(3)),
                   vector([0,0,0],[1,-1,0]/sqrt(2)),
                   vector([0,0,0],[1,1,-2]/sqrt(6)) )$

  Categories: Package draw

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## 52.3 Functions and Variables for pictures

Function: get_pixel (pic,x,y)

  Returns pixel from picture. Coordinates x and y range from 0 to `width-1` and `height-1`, respectively.

  Categories: Package draw

Function: make_level_picture
`make_level_picture` (data)
`make_level_picture` (data,width,height)

  Returns a levels picture object. `make_level_picture (data)` builds the picture object from matrix data. `make_level_picture (data,width,height)` builds the object from a list of numbers; in this case, both the width and the height must be given.

  The returned picture object contains the following four parts:

  1. symbol `level`
  2. image width
  3. image height
  4. an integer array with pixel data ranging from 0 to 255. Argument data must contain only numbers ranged from 0 to 255; negative numbers are substituted by 0, and those which are greater than 255 are set to 255.

  Example:

  Level picture from matrix.

      (%i1) make_level_picture(matrix([3,2,5],[7,-9,3000]));
      (%o1)         picture(level, 3, 2, {Array:  #(3 2 5 7 0 255)})

  Level picture from numeric list.

      (%i1) make_level_picture([-2,0,54,%pi],2,2);
      (%o1)            picture(level, 2, 2, {Array:  #(0 0 54 3)})

  Categories: Package draw

Function: make_rgb_picture (redlevel,greenlevel,bluelevel)

  Returns an rgb-coloured picture object. All three arguments must be levels picture; with red, green and blue levels.

  The returned picture object contains the following four parts:

  1. symbol `rgb`
  2. image width
  3. image height
  4. an integer array of length 3*width*height with pixel data ranging from 0 to 255. Each pixel is represented by three consecutive numbers (red, green, blue).

  Example:

      (%i1) red: make_level_picture(matrix([3,2],[7,260]));
      (%o1)           picture(level, 2, 2, {Array:  #(3 2 7 255)})
      (%i2) green: make_level_picture(matrix([54,23],[73,-9]));
      (%o2)           picture(level, 2, 2, {Array:  #(54 23 73 0)})
      (%i3) blue: make_level_picture(matrix([123,82],[45,32.5698]));
      (%o3)          picture(level, 2, 2, {Array:  #(123 82 45 33)})
      (%i4) make_rgb_picture(red,green,blue);
      (%o4) picture(rgb, 2, 2,
                    {Array:  #(3 54 123 2 23 82 7 73 45 255 0 33)})

  Categories: Package draw

Function: negative_picture (pic)

  Returns the negative of a (level or rgb) picture.

  Categories: Package draw

Function: picture_equalp (x,y)

  Returns `true` in case of equal pictures, and `false` otherwise.

  Categories: Package draw · Predicate functions

Function: picturep (x)

  Returns `true` if the argument is a well formed image, and `false` otherwise.

  Categories: Package draw · Predicate functions

Function: read_xpm (xpm_file)

  Reads a file in xpm and returns a picture object.

  Categories: Package draw

Function: rgb2level (pic)

  Transforms an rgb picture into a level one by averaging the red, green and blue channels.

  Categories: Package draw

Function: take_channel (im,color)

  If argument color is `red`, `green` or `blue`, function `take_channel` returns the corresponding color channel of picture im. Example:

      (%i1) red: make_level_picture(matrix([3,2],[7,260]));
      (%o1)           picture(level, 2, 2, {Array:  #(3 2 7 255)})
      (%i2) green: make_level_picture(matrix([54,23],[73,-9]));
      (%o2)           picture(level, 2, 2, {Array:  #(54 23 73 0)})
      (%i3) blue: make_level_picture(matrix([123,82],[45,32.5698]));
      (%o3)          picture(level, 2, 2, {Array:  #(123 82 45 33)})
      (%i4) make_rgb_picture(red,green,blue);
      (%o4) picture(rgb, 2, 2,
                    {Array:  #(3 54 123 2 23 82 7 73 45 255 0 33)})
      (%i5) take_channel(%,'green);  /* simple quote!!! */
      (%o5)           picture(level, 2, 2, {Array:  #(54 23 73 0)})

  Categories: Package draw

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## 52.4 Functions and Variables for worldmap

This package automatically loads package `draw`.

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### 52.4.1 Variables and Functions

Global variable: boundaries_array

  Default value: `false`

  `boundaries_array` is where the graphic object `geomap` looks for boundaries coordinates.

  Each component of `boundaries_array` is an array of floating point quantities, the coordinates of a polygonal segment or map boundary.

  See also `geomap`.

  Categories: Package draw

Function: numbered_boundaries (nlist)

  Draws a list of polygonal segments (boundaries), labeled by its numbers (`boundaries_array` coordinates). This is of great help when building new geographical entities.

  Example:

  Map of Europe labeling borders with their component number in `boundaries_array`.

      (%i1) load(worldmap)$
      (%i2) european_borders:
                 region_boundaries(-31.81,74.92,49.84,32.06)$
      (%i3) numbered_boundaries(european_borders)$

  Categories: Package draw

Function: make_poly_continent
`make_poly_continent` (continent_name)
`make_poly_continent` (country_list)

  Makes the necessary polygons to draw a colored continent or a list of countries.

  Example:

      (%i1) load(worldmap)$
      (%i2) /* A continent */
            make_poly_continent(Africa)$
      (%i3) apply(draw2d, %)$

      (%i4) /* A list of countries */
            make_poly_continent([Germany,Denmark,Poland])$
      (%i5) apply(draw2d, %)$

  Categories: Package draw

Function: make_poly_country (country_name)

  Makes the necessary polygons to draw a colored country. If islands exist, one country can be defined with more than just one polygon.

  Example:

      (%i1) load(worldmap)$
      (%i2) make_poly_country(India)$
      (%i3) apply(draw2d, %)$

  Categories: Package draw

Function: make_polygon (nlist)

  Returns a `polygon` object from boundary indices. Argument nlist is a list of components of `boundaries_array`.

  Example:

  Bhutan is defined by boundary numbers 171, 173 and 1143, so that `make_polygon([171,173,1143])` appends arrays of coordinates `boundaries_array[171]`, `boundaries_array[173]` and `boundaries_array[1143]` and returns a `polygon` object suited to be plotted by `draw`. To avoid an error message, arrays must be compatible in the sense that any two consecutive arrays have two coordinates in the extremes in common. In this example, the two first components of `boundaries_array[171]` are equal to the last two coordinates of `boundaries_array[173]`, and the two first of `boundaries_array[173]` are equal to the two first of `boundaries_array[1143]`; in conclussion, boundary numbers 171, 173 and 1143 (in this order) are compatible and the colored polygon can be drawn.

      (%i1) load(worldmap)$
      (%i2) Bhutan;
      (%o2)                        [[171, 173, 1143]]
      (%i3) boundaries_array[171];
      (%o3) {Array:
             #(88.750549 27.14727 88.806351 27.25305 88.901367 27.282221
               88.917877 27.321039)}
      (%i4) boundaries_array[173];
      (%o4) {Array:
             #(91.659554 27.76511 91.6008 27.66666 91.598022 27.62499
               91.631348 27.536381 91.765533 27.45694 91.775253 27.4161
               92.007751 27.471939 92.11441 27.28583 92.015259 27.168051
               92.015533 27.08083 92.083313 27.02277 92.112183 26.920271
               92.069977 26.86194 91.997192 26.85194 91.915253 26.893881
               91.916924 26.85416 91.8358 26.863331 91.712479 26.799999
               91.542191 26.80444 91.492188 26.87472 91.418854 26.873329
               91.371353 26.800831 91.307457 26.778049 90.682457 26.77417
               90.392197 26.903601 90.344131 26.894159 90.143044 26.75333
               89.98996 26.73583 89.841919 26.70138 89.618301 26.72694
               89.636093 26.771111 89.360786 26.859989 89.22081 26.81472
               89.110237 26.829161 88.921631 26.98777 88.873016 26.95499
               88.867737 27.080549 88.843307 27.108601 88.750549
               27.14727)}
      (%i5) boundaries_array[1143];
      (%o5) {Array:
             #(91.659554 27.76511 91.666924 27.88888 91.65831 27.94805
               91.338028 28.05249 91.314972 28.096661 91.108856 27.971109
               91.015808 27.97777 90.896927 28.05055 90.382462 28.07972
               90.396088 28.23555 90.366074 28.257771 89.996353 28.32333
               89.83165 28.24888 89.58609 28.139999 89.35997 27.87166
               89.225517 27.795 89.125793 27.56749 88.971077 27.47361
               88.917877 27.321039)}
      (%i6) Bhutan_polygon: make_polygon([171,173,1143])$
      (%i7) draw2d(Bhutan_polygon)$

  Categories: Package draw

Function: region_boundaries (x1,y1,x2,y2)

  Detects polygonal segments of global variable `boundaries_array` fully contained in the rectangle with vertices (x1,y1) -upper left- and (x2,y2) -bottom right-.

  Example:

  Returns segment numbers for plotting southern Italy.

      (%i1) load(worldmap)$
      (%i2) region_boundaries(10.4,41.5,20.7,35.4);
      (%o2)                [1846, 1863, 1864, 1881, 1888, 1894]
      (%i3) draw2d(geomap(%))$

  Categories: Package draw

Function: region_boundaries_plus (x1,y1,x2,y2)

  Detects polygonal segments of global variable `boundaries_array` containing at least one vertex in the rectangle defined by vertices (x1,y1) -upper left- and (x2,y2) -bottom right-.

  Example:

      (%i1) load(worldmap)$
      (%i2) region_boundaries_plus(10.4,41.5,20.7,35.4);
      (%o2) [1060, 1062, 1076, 1835, 1839, 1844, 1846, 1858,
             1861, 1863, 1864, 1871, 1881, 1888, 1894, 1897]
      (%i3) draw2d(geomap(%))$

  Categories: Package draw

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### 52.4.2 Graphic objects

Graphic object: geomap
`geomap` (numlist)
`geomap` (numlist,3Dprojection)

  Draws cartographic maps in 2D and 3D.

  2D

  This function works together with global variable `boundaries_array`.

  Argument numlist is a list containing numbers or lists of numbers. All these numbers must be integers greater or equal than zero, representing the components of global array `boundaries_array`.

  Each component of `boundaries_array` is an array of floating point quantities, the coordinates of a polygonal segment or map boundary.

  `geomap (numlist)` flattens its arguments and draws the associated boundaries in `boundaries_array`.

  This object is affected by the following graphic options: `line_width`, `line_type` and `color`.

  Examples:

  A simple map defined by hand:

      (%i1) load(worldmap)$
      (%i2) /* Vertices of boundary #0: {(1,1),(2,5),(4,3)} */
         ( bnd0: make_array(flonum,6),
           bnd0[0]:1.0, bnd0[1]:1.0, bnd0[2]:2.0,
           bnd0[3]:5.0, bnd0[4]:4.0, bnd0[5]:3.0 )$
      (%i3) /* Vertices of boundary #1: {(4,3),(5,4),(6,4),(5,1)} */
         ( bnd1: make_array(flonum,8),
           bnd1[0]:4.0, bnd1[1]:3.0, bnd1[2]:5.0, bnd1[3]:4.0,
           bnd1[4]:6.0, bnd1[5]:4.0, bnd1[6]:5.0, bnd1[7]:1.0)$
      (%i4) /* Vertices of boundary #2: {(5,1), (3,0), (1,1)} */
         ( bnd2: make_array(flonum,6),
           bnd2[0]:5.0, bnd2[1]:1.0, bnd2[2]:3.0,
           bnd2[3]:0.0, bnd2[4]:1.0, bnd2[5]:1.0 )$
      (%i5) /* Vertices of boundary #3: {(1,1), (4,3)} */
         ( bnd3: make_array(flonum,4),
           bnd3[0]:1.0, bnd3[1]:1.0, bnd3[2]:4.0, bnd3[3]:3.0)$
      (%i6) /* Vertices of boundary #4: {(4,3), (5,1)} */
         ( bnd4: make_array(flonum,4),
           bnd4[0]:4.0, bnd4[1]:3.0, bnd4[2]:5.0, bnd4[3]:1.0)$
      (%i7) /* Pack all together in boundaries_array */
         ( boundaries_array: make_array(any,5),
           boundaries_array[0]: bnd0, boundaries_array[1]: bnd1,
           boundaries_array[2]: bnd2, boundaries_array[3]: bnd3,
           boundaries_array[4]: bnd4 )$
      (%i8) draw2d(geomap([0,1,2,3,4]))$

  The auxiliary package `worldmap` sets the global variable `boundaries_array` to real world boundaries in (longitude, latitude) coordinates. These data are in the public domain and come from https://web.archive.org/web/20100310124019/http://www-cger.nies.go.jp/grid-e/gridtxt/grid19.html. Package `worldmap` defines also boundaries for countries, continents and coastlines as lists with the necessary components of `boundaries_array` (see file `share/draw/worldmap.mac` for more information). Package `worldmap` automatically loads package `worldmap`.

      (%i1) load(worldmap)$
      (%i2) c1: gr2d(geomap([Canada,United_States,
                             Mexico,Cuba]))$
      (%i3) c2: gr2d(geomap(Africa))$
      (%i4) c3: gr2d(geomap([Oceania,China,Japan]))$
      (%i5) c4: gr2d(geomap([France,Portugal,Spain,
                             Morocco,Western_Sahara]))$
      (%i6) draw(columns  = 2,
                 c1,c2,c3,c4)$

  Package `worldmap` is also useful for plotting countries as polygons. In this case, graphic object `geomap` is no longer necessary and the `polygon` object is used instead. Since lists are now used and not arrays, maps rendering will be slower. See also `make_poly_country` and `make_poly_continent` to understand the following code.

      (%i1) load(worldmap)$
      (%i2) mymap: append(
         [color      = white],  /* borders are white */
         [fill_color = red],             make_poly_country(Bolivia),
         [fill_color = cyan],            make_poly_country(Paraguay),
         [fill_color = green],           make_poly_country(Colombia),
         [fill_color = blue],            make_poly_country(Chile),
         [fill_color = "#23ab0f"],       make_poly_country(Brazil),
         [fill_color = goldenrod],       make_poly_country(Argentina),
         [fill_color = "midnight-blue"], make_poly_country(Uruguay))$
      (%i3) apply(draw2d, mymap)$

  3D

  `geomap (numlist)` projects map boundaries on the sphere of radius 1 centered at (0,0,0). It is possible to change the sphere or the projection type by using `geomap (numlist,3Dprojection)`.

  Available 3D projections:

  - `[spherical_projection,x,y,z,r]`: projects map boundaries on the sphere of radius r centered at (x,y,z).

        (%i1) load(worldmap)$
        (%i2) draw3d(geomap(Australia), /* default projection */
                     geomap(Australia,
                            [spherical_projection,2,2,2,3]))$

  - `[cylindrical_projection,x,y,z,r,rc]`: re-projects spherical map boundaries on the cylinder of radius rc and axis passing through the poles of the globe of radius r centered at (x,y,z).

        (%i1) load(worldmap)$
        (%i2) draw3d(geomap([America_coastlines,Eurasia_coastlines],
                            [cylindrical_projection,2,2,2,3,4]))$

  - `[conic_projection,x,y,z,r,alpha]`: re-projects spherical map boundaries on the cones of angle alpha, with axis passing through the poles of the globe of radius r centered at (x,y,z). Both the northern and southern cones are tangent to sphere.

        (%i1) load(worldmap)$
        (%i2) draw3d(geomap(World_coastlines,
                            [conic_projection,0,0,0,1,90]))$

  See also http://riotorto.users.sf.net/gnuplot/geomap for more elaborated examples.

  Categories: Package draw

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# 53. drawdf

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## 53.1 Introduction to drawdf

The function `drawdf` draws the direction field of a first-order Ordinary Differential Equation (ODE) or a system of two autonomous first-order ODE's.

Since this is an additional package, in order to use it you must first load it with `load(drawdf)`. Drawdf is built upon the `draw` package, which requires Gnuplot 4.2.

To plot the direction field of a single ODE, the ODE must be written in the form:

           dy
           -- = F(x,y)
           dx

and the function F should be given as the argument for `drawdf`. If the independent and dependent variables are not x, and y, as in the equation above, then those two variables should be named explicitly in a list given as an argument to the drawdf command (see the examples).

To plot the direction field of a set of two autonomous ODE's, they must be written in the form

           dx             dy
           -- = G(x,y)    -- = F(x,y)
           dt             dt

and the argument for `drawdf` should be a list with the two functions G and F, in that order; namely, the first expression in the list will be taken to be the time derivative of the variable represented on the horizontal axis, and the second expression will be the time derivative of the variable represented on the vertical axis. Those two variables do not have to be x and y, but if they are not, then the second argument given to drawdf must be another list naming the two variables, first the one on the horizontal axis and then the one on the vertical axis.

If only one ODE is given, `drawdf` will implicitly admit `x=t`, and `G(x,y)=1`, transforming the non-autonomous equation into a system of two autonomous equations.

Categories: Differential equations · Plotting Share packages · Package drawdf Package draw

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## 53.2 Functions and Variables for drawdf

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### 53.2.1 Functions

Function: drawdf
`drawdf` (dydx, ...options and objects...)
`drawdf` (dvdu, [u,v], ...options and objects...)
`drawdf` (dvdu, [u,umin,umax], [v,vmin,vmax], ...options and objects...)
`drawdf` ([dxdt,dydt], ...options and objects...)
`drawdf` ([dudt,dvdt], [u,v], ...options and objects...)
`drawdf` ([dudt,dvdt], [u,umin,umax], [v,vmin,vmax], ...options and objects...)

  Function `drawdf` draws a 2D direction field with optional solution curves and other graphics using the `draw` package.

  The first argument specifies the derivative(s), and must be either an expression or a list of two expressions. dydx, dxdt and dydt are expressions that depend on x and y. dvdu, dudt and dvdt are expressions that depend on u and v.

  If the independent and dependent variables are not x and y, then their names must be specified immediately following the derivative(s), either as a list of two names `[`u,v`]`, or as two lists of the form `[`u,umin,umax`]` and `[`v,vmin,vmax`]`.

  The remaining arguments are graphic options, graphic objects, or lists containing graphic options and objects, nested to arbitrary depth. The set of graphic options and objects supported by `drawdf` is a superset of those supported by `draw2d` and `gr2d` from the `draw` package.

  The arguments are interpreted sequentially: graphic options affect all following graphic objects. Furthermore, graphic objects are drawn on the canvas in order specified, and may obscure graphics drawn earlier. Some graphic options affect the global appearance of the scene.

  The additional graphic objects supported by `drawdf` include: `solns_at`, `points_at`, `saddles_at`, `soln_at`, `point_at`, and `saddle_at`.

  The additional graphic options supported by `drawdf` include: `field_degree`, `soln_arrows`, `field_arrows`, `field_grid`, `field_color`, `show_field`, `tstep`, `nsteps`, `duration`, `direction`, `field_tstep`, `field_nsteps`, and `field_duration`.

  Commonly used graphic objects inherited from the `draw` package include: `explicit`, `implicit`, `parametric`, `polygon`, `points`, `vector`, `label`, and all others supported by `draw2d` and `gr2d`.

  Commonly used graphic options inherited from the `draw` package include:
  `points_joined`, `color`, `point_type`, `point_size`, `line_width`, `line_type`, `key`, `title`, `xlabel`, `ylabel`, `user_preamble`, `terminal`, `dimensions`, `file_name`, and all others supported by `draw2d` and `gr2d`.

  See also `draw2d`.

  Users of wxMaxima or Imaxima may optionally use `wxdrawdf`, which is identical to `drawdf` except that the graphics are drawn within the notebook using `wxdraw`.

  To make use of this function, write first `load(drawdf)`.

  Examples:

      (%i1) load(drawdf)$
      (%i2) drawdf(exp(-x)+y)$        /* default vars: x,y */
      (%i3) drawdf(exp(-t)+y, [t,y])$ /* default range: [-10,10] */
      (%i4) drawdf([y,-9*sin(x)-y/5], [x,1,5], [y,-2,2])$

  For backward compatibility, `drawdf` accepts most of the parameters supported by plotdf.

      (%i5) drawdf(2*cos(t)-1+y, [t,y], [t,-5,10], [y,-4,9],
                   [trajectory_at,0,0])$

  `soln_at` and `solns_at` draw solution curves passing through the specified points, using a slightly enhanced 4th-order Runge Kutta numerical integrator.

      (%i6) drawdf(2*cos(t)-1+y, [t,-5,10], [y,-4,9],
                   solns_at([0,0.1],[0,-0.1]),
                   color=blue, soln_at(0,0))$

  `field_degree=2` causes the field to be composed of quadratic splines, based on the first and second derivatives at each grid point. `field_grid=[`COLS,ROWS`]` specifies the number of columns and rows in the grid.

      (%i7) drawdf(2*cos(t)-1+y, [t,-5,10], [y,-4,9],
                   field_degree=2, field_grid=[20,15],
                   solns_at([0,0.1],[0,-0.1]),
                   color=blue, soln_at(0,0))$

  `soln_arrows=true` adds arrows to the solution curves, and (by default) removes them from the direction field. It also changes the default colors to emphasize the solution curves.

      (%i8) drawdf(2*cos(t)-1+y, [t,-5,10], [y,-4,9],
                   soln_arrows=true,
                   solns_at([0,0.1],[0,-0.1],[0,0]))$

  `duration=40` specifies the time duration of numerical integration (default 10). Integration will also stop automatically if the solution moves too far away from the plotted region, or if the derivative becomes complex or infinite. Here we also specify `field_degree=2` to plot quadratic splines. The equations below model a predator-prey system.

      (%i9) drawdf([x*(1-x-y), y*(3/4-y-x/2)], [x,0,1.1], [y,0,1],
                   field_degree=2, duration=40,
                   soln_arrows=true, point_at(1/2,1/2),
                   solns_at([0.1,0.2], [0.2,0.1], [1,0.8], [0.8,1],
                            [0.1,0.1], [0.6,0.05], [0.05,0.4],
                            [1,0.01], [0.01,0.75]))$

  `field_degree='solns` causes the field to be composed of many small solution curves computed by 4th-order Runge Kutta, with better results in this case.

      (%i10) drawdf([x*(1-x-y), y*(3/4-y-x/2)], [x,0,1.1], [y,0,1],
                    field_degree='solns, duration=40,
                    soln_arrows=true, point_at(1/2,1/2),
                    solns_at([0.1,0.2], [0.2,0.1], [1,0.8],
                             [0.8,1], [0.1,0.1], [0.6,0.05],
                             [0.05,0.4], [1,0.01], [0.01,0.75]))$

  `saddles_at` attempts to automatically linearize the equation at each saddle, and to plot a numerical solution corresponding to each eigenvector, including the separatrices. `tstep=0.05` specifies the maximum time step for the numerical integrator (the default is 0.1). Note that smaller time steps will sometimes be used in order to keep the x and y steps small. The equations below model a damped pendulum.

      (%i11) drawdf([y,-9*sin(x)-y/5], tstep=0.05,
                    soln_arrows=true, point_size=0.5,
                    points_at([0,0], [2*%pi,0], [-2*%pi,0]),
                    field_degree='solns,
                    saddles_at([%pi,0], [-%pi,0]))$

  `show_field=false` suppresses the field entirely.

      (%i12) drawdf([y,-9*sin(x)-y/5], tstep=0.05,
                    show_field=false, soln_arrows=true,
                    point_size=0.5,
                    points_at([0,0], [2*%pi,0], [-2*%pi,0]),
                    saddles_at([3*%pi,0], [-3*%pi,0],
                               [%pi,0], [-%pi,0]))$

  `drawdf` passes all unrecognized parameters to `draw2d` or `gr2d`, allowing you to combine the full power of the `draw` package with `drawdf`.

      (%i13) drawdf(x^2+y^2, [x,-2,2], [y,-2,2], field_color=gray,
                    key="soln 1", color=black, soln_at(0,0),
                    key="soln 2", color=red, soln_at(0,1),
                    key="isocline", color=green, line_width=2,
                    nticks=100, parametric(cos(t),sin(t),t,0,2*%pi))$

  `drawdf` accepts nested lists of graphic options and objects, allowing convenient use of makelist and other function calls to generate graphics.

      (%i14) colors : ['red,'blue,'purple,'orange,'green]$
      (%i15) drawdf([x-x*y/2, (x*y - 3*y)/4],
                    [x,2.5,3.5], [y,1.5,2.5],
                    field_color = gray,
                    makelist([ key   = concat("soln",k),
                               color = colors[k],
                               soln_at(3, 2 + k/20) ],
                             k,1,5))$

  Categories: Package drawdf

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# 54. dynamics

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## 54.1 The dynamics package

Package `dynamics` includes functions for 3D visualization, animations, graphical analysis of differential and difference equations and numerical solution of differential equations. The functions for differential equations are described in the section on `Numerical Methods` and the functions to plot the Mandelbrot and Julia sets are described in the section on `Plotting`.

All the functions in this package will be loaded automatically the first time they are used.

Categories: Dynamical systems · Share packages · Package dynamics

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## 54.2 Graphical analysis of discrete dynamical systems

Function: chaosgame ([[x1, y1]…[xm, ym]], [x0, y0], b, n, options, …);

  Implements the so-called chaos game: the initial point (x0, y0) is plotted and then one of the m points [x1, y1]…xm, ym] will be selected at random. The next point plotted will be on the segment from the previous point plotted to the point chosen randomly, at a distance from the random point which will be b times that segment's length. The procedure is repeated n times. The options are the same as for `plot2d`.

  Example. A plot of Sierpinsky's triangle:

      (%i1) chaosgame([[0, 0], [1, 0], [0.5, sqrt(3)/2]], [0.1, 0.1], 1/2,
                       30000, [style, dots]);

  Categories: Package dynamics · Plotting

Function: evolution (F, y0, n, …, options, …);

  Draws n+1 points in a two-dimensional graph, where the horizontal coordinates of the points are the integers 0, 1, 2, ..., n, and the vertical coordinates are the corresponding values y(n) of the sequence defined by the recurrence relation

              y(n+1) = F(y(n))

  With initial value y(0) equal to y0. F must be an expression that depends only on one variable (in the example, it depend on y, but any other variable can be used), y0 must be a real number and n must be a positive integer. This function accepts the same options as `plot2d`.

  Example.

      (%i1) evolution(cos(y), 2, 11);

  Categories: Package dynamics · Plotting

Function: evolution2d ([F, G], [u, v], [u0, y0], n, options, …);

  Shows, in a two-dimensional plot, the first n+1 points in the sequence of points defined by the two-dimensional discrete dynamical system with recurrence relations

              u(n+1) = F(u(n), v(n))    v(n+1) = G(u(n), v(n))

  With initial values u0 and v0. F and G must be two expressions that depend only on two variables, u and v, which must be named explicitly in a list. The options are the same as for `plot2d`.

  Example. Evolution of a two-dimensional discrete dynamical system:

      (%i1) f: 0.6*x*(1+2*x)+0.8*y*(x-1)-y^2-0.9$
      (%i2) g: 0.1*x*(1-6*x+4*y)+0.1*y*(1+9*y)-0.4$
      (%i3) evolution2d([f,g], [x,y], [-0.5,0], 50000, [style,dots]);

  And an enlargement of a small region in that fractal:

      (%i9) evolution2d([f,g], [x,y], [-0.5,0], 300000, [x,-0.8,-0.6],
                        [y,-0.4,-0.2], [style, dots]);

  Categories: Package dynamics · Plotting

Function: ifs ([r1, …, rm], [A1,…, Am], [[x1, y1], …, [xm, ym]], [x0, y0], n, options, …);

  Implements the Iterated Function System method. This method is similar to the method described in the function `chaosgame`. but instead of shrinking the segment from the current point to the randomly chosen point, the 2 components of that segment will be multiplied by the 2 by 2 matrix Ai that corresponds to the point chosen randomly.

  The random choice of one of the m attractive points can be made with a non-uniform probability distribution defined by the weights r1,...,rm. Those weights are given in cumulative form; for instance if there are 3 points with probabilities 0.2, 0.5 and 0.3, the weights r1, r2 and r3 could be 2, 7 and 10. The options are the same as for `plot2d`.

  Example. Barnsley's fern, obtained with 4 matrices and 4 points:

      (%i1) a1: matrix([0.85,0.04],[-0.04,0.85])$
      (%i2) a2: matrix([0.2,-0.26],[0.23,0.22])$
      (%i3) a3: matrix([-0.15,0.28],[0.26,0.24])$
      (%i4) a4: matrix([0,0],[0,0.16])$
      (%i5) p1: [0,1.6]$
      (%i6) p2: [0,1.6]$
      (%i7) p3: [0,0.44]$
      (%i8) p4: [0,0]$
      (%i9) w: [85,92,99,100]$
      (%i10) ifs(w, [a1,a2,a3,a4], [p1,p2,p3,p4], [5,0], 50000, [style,dots]);

  Categories: Package dynamics · Plotting

Function: orbits (F, y0, n1, n2, [x, x0, xf, xstep], options, …);

  Draws the orbits diagram for a family of one-dimensional discrete dynamical systems, with one parameter x; that kind of diagram is used to study the bifurcations of a one-dimensional discrete system.

  The function F(y) defines a sequence with a starting value of y0, as in the case of the function `evolution`, but in this case that function will also depend on a parameter x that will take values in the interval from x0 to xf with increments of xstep. Each value used for the parameter x is shown on the horizontal axis. The vertical axis will show the n2 values of the sequence y(n1+1),..., y(n1+n2+1) obtained after letting the sequence evolve n1 iterations. In addition to the options accepted by `plot2d`, it accepts an option pixels that sets up the maximum number of different points that will be represented in the vertical direction.

  Example. Orbits diagram of the quadratic map, with a parameter a:

      (%i1) orbits(x^2+a, 0, 50, 200, [a, -2, 0.25], [style, dots]);

  To enlarge the region around the lower bifurcation near x `=` -1.25 use:

      (%i2) orbits(x^2+a, 0, 100, 400, [a,-1,-1.53], [x,-1.6,-0.8],
                   [nticks, 400], [style,dots]);

  Categories: Package dynamics · Plotting

Function: staircase (F, y0, n,options,…);

  Draws a staircase diagram for the sequence defined by the recurrence relation

              y(n+1) = F(y(n))

  The interpretation and allowed values of the input parameters is the same as for the function `evolution`. A staircase diagram consists of a plot of the function F(y), together with the line G(y) `=` y. A vertical segment is drawn from the point (y0, y0) on that line until the point where it intersects the function F. From that point a horizontal segment is drawn until it reaches the point (y1, y1) on the line, and the procedure is repeated n times until the point (yn, yn) is reached. The options are the same as for `plot2d`.

  Example.

      (%i1) staircase(cos(y), 1, 11, [y, 0, 1.2]);

  Categories: Package dynamics · Plotting

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## 54.3 Visualization with VTK

Function scene creates 3D images and animations using the Visualization ToolKit (VTK) software. In order to use that function, Xmaxima and VTK should be installed in your system (including the TCL bindings of VTK, which in some system might come in a separate package).

Function: scene (objects, …, options, …);

  Accepts an empty list or a list of several `objects` and `options`. The program launches Xmaxima, which opens an external window representing the given objects in a 3-dimensional space and applying the options given. Each object must belong to one of the following 4 classes: sphere, cube, cylinder or cone (see `Scene objects`). Objects are identified by giving their name or by a list in which the first element is the class name and the following elements are options for that object.

  Example. A hexagonal pyramid with a blue background:

      (%i1) scene(cone, [background,"#9980e5"])$

  By holding down the left button of the mouse while it is moved on the graphics window, the camera can be rotated showing different views of the pyramid. The two plot options `elevation` and `azimuth` can also be used to change the initial orientation of the viewing camera. The camera can be moved by holding the middle mouse button while moving it and holding the right-side mouse button while moving it up or down will zoom in or out.

  Each object option should be a list starting with the option name, followed by its value. The list of allowed options can be found in the `Scene object's options` section.

  Example. This will show a sphere falling to the ground and bouncing off without losing any energy. To start or pause the animation, press the play/pause button.

      (%i1) p: makelist ([0,0,2.1- 9.8*t^2/2], t, 0, 0.64, 0.01)$

      (%i2) p: append (p, reverse(p))$

      (%i3) ball: [sphere, [radius,0.1], [thetaresolution,20],
        [phiresolution,20], [position,0,0,2.1], [color,red],
        [animate,position,p]]$

      (%i4) ground: [cube, [xlength,2], [ylength,2], [zlength,0.2],
        [position,0,0,-0.1],[color,violet]]$

      (%i5) scene (ball, ground, restart)$

  The restart option was used to make the animation restart automatically every time the last point in the position list is reached. The accepted values for the colors are the same as for the `color` option of plot2d.

  Categories: Package dynamics · Plotting

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### 54.3.1 Scene options

Scene option: azimuth [azimuth, angle]

  Default value: `135`

  The rotation of the camera on the horizontal (x, y) plane. angle must be a real number; an angle of 0 means that the camera points in the direction of the y axis and the x axis will appear on the right.

  Categories: Package dynamics · Plotting

Scene option: background [background, color]

  Default value: `black`

  The color of the graphics window's background. It accepts color names or hexadecimal red-green-blue strings (see the `color` option of plot2d).

  Categories: Package dynamics · Plotting

Scene option: elevation [elevation, angle]

  Default value: `30`

  The vertical rotation of the camera. The angle must be a real number; an angle of 0 means that the camera points on the horizontal, and the default angle of 30 means that the camera is pointing 30 degrees down from the horizontal.

  Categories: Package dynamics · Plotting

Scene option: height [height, pixels]

  Default value: `500`

  The height, in pixels, of the graphics window. pixels must be a positive integer number.

  Categories: Package dynamics · Plotting

Scene option: restart [restart, value]

  Default value: `false`

  A true value means that animations will restart automatically when the end of the list is reached. Writing just "restart" is equivalent to [restart, true].

  Categories: Package dynamics · Plotting

Scene option: tstep [tstep, time]

  Default value: `10`

  The amount of time, in mili-seconds, between iterations among consecutive animation frames. time must be a real number.

  Categories: Package dynamics · Plotting

Scene option: width [width, pixels]

  Default value: `500`

  The width, in pixels, of the graphics window. pixels must be a positive integer number.

  Categories: Package dynamics · Plotting

Scene option: windowname [windowtitle, name]

  Default value: `.scene`

  name must be a string that can be used as the name of the Tk window created by Xmaxima for the `scene` graphics. The default value `.scene` implies that a new top level window will be created.

  Categories: Package dynamics · Plotting

Scene option: windowtitle [windowtitle, name]

  Default value: `Xmaxima: scene`

  name must be a string that will be written in the title of the window created by `scene`.

  Categories: Package dynamics · Plotting

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### 54.3.2 Scene objects

Scene object: cone [cone, options]

  Creates a regular pyramid with height equal to 1 and a hexagonal base with vertices 0.5 units away from the axis. Options `height` and `radius` can be used to change those defaults and option `resolution` can be used to change the number of edges of the base; higher values will make it look like a cone. By default, the axis will be along the x axis, the middle point of the axis will be at the origin and the vertex on the positive side of the x axis; use options `orientation` and `center` to change those defaults.

  Example. This shows a pyramid that starts rotating around the z axis when the play button is pressed.

      (%i1) scene([cone, [orientation,0,30,0], [tstep,100],
         [animate,orientation,makelist([0,30,i],i,5,360,5)]], restart)$

  Categories: Package dynamics · Plotting

Scene object: cube [cube, options]

  A cube with edges of 1 unit and faces parallel to the xy, xz and yz planes. The lengths of the three edges can be changed with options `xlength`, `ylength` and `zlength`, turning it into a rectangular box and the faces can be rotated with option `orientation`.

  Categories: Package dynamics · Plotting

Scene object: cylinder [cylinder, options]

  Creates a regular prism with height equal to 1 and a hexagonal base with vertices 0.5 units away from the axis. Options `height` and `radius` can be used to change those defaults and option `resolution` can be used to change the number of edges of the base; higher values will make it look like a cylinder. The default height can be changed with the option `height`. By default, the axis will be along the x axis and the middle point of the axis will be at the origin; use options `orientation` and `center` to change those defaults.

  Categories: Package dynamics · Plotting

Scene object: sphere [sphere, options]

  A sphere with default radius of 0.5 units and center at the origin.

  Categories: Package dynamics · Plotting

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### 54.3.3 Scene object's options

Object option: animation [animation, property, positions]

  property should be one of the following 4 object's properties: `origin`, `scale`, `position` or `orientation` and positions should be a list of points. When the play button is pressed, the object property will be changed sequentially through all the values in the list, at intervals of time given by the option `tstep`. The rewind button can be used to point at the start of the sequence making the animation restart after the play button is pressed again.

  See also `track`.

  Categories: Package dynamics · Plotting

Object option: capping [capping, number]

  Default value: `1`

  In a cone or a cylinder, it defines whether the base (or bases) will be shown. A value of 1 for number makes the base visible and a value of 0 makes it invisible.

  Categories: Package dynamics · Plotting

Object option: center [center, point]

  Default value: `[0, 0, 0]`

  The coordinates of the object's geometric center, with respect to its `position`. point can be a list with 3 real numbers, or 3 real numbers separated by commas. In a cylinder, cone or cube it will be at half its height and in a sphere at its center.

  Categories: Package dynamics · Plotting

Object option: color [color, colorname]

  Default value: `white`

  The color of the object. It accepts color names or hexadecimal red-green-blue strings (see the `color` option of plot2d).

  Categories: Package dynamics · Plotting

Object option: endphi [endphi, angle]

  Default value: `180`

  In a sphere phi is the angle on the vertical plane that passes through the z axis, measured from the positive part of the z axis. angle must be a number between 0 and 180 that sets the final value of phi at which the surface will end. A value smaller than 180 will eliminate a part of the sphere's surface.

  See also `startphi` and `phiresolution`.

  Categories: Package dynamics · Plotting

Object option: endtheta [endtheta, angle]

  Default value: `360`

  In a sphere theta is the angle on the horizontal plane (longitude), measured from the positive part of the x axis. angle must be a number between 0 and 360 that sets the final value of theta at which the surface will end. A value smaller than 360 will eliminate a part of the sphere's surface.

  See also `starttheta` and `thetaresolution`.

  Categories: Package dynamics · Plotting

Object option: height [height, value]

  Default value: `1`

  value must be a positive number which sets the height of a cone or a cylinder.

  Categories: Package dynamics · Plotting

Object option: linewidth [linewidth, value]

  Default value: `1`

  The width of the lines, when option `wireframe` is used. value must be a positive number.

  Categories: Package dynamics · Plotting

Object option: opacity [opacity, value]

  Default value: `1`

  value must be a number between 0 and 1. The lower the number, the more transparent the object will become. The default value of 1 means a completely opaque object.

  Categories: Package dynamics · Plotting

Object option: orientation [orientation, angles]

  Default value: `[0, 0, 0]`

  Three angles by which the object will be rotated with respect to the three axis. angles can be a list with 3 real numbers, or 3 real numbers separated by commas. Example: `[0, 0, 90]` rotates the x axis of the object to the y axis of the reference frame.

  Categories: Package dynamics · Plotting

Object option: origin [origin, point]

  Default value: `[0, 0, 0]`

  The coordinates of the object's origin, with respect to which its other dimensions are defined. point can be a list with 3 real numbers, or 3 real numbers separated by commas.

  Categories: Package dynamics · Plotting

Object option: phiresolution [phiresolution, num]

  Default value: ``

  The number of sub-intervals into which the phi angle interval from `startphi` to `endphi` will be divided. num must be a positive integer.

  See also `startphi` and `endphi`.

  Categories: Package dynamics · Plotting

Object option: points [points]

  Only the vertices of the triangulation used to render the surface will be shown. Example: `[sphere, [points]]`

  See also `surface` and `wireframe`.

  Categories: Package dynamics · Plotting

Object option: pointsize [pointsize, value]

  Default value: `1`

  The size of the points, when option `points` is used. value must be a positive number.

  Categories: Package dynamics · Plotting

Object option: position [position, point]

  Default value: `[0, 0, 0]`

  The coordinates of the object's position. point can be a list with 3 real numbers, or 3 real numbers separated by commas.

  Categories: Package dynamics · Plotting

Object option: radius [radius, value]

  Default value: `0.5`

  The radius or a sphere or the distance from the axis to the base's vertices in a cylinder or a cone. value must be a positive number.

  Categories: Package dynamics · Plotting

Object option: resolution [resolution, number]

  Default value: `6`

  number must be a integer greater than 2 that sets the number of edges in the base of a cone or a cylinder.

  Categories: Package dynamics · Plotting

Object option: scale [scale, factors]

  Default value: `[1, 1, 1]`

  Three numbers by which the object will be scaled with respect to the three axis. factors can be a list with 3 real numbers, or 3 real numbers separated by commas. Example: `[2, 0.5, 1]` enlarges the object to twice its size in the x direction, reduces the dimensions in the y direction to half and leaves the z dimensions unchanged.

  Categories: Package dynamics · Plotting

Object option: startphi [startphi, angle]

  Default value: `0`

  In a sphere phi is the angle on the vertical plane that passes through the z axis, measured from the positive part of the z axis. angle must be a number between 0 and 180 that sets the initial value of phi at which the surface will start. A value bigger than 0 will eliminate a part of the sphere's surface.

  See also `endphi` and `phiresolution`.

  Categories: Package dynamics · Plotting

Object option: starttheta [starttheta, angle]

  Default value: `0`

  In a sphere theta is the angle on the horizontal plane (longitude), measured from the positive part of the x axis. angle must be a number between 0 and 360 that sets the initial value of theta at which the surface will start. A value bigger than 0 will eliminate a part of the sphere's surface.

  See also `endtheta` and `thetaresolution`.

  Categories: Package dynamics · Plotting

Object option: surface [surface]

  The surfaces of the object will be rendered and the lines and points of the triangulation used to build the surface will not be shown. This is the default behavior, which can be changed using either the option `points` or `wireframe`.

  Categories: Package dynamics · Plotting

Object option: thetaresolution [thetaresolution, num]

  Default value: ``

  The number of sub-intervals into which the theta angle interval from `starttheta` to `endtheta` will be divided. num must be a positive integer.

  See also `starttheta` and `endtheta`.

  Categories: Package dynamics · Plotting

Object option: track [track, positions]

  positions should be a list of points. When the play button is pressed, the object position will be changed sequentially through all the points in the list, at intervals of time given by the option `tstep`, leaving behind a track of the object's trajectory. The rewind button can be used to point at the start of the sequence making the animation restart after the play button is pressed again.

  Example. This will show the trajectory of a ball thrown with speed of 5 m/s, at an angle of 45 degrees, when the air resistance can be neglected:

      (%i1) p: makelist ([0,4*t,4*t- 9.8*t^2/2], t, 0, 0.82, 0.01)$

      (%i2) ball: [sphere, [radius,0.1], [color,red], [track,p]]$

      (%i3) ground: [cube, [xlength,2], [ylength,4], [zlength,0.2],
            [position,0,1.5,-0.2],[color,green]]$

      (%i4) scene (ball, ground)$

  See also `animation`.

  Categories: Package dynamics · Plotting

Object option: xlength [xlength, length]

  Default value: `1`

  The height of a cube in the x direction. length must be a positive number. See also `ylength` and `zlength`.

  Categories: Package dynamics · Plotting

Object option: ylength [ylength, length]

  Default value: `1`

  The height of a cube in the y direction. length must be a positive number. See also `xlength` and `zlength`.

  Categories: Package dynamics · Plotting

Object option: zlength [zlength, length]

  Default value: `1`

  The height of a cube in z the direction. length must be a positive number. See also `xlength` and `ylength`.

  Categories: Package dynamics · Plotting

Object option: wireframe [wireframe]

  Only the edges of the triangulation used to render the surface will be shown. Example: `[cube, [wireframe]]`

  See also `surface` and `points`.

  Categories: Package dynamics · Plotting

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# 55. engineering-format

Engineering-format changes the way maxima outputs floating-point numbers to the notation engineers are used to: `a*10^b` with `b` dividable by three.

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## 55.1 Functions and Variables for engineering-format

Option variable: engineering_format_floats

  Default value: `true`

  This variable allows to temporarily switch off engineering-format.

      (%i1) load("engineering-format");
      (%o1)
          /maxima/share/contrib/engineering-format.lisp
      (%i2) float(sin(10)/10000);
      (%o2)                - 54.40211108893698e-6
      (%i3) engineering_format_floats:false$
      (%i4) float(sin(10)/10000);
      (%o4)                - 5.440211108893698e-5

  See also `fpprintprec` and `float`.

  Categories: Display functions · Global flags · Share packages

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## 55.2 Known Bugs

The output routine of SBCL 1.3.0 has a bug that sometimes causes the exponent not to be dividable by three. The value of the displayed number is still valid in this case.

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# 56. ezunits

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## 56.1 Introduction to ezunits

`ezunits` is a package for working with dimensional quantities, including some functions for dimensional analysis. `ezunits` can carry out arithmetic operations on dimensional quantities and unit conversions. The built-in units include Systeme Internationale (SI) and US customary units, and other units can be declared. See also `physical_constants`, a collection of physical constants.

`load(ezunits)` loads this package. `demo(ezunits)` displays several examples. The convenience function `known_units` returns a list of the built-in and user-declared units, while `display_known_unit_conversions` displays the set of known conversions in an easy-to-read format.

An expression a ` b represents a dimensional quantity, with `a` indicating a nondimensional quantity and `b` indicating the dimensional units. A symbol can be used as a unit without declaring it as such; unit symbols need not have any special properties. The quantity and unit of an expression a ` b can be extracted by the `qty` and `units` functions, respectively.

A symbol may be declared to be a dimensional quantity, with specified quantity or specified units or both.

An expression a ` b `` c converts from unit `b` to unit `c`. `ezunits` has built-in conversions for SI base units, SI derived units, and some non-SI units. Unit conversions not already known to `ezunits` can be declared. The unit conversions known to `ezunits` are specified by the global variable `known_unit_conversions`, which comprises built-in and user-defined conversions. Conversions for products, quotients, and powers of units are derived from the set of known unit conversions.

As Maxima generally prefers exact numbers (integers or rationals) to inexact (float or bigfloat), so `ezunits` preserves exact numbers when they appear in dimensional quantities. All built-in unit conversions are expressed in terms of exact numbers; inexact numbers in declared conversions are coerced to exact.

There is no preferred system for display of units; input units are not converted to other units unless conversion is explicitly indicated. `ezunits` recognizes the prefixes m-, k-, M, and G- (for milli-, kilo-, mega-, and giga-) as applied to SI base units and SI derived units, but such prefixes are applied only when indicated by an explicit conversion.

Arithmetic operations on dimensional quantities are carried out by conventional rules for such operations.

`ezunits` does not require that units in a sum have the same dimensions; such terms are not added together, and no error is reported.

`ezunits` includes functions for elementary dimensional analysis, namely the fundamental dimensions and fundamental units of a dimensional quantity, and computation of dimensionless quantities and natural units. The functions for dimensional analysis were adapted from similar functions in another package, written by Barton Willis.

For the purpose of dimensional analysis, a list of fundamental dimensions and an associated list of fundamental units are maintained; by default the fundamental dimensions are length, mass, time, charge, temperature, and quantity, and the fundamental units are the associated SI units, but other fundamental dimensions and units can be declared.

Categories: Physical units · Share packages · Package ezunits

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## 56.2 Introduction to physical_constants

`physical_constants` is a collection of physical constants, copied from CODATA 2006 recommended values (http://physics.nist.gov/constants). `load(physical_constants)` loads this package, and loads `ezunits` also, if it is not already loaded.

A physical constant is represented as a symbol which has a property which is the constant value. The constant value is a dimensional quantity, as represented by `ezunits`. The function `constvalue` fetches the constant value; the constant value is not the ordinary value of the symbol, so symbols of physical constants persist in evaluated expressions until their values are fetched by `constvalue`.

`physical_constants` includes some auxiliary information, namely, a description string for each constant, an estimate of the error of its numerical value, and a property for TeX display. To identify physical constants, each symbol has the `physical_constant` property; `propvars(physical_constant)` therefore shows the list of all such symbols.

`physical_constants` comprises the following constants.

`%c`

  speed of light in vacuum

`%mu_0`

  magnetic constant

`%e_0`

  electric constant

`%Z_0`

  characteristic impedance of vacuum

`%G`

  Newtonian constant of gravitation

`%h`

  Planck constant

`%h_bar`

  Planck constant

`%m_P`

  Planck mass

`%T_P`

  Planck temperature

`%l_P`

  Planck length

`%t_P`

  Planck time

`%%e`

  elementary charge

`%Phi_0`

  magnetic flux quantum

`%G_0`

  conductance quantum

`%K_J`

  Josephson constant

`%R_K`

  von Klitzing constant

`%mu_B`

  Bohr magneton

`%mu_N`

  nuclear magneton

`%alpha`

  fine-structure constant

`%R_inf`

  Rydberg constant

`%a_0`

  Bohr radius

`%E_h`

  Hartree energy

`%ratio_h_me`

  quantum of circulation

`%m_e`

  electron mass

`%N_A`

  Avogadro constant

`%m_u`

  atomic mass constant

`%F`

  Faraday constant

`%R`

  molar gas constant

`%%k`

  Boltzmann constant

`%V_m`

  molar volume of ideal gas

`%n_0`

  Loschmidt constant

`%ratio_S0_R`

  Sackur-Tetrode constant (absolute entropy constant)

`%sigma`

  Stefan-Boltzmann constant

`%c_1`

  first radiation constant

`%c_1L`

  first radiation constant for spectral radiance

`%c_2`

  second radiation constant

`%b`

  Wien displacement law constant

`%b_prime`

  Wien displacement law constant

Reference: http://physics.nist.gov/constants

Examples:

The list of all symbols which have the `physical_constant` property.

    (%i1) load ("physical_constants")$
    (%i2) propvars (physical_constant);
    (%o2) [%c, %mu_0, %e_0, %Z_0, %G, %h, %h_bar, %m_P, %T_P, %l_P,
    %t_P, %%e, %Phi_0, %G_0, %K_J, %R_K, %mu_B, %mu_N, %alpha,
    %R_inf, %a_0, %E_h, %ratio_h_me, %m_e, %N_A, %m_u, %F, %R, %%k,
    %V_m, %n_0, %ratio_S0_R, %sigma, %c_1, %c_1L, %c_2, %b, %b_prime]

Properties of the physical constant `%c`.

    (%i1) load ("physical_constants")$
    (%i2) constantp (%c);
    (%o2)                         true
    (%i3) get (%c, description);
    (%o3)               speed of light in vacuum
    (%i4) constvalue (%c);
                                          m
    (%o4)                     299792458 ` -
                                          s
    (%i5) get (%c, RSU);
    (%o5)                           0
    (%i6) tex (%c);
    $$c$$
    (%o6)                         false

The energy equivalent of 1 pound-mass. The symbol `%c` persists until its value is fetched by `constvalue`.

    (%i1) load ("physical_constants")$
    (%i2) m * %c^2;
                                    2
    (%o2)                         %c  m
    (%i3) %, m = 1 ` lbm;
                                  2
    (%o3)                       %c  ` lbm
    (%i4) constvalue (%);
                                                2
                                           lbm m
    (%o4)              89875517873681764 ` ------
                                              2
                                             s
    (%i5) E : % `` J;
    Computing conversions to base units; may take a moment.
                         366838848464007200
    (%o5)                ------------------ ` J
                                 9
    (%i6) E `` GJ;
                          458548560580009
    (%o6)                 --------------- ` GJ
                             11250000
    (%i7) float (%);
    (%o7)              4.0759872051556356e+7 ` GJ

Categories: Physical units · Share packages · Package physical_constants

[ < ] | [ > ] |  | [ << ] | [ Up ] | [ >> ] |  |  |  |  | [Top] | [Contents] | [Index] | [ ? ]

## 56.3 Functions and Variables for ezunits

Operator: `

  The dimensional quantity operator. An expression a ` b represents a dimensional quantity, with `a` indicating a nondimensional quantity and `b` indicating the dimensional units. A symbol can be used as a unit without declaring it as such; unit symbols need not have any special properties. The quantity and unit of an expression a ` b can be extracted by the `qty` and `units` functions, respectively.

  Arithmetic operations on dimensional quantities are carried out by conventional rules for such operations.

  `ezunits` does not require that units in a sum have the same dimensions; such terms are not added together, and no error is reported.

  `load(ezunits)` enables this operator.

  Examples:

  SI (Systeme Internationale) units.

      (%i1) load ("ezunits")$
      (%i2) foo : 10 ` m;
      (%o2)                        10 ` m
      (%i3) qty (foo);
      (%o3)                          10
      (%i4) units (foo);
      (%o4)                           m
      (%i5) dimensions (foo);
      (%o5)                        length

  "Customary" units.

      (%i1) load ("ezunits")$
      (%i2) bar : x ` acre;
      (%o2)                       x ` acre
      (%i3) dimensions (bar);
                                         2
      (%o3)                        length
      (%i4) fundamental_units (bar);
                                      2
      (%o4)                          m

  Units ad hoc.

      (%i1) load ("ezunits")$
      (%i2) baz : 3 ` sheep + 8 ` goat + 1 ` horse;
      (%o2)           8 ` goat + 3 ` sheep + 1 ` horse
      (%i3) subst ([sheep = 3*goat, horse = 10*goat], baz);
      (%o3)                       27 ` goat
      (%i4) baz2 : 1000`gallon/fortnight;
                                      gallon
      (%o4)                   1000 ` ---------
                                     fortnight
      (%i5) subst (fortnight = 14*day, baz2);
                                500   gallon
      (%o5)                     --- ` ------
                                 7     day

  Arithmetic operations on dimensional quantities.

      (%i1) load ("ezunits")$
      (%i2) 100 ` kg + 200 ` kg;
      (%o2)                       300 ` kg
      (%i3) 100 ` m^3 - 100 ` m^3;
                                        3
      (%o3)                        0 ` m
      (%i4) (10 ` kg) * (17 ` m/s^2);
                                       kg m
      (%o4)                      170 ` ----
                                         2
                                        s
      (%i5) (x ` m) / (y ` s);
                                    x   m
      (%o5)                         - ` -
                                    y   s
      (%i6) (a ` m)^2;
                                    2    2
      (%o6)                        a  ` m

  Categories: Package ezunits

Operator: ``

  The unit conversion operator. An expression a ` b `` c converts from unit `b` to unit `c`. `ezunits` has built-in conversions for SI base units, SI derived units, and some non-SI units. Unit conversions not already known to `ezunits` can be declared. The unit conversions known to `ezunits` are specified by the global variable `known_unit_conversions`, which comprises built-in and user-defined conversions. Conversions for products, quotients, and powers of units are derived from the set of known unit conversions.

  There is no preferred system for display of units; input units are not converted to other units unless conversion is explicitly indicated. `ezunits` does not attempt to simplify units by prefixes (milli-, centi-, deci-, etc) unless such conversion is explicitly indicated.

  `load(ezunits)` enables this operator.

  Examples:

  The set of known unit conversions.

      (%i1) load ("ezunits")$
      (%i2) display2d : false$
      (%i3) known_unit_conversions;
      (%o3) {acre = 4840*yard^2,Btu = 1055*J,cfm = feet^3/minute,
             cm = m/100,day = 86400*s,feet = 381*m/1250,ft = feet,
             g = kg/1000,gallon = 757*l/200,GHz = 1000000000*Hz,
             GOhm = 1000000000*Ohm,GPa = 1000000000*Pa,
             GWb = 1000000000*Wb,Gg = 1000000*kg,Gm = 1000000000*m,
             Gmol = 1000000*mol,Gs = 1000000000*s,ha = hectare,
             hectare = 100*m^2,hour = 3600*s,Hz = 1/s,inch = feet/12,
             km = 1000*m,kmol = 1000*mol,ks = 1000*s,l = liter,
             lbf = pound_force,lbm = pound_mass,liter = m^3/1000,
             metric_ton = Mg,mg = kg/1000000,MHz = 1000000*Hz,
             microgram = kg/1000000000,micrometer = m/1000000,
             micron = micrometer,microsecond = s/1000000,
             mile = 5280*feet,minute = 60*s,mm = m/1000,
             mmol = mol/1000,month = 2629800*s,MOhm = 1000000*Ohm,
             MPa = 1000000*Pa,ms = s/1000,MWb = 1000000*Wb,
             Mg = 1000*kg,Mm = 1000000*m,Mmol = 1000000000*mol,
             Ms = 1000000*s,ns = s/1000000000,ounce = pound_mass/16,
             oz = ounce,Ohm = s*J/C^2,
             pound_force = 32*ft*pound_mass/s^2,
             pound_mass = 200*kg/441,psi = pound_force/inch^2,
             Pa = N/m^2,week = 604800*s,Wb = J/A,yard = 3*feet,
             year = 31557600*s,C = s*A,F = C^2/J,GA = 1000000000*A,
             GC = 1000000000*C,GF = 1000000000*F,GH = 1000000000*H,
             GJ = 1000000000*J,GK = 1000000000*K,GN = 1000000000*N,
             GS = 1000000000*S,GT = 1000000000*T,GV = 1000000000*V,
             GW = 1000000000*W,H = J/A^2,J = m*N,kA = 1000*A,
             kC = 1000*C,kF = 1000*F,kH = 1000*H,kHz = 1000*Hz,
             kJ = 1000*J,kK = 1000*K,kN = 1000*N,kOhm = 1000*Ohm,
             kPa = 1000*Pa,kS = 1000*S,kT = 1000*T,kV = 1000*V,
             kW = 1000*W,kWb = 1000*Wb,mA = A/1000,mC = C/1000,
             mF = F/1000,mH = H/1000,mHz = Hz/1000,mJ = J/1000,
             mK = K/1000,mN = N/1000,mOhm = Ohm/1000,mPa = Pa/1000,
             mS = S/1000,mT = T/1000,mV = V/1000,mW = W/1000,
             mWb = Wb/1000,MA = 1000000*A,MC = 1000000*C,
             MF = 1000000*F,MH = 1000000*H,MJ = 1000000*J,
             MK = 1000000*K,MN = 1000000*N,MS = 1000000*S,
             MT = 1000000*T,MV = 1000000*V,MW = 1000000*W,
             N = kg*m/s^2,R = 5*K/9,S = 1/Ohm,T = J/(m^2*A),V = J/C,
             W = J/s}

  Elementary unit conversions.

      (%i1) load ("ezunits")$
      (%i2) 1 ` ft `` m;
      Computing conversions to base units; may take a moment.
                                  381
      (%o2)                       ---- ` m
                                  1250
      (%i3) %, numer;
      (%o3)                      0.3048 ` m
      (%i4) 1 ` kg `` lbm;
                                  441
      (%o4)                       --- ` lbm
                                  200
      (%i5) %, numer;
      (%o5)                      2.205 ` lbm
      (%i6) 1 ` W `` Btu/hour;
                                 720   Btu
      (%o6)                      --- ` ----
                                 211   hour
      (%i7) %, numer;
                                              Btu
      (%o7)               3.412322274881517 ` ----
                                              hour
      (%i8) 100 ` degC `` degF;
      (%o8)                      212 ` degF
      (%i9) -40 ` degF `` degC;
      (%o9)                     (- 40) ` degC
      (%i10) 1 ` acre*ft `` m^3;
                              60228605349    3
      (%o10)                  ----------- ` m
                               48828125
      (%i11) %, numer;
                                                3
      (%o11)                1233.48183754752 ` m

  Coercing quantities in feet and meters to one or the other.

      (%i1) load ("ezunits")$
      (%i2) 100 ` m + 100 ` ft;
      (%o2)                  100 ` m + 100 ` ft
      (%i3) (100 ` m + 100 ` ft) `` ft;
      Computing conversions to base units; may take a moment.
                                 163100
      (%o3)                      ------ ` ft
                                  381
      (%i4) %, numer;
      (%o4)                428.0839895013123 ` ft
      (%i5) (100 ` m + 100 ` ft) `` m;
                                  3262
      (%o5)                       ---- ` m
                                   25
      (%i6) %, numer;
      (%o6)                      130.48 ` m

  Dimensional analysis to find fundamental dimensions and fundamental units.

      (%i1) load ("ezunits")$
      (%i2) foo : 1 ` acre * ft;
      (%o2)                      1 ` acre ft
      (%i3) dimensions (foo);
                                         3
      (%o3)                        length
      (%i4) fundamental_units (foo);
                                      3
      (%o4)                          m
      (%i5) foo `` m^3;
      Computing conversions to base units; may take a moment.
                              60228605349    3
      (%o5)                   ----------- ` m
                               48828125
      (%i6) %, numer;
                                                3
      (%o6)                 1233.48183754752 ` m

  Declared unit conversions.

      (%i1) load ("ezunits")$
      (%i2) declare_unit_conversion (MMBtu = 10^6*Btu, kW = 1000*W);
      (%o2)                         done
      (%i3) declare_unit_conversion (kWh = kW*hour, MWh = 1000*kWh,
                                     bell = 1800*s);
      (%o3)                         done
      (%i4) 1 ` kW*s `` MWh;
      Computing conversions to base units; may take a moment.
                                   1
      (%o4)                     ------- ` MWh
                                3600000
      (%i5) 1 ` kW/m^2 `` MMBtu/bell/ft^2;
                             1306449      MMBtu
      (%o5)                 ---------- ` --------
                            8242187500          2
                                         bell ft

  Categories: Package ezunits

Function: constvalue (x)

  Shows the value and the units of one of the constants declared by package `physical_constants`, which includes a list of physical constants, or of a new constant declared in package `ezunits` (see `declare_constvalue`).

  Note that constant values as recognized by `constvalue` are separate from values declared by `numerval` and recognized by `constantp`.

  Example:

      (%i1) load ("physical_constants")$
      (%i2) constvalue (%G);
                                           3
                                          m
      (%o2)                    6.67428 ` -----
                                             2
                                         kg s
      (%i3) get ('%G, 'description);
      (%o3)           Newtonian constant of gravitation

  Categories: Package ezunits

Function: declare_constvalue (a, x)

  Declares the value of a constant to be used in package `ezunits`. This function should be loaded with `load(ezunits)`.

  Example:

      (%i1) load ("ezunits")$
      (%i2) declare_constvalue (FOO, 100 ` lbm / acre);
                                       lbm
      (%o2)                      100 ` ----
                                       acre
      (%i3) FOO * (50 ` acre);
      (%o3)                     50 FOO ` acre
      (%i4) constvalue (%);
      (%o4)                      5000 ` lbm

  Categories: Package ezunits

Function: remove_constvalue (a)

  Reverts the effect of `declare_constvalue`. This function should be loaded with `load(ezunits)`.

  Categories: Package ezunits

Function: units (x)

  Returns the units of a dimensional quantity x, or returns 1 if x is nondimensional.

  x may be a literal dimensional expression a ` b, a symbol with declared units via `declare_units`, or an expression containing either or both of those.

  This function should be loaded with `load(ezunits)`.

  Example:

      (%i1) load ("ezunits")$
      (%i2) foo : 100 ` kg;
      (%o2)                       100 ` kg
      (%i3) bar : x ` m/s;
                                        m
      (%o3)                         x ` -
                                        s
      (%i4) units (foo);
      (%o4)                          kg
      (%i5) units (bar);
                                      m
      (%o5)                           -
                                      s
      (%i6) units (foo * bar);
                                    kg m
      (%o6)                         ----
                                     s
      (%i7) units (foo / bar);
                                    kg s
      (%o7)                         ----
                                     m
      (%i8) units (foo^2);
                                       2
      (%o8)                          kg

  Categories: Package ezunits

Function: declare_units (a, u)

  Declares that `units` should return units u for a, where u is an expression. This function should be loaded with `load(ezunits)`.

  Example:

      (%i1) load ("ezunits")$
      (%i2) units (aa);
      (%o2)                           1
      (%i3) declare_units (aa, J);
      (%o3)                           J
      (%i4) units (aa);
      (%o4)                           J
      (%i5) units (aa^2);
                                      2
      (%o5)                          J
      (%i6) foo : 100 ` kg;
      (%o6)                       100 ` kg
      (%i7) units (aa * foo);
      (%o7)                         kg J

  Categories: Package ezunits

Function: qty (x)

  Returns the nondimensional part of a dimensional quantity x, or returns x if x is nondimensional. x may be a literal dimensional expression a ` b, a symbol with declared quantity, or an expression containing either or both of those.

  This function should be loaded with `load(ezunits)`.

  Example:

      (%i1) load ("ezunits")$
      (%i2) foo : 100 ` kg;
      (%o2)                       100 ` kg
      (%i3) qty (foo);
      (%o3)                          100
      (%i4) bar : v ` m/s;
                                        m
      (%o4)                         v ` -
                                        s
      (%i5) foo * bar;
                                        kg m
      (%o5)                     100 v ` ----
                                         s
      (%i6) qty (foo * bar);
      (%o6)                         100 v

  Categories: Package ezunits

Function: declare_qty (a, x)

  Declares that `qty` should return x for symbol a, where x is a nondimensional quantity. This function should be loaded with `load(ezunits)`.

  Example:

      (%i1) load ("ezunits")$
      (%i2) declare_qty (aa, xx);
      (%o2)                          xx
      (%i3) qty (aa);
      (%o3)                          xx
      (%i4) qty (aa^2);
                                       2
      (%o4)                          xx
      (%i5) foo : 100 ` kg;
      (%o5)                       100 ` kg
      (%i6) qty (aa * foo);
      (%o6)                        100 xx

  Categories: Package ezunits

Function: unitp (x)

  Returns `true` if x is a literal dimensional expression, a symbol declared dimensional, or an expression in which the main operator is declared dimensional. `unitp` returns `false` otherwise.

  `load(ezunits)` loads this function.

  Examples:

  `unitp` applied to a literal dimensional expression.

      (%i1) load ("ezunits")$
      (%i2) unitp (100 ` kg);
      (%o2)                         true

  `unitp` applied to a symbol declared dimensional.

      (%i1) load ("ezunits")$
      (%i2) unitp (foo);
      (%o2)                         false
      (%i3) declare (foo, dimensional);
      (%o3)                         done
      (%i4) unitp (foo);
      (%o4)                         true

  `unitp` applied to an expression in which the main operator is declared dimensional.

      (%i1) load ("ezunits")$
      (%i2) unitp (bar (x, y, z));
      (%o2)                         false
      (%i3) declare (bar, dimensional);
      (%o3)                         done
      (%i4) unitp (bar (x, y, z));
      (%o4)                         true

  Categories: Package ezunits

Function: declare_unit_conversion (u = v, ...)

  Appends equations u = v, ... to the list of unit conversions known to the unit conversion operator ``. u and v are both multiplicative terms, in which any variables are units, or both literal dimensional expressions.

  At present, it is necessary to express conversions such that the left-hand side of each equation is a simple unit (not a multiplicative expression) or a literal dimensional expression with the quantity equal to 1 and the unit being a simple unit. This limitation might be relaxed in future versions.

  `known_unit_conversions` is the list of known unit conversions.

  This function should be loaded with `load(ezunits)`.

  Examples:

  Unit conversions expressed by equations of multiplicative terms.

      (%i1) load ("ezunits")$
      (%i2) declare_unit_conversion (nautical_mile = 1852 * m,
                                     fortnight = 14 * day);
      (%o2)                         done
      (%i3) 100 ` nautical_mile / fortnight `` m/s;
      Computing conversions to base units; may take a moment.
                                  463    m
      (%o3)                       ---- ` -
                                  3024   s

  Unit conversions expressed by equations of literal dimensional expressions.

      (%i1) load ("ezunits")$
      (%i2) declare_unit_conversion (1 ` fluid_ounce = 2 ` tablespoon);
      (%o2)                         done
      (%i3) declare_unit_conversion (1 ` tablespoon = 3 ` teaspoon);
      (%o3)                         done
      (%i4) 15 ` fluid_ounce `` teaspoon;
      Computing conversions to base units; may take a moment.
      (%o4)                     90 ` teaspoon

  Categories: Package ezunits

Function: declare_dimensions (a_1, d_1, ..., a_n, d_n)

  Declares a_1, ..., a_n to have dimensions d_1, ..., d_n, respectively.

  Each a_k is a symbol or a list of symbols. If it is a list, then every symbol in a_k is declared to have dimension d_k.

  `load(ezunits)` loads these functions.

  Examples:

      (%i1) load ("ezunits") $
      (%i2) declare_dimensions ([x, y, z], length, [t, u], time);
      (%o2)                         done
      (%i3) dimensions (y^2/u);
                                         2
                                   length
      (%o3)                        -------
                                    time
      (%i4) fundamental_units (y^2/u);
      0 errors, 0 warnings
                                      2
                                     m
      (%o4)                          --
                                     s

  Categories: Package ezunits

Function: remove_dimensions (a_1, ..., a_n)

  Reverts the effect of `declare_dimensions`. This function should be loaded with `load(ezunits)`.

  Categories: Package ezunits

Function: declare_fundamental_dimensions (d_1, d_2, d_3, ...)
Function: remove_fundamental_dimensions (d_1, d_2, d_3, ...)
Global variable: fundamental_dimensions

  `declare_fundamental_dimensions` declares fundamental dimensions. Symbols d_1, d_2, d_3, ... are appended to the list of fundamental dimensions, if they are not already on the list.

  `remove_fundamental_dimensions` reverts the effect of `declare_fundamental_dimensions`.

  `fundamental_dimensions` is the list of fundamental dimensions. By default, the list comprises several physical dimensions.

  `load(ezunits)` loads these functions.

  Examples:

      (%i1) load ("ezunits") $
      (%i2) fundamental_dimensions;
      (%o2) [length, mass, time, current, temperature, quantity]
      (%i3) declare_fundamental_dimensions (money, cattle, happiness);
      (%o3)                         done
      (%i4) fundamental_dimensions;
      (%o4) [length, mass, time, current, temperature, quantity,
                                              money, cattle, happiness]
      (%i5) remove_fundamental_dimensions (cattle, happiness);
      (%o5)                         done
      (%i6) fundamental_dimensions;
      (%o6) [length, mass, time, current, temperature, quantity, money]

  Categories: Package ezunits

Function: declare_fundamental_units (u_1, d_1, ..., u_n, d_n)
Function: remove_fundamental_units (u_1, ..., u_n)

  `declare_fundamental_units` declares u_1, ..., u_n to have dimensions d_1, ..., d_n, respectively. All arguments must be symbols.

  After calling `declare_fundamental_units`, `dimensions(u_k)` returns d_k for each argument u_1, ..., u_n, and `fundamental_units(d_k)` returns u_k for each argument d_1, ..., d_n.

  `remove_fundamental_units` reverts the effect of `declare_fundamental_units`.

  `load(ezunits)` loads these functions.

  Examples:

      (%i1) load ("ezunits") $
      (%i2) declare_fundamental_dimensions (money, cattle, happiness);
      (%o2)                         done
      (%i3) declare_fundamental_units (dollar, money, goat, cattle,
                                       smile, happiness);
      (%o3)                 [dollar, goat, smile]
      (%i4) dimensions (100 ` dollar/goat/km^2);
                                   money
      (%o4)                    --------------
                                            2
                               cattle length
      (%i5) dimensions (x ` smile/kg);
                                  happiness
      (%o5)                       ---------
                                    mass
      (%i6) fundamental_units (money*cattle/happiness);
      0 errors, 0 warnings
                                 dollar goat
      (%o6)                      -----------
                                    smile

  Categories: Package ezunits

Function: dimensions (x)
Function: dimensions_as_list (x)

  `dimensions` returns the dimensions of the dimensional quantity x as an expression comprising products and powers of base dimensions.

  `dimensions_as_list` returns the dimensions of the dimensional quantity x as a list, in which each element is an integer which indicates the power of the corresponding base dimension in the dimensions of x.

  `load(ezunits)` loads these functions.

  Examples:

      (%i1) load ("ezunits")$
      (%i2) dimensions (1000 ` kg*m^2/s^3);
                                      2
                                length  mass
      (%o2)                     ------------
                                       3
                                   time
      (%i3) declare_units (foo, acre*ft/hour);
                                   acre ft
      (%o3)                        -------
                                    hour
      (%i4) dimensions (foo);
                                         3
                                   length
      (%o4)                        -------
                                    time

      (%i1) load ("ezunits")$
      (%i2) fundamental_dimensions;
      (%o2)  [length, mass, time, charge, temperature, quantity]
      (%i3) dimensions_as_list (1000 ` kg*m^2/s^3);
      (%o3)                 [2, 1, - 3, 0, 0, 0]
      (%i4) declare_units (foo, acre*ft/hour);
                                   acre ft
      (%o4)                        -------
                                    hour
      (%i5) dimensions_as_list (foo);
      (%o5)                 [3, 0, - 1, 0, 0, 0]

  Categories: Package ezunits

Function: fundamental_units
`fundamental_units` (x)
`fundamental_units` ()

  `fundamental_units(x)` returns the units associated with the fundamental dimensions of x. as determined by `dimensions(x)`.

  x may be a literal dimensional expression a ` b, a symbol with declared units via `declare_units`, or an expression containing either or both of those.

  `fundamental_units()` returns the list of all known fundamental units, as declared by `declare_fundamental_units`.

  `load(ezunits)` loads this function.

  Examples:

      (%i1) load ("ezunits")$
      (%i2) fundamental_units ();
      (%o2)                 [m, kg, s, A, K, mol]
      (%i3) fundamental_units (100 ` mile/hour);
                                      m
      (%o3)                           -
                                      s
      (%i4) declare_units (aa, g/foot^2);
                                      g
      (%o4)                         -----
                                        2
                                    foot
      (%i5) fundamental_units (aa);
                                     kg
      (%o5)                          --
                                      2
                                     m

  Categories: Package ezunits

Function: dimensionless (L)

  Returns a basis for the dimensionless quantities which can be formed from a list L of dimensional quantities.

  `load(ezunits)` loads this function.

  Examples:

      (%i1) load ("ezunits") $
      (%i2) dimensionless ([x ` m, y ` m/s, z ` s]);
      0 errors, 0 warnings
      0 errors, 0 warnings
                                     y z
      (%o2)                         [---]
                                      x

  Dimensionless quantities derived from fundamental physical quantities. Note that the first element on the list is proportional to the fine-structure constant.

      (%i1) load ("ezunits") $
      (%i2) load ("physical_constants") $
      (%i3) dimensionless([%h_bar, %m_e, %m_P, %%e, %c, %e_0]);
      0 errors, 0 warnings
      0 errors, 0 warnings
                                    2
                                 %%e        %m_e
      (%o3)                [--------------, ----]
                            %c %e_0 %h_bar  %m_P

  Categories: Package ezunits

Function: natural_unit (expr, [v_1, ..., v_n])

  Finds exponents e_1, ..., e_n such that `dimension(expr) = dimension(v_1^e_1 ... v_n^e_n)`.

  `load(ezunits)` loads this function.

  Examples:

  Categories: Package ezunits

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# 57. f90

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## 57.1 Functions and Variables for f90

Function: f90 (expr_1, …, expr_n)

  Prints one or more expressions expr_1, …, expr_n as a Fortran 90 program. Output is printed to the standard output.

  `f90` prints output in the so-called "free form" input format for Fortran 90: there is no special attention to column positions. Long lines are split at a fixed width with the ampersand `&` continuation character.

  `load(f90)` loads this function. See also the function `fortran`.

  Examples:

      (%i1) load ("f90")$
      (%i2) foo : expand ((xxx + yyy + 7)^4);
               4            3         3        2    2             2
      (%o2) yyy  + 4 xxx yyy  + 28 yyy  + 6 xxx  yyy  + 84 xxx yyy
                2        3             2
       + 294 yyy  + 4 xxx  yyy + 84 xxx  yyy + 588 xxx yyy + 1372 yyy
            4         3          2
       + xxx  + 28 xxx  + 294 xxx  + 1372 xxx + 2401
      (%i3) f90 ('foo = foo);
      foo = yyy**4+4*xxx*yyy**3+28*yyy**3+6*xxx**2*yyy**2+84*xxx*yyy**2&
      +294*yyy**2+4*xxx**3*yyy+84*xxx**2*yyy+588*xxx*yyy+1372*yyy+xxx**&
      4+28*xxx**3+294*xxx**2+1372*xxx+2401
      (%o3)                         false

  Multiple expressions. Capture standard output into a file via the `with_stdout` function.

      (%i1) load ("f90")$
      (%i2) foo : sin (3*x + 1) - cos (7*x - 2);
      (%o2)              sin(3 x + 1) - cos(7 x - 2)
      (%i3) with_stdout ("foo.f90",
                         f90 (x=0.25, y=0.625, 'foo=foo, 'stop, 'end));
      (%o3)                         false
      (%i4) printfile ("foo.f90");
      x = 0.25
      y = 0.625
      foo = sin(3*x+1)-cos(7*x-2)
      stop
      end
      (%o4)                        foo.f90

  Categories: Translation and compilation · Share packages · Package f90

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# 58. finance

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## 58.1 Introduction to finance

This is the Finance Package (Ver 0.1).

In all the functions, rate is the compound interest rate, num is the number of periods and must be positive and flow refers to cash flow so if you have an Output the flow is negative and positive for Inputs.

Note that before using the functions defined in this package, you have to load it writing `load(finance)$`.

Author: Nicolas Guarin Zapata.

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## 58.2 Functions and Variables for finance

Function: days360 (year1,month1,day1,year2,month2,day2)

  Calculates the distance between 2 dates, assuming 360 days years, 30 days months.

  Example:

      (%i1) load(finance)$
      (%i2) days360(2008,12,16,2007,3,25);
      (%o2)                      - 621

  Categories: Package finance

Function: fv (rate,PV,num)

  We can calculate the future value of a Present one given a certain interest rate. rate is the interest rate, PV is the present value and num is the number of periods.

  Example:

      (%i1) load(finance)$
      (%i2) fv(0.12,1000,3);
      (%o2)                     1404.928

  Categories: Package finance

Function: pv (rate,FV,num)

  We can calculate the present value of a Future one given a certain interest rate. rate is the interest rate, FV is the future value and num is the number of periods.

  Example:

      (%i1) load(finance)$
      (%i2) pv(0.12,1000,3);
      (%o2)                711.7802478134108

  Categories: Package finance

Function: graph_flow (val)

  Plots the money flow in a time line, the positive values are in blue and upside; the negative ones are in red and downside. The direction of the flow is given by the sign of the value. val is a list of flow values.

  Example:

      (%i1) load(finance)$
      (%i2) graph_flow([-5000,-3000,800,1300,1500,2000])$

  Categories: Package finance

Function: annuity_pv (rate,PV,num)

  We can calculate the annuity knowing the present value (like an amount), it is a constant and periodic payment. rate is the interest rate, PV is the present value and num is the number of periods.

  Example:

      (%i1) load(finance)$
      (%i2) annuity_pv(0.12,5000,10);
      (%o2)                884.9208207992202

  Categories: Package finance

Function: annuity_fv (rate,FV,num)

  We can calculate the annuity knowing the desired value (future value), it is a constant and periodic payment. rate is the interest rate, FV is the future value and num is the number of periods.

  Example:

      (%i1) load(finance)$
      (%i2) annuity_fv(0.12,65000,10);
      (%o2)                3703.970670389863

  Categories: Package finance

Function: geo_annuity_pv (rate,growing_rate,PV,num)

  We can calculate the annuity knowing the present value (like an amount), in a growing periodic payment. rate is the interest rate, growing_rate is the growing rate, PV is the present value and num is the number of periods.

  Example:

      (%i1) load(finance)$
      (%i2) geo_annuity_pv(0.14,0.05,5000,10);
      (%o2)                802.6888176505123

  Categories: Package finance

Function: geo_annuity_fv (rate,growing_rate,FV,num)

  We can calculate the annuity knowing the desired value (future value), in a growing periodic payment. rate is the interest rate, growing_rate is the growing rate, FV is the future value and num is the number of periods.

  Example:

      (%i1) load(finance)$
      (%i2) geo_annuity_fv(0.14,0.05,5000,10);
      (%o2)                216.5203395312695

  Categories: Package finance

Function: amortization (rate,amount,num)

  Amortization table determined by a specific rate. rate is the interest rate, amount is the amount value, and num is the number of periods.

  Example:

      (%i1) load(finance)$
      (%i2) amortization(0.05,56000,12)$
            "n"    "Balance"     "Interest"   "Amortization"  "Payment"
           0.000     56000.000         0.000         0.000         0.000
           1.000     52481.777      2800.000      3518.223      6318.223
           2.000     48787.643      2624.089      3694.134      6318.223
           3.000     44908.802      2439.382      3878.841      6318.223
           4.000     40836.019      2245.440      4072.783      6318.223
           5.000     36559.597      2041.801      4276.422      6318.223
           6.000     32069.354      1827.980      4490.243      6318.223
           7.000     27354.599      1603.468      4714.755      6318.223
           8.000     22404.106      1367.730      4950.493      6318.223
           9.000     17206.088      1120.205      5198.018      6318.223
          10.000     11748.170       860.304      5457.919      6318.223
          11.000      6017.355       587.408      5730.814      6318.223
          12.000         0.000       300.868      6017.355      6318.223

  Categories: Package finance

Function: arit_amortization (rate,increment,amount,num)

  The amortization table determined by a specific rate and with growing payment can be claculated by `arit_amortization`. Notice that the payment is not constant, it presents an arithmetic growing, increment is then the difference between two consecutive rows in the "Payment" column. rate is the interest rate, increment is the increment, amount is the amount value, and num is the number of periods.

  Example:

      (%i1) load(finance)$
      (%i2) arit_amortization(0.05,1000,56000,12)$
            "n"    "Balance"     "Interest"   "Amortization"  "Payment"
           0.000     56000.000         0.000         0.000         0.000
           1.000     57403.679      2800.000     -1403.679      1396.321
           2.000     57877.541      2870.184      -473.863      2396.321
           3.000     57375.097      2893.877       502.444      3396.321
           4.000     55847.530      2868.755      1527.567      4396.321
           5.000     53243.586      2792.377      2603.945      5396.321
           6.000     49509.443      2662.179      3734.142      6396.321
           7.000     44588.594      2475.472      4920.849      7396.321
           8.000     38421.703      2229.430      6166.892      8396.321
           9.000     30946.466      1921.085      7475.236      9396.321
          10.000     22097.468      1547.323      8848.998     10396.321
          11.000     11806.020      1104.873     10291.448     11396.321
          12.000        -0.000       590.301     11806.020     12396.321

  Categories: Package finance

Function: geo_amortization (rate,growing_rate,amount,num)

  The amortization table determined by rate, amount, and number of periods can be found by `geo_amortization`. Notice that the payment is not constant, it presents a geometric growing, growing_rate is then the quotient between two consecutive rows in the "Payment" column. rate is the interest rate, amount is the amount value, and num is the number of periods.

  Example:

      (%i1) load(finance)$
      (%i2) geo_amortization(0.05,0.03,56000,12)$
            "n"    "Balance"     "Interest"   "Amortization"  "Payment"
           0.000     56000.000         0.000         0.000         0.000
           1.000     53365.296      2800.000      2634.704      5434.704
           2.000     50435.816      2668.265      2929.480      5597.745
           3.000     47191.930      2521.791      3243.886      5765.677
           4.000     43612.879      2359.596      3579.051      5938.648
           5.000     39676.716      2180.644      3936.163      6116.807
           6.000     35360.240      1983.836      4316.475      6300.311
           7.000     30638.932      1768.012      4721.309      6489.321
           8.000     25486.878      1531.947      5152.054      6684.000
           9.000     19876.702      1274.344      5610.176      6884.520
          10.000     13779.481       993.835      6097.221      7091.056
          11.000      7164.668       688.974      6614.813      7303.787
          12.000         0.000       358.233      7164.668      7522.901

  Categories: Package finance

Function: saving (rate,amount,num)

  The table that represents the values in a constant and periodic saving can be found by `saving`. amount represents the desired quantity and num the number of periods to save.

  Example:

      (%i1) load(finance)$
      (%i2) saving(0.15,12000,15)$
            "n"    "Balance"     "Interest"   "Payment"
           0.000         0.000         0.000         0.000
           1.000       252.205         0.000       252.205
           2.000       542.240        37.831       252.205
           3.000       875.781        81.336       252.205
           4.000      1259.352       131.367       252.205
           5.000      1700.460       188.903       252.205
           6.000      2207.733       255.069       252.205
           7.000      2791.098       331.160       252.205
           8.000      3461.967       418.665       252.205
           9.000      4233.467       519.295       252.205
          10.000      5120.692       635.020       252.205
          11.000      6141.000       768.104       252.205
          12.000      7314.355       921.150       252.205
          13.000      8663.713      1097.153       252.205
          14.000     10215.474      1299.557       252.205
          15.000     12000.000      1532.321       252.205

  Categories: Package finance

Function: npv (rate,val)

  Calculates the present value of a value series to evaluate the viability in a project. val is a list of varying cash flows.

  Example:

      (%i1) load(finance)$
      (%i2) npv(0.25,[100,500,323,124,300]);
      (%o2)                714.4703999999999

  Categories: Package finance

Function: irr (val,IO)

  IRR (Internal Rate of Return) is the value of rate which makes Net Present Value zero. flowValues is a list of varying cash flows, I0 is the initial investment.

  Example:

      (%i1) load(finance)$
      (%i2) res:irr([-5000,0,800,1300,1500,2000],0)$
      (%i3) rhs(res[1][1]);
      (%o3)                .03009250374237132

  Categories: Package finance

Function: benefit_cost (rate,input,output)

  Calculates the ratio Benefit/Cost. Benefit is the Net Present Value (NPV) of the inputs, and Cost is the Net Present Value (NPV) of the outputs. Notice that if there is not an input or output value in a specific period, the input/output would be a zero for that period. rate is the interest rate, input is a list of input values, and output is a list of output values.

  Example:

      (%i1) load(finance)$
      (%i2) benefit_cost(0.24,[0,300,500,150],[100,320,0,180]);
      (%o2)               1.427249324905784

  Categories: Package finance

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# 59. fractals

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## 59.1 Introduction to fractals

This package defines some well known fractals:

- with random IFS (Iterated Function System): the Sierpinsky triangle, a Tree and a Fern

- Complex Fractals: the Mandelbrot and Julia Sets

- the Koch snowflake sets

- Peano maps: the Sierpinski and Hilbert maps

Author: José Ramírez Labrador.

For questions, suggestions and bugs, please feel free to contact me at

pepe DOT ramirez AAATTT uca DOT es

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## 59.2 Definitions for IFS fractals

Some fractals can be generated by iterative applications of contractive affine transformations in a random way; see

Hoggar S. G., "Mathematics for computer graphics", Cambridge University Press 1994.

We define a list with several contractive affine transformations, and we randomly select the transformation in a recursive way. The probability of the choice of a transformation must be related with the contraction ratio.

You can change the transformations and find another fractal

Function: sierpinskiale (n)

  Sierpinski Triangle: 3 contractive maps; .5 contraction constant and translations; all maps have the same contraction ratio. Argument n must be great enougth, 10000 or greater.

  Example:

      (%i1) load(fractals)$
      (%i2) n: 10000$
      (%i3) plot2d([discrete,sierpinskiale(n)], [style,dots])$

  Categories: Package fractals

Function: treefale (n)

  3 contractive maps all with the same contraction ratio. Argument n must be great enougth, 10000 or greater.

  Example:

      (%i1) load(fractals)$
      (%i2) n: 10000$
      (%i3) plot2d([discrete,treefale(n)], [style,dots])$

  Categories: Package fractals

Function: fernfale (n)

  4 contractive maps, the probability to choice a transformation must be related with the contraction ratio. Argument n must be great enougth, 10000 or greater.

  Example:

      (%i1) load(fractals)$
      (%i2) n: 10000$
      (%i3) plot2d([discrete,fernfale(n)], [style,dots])$

  Categories: Package fractals

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## 59.3 Definitions for complex fractals

Function: mandelbrot_set (x, y)

  Mandelbrot set.

  Example:

  This program is time consuming because it must make a lot of operations; the computing time is also related with the number of grid points.

      (%i1) load(fractals)$
      (%i2) plot3d (mandelbrot_set, [x, -2.5, 1], [y, -1.5, 1.5],
                      [gnuplot_preamble, "set view map"],
                      [gnuplot_pm3d, true],
                      [grid, 150, 150])$

  Categories: Package fractals

Function: julia_set (x, y)

  Julia sets.

  This program is time consuming because it must make a lot of operations; the computing time is also related with the number of grid points.

  Example:

      (%i1) load(fractals)$
      (%i2) plot3d (julia_set, [x, -2, 1], [y, -1.5, 1.5],
                      [gnuplot_preamble, "set view map"],
                      [gnuplot_pm3d, true],
                      [grid, 150, 150])$

  See also `julia_parameter`.

  Categories: Package fractals

Optional variable: julia_parameter

  Default value: `%i`

  Complex parameter for Julia fractals. Its default value is `%i`; we suggest the values `-.745+%i*.113002`, `-.39054-%i*.58679`, `-.15652+%i*1.03225`, `-.194+%i*.6557` and `.011031-%i*.67037`.

  Categories: Package fractals

Function: julia_sin (x, y)

  While function `julia_set` implements the transformation `julia_parameter+z^2`, function `julia_sin` implements `julia_parameter*sin(z)`. See source code for more details.

  This program runs slowly because it calculates a lot of sines.

  Example:

  This program is time consuming because it must make a lot of operations; the computing time is also related with the number of grid points.

      (%i1) load(fractals)$
      (%i2) julia_parameter:1+.1*%i$
      (%i3) plot3d (julia_sin, [x, -2, 2], [y, -3, 3],
                      [gnuplot_preamble, "set view map"],
                      [gnuplot_pm3d, true],
                      [grid, 150, 150])$

  See also `julia_parameter`.

  Categories: Package fractals

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## 59.4 Definitions for Koch snowflakes

Function: snowmap (ent, nn)

  Koch snowflake sets. Function `snowmap` plots the snow Koch map over the vertex of an initial closed polygonal, in the complex plane. Here the orientation of the polygon is important. Argument nn is the number of recursive applications of Koch transformation; nn must be small (5 or 6).

  Examples:

      (%i1) load(fractals)$
      (%i2) plot2d([discrete,
                    snowmap([1,exp(%i*%pi*2/3),exp(-%i*%pi*2/3),1],4)])$
      (%i3) plot2d([discrete,
                    snowmap([1,exp(-%i*%pi*2/3),exp(%i*%pi*2/3),1],4)])$
      (%i4) plot2d([discrete, snowmap([0,1,1+%i,%i,0],4)])$
      (%i5) plot2d([discrete, snowmap([0,%i,1+%i,1,0],4)])$

  Categories: Package fractals

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## 59.5 Definitions for Peano maps

Continuous curves that cover an area. Warning: the number of points exponentially grows with n.

Function: hilbertmap (nn)

  Hilbert map. Argument nn must be small (5, for example). Maxima can crash if nn is 7 or greater.

  Example:

      (%i1) load(fractals)$
      (%i2) plot2d([discrete,hilbertmap(6)])$

  Categories: Package fractals

Function: sierpinskimap (nn)

  Sierpinski map. Argument nn must be small (5, for example). Maxima can crash if nn is 7 or greater.

  Example:

      (%i1) load(fractals)$
      (%i2) plot2d([discrete,sierpinskimap(6)])$

  Categories: Package fractals

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# 60. ggf

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## 60.1 Functions and Variables for ggf

Option variable: GGFINFINITY

  Default value: 3

  This is an option variable for function `ggf`.

  When computing the continued fraction of the generating function, a partial quotient having a degree (strictly) greater than GGFINFINITY will be discarded and the current convergent will be considered as the exact value of the generating function; most often the degree of all partial quotients will be 0 or 1; if you use a greater value, then you should give enough terms in order to make the computation accurate enough.

  See also `ggf`.

  Categories: Package ggf

Option variable: GGFCFMAX

  Default value: 3

  This is an option variable for function `ggf`.

  When computing the continued fraction of the generating function, if no good result has been found (see the GGFINFINITY flag) after having computed GGFCFMAX partial quotients, the generating function will be considered as not being a fraction of two polynomials and the function will exit. Put freely a greater value for more complicated generating functions.

  See also `ggf`.

  Categories: Package ggf

Function: ggf (l)

  Compute the generating function (if it is a fraction of two polynomials) of a sequence, its first terms being given. l is a list of numbers.

  The solution is returned as a fraction of two polynomials. If no solution has been found, it returns with `done`.

  This function is controlled by global variables GGFINFINITY and GGFCFMAX. See also GGFINFINITY and GGFCFMAX.

  To use this function write first `load("ggf")`.

  Categories: Generating functions · Share packages · Package ggf

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# 61. graphs

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## 61.1 Introduction to graphs

The `graphs` package provides graph and digraph data structure for Maxima. Graphs and digraphs are simple (have no multiple edges nor loops), although digraphs can have a directed edge from u to v and a directed edge from v to u.

Internally graphs are represented by adjacency lists and implemented as a lisp structures. Vertices are identified by their ids (an id is an integer). Edges/arcs are represented by lists of length 2. Labels can be assigned to vertices of graphs/digraphs and weights can be assigned to edges/arcs of graphs/digraphs.

There is a `draw_graph` function for drawing graphs. Graphs are drawn using a force based vertex positioning algorithm. `draw_graph` can also use graphviz programs available from http://www.graphviz.org. `draw_graph` is based on the maxima `draw` package.

To use the `graphs` package, first load it with `load(graphs)`.

Categories: Share packages · Package graphs

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## 61.2 Functions and Variables for graphs

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### 61.2.1 Building graphs

Function: create_graph
`create_graph` (v_list, e_list)
`create_graph` (n, e_list)
`create_graph` (v_list, e_list, directed)

  Creates a new graph on the set of vertices v_list and with edges e_list.

  v_list is a list of vertices (`[v1, v2,..., vn]`) or a list of vertices together with vertex labels (`[[v1,l1], [v2,l2],..., [vn,ln]]`).

  n is the number of vertices. Vertices will be identified by integers from 0 to n-1.

  e_list is a list of edges (`[e1, e2,..., em]`) or a list of edges together with edge-weights (`[[e1, w1], ..., [em, wm]]`).

  If directed is not `false`, a directed graph will be returned.

  Example 1: create a cycle on 3 vertices:

      (%i1) load ("graphs")$
      (%i2) g : create_graph([1,2,3], [[1,2], [2,3], [1,3]])$
      (%i3) print_graph(g)$
      Graph on 3 vertices with 3 edges.
      Adjacencies:
        3 :  1  2
        2 :  3  1
        1 :  3  2

  Example 2: create a cycle on 3 vertices with edge weights:

      (%i1) load ("graphs")$
      (%i2) g : create_graph([1,2,3], [[[1,2], 1.0], [[2,3], 2.0],
                                [[1,3], 3.0]])$
      (%i3) print_graph(g)$
      Graph on 3 vertices with 3 edges.
      Adjacencies:
        3 :  1  2
        2 :  3  1
        1 :  3  2

  Example 3: create a directed graph:

      (%i1) load ("graphs")$
      (%i2) d : create_graph(
              [1,2,3,4],
              [
               [1,3], [1,4],
               [2,3], [2,4]
              ],
              'directed = true)$
      (%i3) print_graph(d)$
      Digraph on 4 vertices with 4 arcs.
      Adjacencies:
        4 :
        3 :
        2 :  4  3
        1 :  4  3

  Categories: Package graphs

Function: copy_graph (g)

  Returns a copy of the graph g.

  Categories: Package graphs · Package graphs - constructions

Function: circulant_graph (n, d)

  Returns the circulant graph with parameters n and d.

  Example:

      (%i1) load ("graphs")$
      (%i2) g : circulant_graph(10, [1,3])$
      (%i3) print_graph(g)$
      Graph on 10 vertices with 20 edges.
      Adjacencies:
        9 :  2  6  0  8
        8 :  1  5  9  7
        7 :  0  4  8  6
        6 :  9  3  7  5
        5 :  8  2  6  4
        4 :  7  1  5  3
        3 :  6  0  4  2
        2 :  9  5  3  1
        1 :  8  4  2  0
        0 :  7  3  9  1

  Categories: Package graphs · Package graphs - constructions

Function: clebsch_graph ()

  Returns the Clebsch graph.

  Categories: Package graphs · Package graphs - constructions

Function: complement_graph (g)

  Returns the complement of the graph g.

  Categories: Package graphs · Package graphs - constructions

Function: complete_bipartite_graph (n, m)

  Returns the complete bipartite graph on n+m vertices.

  Categories: Package graphs · Package graphs - constructions

Function: complete_graph (n)

  Returns the complete graph on n vertices.

  Categories: Package graphs · Package graphs - constructions

Function: cycle_digraph (n)

  Returns the directed cycle on n vertices.

  Categories: Package graphs · Package graphs - constructions

Function: cycle_graph (n)

  Returns the cycle on n vertices.

  Categories: Package graphs · Package graphs - constructions

Function: cuboctahedron_graph (n)

  Returns the cuboctahedron graph.

  Categories: Package graphs · Package graphs - constructions

Function: cube_graph (n)

  Returns the n-dimensional cube.

  Categories: Package graphs · Package graphs - constructions

Function: dodecahedron_graph ()

  Returns the dodecahedron graph.

  Categories: Package graphs · Package graphs - constructions

Function: empty_graph (n)

  Returns the empty graph on n vertices.

  Categories: Package graphs · Package graphs - constructions

Function: flower_snark (n)

  Returns the flower graph on 4n vertices.

  Example:

      (%i1) load ("graphs")$
      (%i2) f5 : flower_snark(5)$
      (%i3) chromatic_index(f5);
      (%o3)                           4

  Categories: Package graphs · Package graphs - constructions

Function: from_adjacency_matrix (A)

  Returns the graph represented by its adjacency matrix A.

  Categories: Package graphs · Package graphs - constructions

Function: frucht_graph ()

  Returns the Frucht graph.

  Categories: Package graphs · Package graphs - constructions

Function: graph_product (g1, g1)

  Returns the direct product of graphs g1 and g2.

  Example:

      (%i1) load ("graphs")$
      (%i2) grid : graph_product(path_graph(3), path_graph(4))$
      (%i3) draw_graph(grid)$

  Categories: Package graphs · Package graphs - constructions

Function: graph_union (g1, g1)

  Returns the union (sum) of graphs g1 and g2.

  Categories: Package graphs · Package graphs - constructions

Function: grid_graph (n, m)

  Returns the n x m grid.

  Categories: Package graphs · Package graphs - constructions

Function: great_rhombicosidodecahedron_graph ()

  Returns the great rhombicosidodecahedron graph.

  Categories: Package graphs · Package graphs - constructions

Function: great_rhombicuboctahedron_graph ()

  Returns the great rhombicuboctahedron graph.

  Categories: Package graphs · Package graphs - constructions

Function: grotzch_graph ()

  Returns the Grotzch graph.

  Categories: Package graphs · Package graphs - constructions

Function: heawood_graph ()

  Returns the Heawood graph.

  Categories: Package graphs · Package graphs - constructions

Function: icosahedron_graph ()

  Returns the icosahedron graph.

  Categories: Package graphs · Package graphs - constructions

Function: icosidodecahedron_graph ()

  Returns the icosidodecahedron graph.

  Categories: Package graphs · Package graphs - constructions

Function: induced_subgraph (V, g)

  Returns the graph induced on the subset V of vertices of the graph g.

  Example:

      (%i1) load ("graphs")$
      (%i2) p : petersen_graph()$
      (%i3) V : [0,1,2,3,4]$
      (%i4) g : induced_subgraph(V, p)$
      (%i5) print_graph(g)$
      Graph on 5 vertices with 5 edges.
      Adjacencies:
        4 :  3  0
        3 :  2  4
        2 :  1  3
        1 :  0  2
        0 :  1  4

  Categories: Package graphs · Package graphs - constructions

Function: line_graph (g)

  Returns the line graph of the graph g.

  Categories: Package graphs · Package graphs - constructions

Function: make_graph
`make_graph` (vrt, f)
`make_graph` (vrt, f, oriented)

  Creates a graph using a predicate function f.

  vrt is a list/set of vertices or an integer. If vrt is an integer, then vertices of the graph will be integers from 1 to vrt.

  f is a predicate function. Two vertices a and b will be connected if `f(a,b)=true`.

  If directed is not false, then the graph will be directed.

  Example 1:

      (%i1) load(graphs)$
      (%i2) g : make_graph(powerset({1,2,3,4,5}, 2), disjointp)$
      (%i3) is_isomorphic(g, petersen_graph());
      (%o3)                         true
      (%i4) get_vertex_label(1, g);
      (%o4)                        {1, 2}

  Example 2:

      (%i1) load(graphs)$
      (%i2) f(i, j) := is (mod(j, i)=0)$
      (%i3) g : make_graph(20, f, directed=true)$
      (%i4) out_neighbors(4, g);
      (%o4)                    [8, 12, 16, 20]
      (%i5) in_neighbors(18, g);
      (%o5)                    [1, 2, 3, 6, 9]

  Categories: Package graphs · Package graphs - constructions

Function: mycielski_graph (g)

  Returns the mycielskian graph of the graph g.

  Categories: Package graphs · Package graphs - constructions

Function: new_graph ()

  Returns the graph with no vertices and no edges.

  Categories: Package graphs · Package graphs - constructions

Function: path_digraph (n)

  Returns the directed path on n vertices.

  Categories: Package graphs · Package graphs - constructions

Function: path_graph (n)

  Returns the path on n vertices.

  Categories: Package graphs · Package graphs - constructions

Function: petersen_graph
`petersen_graph` ()
`petersen_graph` (n, d)

  Returns the petersen graph P_{n,d}. The default values for n and d are `n=5` and `d=2`.

  Categories: Package graphs · Package graphs - constructions

Function: random_bipartite_graph (a, b, p)

  Returns a random bipartite graph on `a+b` vertices. Each edge is present with probability p.

  Categories: Package graphs · Package graphs - constructions

Function: random_digraph (n, p)

  Returns a random directed graph on n vertices. Each arc is present with probability p.

  Categories: Package graphs · Package graphs - constructions

Function: random_regular_graph
`random_regular_graph` (n)
`random_regular_graph` (n, d)

  Returns a random d-regular graph on n vertices. The default value for d is `d=3`.

  Categories: Package graphs · Package graphs - constructions

Function: random_graph (n, p)

  Returns a random graph on n vertices. Each edge is present with probability p.

  Categories: Package graphs · Package graphs - constructions

Function: random_graph1 (n, m)

  Returns a random graph on n vertices and random m edges.

  Categories: Package graphs · Package graphs - constructions

Function: random_network (n, p, w)

  Returns a random network on n vertices. Each arc is present with probability p and has a weight in the range `[0,w]`. The function returns a list `[network, source, sink]`.

  Example:

      (%i1) load ("graphs")$
      (%i2) [net, s, t] : random_network(50, 0.2, 10.0);
      (%o2)                   [DIGRAPH, 50, 51]
      (%i3) max_flow(net, s, t)$
      (%i4) first(%);
      (%o4)                   27.65981397932507

  Categories: Package graphs · Package graphs - constructions

Function: random_tournament (n)

  Returns a random tournament on n vertices.

  Categories: Package graphs · Package graphs - constructions

Function: random_tree (n)

  Returns a random tree on n vertices.

  Categories: Package graphs · Package graphs - constructions

Function: small_rhombicosidodecahedron_graph ()

  Returns the small rhombicosidodecahedron graph.

  Categories: Package graphs · Package graphs - constructions

Function: small_rhombicuboctahedron_graph ()

  Returns the small rhombicuboctahedron graph.

  Categories: Package graphs · Package graphs - constructions

Function: snub_cube_graph ()

  Returns the snub cube graph.

  Categories: Package graphs · Package graphs - constructions

Function: snub_dodecahedron_graph ()

  Returns the snub dodecahedron graph.

  Categories: Package graphs · Package graphs - constructions

Function: truncated_cube_graph ()

  Returns the truncated cube graph.

  Categories: Package graphs · Package graphs - constructions

Function: truncated_dodecahedron_graph ()

  Returns the truncated dodecahedron graph.

  Categories: Package graphs · Package graphs - constructions

Function: truncated_icosahedron_graph ()

  Returns the truncated icosahedron graph.

  Categories: Package graphs · Package graphs - constructions

Function: truncated_tetrahedron_graph ()

  Returns the truncated tetrahedron graph.

  Categories: Package graphs · Package graphs - constructions

Function: tutte_graph ()

  Returns the Tutte graph.

  Categories: Package graphs · Package graphs - constructions

Function: underlying_graph (g)

  Returns the underlying graph of the directed graph g.

  Categories: Package graphs · Package graphs - constructions

Function: wheel_graph (n)

  Returns the wheel graph on n+1 vertices.

  Categories: Package graphs · Package graphs - constructions

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### 61.2.2 Graph properties

Function: adjacency_matrix (gr)

  Returns the adjacency matrix of the graph gr.

  Example:

      (%i1) load ("graphs")$
      (%i2) c5 : cycle_graph(4)$
      (%i3) adjacency_matrix(c5);
                               [ 0  1  0  1 ]
                               [            ]
                               [ 1  0  1  0 ]
      (%o3)                    [            ]
                               [ 0  1  0  1 ]
                               [            ]
                               [ 1  0  1  0 ]

  Categories: Package graphs · Package graphs - properties

Function: average_degree (gr)

  Returns the average degree of vertices in the graph gr.

  Example:

      (%i1) load ("graphs")$
      (%i2) average_degree(grotzch_graph());
                                     40
      (%o2)                          --
                                     11

  Categories: Package graphs · Package graphs - properties

Function: biconnected_components (gr)

  Returns the (vertex sets of) 2-connected components of the graph gr.

  Example:

      (%i1) load ("graphs")$
      (%i2) g : create_graph(
                  [1,2,3,4,5,6,7],
                  [
                   [1,2],[2,3],[2,4],[3,4],
                   [4,5],[5,6],[4,6],[6,7]
                  ])$
      (%i3) biconnected_components(g);
      (%o3)        [[6, 7], [4, 5, 6], [1, 2], [2, 3, 4]]

  Categories: Package graphs · Package graphs - properties

Function: bipartition (gr)

  Returns a bipartition of the vertices of the graph gr or an empty list if gr is not bipartite.

  Example:

      (%i1) load ("graphs")$
      (%i2) h : heawood_graph()$
      (%i3) [A,B]:bipartition(h);
      (%o3)  [[8, 12, 6, 10, 0, 2, 4], [13, 5, 11, 7, 9, 1, 3]]
      (%i4) draw_graph(h, show_vertices=A, program=circular)$

  Categories: Package graphs · Package graphs - properties

Function: chromatic_index (gr)

  Returns the chromatic index of the graph gr.

  Example:

      (%i1) load ("graphs")$
      (%i2) p : petersen_graph()$
      (%i3) chromatic_index(p);
      (%o3)                           4

  Categories: Package graphs · Package graphs - properties

Function: chromatic_number (gr)

  Returns the chromatic number of the graph gr.

  Example:

      (%i1) load ("graphs")$
      (%i2) chromatic_number(cycle_graph(5));
      (%o2)                           3
      (%i3) chromatic_number(cycle_graph(6));
      (%o3)                           2

  Categories: Package graphs · Package graphs - properties

Function: clear_edge_weight (e, gr)

  Removes the weight of the edge e in the graph gr.

  Example:

      (%i1) load ("graphs")$
      (%i2) g : create_graph(3, [[[0,1], 1.5], [[1,2], 1.3]])$
      (%i3) get_edge_weight([0,1], g);
      (%o3)                          1.5
      (%i4) clear_edge_weight([0,1], g)$
      (%i5) get_edge_weight([0,1], g);
      (%o5)                           1

  Categories: Package graphs · Package graphs - properties

Function: clear_vertex_label (v, gr)

  Removes the label of the vertex v in the graph gr.

  Example:

      (%i1) load ("graphs")$
      (%i2) g : create_graph([[0,"Zero"], [1, "One"]], [[0,1]])$
      (%i3) get_vertex_label(0, g);
      (%o3)                         Zero
      (%i4) clear_vertex_label(0, g);
      (%o4)                         done
      (%i5) get_vertex_label(0, g);
      (%o5)                         false

  Categories: Package graphs · Package graphs - properties

Function: connected_components (gr)

  Returns the (vertex sets of) connected components of the graph gr.

  Example:

      (%i1) load ("graphs")$
      (%i2) g: graph_union(cycle_graph(5), path_graph(4))$
      (%i3) connected_components(g);
      (%o3)            [[1, 2, 3, 4, 0], [8, 7, 6, 5]]

  Categories: Package graphs · Package graphs - properties

Function: diameter (gr)

  Returns the diameter of the graph gr.

  Example:

      (%i1) load ("graphs")$
      (%i2) diameter(dodecahedron_graph());
      (%o2)                           5

  Categories: Package graphs · Package graphs - properties

Function: edge_coloring (gr)

  Returns an optimal coloring of the edges of the graph gr.

  The function returns the chromatic index and a list representing the coloring of the edges of gr.

  Example:

      (%i1) load ("graphs")$
      (%i2) p : petersen_graph()$
      (%i3) [ch_index, col] : edge_coloring(p);
      (%o3) [4, [[[0, 5], 3], [[5, 7], 1], [[0, 1], 1], [[1, 6], 2],
      [[6, 8], 1], [[1, 2], 3], [[2, 7], 4], [[7, 9], 2], [[2, 3], 2],
      [[3, 8], 3], [[5, 8], 2], [[3, 4], 1], [[4, 9], 4], [[6, 9], 3],
      [[0, 4], 2]]]
      (%i4) assoc([0,1], col);
      (%o4)                           1
      (%i5) assoc([0,5], col);
      (%o5)                           3

  Categories: Package graphs · Package graphs - properties

Function: degree_sequence (gr)

  Returns the list of vertex degrees of the graph gr.

  Example:

      (%i1) load ("graphs")$
      (%i2) degree_sequence(random_graph(10, 0.4));
      (%o2)            [2, 2, 2, 2, 2, 2, 3, 3, 3, 3]

  Categories: Package graphs · Package graphs - properties

Function: edge_connectivity (gr)

  Returns the edge-connectivity of the graph gr.

  See also `min_edge_cut`.

  Categories: Package graphs · Package graphs - properties

Function: edges (gr)

  Returns the list of edges (arcs) in a (directed) graph gr.

  Example:

      (%i1) load ("graphs")$
      (%i2) edges(complete_graph(4));
      (%o2)   [[2, 3], [1, 3], [1, 2], [0, 3], [0, 2], [0, 1]]

  Categories: Package graphs · Package graphs - properties

Function: get_edge_weight
`get_edge_weight` (e, gr)
`get_edge_weight` (e, gr, ifnot)

  Returns the weight of the edge e in the graph gr.

  If there is no weight assigned to the edge, the function returns 1. If the edge is not present in the graph, the function signals an error or returns the optional argument ifnot.

  Example:

      (%i1) load ("graphs")$
      (%i2) c5 : cycle_graph(5)$
      (%i3) get_edge_weight([1,2], c5);
      (%o3)                           1
      (%i4) set_edge_weight([1,2], 2.0, c5);
      (%o4)                         done
      (%i5) get_edge_weight([1,2], c5);
      (%o5)                          2.0

  Categories: Package graphs · Package graphs - properties

Function: get_vertex_label (v, gr)

  Returns the label of the vertex v in the graph gr.

  Example:

      (%i1) load ("graphs")$
      (%i2) g : create_graph([[0,"Zero"], [1, "One"]], [[0,1]])$
      (%i3) get_vertex_label(0, g);
      (%o3)                         Zero

  Categories: Package graphs · Package graphs - properties

Function: graph_charpoly (gr, x)

  Returns the characteristic polynomial (in variable x) of the graph gr.

  Example:

      (%i1) load ("graphs")$
      (%i2) p : petersen_graph()$
      (%i3) graph_charpoly(p, x), factor;
                                         5        4
      (%o3)               (x - 3) (x - 1)  (x + 2)

  Categories: Package graphs · Package graphs - properties

Function: graph_center (gr)

  Returns the center of the graph gr.

  Example:

      (%i1) load ("graphs")$
      (%i2) g : grid_graph(5,5)$
      (%i3) graph_center(g);
      (%o3)                         [12]

  Categories: Package graphs · Package graphs - properties

Function: graph_eigenvalues (gr)

  Returns the eigenvalues of the graph gr. The function returns eigenvalues in the same format as maxima `eigenvalues` function.

  Example:

      (%i1) load ("graphs")$
      (%i2) p : petersen_graph()$
      (%i3) graph_eigenvalues(p);
      (%o3)               [[3, - 2, 1], [1, 4, 5]]

  Categories: Package graphs · Package graphs - properties

Function: graph_periphery (gr)

  Returns the periphery of the graph gr.

  Example:

      (%i1) load ("graphs")$
      (%i2) g : grid_graph(5,5)$
      (%i3) graph_periphery(g);
      (%o3)                    [24, 20, 4, 0]

  Categories: Package graphs · Package graphs - properties

Function: graph_size (gr)

  Returns the number of edges in the graph gr.

  Example:

      (%i1) load ("graphs")$
      (%i2) p : petersen_graph()$
      (%i3) graph_size(p);
      (%o3)                          15

  Categories: Package graphs · Package graphs - properties

Function: graph_order (gr)

  Returns the number of vertices in the graph gr.

  Example:

      (%i1) load ("graphs")$
      (%i2) p : petersen_graph()$
      (%i3) graph_order(p);
      (%o3)                          10

  Categories: Package graphs · Package graphs - properties

Function: girth (gr)

  Returns the length of the shortest cycle in gr.

  Example:

      (%i1) load ("graphs")$
      (%i2) g : heawood_graph()$
      (%i3) girth(g);
      (%o3)                           6

  Categories: Package graphs · Package graphs - properties

Function: hamilton_cycle (gr)

  Returns the Hamilton cycle of the graph gr or an empty list if gr is not hamiltonian.

  Example:

      (%i1) load ("graphs")$
      (%i2) c : cube_graph(3)$
      (%i3) hc : hamilton_cycle(c);
      (%o3)              [7, 3, 2, 6, 4, 0, 1, 5, 7]
      (%i4) draw_graph(c, show_edges=vertices_to_cycle(hc))$

  Categories: Package graphs · Package graphs - properties

Function: hamilton_path (gr)

  Returns the Hamilton path of the graph gr or an empty list if gr does not have a Hamilton path.

  Example:

      (%i1) load ("graphs")$
      (%i2) p : petersen_graph()$
      (%i3) hp : hamilton_path(p);
      (%o3)            [0, 5, 7, 2, 1, 6, 8, 3, 4, 9]
      (%i4) draw_graph(p, show_edges=vertices_to_path(hp))$

  Categories: Package graphs · Package graphs - properties

Function: isomorphism (gr1, gr2)

  Returns a an isomorphism between graphs/digraphs gr1 and gr2. If gr1 and gr2 are not isomorphic, it returns an empty list.

  Example:

      (%i1) load ("graphs")$
      (%i2) clk5:complement_graph(line_graph(complete_graph(5)))$
      (%i3) isomorphism(clk5, petersen_graph());
      (%o3) [9 -> 0, 2 -> 1, 6 -> 2, 5 -> 3, 0 -> 4, 1 -> 5, 3 -> 6,
                                                4 -> 7, 7 -> 8, 8 -> 9]

  Categories: Package graphs · Package graphs - properties

Function: in_neighbors (v, gr)

  Returns the list of in-neighbors of the vertex v in the directed graph gr.

  Example:

      (%i1) load ("graphs")$
      (%i2) p : path_digraph(3)$
      (%i3) in_neighbors(2, p);
      (%o3)                          [1]
      (%i4) out_neighbors(2, p);
      (%o4)                          []

  Categories: Package graphs · Package graphs - properties

Function: is_biconnected (gr)

  Returns `true` if gr is 2-connected and `false` otherwise.

  Example:

      (%i1) load ("graphs")$
      (%i2) is_biconnected(cycle_graph(5));
      (%o2)                         true
      (%i3) is_biconnected(path_graph(5));
      (%o3)                         false

  Categories: Package graphs · Package graphs - properties

Function: is_bipartite (gr)

  Returns `true` if gr is bipartite (2-colorable) and `false` otherwise.

  Example:

      (%i1) load ("graphs")$
      (%i2) is_bipartite(petersen_graph());
      (%o2)                         false
      (%i3) is_bipartite(heawood_graph());
      (%o3)                         true

  Categories: Package graphs · Package graphs - properties

Function: is_connected (gr)

  Returns `true` if the graph gr is connected and `false` otherwise.

  Example:

      (%i1) load ("graphs")$
      (%i2) is_connected(graph_union(cycle_graph(4), path_graph(3)));
      (%o2)                         false

  Categories: Package graphs · Package graphs - properties

Function: is_digraph (gr)

  Returns `true` if gr is a directed graph and `false` otherwise.

  Example:

      (%i1) load ("graphs")$
      (%i2) is_digraph(path_graph(5));
      (%o2)                         false
      (%i3) is_digraph(path_digraph(5));
      (%o3)                         true

  Categories: Package graphs · Package graphs - properties

Function: is_edge_in_graph (e, gr)

  Returns `true` if e is an edge (arc) in the (directed) graph g and `false` otherwise.

  Example:

      (%i1) load ("graphs")$
      (%i2) c4 : cycle_graph(4)$
      (%i3) is_edge_in_graph([2,3], c4);
      (%o3)                         true
      (%i4) is_edge_in_graph([3,2], c4);
      (%o4)                         true
      (%i5) is_edge_in_graph([2,4], c4);
      (%o5)                         false
      (%i6) is_edge_in_graph([3,2], cycle_digraph(4));
      (%o6)                         false

  Categories: Package graphs · Package graphs - properties

Function: is_graph (gr)

  Returns `true` if gr is a graph and `false` otherwise.

  Example:

      (%i1) load ("graphs")$
      (%i2) is_graph(path_graph(5));
      (%o2)                         true
      (%i3) is_graph(path_digraph(5));
      (%o3)                         false

  Categories: Package graphs · Package graphs - properties

Function: is_graph_or_digraph (gr)

  Returns `true` if gr is a graph or a directed graph and `false` otherwise.

  Example:

      (%i1) load ("graphs")$
      (%i2) is_graph_or_digraph(path_graph(5));
      (%o2)                         true
      (%i3) is_graph_or_digraph(path_digraph(5));
      (%o3)                         true

  Categories: Package graphs · Package graphs - properties

Function: is_isomorphic (gr1, gr2)

  Returns `true` if graphs/digraphs gr1 and gr2 are isomorphic and `false` otherwise.

  See also `isomorphism`.

  Example:

      (%i1) load ("graphs")$
      (%i2) clk5:complement_graph(line_graph(complete_graph(5)))$
      (%i3) is_isomorphic(clk5, petersen_graph());
      (%o3)                         true

  Categories: Package graphs · Package graphs - properties

Function: is_planar (gr)

  Returns `true` if gr is a planar graph and `false` otherwise.

  The algorithm used is the Demoucron's algorithm, which is a quadratic time algorithm.

  Example:

      (%i1) load ("graphs")$
      (%i2) is_planar(dodecahedron_graph());
      (%o2)                         true
      (%i3) is_planar(petersen_graph());
      (%o3)                         false
      (%i4) is_planar(petersen_graph(10,2));
      (%o4)                         true

  Categories: Package graphs · Package graphs - properties

Function: is_sconnected (gr)

  Returns `true` if the directed graph gr is strongly connected and `false` otherwise.

  Example:

      (%i1) load ("graphs")$
      (%i2) is_sconnected(cycle_digraph(5));
      (%o2)                         true
      (%i3) is_sconnected(path_digraph(5));
      (%o3)                         false

  Categories: Package graphs · Package graphs - properties

Function: is_vertex_in_graph (v, gr)

  Returns `true` if v is a vertex in the graph g and `false` otherwise.

  Example:

      (%i1) load ("graphs")$
      (%i2) c4 : cycle_graph(4)$
      (%i3) is_vertex_in_graph(0, c4);
      (%o3)                         true
      (%i4) is_vertex_in_graph(6, c4);
      (%o4)                         false

  Categories: Package graphs · Package graphs - properties

Function: is_tree (gr)

  Returns `true` if gr is a tree and `false` otherwise.

  Example:

      (%i1) load ("graphs")$
      (%i2) is_tree(random_tree(4));
      (%o2)                         true
      (%i3) is_tree(graph_union(random_tree(4), random_tree(5)));
      (%o3)                         false

  Categories: Package graphs · Package graphs - properties

Function: laplacian_matrix (gr)

  Returns the laplacian matrix of the graph gr.

  Example:

      (%i1) load ("graphs")$
      (%i2) laplacian_matrix(cycle_graph(5));
                         [  2   - 1   0    0   - 1 ]
                         [                         ]
                         [ - 1   2   - 1   0    0  ]
                         [                         ]
      (%o2)              [  0   - 1   2   - 1   0  ]
                         [                         ]
                         [  0    0   - 1   2   - 1 ]
                         [                         ]
                         [ - 1   0    0   - 1   2  ]

  Categories: Package graphs · Package graphs - properties

Function: max_clique (gr)

  Returns a maximum clique of the graph gr.

  Example:

      (%i1) load ("graphs")$
      (%i2) g : random_graph(100, 0.5)$
      (%i3) max_clique(g);
      (%o3)          [6, 12, 31, 36, 52, 59, 62, 63, 80]

  Categories: Package graphs · Package graphs - properties

Function: max_degree (gr)

  Returns the maximal degree of vertices of the graph gr and a vertex of maximal degree.

  Example:

      (%i1) load ("graphs")$
      (%i2) g : random_graph(100, 0.02)$
      (%i3) max_degree(g);
      (%o3)                        [6, 79]
      (%i4) vertex_degree(95, g);
      (%o4)                           2

  Categories: Package graphs · Package graphs - properties

Function: max_flow (net, s, t)

  Returns a maximum flow through the network net with the source s and the sink t.

  The function returns the value of the maximal flow and a list representing the weights of the arcs in the optimal flow.

  Example:

      (%i1) load ("graphs")$
      (%i2) net : create_graph(
        [1,2,3,4,5,6],
        [[[1,2], 1.0],
         [[1,3], 0.3],
         [[2,4], 0.2],
         [[2,5], 0.3],
         [[3,4], 0.1],
         [[3,5], 0.1],
         [[4,6], 1.0],
         [[5,6], 1.0]],
        directed=true)$
      (%i3) [flow_value, flow] : max_flow(net, 1, 6);
      (%o3) [0.7, [[[1, 2], 0.5], [[1, 3], 0.2], [[2, 4], 0.2],
      [[2, 5], 0.3], [[3, 4], 0.1], [[3, 5], 0.1], [[4, 6], 0.3],
      [[5, 6], 0.4]]]
      (%i4) fl : 0$
      (%i5) for u in out_neighbors(1, net)
           do fl : fl + assoc([1, u], flow)$
      (%i6) fl;
      (%o6)                          0.7

  Categories: Package graphs · Package graphs - properties

Function: max_independent_set (gr)

  Returns a maximum independent set of the graph gr.

  Example:

      (%i1) load ("graphs")$
      (%i2) d : dodecahedron_graph()$
      (%i3) mi : max_independent_set(d);
      (%o3)             [0, 3, 5, 9, 10, 11, 18, 19]
      (%i4) draw_graph(d, show_vertices=mi)$

  Categories: Package graphs · Package graphs - properties

Function: max_matching (gr)

  Returns a maximum matching of the graph gr.

  Example:

      (%i1) load ("graphs")$
      (%i2) d : dodecahedron_graph()$
      (%i3) m : max_matching(d);
      (%o3) [[5, 7], [8, 9], [6, 10], [14, 19], [13, 18], [12, 17],
                                     [11, 16], [0, 15], [3, 4], [1, 2]]
      (%i4) draw_graph(d, show_edges=m)$

  Categories: Package graphs · Package graphs - properties

Function: min_degree (gr)

  Returns the minimum degree of vertices of the graph gr and a vertex of minimum degree.

  Example:

      (%i1) load ("graphs")$
      (%i2) g : random_graph(100, 0.1)$
      (%i3) min_degree(g);
      (%o3)                        [3, 49]
      (%i4) vertex_degree(21, g);
      (%o4)                           9

  Categories: Package graphs · Package graphs - properties

Function: min_edge_cut (gr)

  Returns the minimum edge cut in the graph gr.

  See also `edge_connectivity`.

  Categories: Package graphs · Package graphs - properties

Function: min_vertex_cover (gr)

  Returns the minimum vertex cover of the graph gr.

  Categories: Package graphs · Package graphs - properties

Function: min_vertex_cut (gr)

  Returns the minimum vertex cut in the graph gr.

  See also `vertex_connectivity`.

  Categories: Package graphs · Package graphs - properties

Function: minimum_spanning_tree (gr)

  Returns the minimum spanning tree of the graph gr.

  Example:

      (%i1) load ("graphs")$
      (%i2) g : graph_product(path_graph(10), path_graph(10))$
      (%i3) t : minimum_spanning_tree(g)$
      (%i4) draw_graph(g, show_edges=edges(t))$

  Categories: Package graphs · Package graphs - properties

Function: neighbors (v, gr)

  Returns the list of neighbors of the vertex v in the graph gr.

  Example:

      (%i1) load ("graphs")$
      (%i2) p : petersen_graph()$
      (%i3) neighbors(3, p);
      (%o3)                       [4, 8, 2]

  Categories: Package graphs · Package graphs - properties

Function: odd_girth (gr)

  Returns the length of the shortest odd cycle in the graph gr.

  Example:

      (%i1) load ("graphs")$
      (%i2) g : graph_product(cycle_graph(4), cycle_graph(7))$
      (%i3) girth(g);
      (%o3)                           4
      (%i4) odd_girth(g);
      (%o4)                           7

  Categories: Package graphs · Package graphs - properties

Function: out_neighbors (v, gr)

  Returns the list of out-neighbors of the vertex v in the directed graph gr.

  Example:

      (%i1) load ("graphs")$
      (%i2) p : path_digraph(3)$
      (%i3) in_neighbors(2, p);
      (%o3)                          [1]
      (%i4) out_neighbors(2, p);
      (%o4)                          []

  Categories: Package graphs · Package graphs - properties

Function: planar_embedding (gr)

  Returns the list of facial walks in a planar embedding of gr and `false` if gr is not a planar graph.

  The graph gr must be biconnected.

  The algorithm used is the Demoucron's algorithm, which is a quadratic time algorithm.

  Example:

      (%i1) load ("graphs")$
      (%i2) planar_embedding(grid_graph(3,3));
      (%o2) [[3, 6, 7, 8, 5, 2, 1, 0], [4, 3, 0, 1], [3, 4, 7, 6],
                                            [8, 7, 4, 5], [1, 2, 5, 4]]

  Categories: Package graphs · Package graphs - properties

Function: print_graph (gr)

  Prints some information about the graph gr.

  Example:

      (%i1) load ("graphs")$
      (%i2) c5 : cycle_graph(5)$
      (%i3) print_graph(c5)$
      Graph on 5 vertices with 5 edges.
      Adjacencies:
        4 :  0  3
        3 :  4  2
        2 :  3  1
        1 :  2  0
        0 :  4  1
      (%i4) dc5 : cycle_digraph(5)$
      (%i5) print_graph(dc5)$
      Digraph on 5 vertices with 5 arcs.
      Adjacencies:
        4 :  0
        3 :  4
        2 :  3
        1 :  2
        0 :  1
      (%i6) out_neighbors(0, dc5);
      (%o6)                          [1]

  Categories: Package graphs

Function: radius (gr)

  Returns the radius of the graph gr.

  Example:

      (%i1) load ("graphs")$
      (%i2) radius(dodecahedron_graph());
      (%o2)                           5

  Categories: Package graphs · Package graphs - properties

Function: set_edge_weight (e, w, gr)

  Assigns the weight w to the edge e in the graph gr.

  Example:

      (%i1) load ("graphs")$
      (%i2) g : create_graph([1, 2], [[[1,2], 1.2]])$
      (%i3) get_edge_weight([1,2], g);
      (%o3)                          1.2
      (%i4) set_edge_weight([1,2], 2.1, g);
      (%o4)                         done
      (%i5) get_edge_weight([1,2], g);
      (%o5)                          2.1

  Categories: Package graphs · Package graphs - properties

Function: set_vertex_label (v, l, gr)

  Assigns the label l to the vertex v in the graph gr.

  Example:

      (%i1) load ("graphs")$
      (%i2) g : create_graph([[1, "One"], [2, "Two"]], [[1,2]])$
      (%i3) get_vertex_label(1, g);
      (%o3)                          One
      (%i4) set_vertex_label(1, "oNE", g);
      (%o4)                         done
      (%i5) get_vertex_label(1, g);
      (%o5)                          oNE

  Categories: Package graphs · Package graphs - properties

Function: shortest_path (u, v, gr)

  Returns the shortest path from u to v in the graph gr.

  Example:

      (%i1) load ("graphs")$
      (%i2) d : dodecahedron_graph()$
      (%i3) path : shortest_path(0, 7, d);
      (%o3)                   [0, 1, 19, 13, 7]
      (%i4) draw_graph(d, show_edges=vertices_to_path(path))$

  Categories: Package graphs · Package graphs - properties

Function: shortest_weighted_path (u, v, gr)

  Returns the length of the shortest weighted path and the shortest weighted path from u to v in the graph gr.

  The length of a weighted path is the sum of edge weights of edges in the path. If an edge has no weight, then it has a default weight 1.

  Example:

      (%i1) load ("graphs")$
      (%i2) g: petersen_graph(20, 2)$
      (%i3) for e in edges(g) do set_edge_weight(e, random(1.0), g)$
      (%i4) shortest_weighted_path(0, 10, g);
      (%o4) [2.575143920268482, [0, 20, 38, 36, 34, 32, 30, 10]]

  Categories: Package graphs · Package graphs - properties

Function: strong_components (gr)

  Returns the strong components of a directed graph gr.

  Example:

      (%i1) load ("graphs")$
      (%i2) t : random_tournament(4)$
      (%i3) strong_components(t);
      (%o3)                 [[1], [0], [2], [3]]
      (%i4) vertex_out_degree(3, t);
      (%o4)                           3

  Categories: Package graphs · Package graphs - properties

Function: topological_sort (dag)

  Returns a topological sorting of the vertices of a directed graph dag or an empty list if dag is not a directed acyclic graph.

  Example:

      (%i1) load ("graphs")$
      (%i2) g:create_graph(
               [1,2,3,4,5],
               [
                [1,2], [2,5], [5,3],
                [5,4], [3,4], [1,3]
               ],
               directed=true)$
      (%i3) topological_sort(g);
      (%o3)                    [1, 2, 5, 3, 4]

  Categories: Package graphs · Package graphs - properties

Function: vertex_connectivity (g)

  Returns the vertex connectivity of the graph g.

  See also `min_vertex_cut`.

  Categories: Package graphs · Package graphs - properties

Function: vertex_degree (v, gr)

  Returns the degree of the vertex v in the graph gr.

  Categories: Package graphs · Package graphs - properties

Function: vertex_distance (u, v, gr)

  Returns the length of the shortest path between u and v in the (directed) graph gr.

  Example:

      (%i1) load ("graphs")$
      (%i2) d : dodecahedron_graph()$
      (%i3) vertex_distance(0, 7, d);
      (%o3)                           4
      (%i4) shortest_path(0, 7, d);
      (%o4)                   [0, 1, 19, 13, 7]

  Categories: Package graphs · Package graphs - properties

Function: vertex_eccentricity (v, gr)

  Returns the eccentricity of the vertex v in the graph gr.

  Example:

      (%i1) load ("graphs")$
      (%i2) g:cycle_graph(7)$
      (%i3) vertex_eccentricity(0, g);
      (%o3)                           3

  Categories: Package graphs · Package graphs - properties

Function: vertex_in_degree (v, gr)

  Returns the in-degree of the vertex v in the directed graph gr.

  Example:

      (%i1) load ("graphs")$
      (%i2) p5 : path_digraph(5)$
      (%i3) print_graph(p5)$
      Digraph on 5 vertices with 4 arcs.
      Adjacencies:
        4 :
        3 :  4
        2 :  3
        1 :  2
        0 :  1
      (%i4) vertex_in_degree(4, p5);
      (%o4)                           1
      (%i5) in_neighbors(4, p5);
      (%o5)                          [3]

  Categories: Package graphs · Package graphs - properties

Function: vertex_out_degree (v, gr)

  Returns the out-degree of the vertex v in the directed graph gr.

  Example:

      (%i1) load ("graphs")$
      (%i2) t : random_tournament(10)$
      (%i3) vertex_out_degree(0, t);
      (%o3)                           2
      (%i4) out_neighbors(0, t);
      (%o4)                        [7, 1]

  Categories: Package graphs · Package graphs - properties

Function: vertices (gr)

  Returns the list of vertices in the graph gr.

  Example:

      (%i1) load ("graphs")$
      (%i2) vertices(complete_graph(4));
      (%o2)                     [3, 2, 1, 0]

  Categories: Package graphs · Package graphs - properties

Function: vertex_coloring (gr)

  Returns an optimal coloring of the vertices of the graph gr.

  The function returns the chromatic number and a list representing the coloring of the vertices of gr.

  Example:

      (%i1) load ("graphs")$
      (%i2) p:petersen_graph()$
      (%i3) vertex_coloring(p);
      (%o3) [3, [[0, 2], [1, 3], [2, 2], [3, 3], [4, 1], [5, 3],
                                       [6, 1], [7, 1], [8, 2], [9, 2]]]

  Categories: Package graphs · Package graphs - properties

Function: wiener_index (gr)

  Returns the Wiener index of the graph gr.

  Example:

      (%i2) wiener_index(dodecahedron_graph());
      (%o2)                          500

  Categories: Package graphs · Package graphs - properties

[ < ] | [ > ] |  | [ << ] | [ Up ] | [ >> ] |  |  |  |  | [Top] | [Contents] | [Index] | [ ? ]

### 61.2.3 Modifying graphs

Function: add_edge (e, gr)

  Adds the edge e to the graph gr.

  Example:

      (%i1) load ("graphs")$
      (%i2) p : path_graph(4)$
      (%i3) neighbors(0, p);
      (%o3)                          [1]
      (%i4) add_edge([0,3], p);
      (%o4)                         done
      (%i5) neighbors(0, p);
      (%o5)                        [3, 1]

  Categories: Package graphs · Package graphs - modifications

Function: add_edges (e_list, gr)

  Adds all edges in the list e_list to the graph gr.

  Example:

      (%i1) load ("graphs")$
      (%i2) g : empty_graph(3)$
      (%i3) add_edges([[0,1],[1,2]], g)$
      (%i4) print_graph(g)$
      Graph on 3 vertices with 2 edges.
      Adjacencies:
        2 :  1
        1 :  2  0
        0 :  1

  Categories: Package graphs · Package graphs - modifications

Function: add_vertex (v, gr)

  Adds the vertex v to the graph gr.

  Example:

      (%i1) load ("graphs")$
      (%i2) g : path_graph(2)$
      (%i3) add_vertex(2, g)$
      (%i4) print_graph(g)$
      Graph on 3 vertices with 1 edges.
      Adjacencies:
        2 :
        1 :  0
        0 :  1

  Categories: Package graphs · Package graphs - modifications

Function: add_vertices (v_list, gr)

  Adds all vertices in the list v_list to the graph gr.

  Categories: Package graphs · Package graphs - modifications

Function: connect_vertices (v_list, u_list, gr)

  Connects all vertices from the list v_list with the vertices in the list u_list in the graph gr.

  v_list and u_list can be single vertices or lists of vertices.

  Example:

      (%i1) load ("graphs")$
      (%i2) g : empty_graph(4)$
      (%i3) connect_vertices(0, [1,2,3], g)$
      (%i4) print_graph(g)$
      Graph on 4 vertices with 3 edges.
      Adjacencies:
        3 :  0
        2 :  0
        1 :  0
        0 :  3  2  1

  Categories: Package graphs · Package graphs - modifications

Function: contract_edge (e, gr)

  Contracts the edge e in the graph gr.

  Example:

      (%i1) load ("graphs")$
      (%i2) g: create_graph(
            8, [[0,3],[1,3],[2,3],[3,4],[4,5],[4,6],[4,7]])$
      (%i3) print_graph(g)$
      Graph on 8 vertices with 7 edges.
      Adjacencies:
        7 :  4
        6 :  4
        5 :  4
        4 :  7  6  5  3
        3 :  4  2  1  0
        2 :  3
        1 :  3
        0 :  3
      (%i4) contract_edge([3,4], g)$
      (%i5) print_graph(g)$
      Graph on 7 vertices with 6 edges.
      Adjacencies:
        7 :  3
        6 :  3
        5 :  3
        3 :  5  6  7  2  1  0
        2 :  3
        1 :  3
        0 :  3

  Categories: Package graphs · Package graphs - modifications

Function: remove_edge (e, gr)

  Removes the edge e from the graph gr.

  Example:

      (%i1) load ("graphs")$
      (%i2) c3 : cycle_graph(3)$
      (%i3) remove_edge([0,1], c3)$
      (%i4) print_graph(c3)$
      Graph on 3 vertices with 2 edges.
      Adjacencies:
        2 :  0  1
        1 :  2
        0 :  2

  Categories: Package graphs · Package graphs - modifications

Function: remove_vertex (v, gr)

  Removes the vertex v from the graph gr.

  Categories: Package graphs

[ < ] | [ > ] |  | [ << ] | [ Up ] | [ >> ] |  |  |  |  | [Top] | [Contents] | [Index] | [ ? ]

### 61.2.4 Reading and writing to files

Function: dimacs_export
`dimacs_export` (gr, fl)
`dimacs_export` (gr, fl, comment1, ..., commentn)

  Exports the graph into the file fl in the DIMACS format. Optional comments will be added to the top of the file.

  Categories: Package graphs · Package graphs - io

Function: dimacs_import (fl)

  Returns the graph from file fl in the DIMACS format.

  Categories: Package graphs · Package graphs - io

Function: graph6_decode (str)

  Returns the graph encoded in the graph6 format in the string str.

  Categories: Package graphs · Package graphs - io

Function: graph6_encode (gr)

  Returns a string which encodes the graph gr in the graph6 format.

  Categories: Package graphs · Package graphs - io

Function: graph6_export (gr_list, fl)

  Exports graphs in the list gr_list to the file fl in the graph6 format.

  Categories: Package graphs · Package graphs - io

Function: graph6_import (fl)

  Returns a list of graphs from the file fl in the graph6 format.

  Categories: Package graphs · Package graphs - io

Function: sparse6_decode (str)

  Returns the graph encoded in the sparse6 format in the string str.

  Categories: Package graphs · Package graphs - io

Function: sparse6_encode (gr)

  Returns a string which encodes the graph gr in the sparse6 format.

  Categories: Package graphs · Package graphs - io

Function: sparse6_export (gr_list, fl)

  Exports graphs in the list gr_list to the file fl in the sparse6 format.

  Categories: Package graphs · Package graphs - io

Function: sparse6_import (fl)

  Returns a list of graphs from the file fl in the sparse6 format.

  Categories: Package graphs · Package graphs - io

[ < ] | [ > ] |  | [ << ] | [ Up ] | [ >> ] |  |  |  |  | [Top] | [Contents] | [Index] | [ ? ]

### 61.2.5 Visualization

Function: draw_graph
`draw_graph` (graph)
`draw_graph` (graph, option1, ..., optionk)

  Draws the graph using the draw package.

  The algorithm used to position vertices is specified by the optional argument program. The default value is `program=spring_embedding`. draw_graph can also use the graphviz programs for positioning vertices, but graphviz must be installed separately.

  Example 1:

      (%i1) load ("graphs")$
      (%i2) g:grid_graph(10,10)$
      (%i3) m:max_matching(g)$
      (%i4) draw_graph(g,
         spring_embedding_depth=100,
         show_edges=m, edge_type=dots,
         vertex_size=0)$

  Example 2:

      (%i1) load ("graphs")$
      (%i2) g:create_graph(16,
          [
           [0,1],[1,3],[2,3],[0,2],[3,4],[2,4],
           [5,6],[6,4],[4,7],[6,7],[7,8],[7,10],[7,11],
           [8,10],[11,10],[8,9],[11,12],[9,15],[12,13],
           [10,14],[15,14],[13,14]
          ])$
      (%i3) t:minimum_spanning_tree(g)$
      (%i4) draw_graph(
          g,
          show_edges=edges(t),
          show_edge_width=4,
          show_edge_color=green,
          vertex_type=filled_square,
          vertex_size=2
          )$

  Example 3:

      (%i1) load ("graphs")$
      (%i2) g:create_graph(16,
          [
           [0,1],[1,3],[2,3],[0,2],[3,4],[2,4],
           [5,6],[6,4],[4,7],[6,7],[7,8],[7,10],[7,11],
           [8,10],[11,10],[8,9],[11,12],[9,15],[12,13],
           [10,14],[15,14],[13,14]
          ])$
      (%i3) mi : max_independent_set(g)$
      (%i4) draw_graph(
          g,
          show_vertices=mi,
          show_vertex_type=filled_up_triangle,
          show_vertex_size=2,
          edge_color=cyan,
          edge_width=3,
          show_id=true,
          text_color=brown
          )$

  Example 4:

      (%i1) load ("graphs")$
      (%i2) net : create_graph(
          [0,1,2,3,4,5],
          [
           [[0,1], 3], [[0,2], 2],
           [[1,3], 1], [[1,4], 3],
           [[2,3], 2], [[2,4], 2],
           [[4,5], 2], [[3,5], 2]
          ],
          directed=true
          )$
      (%i3) draw_graph(
          net,
          show_weight=true,
          vertex_size=0,
          show_vertices=[0,5],
          show_vertex_type=filled_square,
          head_length=0.2,
          head_angle=10,
          edge_color="dark-green",
          text_color=blue
          )$

  Example 5:

      (%i1) load(graphs)$
      (%i2) g: petersen_graph(20, 2);
      (%o2)                         GRAPH
      (%i3) draw_graph(g, redraw=true, program=planar_embedding);
      (%o3)                         done

  Example 6:

      (%i1) load(graphs)$
      (%i2) t: tutte_graph();
      (%o2)                         GRAPH
      (%i3) draw_graph(t, redraw=true,
                          fixed_vertices=[1,2,3,4,5,6,7,8,9]);
      (%o3)                         done

  Categories: Package graphs

Option variable: draw_graph_program

  Default value: spring_embedding

  The default value for the program used to position vertices in `draw_graph` program.

  Categories: Package graphs · Package graphs - draw_graphs options

draw_graph option: show_id

  Default value: false

  If true then ids of the vertices are displayed.

  Categories: Package graphs · Package graphs - draw_graphs options

draw_graph option: show_label

  Default value: false

  If true then labels of the vertices are displayed.

  Categories: Package graphs · Package graphs - draw_graphs options

draw_graph option: label_alignment

  Default value: center

  Determines how to align the labels/ids of the vertices. Can be `left`, `center` or `right`.

  Categories: Package graphs · Package graphs - draw_graphs options

draw_graph option: show_weight

  Default value: false

  If true then weights of the edges are displayed.

  Categories: Package graphs · Package graphs - draw_graphs options

draw_graph option: vertex_type

  Default value: circle

  Defines how vertices are displayed. See the point_type option for the `draw` package for possible values.

  Categories: Package graphs · Package graphs - draw_graphs options

draw_graph option: vertex_size

  The size of vertices.

  Categories: Package graphs · Package graphs - draw_graphs options

draw_graph option: vertex_color

  The color used for displaying vertices.

  Categories: Package graphs · Package graphs - draw_graphs options

draw_graph option: show_vertices

  Default value: []

  Display selected vertices in the using a different color.

  Categories: Package graphs · Package graphs - draw_graphs options

draw_graph option: show_vertex_type

  Defines how vertices specified in show_vertices are displayed. See the point_type option for the `draw` package for possible values.

  Categories: Package graphs · Package graphs - draw_graphs options

draw_graph option: show_vertex_size

  The size of vertices in show_vertices.

  Categories: Package graphs · Package graphs - draw_graphs options

draw_graph option: show_vertex_color

  The color used for displaying vertices in the show_vertices list.

  Categories: Package graphs · Package graphs - draw_graphs options

draw_graph option: vertex_partition

  Default value: []

  A partition `[[v1,v2,...],...,[vk,...,vn]]` of the vertices of the graph. The vertices of each list in the partition will be drawn in a different color.

  Categories: Package graphs · Package graphs - draw_graphs options

draw_graph option: vertex_coloring

  Specifies coloring of the vertices. The coloring col must be specified in the format as returned by vertex_coloring.

  Categories: Package graphs · Package graphs - draw_graphs options

draw_graph option: edge_color

  The color used for displaying edges.

  Categories: Package graphs · Package graphs - draw_graphs options

draw_graph option: edge_width

  The width of edges.

  Categories: Package graphs · Package graphs - draw_graphs options

draw_graph option: edge_type

  Defines how edges are displayed. See the line_type option for the `draw` package.

  Categories: Package graphs · Package graphs - draw_graphs options

draw_graph option: show_edges

  Display edges specified in the list e_list using a different color.

  Categories: Package graphs · Package graphs - draw_graphs options

draw_graph option: show_edge_color

  The color used for displaying edges in the show_edges list.

  Categories: Package graphs · Package graphs - draw_graphs options

draw_graph option: show_edge_width

  The width of edges in show_edges.

  Categories: Package graphs · Package graphs - draw_graphs options

draw_graph option: show_edge_type

  Defines how edges in show_edges are displayed. See the line_type option for the `draw` package.

  Categories: Package graphs · Package graphs - draw_graphs options

draw_graph option: edge_partition

  A partition `[[e1,e2,...],...,[ek,...,em]]` of edges of the graph. The edges of each list in the partition will be drawn using a different color.

  Categories: Package graphs · Package graphs - draw_graphs options

draw_graph option: edge_coloring

  The coloring of edges. The coloring must be specified in the format as returned by the function edge_coloring.

  Categories: Package graphs · Package graphs - draw_graphs options

draw_graph option: redraw

  Default value: false

  If `true`, vertex positions are recomputed even if the positions have been saved from a previous drawing of the graph.

  Categories: Package graphs · Package graphs - draw_graphs options

draw_graph option: head_angle

  Default value: 15

  The angle for the arrows displayed on arcs (in directed graphs).

  Categories: Package graphs · Package graphs - draw_graphs options

draw_graph option: head_length

  Default value: 0.1

  The length for the arrows displayed on arcs (in directed graphs).

  Categories: Package graphs · Package graphs - draw_graphs options

draw_graph option: spring_embedding_depth

  Default value: 50

  The number of iterations in the spring embedding graph drawing algorithm.

  Categories: Package graphs · Package graphs - draw_graphs options

draw_graph option: terminal

  The terminal used for drawing (see the terminal option in the `draw` package).

  Categories: Package graphs · Package graphs - draw_graphs options

draw_graph option: file_name

  The filename of the drawing if terminal is not screen.

  Categories: Package graphs · Package graphs - draw_graphs options

draw_graph option: program

  Defines the program used for positioning vertices of the graph. Can be one of the graphviz programs (dot, neato, twopi, circ, fdp), circular, spring_embedding or planar_embedding. planar_embedding is only available for 2-connected planar graphs. When `program=spring_embedding`, a set of vertices with fixed position can be specified with the fixed_vertices option.

  Categories: Package graphs · Package graphs - draw_graphs options

draw_graph option: fixed_vertices

  Specifies a list of vertices which will have positions fixed along a regular polygon. Can be used when `program=spring_embedding`.

  Categories: Package graphs · Package graphs - draw_graphs options

Function: vertices_to_path (v_list)

  Converts a list v_list of vertices to a list of edges of the path defined by v_list.

  Categories: Package graphs

Function: vertices_to_cycle (v_list)

  Converts a list v_list of vertices to a list of edges of the cycle defined by v_list.

  Categories: Package graphs

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# 62. grobner

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## 62.1 Introduction to grobner

`grobner` is a package for working with Groebner bases in Maxima.

To use the following functions you must load the ``grobner.lisp'` package.

    load(grobner);

A demo can be started by

    demo("grobner.demo");

or

    batch("grobner.demo")

Some of the calculation in the demo will take a lot of time therefore the output ``grobner-demo.output'` of the demo can be found in the same directory as the demo file.

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### 62.1.1 Notes on the grobner package

The package was written by

Marek Rychlik

http://alamos.math.arizona.edu

and is released 2002-05-24 under the terms of the General Public License(GPL) (see file ``grobner.lisp'`). This documentation was extracted from the files

``README'`, ``grobner.lisp'`, ``grobner.demo'`, ``grobner-demo.output'`

by Günter Nowak. Suggestions for improvement of the documentation can be discussed at the maxima-mailing-list maxima@math.utexas.edu. The code is a little bit out of date now. Modern implementation use the fast F4 algorithm described in

    A new efficient algorithm for computing Gröbner bases (F4)
    Jean-Charles Faugère
    LIP6/CNRS Université Paris VI
    January 20, 1999

Categories: Groebner bases · Share packages · Package grobner

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### 62.1.2 Implementations of admissible monomial orders in grobner

- `lex`

  pure lexicographic, default order for monomial comparisons

- `grlex`

  total degree order, ties broken by lexicographic

- `grevlex`

  total degree, ties broken by reverse lexicographic

- `invlex`

  inverse lexicographic order

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## 62.2 Functions and Variables for grobner

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### 62.2.1 Global switches for grobner

Option variable: poly_monomial_order

  Default value: `lex`

  This global switch controls which monomial order is used in polynomial and Groebner Bases calculations. If not set, `lex` will be used.

  Categories: Package grobner

Option variable: poly_coefficient_ring

  Default value: `expression_ring`

  This switch indicates the coefficient ring of the polynomials that will be used in grobner calculations. If not set, maxima's general expression ring will be used. This variable may be set to `ring_of_integers` if desired.

  Categories: Package grobner

Option variable: poly_primary_elimination_order

  Default value: `false`

  Name of the default order for eliminated variables in elimination-based functions. If not set, `lex` will be used.

  Categories: Package grobner

Option variable: poly_secondary_elimination_order

  Default value: `false`

  Name of the default order for kept variables in elimination-based functions. If not set, `lex` will be used.

  Categories: Package grobner

Option variable: poly_elimination_order

  Default value: `false`

  Name of the default elimination order used in elimination calculations. If set, it overrides the settings in variables `poly_primary_elimination_order` and `poly_secondary_elimination_order`. The user must ensure that this is a true elimination order valid for the number of eliminated variables.

  Categories: Package grobner

Option variable: poly_return_term_list

  Default value: `false`

  If set to `true`, all functions in this package will return each polynomial as a list of terms in the current monomial order rather than a maxima general expression.

  Categories: Package grobner

Option variable: poly_grobner_debug

  Default value: `false`

  If set to `true`, produce debugging and tracing output.

  Categories: Package grobner

Option variable: poly_grobner_algorithm

  Default value: `buchberger`

  Possible values:

  The name of the algorithm used to find the Groebner Bases.

  Categories: Package grobner

Option variable: poly_top_reduction_only

  Default value: `false`

  If not `false`, use top reduction only whenever possible. Top reduction means that division algorithm stops after the first reduction.

  Categories: Package grobner

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### 62.2.2 Simple operators in grobner

`poly_add`, `poly_subtract`, `poly_multiply` and `poly_expt` are the arithmetical operations on polynomials. These are performed using the internal representation, but the results are converted back to the maxima general form.

Function: poly_add (poly1, poly2, varlist)

  Adds two polynomials poly1 and poly2.

      (%i1) poly_add(z+x^2*y,x-z,[x,y,z]);
                                          2
      (%o1)                              x  y + x

  Categories: Package grobner

Function: poly_subtract (poly1, poly2, varlist)

  Subtracts a polynomial poly2 from poly1.

      (%i1) poly_subtract(z+x^2*y,x-z,[x,y,z]);
                                            2
      (%o1)                          2 z + x  y - x

  Categories: Package grobner

Function: poly_multiply (poly1, poly2, varlist)

  Returns the product of polynomials poly1 and poly2.

      (%i2) poly_multiply(z+x^2*y,x-z,[x,y,z])-(z+x^2*y)*(x-z),expand;
      (%o1)                                  0

  Categories: Package grobner

Function: poly_s_polynomial (poly1, poly2, varlist)

  Returns the syzygy polynomial (S-polynomial) of two polynomials poly1 and poly2.

  Categories: Package grobner

Function: poly_primitive_part (poly1, varlist)

  Returns the polynomial poly divided by the GCD of its coefficients.

      (%i1) poly_primitive_part(35*y+21*x,[x,y]);
      (%o1)                              5 y + 3 x

  Categories: Package grobner

Function: poly_normalize (poly, varlist)

  Returns the polynomial poly divided by the leading coefficient. It assumes that the division is possible, which may not always be the case in rings which are not fields.

  Categories: Package grobner

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### 62.2.3 Other functions in grobner

Function: poly_expand (poly, varlist)

  This function parses polynomials to internal form and back. It is equivalent to `expand(poly)` if poly parses correctly to a polynomial. If the representation is not compatible with a polynomial in variables varlist, the result is an error. It can be used to test whether an expression correctly parses to the internal representation. The following examples illustrate that indexed and transcendental function variables are allowed.

      (%i1) poly_expand((x-y)*(y+x),[x,y]);
                                           2    2
      (%o1)                               x  - y
      (%i2) poly_expand((y+x)^2,[x,y]);
                                      2            2
      (%o2)                          y  + 2 x y + x
      (%i3) poly_expand((y+x)^5,[x,y]);
                        5      4         2  3       3  2      4      5
      (%o3)            y  + 5 x y  + 10 x  y  + 10 x  y  + 5 x  y + x
      (%i4) poly_expand(-1-x*exp(y)+x^2/sqrt(y),[x]);
                                                2
                                        y      x
      (%o4)                       - x %e  + ------- - 1
                                             sqrt(y)

      (%i5) poly_expand(-1-sin(x)^2+sin(x),[sin(x)]);
                                      2
      (%o5)                      - sin (x) + sin(x) - 1

  Categories: Package grobner

Function: poly_expt (poly, number, varlist)

  exponentitates poly by a positive integer number. If number is not a positive integer number an error will be raised.

      (%i1) poly_expt(x-y,3,[x,y])-(x-y)^3,expand;
      (%o1)                                  0

  Categories: Package grobner

Function: poly_content (poly. varlist)

  `poly_content` extracts the GCD of its coefficients

      (%i1) poly_content(35*y+21*x,[x,y]);
      (%o1)                                  7

  Categories: Package grobner

Function: poly_pseudo_divide (poly, polylist, varlist)

  Pseudo-divide a polynomial poly by the list of n polynomials polylist. Return multiple values. The first value is a list of quotients a. The second value is the remainder r. The third argument is a scalar coefficient c, such that c*poly can be divided by polylist within the ring of coefficients, which is not necessarily a field. Finally, the fourth value is an integer count of the number of reductions performed. The resulting objects satisfy the equation:

  c*poly=sum(a[i]*polylist[i],i=1...n)+r.

  Categories: Package grobner

Function: poly_exact_divide (poly1, poly2, varlist)

  Divide a polynomial poly1 by another polynomial poly2. Assumes that exact division with no remainder is possible. Returns the quotient.

  Categories: Package grobner

Function: poly_normal_form (poly, polylist, varlist)

  `poly_normal_form` finds the normal form of a polynomial poly with respect to a set of polynomials polylist.

  Categories: Package grobner

Function: poly_buchberger_criterion (polylist, varlist)

  Returns `true` if polylist is a Groebner basis with respect to the current term order, by using the Buchberger criterion: for every two polynomials h1 and h2 in polylist the S-polynomial S(h1,h2) reduces to 0 modulo polylist.

  Categories: Package grobner

Function: poly_buchberger (polylist_fl varlist)

  `poly_buchberger` performs the Buchberger algorithm on a list of polynomials and returns the resulting Groebner basis.

  Categories: Package grobner

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### 62.2.4 Standard postprocessing of Groebner Bases

The k-th elimination Ideal I_k of an Ideal I over K[ x[1],...,x[n] ] is the ideal intersect(I, K[ x[k+1],...,x[n] ]).
The colon ideal I:J is the ideal {h|for all w in J: w*h in I}.
The ideal I:p^inf is the ideal {h| there is a n in N: p^n*h in I}.
The ideal I:J^inf is the ideal {h| there is a n in N and a p in J: p^n*h in I}.
The radical ideal sqrt(I) is the ideal {h| there is a n in N : h^n in I }.

Function: poly_reduction (polylist, varlist)

  `poly_reduction` reduces a list of polynomials polylist, so that each polynomial is fully reduced with respect to the other polynomials.

  Categories: Package grobner

Function: poly_minimization (polylist, varlist)

  Returns a sublist of the polynomial list polylist spanning the same monomial ideal as polylist but minimal, i.e. no leading monomial of a polynomial in the sublist divides the leading monomial of another polynomial.

  Categories: Package grobner

Function: poly_normalize_list (polylist, varlist)

  `poly_normalize_list` applies `poly_normalize` to each polynomial in the list. That means it divides every polynomial in a list polylist by its leading coefficient.

  Categories: Package grobner

Function: poly_grobner (polylist, varlist)

  Returns a Groebner basis of the ideal span by the polynomials polylist. Affected by the global flags.

  Categories: Package grobner

Function: poly_reduced_grobner (polylist, varlist)

  Returns a reduced Groebner basis of the ideal span by the polynomials polylist. Affected by the global flags.

  Categories: Package grobner

Function: poly_depends_p (poly, var, varlist)

  `poly_depends` tests whether a polynomial depends on a variable var.

  Categories: Package grobner · Predicate functions

Function: poly_elimination_ideal (polylist, number, varlist)

  `poly_elimination_ideal` returns the grobner basis of the number-th elimination ideal of an ideal specified as a list of generating polynomials (not necessarily Groebner basis).

  Categories: Package grobner

Function: poly_colon_ideal (polylist1, polylist2, varlist)

  Returns the reduced Groebner basis of the colon ideal

  I(polylist1):I(polylist2)

  where polylist1 and polylist2 are two lists of polynomials.

  Categories: Package grobner

Function: poly_ideal_intersection (polylist1, polylist2, varlist)

  `poly_ideal_intersection` returns the intersection of two ideals.

  Categories: Package grobner

Function: poly_lcm (poly1, poly2, varlist)

  Returns the lowest common multiple of poly1 and poly2.

  Categories: Package grobner

Function: poly_gcd (poly1, poly2, varlist)

  Returns the greatest common divisor of poly1 and poly2.

  See also `ezgcd`, `gcd`, `gcdex`, and `gcdivide`.

  Example:

      (%i1) p1:6*x^3+19*x^2+19*x+6;
                              3       2
      (%o1)                6 x  + 19 x  + 19 x + 6
      (%i2) p2:6*x^5+13*x^4+12*x^3+13*x^2+6*x;
                        5       4       3       2
      (%o2)          6 x  + 13 x  + 12 x  + 13 x  + 6 x
      (%i3) poly_gcd(p1, p2, [x]);
                                  2
      (%o3)                    6 x  + 13 x + 6

  Categories: Package grobner

Function: poly_grobner_equal (polylist1, polylist2, varlist)

  `poly_grobner_equal` tests whether two Groebner Bases generate the same ideal. Returns `true` if two lists of polynomials polylist1 and polylist2, assumed to be Groebner Bases, generate the same ideal, and `false` otherwise. This is equivalent to checking that every polynomial of the first basis reduces to 0 modulo the second basis and vice versa. Note that in the example below the first list is not a Groebner basis, and thus the result is `false`.

      (%i1) poly_grobner_equal([y+x,x-y],[x,y],[x,y]);
      (%o1)                         false

  Categories: Package grobner

Function: poly_grobner_subsetp (polylist1, polylist2, varlist)

  `poly_grobner_subsetp` tests whether an ideal generated by polylist1 is contained in the ideal generated by polylist2. For this test to always succeed, polylist2 must be a Groebner basis.

  Categories: Package grobner · Predicate functions

Function: poly_grobner_member (poly, polylist, varlist)

  Returns `true` if a polynomial poly belongs to the ideal generated by the polynomial list polylist, which is assumed to be a Groebner basis. Returns `false` otherwise.

  `poly_grobner_member` tests whether a polynomial belongs to an ideal generated by a list of polynomials, which is assumed to be a Groebner basis. Equivalent to `normal_form` being 0.

  Categories: Package grobner

Function: poly_ideal_saturation1 (polylist, poly, varlist)

  Returns the reduced Groebner basis of the saturation of the ideal

  I(polylist):poly^inf

  Geometrically, over an algebraically closed field, this is the set of polynomials in the ideal generated by polylist which do not identically vanish on the variety of poly.

  Categories: Package grobner

Function: poly_ideal_saturation (polylist1, polylist2, varlist)

  Returns the reduced Groebner basis of the saturation of the ideal

  I(polylist1):I(polylist2)^inf

  Geometrically, over an algebraically closed field, this is the set of polynomials in the ideal generated by polylist1 which do not identically vanish on the variety of polylist2.

  Categories: Package grobner

Function: poly_ideal_polysaturation1 (polylist1, polylist2, varlist)

  polylist2 ist a list of n polynomials `[poly1,...,polyn]`. Returns the reduced Groebner basis of the ideal

  I(polylist):poly1^inf:...:polyn^inf

  obtained by a sequence of successive saturations in the polynomials of the polynomial list polylist2 of the ideal generated by the polynomial list polylist1.

  Categories: Package grobner

Function: poly_ideal_polysaturation (polylist, polylistlist, varlist)

  polylistlist is a list of n list of polynomials `[polylist1,...,polylistn]`. Returns the reduced Groebner basis of the saturation of the ideal

  I(polylist):I(polylist_1)^inf:...:I(polylist_n)^inf

  Categories: Package grobner

Function: poly_saturation_extension (poly, polylist, varlist1, varlist2)

  `poly_saturation_extension` implements the famous Rabinowitz trick.

  Categories: Package grobner

Function: poly_polysaturation_extension (poly, polylist, varlist1, varlist2)

  Categories: Package grobner

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# 63. impdiff

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## 63.1 Functions and Variables for impdiff

Function: implicit_derivative (f,indvarlist,orderlist,depvar)

  This subroutine computes implicit derivatives of multivariable functions. f is an array function, the indexes are the derivative degree in the indvarlist order; indvarlist is the independent variable list; orderlist is the order desired; and depvar is the dependent variable.

  To use this function write first `load("impdiff")`.

  Categories: Differential calculus · Share packages · Package impdiff

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# 64. interpol

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## 64.1 Introduction to interpol

Package `interpol` defines the Lagrangian, the linear and the cubic splines methods for polynomial interpolation.

For comments, bugs or suggestions, please contact me at 'mario AT edu DOT xunta DOT es'.

Categories: Numerical methods · Share packages · Package interpol

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## 64.2 Functions and Variables for interpol

Function: lagrange
`lagrange` (points)
`lagrange` (points, option)

  Computes the polynomial interpolation by the Lagrangian method. Argument points must be either:

  - a two column matrix, `p:matrix([2,4],[5,6],[9,3])`,
  - a list of pairs, `p: [[2,4],[5,6],[9,3]]`,
  - a list of numbers, `p: [4,6,3]`, in which case the abscissas will be assigned automatically to 1, 2, 3, etc.

  In the first two cases the pairs are ordered with respect to the first coordinate before making computations.

  With the option argument it is possible to select the name for the independent variable, which is `'x` by default; to define another one, write something like `varname='z`.

  Note that when working with high degree polynomials, floating point evaluations are unstable.

  See also `linearinterpol`, `cspline`, and `ratinterpol`.

  Examples:

      (%i1) load(interpol)$
      (%i2) p:[[7,2],[8,2],[1,5],[3,2],[6,7]]$
      (%i3) lagrange(p);
             (x - 7) (x - 6) (x - 3) (x - 1)
      (%o3)  -------------------------------
                           35
         (x - 8) (x - 6) (x - 3) (x - 1)
       - -------------------------------
                       12
         7 (x - 8) (x - 7) (x - 3) (x - 1)
       + ---------------------------------
                        30
         (x - 8) (x - 7) (x - 6) (x - 1)
       - -------------------------------
                       60
         (x - 8) (x - 7) (x - 6) (x - 3)
       + -------------------------------
                       84
      (%i4) f(x):=''%;
                     (x - 7) (x - 6) (x - 3) (x - 1)
      (%o4)  f(x) := -------------------------------
                                   35
         (x - 8) (x - 6) (x - 3) (x - 1)
       - -------------------------------
                       12
         7 (x - 8) (x - 7) (x - 3) (x - 1)
       + ---------------------------------
                        30
         (x - 8) (x - 7) (x - 6) (x - 1)
       - -------------------------------
                       60
         (x - 8) (x - 7) (x - 6) (x - 3)
       + -------------------------------
                       84
      (%i5) /* Evaluate the polynomial at some points */
            expand(map(f,[2.3,5/7,%pi]));
                                        4          3           2
                          919062  73 %pi    701 %pi    8957 %pi
      (%o5)  [- 1.567535, ------, ------- - -------- + ---------
                          84035     420       210         420
                                                   5288 %pi   186
                                                 - -------- + ---]
                                                     105       5
      (%i6) %,numer;
      (%o6) [- 1.567535, 10.9366573451538, 2.89319655125692]
      (%i7) load(draw)$  /* load draw package */
      (%i8) /* Plot the polynomial together with points */
            draw2d(
              color      = red,
              key        = "Lagrange polynomial",
              explicit(f(x),x,0,10),
              point_size = 3,
              color      = blue,
              key        = "Sample points",
              points(p))$
      (%i9) /* Change variable name */
            lagrange(p, varname=w);
             (w - 7) (w - 6) (w - 3) (w - 1)
      (%o9)  -------------------------------
                           35
         (w - 8) (w - 6) (w - 3) (w - 1)
       - -------------------------------
                       12
         7 (w - 8) (w - 7) (w - 3) (w - 1)
       + ---------------------------------
                        30
         (w - 8) (w - 7) (w - 6) (w - 1)
       - -------------------------------
                       60
         (w - 8) (w - 7) (w - 6) (w - 3)
       + -------------------------------
                       84

  Categories: Package interpol

Function: charfun2 (x, a, b)

  Returns `true` if number x belongs to the interval [a, b), and `false` otherwise.

  Categories: Package interpol

Function: linearinterpol
`linearinterpol` (points)
`linearinterpol` (points, option)

  Computes the polynomial interpolation by the linear method. Argument points must be either:

  - a two column matrix, `p:matrix([2,4],[5,6],[9,3])`,
  - a list of pairs, `p: [[2,4],[5,6],[9,3]]`,
  - a list of numbers, `p: [4,6,3]`, in which case the abscissas will be assigned automatically to 1, 2, 3, etc.

  In the first two cases the pairs are ordered with respect to the first coordinate before making computations.

  With the option argument it is possible to select the name for the independent variable, which is `'x` by default; to define another one, write something like `varname='z`.

  See also `lagrange`, `cspline`, and `ratinterpol`.

  Examples:

      (%i1) load(interpol)$
      (%i2) p: matrix([7,2],[8,3],[1,5],[3,2],[6,7])$
      (%i3) linearinterpol(p);
              13   3 x
      (%o3)  (-- - ---) charfun2(x, minf, 3)
              2     2
       + (x - 5) charfun2(x, 7, inf) + (37 - 5 x) charfun2(x, 6, 7)
          5 x
       + (--- - 3) charfun2(x, 3, 6)
           3

      (%i4) f(x):=''%;
                      13   3 x
      (%o4)  f(x) := (-- - ---) charfun2(x, minf, 3)
                      2     2
       + (x - 5) charfun2(x, 7, inf) + (37 - 5 x) charfun2(x, 6, 7)
          5 x
       + (--- - 3) charfun2(x, 3, 6)
           3
      (%i5)  /* Evaluate the polynomial at some points */
             map(f,[7.3,25/7,%pi]);
                                  62  5 %pi
      (%o5)                 [2.3, --, ----- - 3]
                                  21    3
      (%i6) %,numer;
      (%o6)  [2.3, 2.952380952380953, 2.235987755982989]
      (%i7) load(draw)$  /* load draw package */
      (%i8)  /* Plot the polynomial together with points */
             draw2d(
               color      = red,
               key        = "Linear interpolator",
               explicit(f(x),x,-5,20),
               point_size = 3,
               color      = blue,
               key        = "Sample points",
               points(args(p)))$
      (%i9)  /* Change variable name */
             linearinterpol(p, varname='s);
             13   3 s
      (%o9) (-- - ---) charfun2(s, minf, 3)
             2     2
       + (s - 5) charfun2(s, 7, inf) + (37 - 5 s) charfun2(s, 6, 7)
          5 s
       + (--- - 3) charfun2(s, 3, 6)
           3

  Categories: Package interpol

Function: cspline
`cspline` (points)
`cspline` (points, option1, option2, ...)

  Computes the polynomial interpolation by the cubic splines method. Argument points must be either:

  - a two column matrix, `p:matrix([2,4],[5,6],[9,3])`,
  - a list of pairs, `p: [[2,4],[5,6],[9,3]]`,
  - a list of numbers, `p: [4,6,3]`, in which case the abscissas will be assigned automatically to 1, 2, 3, etc.

  In the first two cases the pairs are ordered with respect to the first coordinate before making computations.

  There are three options to fit specific needs:

  - `'d1`, default `'unknown`, is the first derivative at x_1; if it is `'unknown`, the second derivative at x_1 is made equal to 0 (natural cubic spline); if it is equal to a number, the second derivative is calculated based on this number.
  - `'dn`, default `'unknown`, is the first derivative at x_n; if it is `'unknown`, the second derivative at x_n is made equal to 0 (natural cubic spline); if it is equal to a number, the second derivative is calculated based on this number.
  - `'varname`, default `'x`, is the name of the independent variable.

  See also `lagrange`, `linearinterpol`, and `ratinterpol`.

  Examples:

      (%i1) load(interpol)$
      (%i2) p:[[7,2],[8,2],[1,5],[3,2],[6,7]]$
      (%i3) /* Unknown first derivatives at the extremes
               is equivalent to natural cubic splines */
            cspline(p);
                    3         2
              1159 x    1159 x    6091 x   8283
      (%o3)  (------- - ------- - ------ + ----) charfun2(x, minf, 3)
               3288      1096      3288    1096
                  3         2
            2587 x    5174 x    494117 x   108928
       + (- ------- + ------- - -------- + ------) charfun2(x, 7, inf)
             1644       137       1644      137
                3          2
          4715 x    15209 x    579277 x   199575
       + (------- - -------- + -------- - ------) charfun2(x, 6, 7)
           1644       274        1644      274
                  3         2
            3287 x    2223 x    48275 x   9609
       + (- ------- + ------- - ------- + ----) charfun2(x, 3, 6)
             4932       274      1644     274

      (%i4) f(x):=''%$
      (%i5) /* Some evaluations */
            map(f,[2.3,5/7,%pi]), numer;
      (%o5) [1.991460766423356, 5.823200187269903, 2.227405312429507]
      (%i6) load(draw)$  /* load draw package */
      (%i7) /* Plotting interpolating function */
            draw2d(
              color      = red,
              key        = "Cubic splines",
              explicit(f(x),x,0,10),
              point_size = 3,
              color      = blue,
              key        = "Sample points",
              points(p))$
      (%i8) /* New call, but giving values at the derivatives */
            cspline(p,d1=0,dn=0);
                    3          2
              1949 x    11437 x    17027 x   1247
      (%o8)  (------- - -------- + ------- + ----) charfun2(x, minf, 3)
               2256       2256      2256     752
                  3          2
            1547 x    35581 x    68068 x   173546
       + (- ------- + -------- - ------- + ------) charfun2(x, 7, inf)
              564       564        141      141
               3          2
          607 x    35147 x    55706 x   38420
       + (------ - -------- + ------- - -----) charfun2(x, 6, 7)
           188       564        141      47
                  3         2
            3895 x    1807 x    5146 x   2148
       + (- ------- + ------- - ------ + ----) charfun2(x, 3, 6)
             5076       188      141      47
      (%i8) /* Defining new interpolating function */
            g(x):=''%$
      (%i9) /* Plotting both functions together */
            draw2d(
              color      = black,
              key        = "Cubic splines (default)",
              explicit(f(x),x,0,10),
              color      = red,
              key        = "Cubic splines (d1=0,dn=0)",
              explicit(g(x),x,0,10),
              point_size = 3,
              color      = blue,
              key        = "Sample points",
              points(p))$

  Categories: Package interpol

Function: ratinterpol
`ratinterpol` (points, numdeg)
`ratinterpol` (points, numdeg, option1)

  Generates a rational interpolator for data given by points and the degree of the numerator being equal to numdeg; the degree of the denominator is calculated automatically. Argument points must be either:

  - a two column matrix, `p:matrix([2,4],[5,6],[9,3])`,
  - a list of pairs, `p: [[2,4],[5,6],[9,3]]`,
  - a list of numbers, `p: [4,6,3]`, in which case the abscissas will be assigned automatically to 1, 2, 3, etc.

  In the first two cases the pairs are ordered with respect to the first coordinate before making computations.

  There is one option to fit specific needs:

  - `'varname`, default `'x`, is the name of the independent variable.

  See also `lagrange`, `linearinterpol`, `cspline`, `minpack_lsquares`, and lbfgs

  Examples:

      (%i1) load(interpol)$
      (%i2) load(draw)$
      (%i3) p:[[7.2,2.5],[8.5,2.1],[1.6,5.1],[3.4,2.4],[6.7,7.9]]$
      (%i4) for k:0 thru length(p)-1 do
              draw2d(
                explicit(ratinterpol(p,k),x,0,9),
                point_size = 3,
                points(p),
                title = concat("Degree of numerator = ",k),
                yrange=[0,10])$

  Categories: Package interpol

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# 65. lapack

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## 65.1 Introduction to lapack

`lapack` is a Common Lisp translation (via the program `f2cl`) of the Fortran library LAPACK, as obtained from the SLATEC project.

Categories: Numerical methods · Share packages · Package lapack

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## 65.2 Functions and Variables for lapack

Function: dgeev
`dgeev` (A)
`dgeev` (A, right_p, left_p)

  Computes the eigenvalues and, optionally, the eigenvectors of a matrix A. All elements of A must be integer or floating point numbers. A must be square (same number of rows and columns). A might or might not be symmetric.

  `dgeev(A)` computes only the eigenvalues of A. `dgeev(A, right_p, left_p)` computes the eigenvalues of A and the right eigenvectors when right_p = `true` and the left eigenvectors when left_p = `true`.

  A list of three items is returned. The first item is a list of the eigenvalues. The second item is `false` or the matrix of right eigenvectors. The third item is `false` or the matrix of left eigenvectors.

  The right eigenvector v(j) (the j-th column of the right eigenvector matrix) satisfies

  A . v(j) = lambda(j) . v(j)

  where lambda(j) is the corresponding eigenvalue. The left eigenvector u(j) (the j-th column of the left eigenvector matrix) satisfies

  u(j)**H . A = lambda(j) . u(j)**H

  where u(j)**H denotes the conjugate transpose of u(j). The Maxima function `ctranspose` computes the conjugate transpose.

  The computed eigenvectors are normalized to have Euclidean norm equal to 1, and largest component has imaginary part equal to zero.

  Example:

      (%i1) load ("lapack")$
      (%i2) fpprintprec : 6;
      (%o2)                           6
      (%i3) M : matrix ([9.5, 1.75], [3.25, 10.45]);
                               [ 9.5   1.75  ]
      (%o3)                    [             ]
                               [ 3.25  10.45 ]
      (%i4) dgeev (M);
      (%o4)          [[7.54331, 12.4067], false, false]
      (%i5) [L, v, u] : dgeev (M, true, true);
                                 [ - .666642  - .515792 ]
      (%o5) [[7.54331, 12.4067], [                      ],
                                 [  .745378   - .856714 ]
                                              [ - .856714  - .745378 ]
                                              [                      ]]
                                              [  .515792   - .666642 ]
      (%i6) D : apply (diag_matrix, L);
                            [ 7.54331     0    ]
      (%o6)                 [                  ]
                            [    0     12.4067 ]
      (%i7) M . v - v . D;
                      [      0.0       - 8.88178E-16 ]
      (%o7)           [                              ]
                      [ - 8.88178E-16       0.0      ]
      (%i8) transpose (u) . M - D . transpose (u);
                           [ 0.0  - 4.44089E-16 ]
      (%o8)                [                    ]
                           [ 0.0       0.0      ]

  Categories: Package lapack

Function: dgeqrf (A)

  Computes the QR decomposition of the matrix A. All elements of A must be integer or floating point numbers. A may or may not have the same number of rows and columns.

  A list of two items is returned. The first item is the matrix Q, which is a square, orthonormal matrix which has the same number of rows as A. The second item is the matrix R, which is the same size as A, and which has all elements equal to zero below the diagonal. The product `Q . R`, where "." is the noncommutative multiplication operator, is equal to A (ignoring floating point round-off errors).

      (%i1) load ("lapack") $
      (%i2) fpprintprec : 6 $
      (%i3) M : matrix ([1, -3.2, 8], [-11, 2.7, 5.9]) $
      (%i4) [q, r] : dgeqrf (M);
             [ - .0905357  .995893  ]
      (%o4) [[                      ],
             [  .995893    .0905357 ]
                                     [ - 11.0454   2.97863   5.15148 ]
                                     [                               ]]
                                     [     0      - 2.94241  8.50131 ]
      (%i5) q . r - M;
               [ - 7.77156E-16   1.77636E-15   - 8.88178E-16 ]
      (%o5)    [                                             ]
               [      0.0       - 1.33227E-15   8.88178E-16  ]
      (%i6) mat_norm (%, 1);
      (%o6)                      3.10862E-15

  Categories: Package lapack

Function: dgesv (A, b)

  Computes the solution x of the linear equation A x = b, where A is a square matrix, and b is a matrix of the same number of rows as A and any number of columns. The return value x is the same size as b.

  The elements of A and b must evaluate to real floating point numbers via `float`; thus elements may be any numeric type, symbolic numerical constants, or expressions which evaluate to floats. The elements of x are always floating point numbers. All arithmetic is carried out as floating point operations.

  `dgesv` computes the solution via the LU decomposition of A.

  Examples:

  `dgesv` computes the solution of the linear equation A x = b.

      (%i1) A : matrix ([1, -2.5], [0.375, 5]);
                                     [   1    - 2.5 ]
      (%o1)                          [              ]
                                     [ 0.375    5   ]
      (%i2) b : matrix ([1.75], [-0.625]);
                                        [  1.75   ]
      (%o2)                             [         ]
                                        [ - 0.625 ]
      (%i3) x : dgesv (A, b);
                                  [  1.210526315789474  ]
      (%o3)                       [                     ]
                                  [ - 0.215789473684211 ]
      (%i4) dlange (inf_norm, b - A.x);
      (%o4)                                 0.0

  b is a matrix with the same number of rows as A and any number of columns. x is the same size as b.

      (%i1) A : matrix ([1, -0.15], [1.82, 2]);
                                     [  1    - 0.15 ]
      (%o1)                          [              ]
                                     [ 1.82    2    ]
      (%i2) b : matrix ([3.7, 1, 8], [-2.3, 5, -3.9]);
                                    [  3.7   1    8   ]
      (%o2)                         [                 ]
                                    [ - 2.3  5  - 3.9 ]
      (%i3) x : dgesv (A, b);
            [  3.103827540695117  1.20985481742191    6.781786185657722 ]
      (%o3) [                                                           ]
            [ -3.974483062032557  1.399032116146062  -8.121425428948527 ]
      (%i4) dlange (inf_norm, b - A . x);
      (%o4)                       1.1102230246251565E-15

  The elements of A and b must evaluate to real floating point numbers.

      (%i1) A : matrix ([5, -%pi], [1b0, 11/17]);
                                     [   5    - %pi ]
                                     [              ]
      (%o1)                          [         11   ]
                                     [ 1.0b0   --   ]
                                     [         17   ]
      (%i2) b : matrix ([%e], [sin(1)]);
                                        [   %e   ]
      (%o2)                             [        ]
                                        [ sin(1) ]
      (%i3) x : dgesv (A, b);
                                   [ 0.690375643155986 ]
      (%o3)                        [                   ]
                                   [ 0.233510982552952 ]
      (%i4) dlange (inf_norm, b - A . x);
      (%o4)                        2.220446049250313E-16

  Categories: Package lapack · Linear equations

Function: dgesvd
`dgesvd` (A)
`dgesvd` (A, left_p, right_p)

  Computes the singular value decomposition (SVD) of a matrix A, comprising the singular values and, optionally, the left and right singular vectors. All elements of A must be integer or floating point numbers. A might or might not be square (same number of rows and columns).

  Let m be the number of rows, and n the number of columns of A. The singular value decomposition of A comprises three matrices, U, Sigma, and V^T, such that

  A = U . Sigma . V^T

  where U is an m-by-m unitary matrix, Sigma is an m-by-n diagonal matrix, and V^T is an n-by-n unitary matrix.

  Let sigma[i] be a diagonal element of Sigma, that is, Sigma[i, i] = sigma[i]. The elements sigma[i] are the so-called singular values of A; these are real and nonnegative, and returned in descending order. The first min(m, n) columns of U and V are the left and right singular vectors of A. Note that `dgesvd` returns the transpose of V, not V itself.

  `dgesvd(A)` computes only the singular values of A. `dgesvd(A, left_p, right_p)` computes the singular values of A and the left singular vectors when left_p = `true` and the right singular vectors when right_p = `true`.

  A list of three items is returned. The first item is a list of the singular values. The second item is `false` or the matrix of left singular vectors. The third item is `false` or the matrix of right singular vectors.

  Example:

      (%i1) load ("lapack")$
      (%i2) fpprintprec : 6;
      (%o2)                           6
      (%i3) M: matrix([1, 2, 3], [3.5, 0.5, 8], [-1, 2, -3], [4, 9, 7]);
                              [  1    2    3  ]
                              [               ]
                              [ 3.5  0.5   8  ]
      (%o3)                   [               ]
                              [ - 1   2   - 3 ]
                              [               ]
                              [  4    9    7  ]
      (%i4) dgesvd (M);
      (%o4)      [[14.4744, 6.38637, .452547], false, false]
      (%i5) [sigma, U, VT] : dgesvd (M, true, true);
      (%o5) [[14.4744, 6.38637, .452547],
      [ - .256731  .00816168   .959029    - .119523 ]
      [                                             ]
      [ - .526456   .672116   - .206236   - .478091 ]
      [                                             ],
      [  .107997   - .532278  - .0708315  - 0.83666 ]
      [                                             ]
      [ - .803287  - .514659  - .180867    .239046  ]
      [ - .374486  - .538209  - .755044 ]
      [                                 ]
      [  .130623   - .836799   0.5317   ]]
      [                                 ]
      [ - .917986   .100488    .383672  ]
      (%i6) m : length (U);
      (%o6)                           4
      (%i7) n : length (VT);
      (%o7)                           3
      (%i8) Sigma:
              genmatrix(lambda ([i, j], if i=j then sigma[i] else 0),
                        m, n);
                        [ 14.4744     0        0    ]
                        [                           ]
                        [    0     6.38637     0    ]
      (%o8)             [                           ]
                        [    0        0     .452547 ]
                        [                           ]
                        [    0        0        0    ]
      (%i9) U . Sigma . VT - M;
                [  1.11022E-15        0.0       1.77636E-15 ]
                [                                           ]
                [  1.33227E-15    1.66533E-15       0.0     ]
      (%o9)     [                                           ]
                [ - 4.44089E-16  - 8.88178E-16  4.44089E-16 ]
                [                                           ]
                [  8.88178E-16    1.77636E-15   8.88178E-16 ]
      (%i10) transpose (U) . U;
             [     1.0      5.55112E-17    2.498E-16     2.77556E-17  ]
             [                                                        ]
             [ 5.55112E-17      1.0       5.55112E-17    4.16334E-17  ]
      (%o10) [                                                        ]
             [  2.498E-16   5.55112E-17       1.0       - 2.08167E-16 ]
             [                                                        ]
             [ 2.77556E-17  4.16334E-17  - 2.08167E-16       1.0      ]
      (%i11) VT . transpose (VT);
                [      1.0           0.0      - 5.55112E-17 ]
                [                                           ]
      (%o11)    [      0.0           1.0       5.55112E-17  ]
                [                                           ]
                [ - 5.55112E-17  5.55112E-17       1.0      ]

  Categories: Package lapack

Function: dlange (norm, A)
Function: zlange (norm, A)

  Computes a norm or norm-like function of the matrix A.

  `max`

    Compute max(abs(A(i, j))) where i and j range over the rows and columns, respectively, of A. Note that this function is not a proper matrix norm.

  `one_norm`

    Compute the L[1] norm of A, that is, the maximum of the sum of the absolute value of elements in each column.

  `inf_norm`

    Compute the L[inf] norm of A, that is, the maximum of the sum of the absolute value of elements in each row.

  `frobenius`

    Compute the Frobenius norm of A, that is, the square root of the sum of squares of the matrix elements.

  Categories: Package lapack

Function: dgemm
`dgemm` (A, B)
`dgemm` (A, B, options)

  Compute the product of two matrices and optionally add the product to a third matrix.

  In the simplest form, `dgemm(A, B)` computes the product of the two real matrices, A and B.

  In the second form, `dgemm` computes the alpha * A * B + beta * C where A, B, C are real matrices of the appropriate sizes and alpha and beta are real numbers. Optionally, A and/or B can be transposed before computing the product. The extra parameters are specifed by optional keyword arguments: The keyword arguments are optional and may be specified in any order. They all take the form `key=val`. The keyword arguments are:

  `C`

    The matrix C that should be added. The default is `false`, which means no matrix is added.

  `alpha`

    The product of A and B is multiplied by this value. The default is 1.

  `beta`

    If a matrix C is given, this value multiplies C before it is added. The default value is 0, which implies that C is not added, even if C is given. Hence, be sure to specify a non-zero value for beta.

  `transpose_a`

    If `true`, the transpose of A is used instead of A for the product. The default is `false`.

  `transpose_b`

    If `true`, the transpose of B is used instead of B for the product. The default is `false`.

      (%i1) load ("lapack")$
      (%i2) A : matrix([1,2,3],[4,5,6],[7,8,9]);
                                        [ 1  2  3 ]
                                        [         ]
      (%o2)                             [ 4  5  6 ]
                                        [         ]
                                        [ 7  8  9 ]
      (%i3) B : matrix([-1,-2,-3],[-4,-5,-6],[-7,-8,-9]);
                                     [ - 1  - 2  - 3 ]
                                     [               ]
      (%o3)                          [ - 4  - 5  - 6 ]
                                     [               ]
                                     [ - 7  - 8  - 9 ]
      (%i4) C : matrix([3,2,1],[6,5,4],[9,8,7]);
                                        [ 3  2  1 ]
                                        [         ]
      (%o4)                             [ 6  5  4 ]
                                        [         ]
                                        [ 9  8  7 ]
      (%i5) dgemm(A,B);
                               [ - 30.0   - 36.0   - 42.0  ]
                               [                           ]
      (%o5)                    [ - 66.0   - 81.0   - 96.0  ]
                               [                           ]
                               [ - 102.0  - 126.0  - 150.0 ]
      (%i6) A . B;
                                  [ - 30   - 36   - 42  ]
                                  [                     ]
      (%o6)                       [ - 66   - 81   - 96  ]
                                  [                     ]
                                  [ - 102  - 126  - 150 ]
      (%i7) dgemm(A,B,transpose_a=true);
                               [ - 66.0  - 78.0   - 90.0  ]
                               [                          ]
      (%o7)                    [ - 78.0  - 93.0   - 108.0 ]
                               [                          ]
                               [ - 90.0  - 108.0  - 126.0 ]
      (%i8) transpose(A) . B;
                                 [ - 66  - 78   - 90  ]
                                 [                    ]
      (%o8)                      [ - 78  - 93   - 108 ]
                                 [                    ]
                                 [ - 90  - 108  - 126 ]
      (%i9) dgemm(A,B,c=C,beta=1);
                               [ - 27.0  - 34.0   - 41.0  ]
                               [                          ]
      (%o9)                    [ - 60.0  - 76.0   - 92.0  ]
                               [                          ]
                               [ - 93.0  - 118.0  - 143.0 ]
      (%i10) A . B + C;
                                  [ - 27  - 34   - 41  ]
                                  [                    ]
      (%o10)                      [ - 60  - 76   - 92  ]
                                  [                    ]
                                  [ - 93  - 118  - 143 ]
      (%i11) dgemm(A,B,c=C,beta=1, alpha=-1);
                                  [ 33.0   38.0   43.0  ]
                                  [                     ]
      (%o11)                      [ 72.0   86.0   100.0 ]
                                  [                     ]
                                  [ 111.0  134.0  157.0 ]
      (%i12) -A . B + C;
                                     [ 33   38   43  ]
                                     [               ]
      (%o12)                         [ 72   86   100 ]
                                     [               ]
                                     [ 111  134  157 ]

  Categories: Package lapack

Function: zgeev
`zgeev` (A)
`zgeev` (A, right_p, left_p)

  Like `dgeev`, but the matrix A is complex.

  Categories: Package lapack

Function: zheev
`zheev` (A)
`zheev` (A, eigvec_p)

  Like `zheev`, but the matrix A is assumed to be a square complex Hermitian matrix. If eigvec_p is `true`, then the eigenvectors of the matrix are also computed.

  No check is made that the matrix A is, in fact, Hermitian.

  A list of two items is returned, as in `dgeev`: a list of eigenvalues, and `false` or the matrix of the eigenvectors.

  Categories: Package lapack

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# 66. lbfgs

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## 66.1 Introduction to lbfgs

`lbfgs` is an implementation of the L-BFGS algorithm [1] to solve unconstrained minimization problems via a limited-memory quasi-Newton (BFGS) algorithm. It is called a limited-memory method because a low-rank approximation of the Hessian matrix inverse is stored instead of the entire Hessian inverse. The program was originally written in Fortran [2] by Jorge Nocedal, incorporating some functions originally written by Jorge J. Moré and David J. Thuente, and translated into Lisp automatically via the program `f2cl`. The Maxima package `lbfgs` comprises the translated code plus an interface function which manages some details.

References:

[1] D. Liu and J. Nocedal. "On the limited memory BFGS method for large scale optimization". Mathematical Programming B 45:503-528 (1989)

[2] http://netlib.org/opt/lbfgs_um.shar

Categories: Numerical methods · Optimization · Share packages · Package lbfgs

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## 66.2 Functions and Variables for lbfgs

Function: lbfgs
`lbfgs` (FOM, X, X0, epsilon, iprint)
`lbfgs` ([FOM, grad] X, X0, epsilon, iprint)

  Finds an approximate solution of the unconstrained minimization of the figure of merit FOM over the list of variables X, starting from initial estimates X0, such that norm(grad(FOM)) < epsilon*max(1, norm(X)).

  grad, if present, is the gradient of FOM with respect to the variables X. grad may be a list or a function that returns a list, with one element for each element of X. If not present, the gradient is computed automatically by symbolic differentiation. If FOM is a function, the gradient grad must be supplied by the user.

  The algorithm applied is a limited-memory quasi-Newton (BFGS) algorithm [1]. It is called a limited-memory method because a low-rank approximation of the Hessian matrix inverse is stored instead of the entire Hessian inverse. Each iteration of the algorithm is a line search, that is, a search along a ray in the variables X, with the search direction computed from the approximate Hessian inverse. The FOM is always decreased by a successful line search. Usually (but not always) the norm of the gradient of FOM also decreases.

  iprint controls progress messages printed by `lbfgs`.

  `iprint[1]`

    `iprint[1]` controls the frequency of progress messages.

    `iprint[1] < 0`

      No progress messages.

    `iprint[1] = 0`

      Messages at the first and last iterations.

    `iprint[1] > 0`

      Print a message every `iprint[1]` iterations.

  `iprint[2]`

    `iprint[2]` controls the verbosity of progress messages.

    `iprint[2] = 0`

      Print out iteration count, number of evaluations of FOM, value of FOM, norm of the gradient of FOM, and step length.

    `iprint[2] = 1`

      Same as `iprint[2] = 0`, plus X0 and the gradient of FOM evaluated at X0.

    `iprint[2] = 2`

      Same as `iprint[2] = 1`, plus values of X at each iteration.

    `iprint[2] = 3`

      Same as `iprint[2] = 2`, plus the gradient of FOM at each iteration.

  The columns printed by `lbfgs` are the following.

  `I`

    Number of iterations. It is incremented for each line search.

  `NFN`

    Number of evaluations of the figure of merit.

  `FUNC`

    Value of the figure of merit at the end of the most recent line search.

  `GNORM`

    Norm of the gradient of the figure of merit at the end of the most recent line search.

  `STEPLENGTH`

    An internal parameter of the search algorithm.

  Additional information concerning details of the algorithm are found in the comments of the original Fortran code [2].

  See also `lbfgs_nfeval_max` and `lbfgs_ncorrections`.

  References:

  [1] D. Liu and J. Nocedal. "On the limited memory BFGS method for large scale optimization". Mathematical Programming B 45:503-528 (1989)

  [2] http://netlib.org/opt/lbfgs_um.shar

  Examples:

  The same FOM as computed by FGCOMPUTE in the program sdrive.f in the LBFGS package from Netlib. Note that the variables in question are subscripted variables. The FOM has an exact minimum equal to zero at u[k] = 1 for k = 1, ..., 8.

      (%i1) load ("lbfgs")$
      (%i2) t1[j] := 1 - u[j];
      (%o2)                     t1  := 1 - u
                                  j         j
      (%i3) t2[j] := 10*(u[j + 1] - u[j]^2);
                                                2
      (%o3)                t2  := 10 (u      - u )
                             j         j + 1    j
      (%i4) n : 8;
      (%o4)                           8
      (%i5) FOM : sum (t1[2*j - 1]^2 + t2[2*j - 1]^2, j, 1, n/2);
                       2 2           2              2 2           2
      (%o5) 100 (u  - u )  + (1 - u )  + 100 (u  - u )  + (1 - u )
                  8    7           7           6    5           5
                           2 2           2              2 2           2
              + 100 (u  - u )  + (1 - u )  + 100 (u  - u )  + (1 - u )
                      4    3           3           2    1           1
      (%i6) lbfgs (FOM, '[u[1],u[2],u[3],u[4],u[5],u[6],u[7],u[8]],
             [-1.2, 1, -1.2, 1, -1.2, 1, -1.2, 1], 1e-3, [1, 0]);
      *************************************************
        N=    8   NUMBER OF CORRECTIONS=25
             INITIAL VALUES
       F=  9.680000000000000D+01   GNORM=  4.657353755084533D+02
      *************************************************

       I NFN   FUNC                    GNORM                   STEPLENGTH

       1   3   1.651479526340304D+01   4.324359291335977D+00   7.926153934390631D-04
       2   4   1.650209316638371D+01   3.575788161060007D+00   1.000000000000000D+00
       3   5   1.645461701312851D+01   6.230869903601577D+00   1.000000000000000D+00
       4   6   1.636867301275588D+01   1.177589920974980D+01   1.000000000000000D+00
       5   7   1.612153014409201D+01   2.292797147151288D+01   1.000000000000000D+00
       6   8   1.569118407390628D+01   3.687447158775571D+01   1.000000000000000D+00
       7   9   1.510361958398942D+01   4.501931728123679D+01   1.000000000000000D+00
       8  10   1.391077875774293D+01   4.526061463810630D+01   1.000000000000000D+00
       9  11   1.165625686278198D+01   2.748348965356907D+01   1.000000000000000D+00
      10  12   9.859422687859144D+00   2.111494974231706D+01   1.000000000000000D+00
      11  13   7.815442521732282D+00   6.110762325764183D+00   1.000000000000000D+00
      12  15   7.346380905773044D+00   2.165281166715009D+01   1.285316401779678D-01
      13  16   6.330460634066464D+00   1.401220851761508D+01   1.000000000000000D+00
      14  17   5.238763939854303D+00   1.702473787619218D+01   1.000000000000000D+00
      15  18   3.754016790406625D+00   7.981845727632704D+00   1.000000000000000D+00
      16  20   3.001238402313225D+00   3.925482944745832D+00   2.333129631316462D-01
      17  22   2.794390709722064D+00   8.243329982586480D+00   2.503577283802312D-01
      18  23   2.563783562920545D+00   1.035413426522664D+01   1.000000000000000D+00
      19  24   2.019429976373283D+00   1.065187312340952D+01   1.000000000000000D+00
      20  25   1.428003167668592D+00   2.475962450735100D+00   1.000000000000000D+00
      21  27   1.197874264859232D+00   8.441707983339661D+00   4.303451060697367D-01
      22  28   9.023848942003913D-01   1.113189216665625D+01   1.000000000000000D+00
      23  29   5.508226405855795D-01   2.380830599637816D+00   1.000000000000000D+00
      24  31   3.902893258879521D-01   5.625595817143044D+00   4.834988416747262D-01
      25  32   3.207542206881058D-01   1.149444645298493D+01   1.000000000000000D+00
      26  33   1.874468266118200D-01   3.632482152347445D+00   1.000000000000000D+00
      27  34   9.575763380282112D-02   4.816497449000391D+00   1.000000000000000D+00
      28  35   4.085145106760390D-02   2.087009347116811D+00   1.000000000000000D+00
      29  36   1.931106005512628D-02   3.886818624052740D+00   1.000000000000000D+00
      30  37   6.894000636920714D-03   3.198505769992936D+00   1.000000000000000D+00
      31  38   1.443296008850287D-03   1.590265460381961D+00   1.000000000000000D+00
      32  39   1.571766574930155D-04   3.098257002223532D-01   1.000000000000000D+00
      33  40   1.288011779655132D-05   1.207784334505595D-02   1.000000000000000D+00
      34  41   1.806140190993455D-06   4.587890258846915D-02   1.000000000000000D+00
      35  42   1.769004612050548D-07   1.790537363138099D-02   1.000000000000000D+00
      36  43   3.312164244118216D-10   6.782068546986653D-04   1.000000000000000D+00

       THE MINIMIZATION TERMINATED WITHOUT DETECTING ERRORS.
       IFLAG = 0
      (%o6) [u  = 1.000005339816132, u  = 1.000009942840108,
              1                       2
      u  = 1.000005339816132, u  = 1.000009942840108,
       3                       4
      u  = 1.000005339816132, u  = 1.000009942840108,
       5                       6
      u  = 1.000005339816132, u  = 1.000009942840108]
       7                       8

  A regression problem. The FOM is the mean square difference between the predicted value F(X[i]) and the observed value Y[i]. The function F is a bounded monotone function (a so-called "sigmoidal" function). In this example, `lbfgs` computes approximate values for the parameters of F and `plot2d` displays a comparison of F with the observed data.

      (%i1) load ("lbfgs")$
      (%i2) FOM : '((1/length(X))*sum((F(X[i]) - Y[i])^2, i, 1,
                                                      length(X)));
                                     2
                     sum((F(X ) - Y ) , i, 1, length(X))
                             i     i
      (%o2)          -----------------------------------
                                  length(X)
      (%i3) X : [1, 2, 3, 4, 5];
      (%o3)                    [1, 2, 3, 4, 5]
      (%i4) Y : [0, 0.5, 1, 1.25, 1.5];
      (%o4)                [0, 0.5, 1, 1.25, 1.5]
      (%i5) F(x) := A/(1 + exp(-B*(x - C)));
                                         A
      (%o5)            F(x) := ----------------------
                               1 + exp((- B) (x - C))
      (%i6) ''FOM;
                      A               2            A                2
      (%o6) ((----------------- - 1.5)  + (----------------- - 1.25)
                - B (5 - C)                  - B (4 - C)
              %e            + 1            %e            + 1
                  A             2            A               2
       + (----------------- - 1)  + (----------------- - 0.5)
            - B (3 - C)                - B (2 - C)
          %e            + 1          %e            + 1
                   2
                  A
       + --------------------)/5
            - B (1 - C)     2
         (%e            + 1)
      (%i7) estimates : lbfgs (FOM, '[A, B, C], [1, 1, 1], 1e-4, [1, 0]);
      *************************************************
        N=    3   NUMBER OF CORRECTIONS=25
             INITIAL VALUES
       F=  1.348738534246918D-01   GNORM=  2.000215531936760D-01
      *************************************************

      I  NFN  FUNC                    GNORM                   STEPLENGTH
      1    3  1.177820636622582D-01   9.893138394953992D-02   8.554435968992371D-01
      2    6  2.302653892214013D-02   1.180098521565904D-01   2.100000000000000D+01
      3    8  1.496348495303004D-02   9.611201567691624D-02   5.257340567840710D-01
      4    9  7.900460841091138D-03   1.325041647391314D-02   1.000000000000000D+00
      5   10  7.314495451266914D-03   1.510670810312226D-02   1.000000000000000D+00
      6   11  6.750147275936668D-03   1.914964958023037D-02   1.000000000000000D+00
      7   12  5.850716021108202D-03   1.028089194579382D-02   1.000000000000000D+00
      8   13  5.778664230657800D-03   3.676866074532179D-04   1.000000000000000D+00
      9   14  5.777818823650780D-03   3.010740179797108D-04   1.000000000000000D+00

       THE MINIMIZATION TERMINATED WITHOUT DETECTING ERRORS.
       IFLAG = 0
      (%o7) [A = 1.461933911464101, B = 1.601593973254801,
                                                 C = 2.528933072164855]
      (%i8) plot2d ([F(x), [discrete, X, Y]], [x, -1, 6]), ''estimates;
      (%o8)

  Gradient of FOM is specified (instead of computing it automatically). Both the FOM and its gradient are passed as functions to `lbfgs`.

      (%i1) load ("lbfgs")$
      (%i2) F(a, b, c) := (a - 5)^2 + (b - 3)^4 + (c - 2)^6$
      (%i3) define(F_grad(a, b, c),
                   map (lambda ([x], diff (F(a, b, c), x)), [a, b, c]))$
      (%i4) estimates : lbfgs ([F, F_grad],
                         [a, b, c], [0, 0, 0], 1e-4, [1, 0]);
      *************************************************
        N=    3   NUMBER OF CORRECTIONS=25
             INITIAL VALUES
       F=  1.700000000000000D+02   GNORM=  2.205175729958953D+02
      *************************************************

         I  NFN     FUNC                    GNORM                   STEPLENGTH

         1    2     6.632967565917637D+01   6.498411132518770D+01   4.534785987412505D-03
         2    3     4.368890936228036D+01   3.784147651974131D+01   1.000000000000000D+00
         3    4     2.685298972775191D+01   1.640262125898520D+01   1.000000000000000D+00
         4    5     1.909064767659852D+01   9.733664001790506D+00   1.000000000000000D+00
         5    6     1.006493272061515D+01   6.344808151880209D+00   1.000000000000000D+00
         6    7     1.215263596054292D+00   2.204727876126877D+00   1.000000000000000D+00
         7    8     1.080252896385329D-02   1.431637116951845D-01   1.000000000000000D+00
         8    9     8.407195124830860D-03   1.126344579730008D-01   1.000000000000000D+00
         9   10     5.022091686198525D-03   7.750731829225275D-02   1.000000000000000D+00
        10   11     2.277152808939775D-03   5.032810859286796D-02   1.000000000000000D+00
        11   12     6.489384688303218D-04   1.932007150271009D-02   1.000000000000000D+00
        12   13     2.075791943844547D-04   6.964319310814365D-03   1.000000000000000D+00
        13   14     7.349472666162258D-05   4.017449067849554D-03   1.000000000000000D+00
        14   15     2.293617477985238D-05   1.334590390856715D-03   1.000000000000000D+00
        15   16     7.683645404048675D-06   6.011057038099202D-04   1.000000000000000D+00

       THE MINIMIZATION TERMINATED WITHOUT DETECTING ERRORS.
       IFLAG = 0
      (%o4) [a = 5.000086823042934, b = 3.052395429705181,
                                                 c = 1.927980629919583]

  Categories: Package lbfgs

Variable: lbfgs_nfeval_max

  Default value: 100

  `lbfgs_nfeval_max` is the maximum number of evaluations of the figure of merit (FOM) in `lbfgs`. When `lbfgs_nfeval_max` is reached, `lbfgs` returns the result of the last successful line search.

  Categories: Package lbfgs

Variable: lbfgs_ncorrections

  Default value: 25

  `lbfgs_ncorrections` is the number of corrections applied to the approximate inverse Hessian matrix which is maintained by `lbfgs`.

  Categories: Package lbfgs

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# 67. lindstedt

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## 67.1 Functions and Variables for lindstedt

Function: Lindstedt (eq,pvar,torder,ic)

  This is a first pass at a Lindstedt code. It can solve problems with initial conditions entered, which can be arbitrary constants, (just not %k1 and %k2) where the initial conditions on the perturbation equations are z[i]=0, z'[i]=0 for i>0. ic is the list of initial conditions.

  Problems occur when initial conditions are not given, as the constants in the perturbation equations are the same as the zero order equation solution. Also, problems occur when the initial conditions for the perturbation equations are not z[i]=0, z'[i]=0 for i>0, such as the Van der Pol equation.

  Example:

      (%i1) load("makeOrders")$

      (%i2) load("lindstedt")$

      (%i3) Lindstedt('diff(x,t,2)+x-(e*x^3)/6,e,2,[1,0]);
                2
               e  (cos(5 T) - 24 cos(3 T) + 23 cos(T))
      (%o3) [[[---------------------------------------
                                36864
         e (cos(3 T) - cos(T))
       - --------------------- + cos(T)],
                  192
                2
             7 e    e
      T = (- ---- - -- + 1) t]]
             3072   16

  To use this function write first `load("makeOrders")` and `load("lindstedt")`.

  Categories: Differential equations · Share packages · Package lindstedt

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# 68. linearalgebra

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## 68.1 Introduction to linearalgebra

`linearalgebra` is a collection of functions for linear algebra.

Example:

    (%i1) M : matrix ([1, 2], [1, 2]);
                                [ 1  2 ]
    (%o1)                       [      ]
                                [ 1  2 ]
    (%i2) nullspace (M);
                                   [  1  ]
                                   [     ]
    (%o2)                     span([   1 ])
                                   [ - - ]
                                   [   2 ]
    (%i3) columnspace (M);
                                    [ 1 ]
    (%o3)                      span([   ])
                                    [ 1 ]
    (%i4) ptriangularize (M - z*ident(2), z);
                             [ 1   2 - z   ]
    (%o4)                    [             ]
                             [           2 ]
                             [ 0  3 z - z  ]
    (%i5) M : matrix ([1, 2, 3], [4, 5, 6], [7, 8, 9]) - z*ident(3);
                         [ 1 - z    2      3   ]
                         [                     ]
    (%o5)                [   4    5 - z    6   ]
                         [                     ]
                         [   7      8    9 - z ]
    (%i6) MM : ptriangularize (M, z);
                  [ 4  5 - z            6            ]
                  [                                  ]
                  [                2                 ]
                  [     66        z    102 z   132   ]
                  [ 0   --      - -- + ----- + ---   ]
    (%o6)         [     49        7     49     49    ]
                  [                                  ]
                  [               3        2         ]
                  [           49 z    245 z    147 z ]
                  [ 0    0    ----- - ------ - ----- ]
                  [            264      88      44   ]
    (%i7) algebraic : true;
    (%o7)                         true
    (%i8) tellrat (MM [3, 3]);
                             3       2
    (%o8)                  [z  - 15 z  - 18 z]
    (%i9) MM : ratsimp (MM);
                   [ 4  5 - z           6           ]
                   [                                ]
                   [                2               ]
    (%o9)          [     66      7 z  - 102 z - 132 ]
                   [ 0   --    - ------------------ ]
                   [     49              49         ]
                   [                                ]
                   [ 0    0             0           ]
    (%i10) nullspace (MM);
                            [        1         ]
                            [                  ]
                            [   2              ]
                            [  z  - 14 z - 16  ]
                            [  --------------  ]
    (%o10)             span([        8         ])
                            [                  ]
                            [    2             ]
                            [   z  - 18 z - 12 ]
                            [ - -------------- ]
                            [         12       ]
    (%i11) M : matrix ([1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 11, 12],
                       [13, 14, 15, 16]);
                           [ 1   2   3   4  ]
                           [                ]
                           [ 5   6   7   8  ]
    (%o11)                 [                ]
                           [ 9   10  11  12 ]
                           [                ]
                           [ 13  14  15  16 ]
    (%i12) columnspace (M);
                               [ 1  ]  [ 2  ]
                               [    ]  [    ]
                               [ 5  ]  [ 6  ]
    (%o12)                span([    ], [    ])
                               [ 9  ]  [ 10 ]
                               [    ]  [    ]
                               [ 13 ]  [ 14 ]
    (%i13) apply ('orthogonal_complement, args (nullspace (transpose (M))));
                               [ 0 ]  [  1  ]
                               [   ]  [     ]
                               [ 1 ]  [  0  ]
    (%o13)                span([   ], [     ])
                               [ 2 ]  [ - 1 ]
                               [   ]  [     ]
                               [ 3 ]  [ - 2 ]

Categories: Linear algebra · Share packages · Package linearalgebra

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## 68.2 Functions and Variables for linearalgebra

Function: addmatrices (f, M_1, …, M_n)

  Using the function f as the addition function, return the sum of the matrices M_1, …, M_n. The function f must accept any number of arguments (a Maxima nary function).

  Examples:

      (%i1) m1 : matrix([1,2],[3,4])$
      (%i2) m2 : matrix([7,8],[9,10])$
      (%i3) addmatrices('max,m1,m2);
      (%o3) matrix([7,8],[9,10])
      (%i4) addmatrices('max,m1,m2,5*m1);
      (%o4) matrix([7,10],[15,20])

  Categories: Package linearalgebra

Function: blockmatrixp (M)

  Return true if and only if M is a matrix and every entry of M is a matrix.

  Categories: Package linearalgebra · Predicate functions

Function: columnop (M, i, j, theta)

  If M is a matrix, return the matrix that results from doing the column operation `C_i <- C_i - theta * C_j`. If M doesn't have a row i or j, signal an error.

  Categories: Package linearalgebra

Function: columnswap (M, i, j)

  If M is a matrix, swap columns i and j. If M doesn't have a column i or j, signal an error.

  Categories: Package linearalgebra

Function: columnspace (M)

  If M is a matrix, return `span (v_1, ..., v_n)`, where the set `{v_1, ..., v_n}` is a basis for the column space of M. The span of the empty set is `{0}`. Thus, when the column space has only one member, return `span ()`.

  Categories: Package linearalgebra

Function: cholesky
`cholesky` (M)
`cholesky` (M, field)

  Return the Cholesky factorization of the matrix selfadjoint (or hermitian) matrix M. The second argument defaults to 'generalring.' For a description of the possible values for field, see `lu_factor`.

  Categories: Matrix decompositions · Package linearalgebra

Function: ctranspose (M)

  Return the complex conjugate transpose of the matrix M. The function `ctranspose` uses `matrix_element_transpose` to transpose each matrix element.

  Categories: Package linearalgebra

Function: diag_matrix (d_1, d_2, …, d_n)

  Return a diagonal matrix with diagonal entries d_1, d_2, …, d_n. When the diagonal entries are matrices, the zero entries of the returned matrix are zero matrices of the appropriate size; for example:

      (%i1) diag_matrix(diag_matrix(1,2),diag_matrix(3,4));

                                  [ [ 1  0 ]  [ 0  0 ] ]
                                  [ [      ]  [      ] ]
                                  [ [ 0  2 ]  [ 0  0 ] ]
      (%o1)                       [                    ]
                                  [ [ 0  0 ]  [ 3  0 ] ]
                                  [ [      ]  [      ] ]
                                  [ [ 0  0 ]  [ 0  4 ] ]
      (%i2) diag_matrix(p,q);

                                         [ p  0 ]
      (%o2)                              [      ]
                                         [ 0  q ]

  Categories: Package linearalgebra

Function: dotproduct (u, v)

  Return the dotproduct of vectors u and v. This is the same as `conjugate (transpose (u)) . v`. The arguments u and v must be column vectors.

  Categories: Package linearalgebra

Function: eigens_by_jacobi
`eigens_by_jacobi` (A)
`eigens_by_jacobi` (A, field_type)

  Computes the eigenvalues and eigenvectors of A by the method of Jacobi rotations. A must be a symmetric matrix (but it need not be positive definite nor positive semidefinite). field_type indicates the computational field, either `floatfield` or `bigfloatfield`. If field_type is not specified, it defaults to `floatfield`.

  The elements of A must be numbers or expressions which evaluate to numbers via `float` or `bfloat` (depending on field_type).

  Examples:

      (%i1) S: matrix([1/sqrt(2), 1/sqrt(2)],[-1/sqrt(2), 1/sqrt(2)]);
                           [     1         1    ]
                           [  -------   ------- ]
                           [  sqrt(2)   sqrt(2) ]
      (%o1)                [                    ]
                           [      1        1    ]
                           [ - -------  ------- ]
                           [   sqrt(2)  sqrt(2) ]
      (%i2) L : matrix ([sqrt(3), 0], [0, sqrt(5)]);
                            [ sqrt(3)     0    ]
      (%o2)                 [                  ]
                            [    0     sqrt(5) ]
      (%i3) M : S . L . transpose (S);
                  [ sqrt(5)   sqrt(3)  sqrt(5)   sqrt(3) ]
                  [ ------- + -------  ------- - ------- ]
                  [    2         2        2         2    ]
      (%o3)       [                                      ]
                  [ sqrt(5)   sqrt(3)  sqrt(5)   sqrt(3) ]
                  [ ------- - -------  ------- + ------- ]
                  [    2         2        2         2    ]
      (%i4) eigens_by_jacobi (M);
      The largest percent change was 0.1454972243679
      The largest percent change was 0.0
      number of sweeps: 2
      number of rotations: 1
      (%o4) [[1.732050807568877, 2.23606797749979],
                              [  0.70710678118655   0.70710678118655 ]
                              [                                      ]]
                              [ - 0.70710678118655  0.70710678118655 ]
      (%i5) float ([[sqrt(3), sqrt(5)], S]);
      (%o5) [[1.732050807568877, 2.23606797749979],
                              [  0.70710678118655   0.70710678118655 ]
                              [                                      ]]
                              [ - 0.70710678118655  0.70710678118655 ]
      (%i6) eigens_by_jacobi (M, bigfloatfield);
      The largest percent change was 1.454972243679028b-1
      The largest percent change was 0.0b0
      number of sweeps: 2
      number of rotations: 1
      (%o6) [[1.732050807568877b0, 2.23606797749979b0],
                      [  7.071067811865475b-1   7.071067811865475b-1 ]
                      [                                              ]]
                      [ - 7.071067811865475b-1  7.071067811865475b-1 ]

  Categories: Matrix decompositions · Package linearalgebra

Function: get_lu_factors (x)

  When `x = lu_factor (A)`, then `get_lu_factors` returns a list of the form `[P, L, U]`, where P is a permutation matrix, L is lower triangular with ones on the diagonal, and U is upper triangular, and `A = P L U`.

  Categories: Package linearalgebra

Function: hankel
`hankel` (col)
`hankel` (col, row)

  Return a Hankel matrix H. The first column of H is col; except for the first entry, the last row of H is row. The default for row is the zero vector with the same length as col.

  Categories: Package linearalgebra

Function: hessian (f, x)

  Returns the Hessian matrix of f with respect to the list of variables x. The `(i, j)`-th element of the Hessian matrix is `diff(f, x[i], 1, x[j], 1)`.

  Examples:

      (%i1) hessian (x * sin (y), [x, y]);
                           [   0       cos(y)   ]
      (%o1)                [                    ]
                           [ cos(y)  - x sin(y) ]
      (%i2) depends (F, [a, b]);
      (%o2)                       [F(a, b)]
      (%i3) hessian (F, [a, b]);
                              [   2      2   ]
                              [  d F    d F  ]
                              [  ---   ----- ]
                              [    2   da db ]
                              [  da          ]
      (%o3)                   [              ]
                              [   2      2   ]
                              [  d F    d F  ]
                              [ -----   ---  ]
                              [ da db     2  ]
                              [         db   ]

  Categories: Differential calculus · Package linearalgebra

Function: hilbert_matrix (n)

  Return the n by n Hilbert matrix. When n isn't a positive integer, signal an error.

  Categories: Package linearalgebra

Function: identfor
`identfor` (M)
`identfor` (M, fld)

  Return an identity matrix that has the same shape as the matrix M. The diagonal entries of the identity matrix are the multiplicative identity of the field fld; the default for fld is generalring.

  The first argument M should be a square matrix or a non-matrix. When M is a matrix, each entry of M can be a square matrix - thus M can be a blocked Maxima matrix. The matrix can be blocked to any (finite) depth.

  See also `zerofor`

  Categories: Package linearalgebra

Function: invert_by_lu (M, (rng generalring))

  Invert a matrix M by using the LU factorization. The LU factorization is done using the ring rng.

  Categories: Package linearalgebra

Function: jacobian (f, x)

  Returns the Jacobian matrix of the list of functions f with respect to the list of variables x. The `(i, j)`-th element of the Jacobian matrix is `diff(f[i], x[j])`.

  Examples:

      (%i1) jacobian ([sin (u - v), sin (u * v)], [u, v]);
                        [ cos(v - u)  - cos(v - u) ]
      (%o1)             [                          ]
                        [ v cos(u v)   u cos(u v)  ]
      (%i2) depends ([F, G], [y, z]);
      (%o2)                  [F(y, z), G(y, z)]
      (%i3) jacobian ([F, G], [y, z]);
                                 [ dF  dF ]
                                 [ --  -- ]
                                 [ dy  dz ]
      (%o3)                      [        ]
                                 [ dG  dG ]
                                 [ --  -- ]
                                 [ dy  dz ]

  Categories: Differential calculus · Package linearalgebra

Function: kronecker_product (A, B)

  Return the Kronecker product of the matrices A and B.

  Categories: Package linearalgebra

Function: listp
`listp` (e, p)
`listp` (e)

  Given an optional argument p, return `true` if e is a Maxima list and p evaluates to `true` for every list element. When `listp` is not given the optional argument, return `true` if e is a Maxima list. In all other cases, return `false`.

  Categories: Package linearalgebra · Predicate functions

Function: locate_matrix_entry (M, r_1, c_1, r_2, c_2, f, rel)

  The first argument must be a matrix; the arguments r_1 through c_2 determine a sub-matrix of M that consists of rows r_1 through r_2 and columns c_1 through c_2.

  Find a entry in the sub-matrix M that satisfies some property. Three cases:

  (1) `rel = 'bool` and f a predicate:

  Scan the sub-matrix from left to right then top to bottom, and return the index of the first entry that satisfies the predicate f. If no matrix entry satisfies f, return `false`.

  (2) `rel = 'max` and f real-valued:

  Scan the sub-matrix looking for an entry that maximizes f. Return the index of a maximizing entry.

  (3) `rel = 'min` and f real-valued:

  Scan the sub-matrix looking for an entry that minimizes f. Return the index of a minimizing entry.

  Categories: Package linearalgebra

Function: lu_backsub (M, b)

  When `M = lu_factor (A, field)`, then `lu_backsub (M, b)` solves the linear system `A x = b`.

  Categories: Package linearalgebra

Function: lu_factor (M, field)

  Return a list of the form `[LU, perm, fld]`, or `[LU, perm, fld, lower-cnd upper-cnd]`, where

  (1) The matrix LU contains the factorization of M in a packed form. Packed form means three things: First, the rows of LU are permuted according to the list perm. If, for example, perm is the list `[3,2,1]`, the actual first row of the LU factorization is the third row of the matrix LU. Second, the lower triangular factor of m is the lower triangular part of LU with the diagonal entries replaced by all ones. Third, the upper triangular factor of M is the upper triangular part of LU.

  (2) When the field is either `floatfield` or `complexfield`, the numbers lower-cnd and upper-cnd are lower and upper bounds for the infinity norm condition number of M. For all fields, the condition number might not be estimated; for such fields, `lu_factor` returns a two item list. Both the lower and upper bounds can differ from their true values by arbitrarily large factors. (See also `mat_cond`.)

  The argument M must be a square matrix.

  The optional argument fld must be a symbol that determines a ring or field. The pre-defined fields and rings are:

  (a) `generalring` - the ring of Maxima expressions,

  (b) `floatfield` - the field of floating point numbers of the type double,

  (c) `complexfield` - the field of complex floating point numbers of the type double,

  (d) `crering` - the ring of Maxima CRE expressions,

  (e) `rationalfield` - the field of rational numbers,

  (f) `runningerror` - track the all floating point rounding errors,

  (g) `noncommutingring` - the ring of Maxima expressions where multiplication is the non-commutative dot operator.

  When the field is `floatfield`, `complexfield`, or `runningerror`, the algorithm uses partial pivoting; for all other fields, rows are switched only when needed to avoid a zero pivot.

  Floating point addition arithmetic isn't associative, so the meaning of 'field' differs from the mathematical definition.

  A member of the field `runningerror` is a two member Maxima list of the form `[x,n]`,where x is a floating point number and `n` is an integer. The relative difference between the 'true' value of `x` and `x` is approximately bounded by the machine epsilon times `n`. The running error bound drops some terms that of the order the square of the machine epsilon.

  There is no user-interface for defining a new field. A user that is familiar with Common Lisp should be able to define a new field. To do this, a user must define functions for the arithmetic operations and functions for converting from the field representation to Maxima and back. Additionally, for ordered fields (where partial pivoting will be used), a user must define functions for the magnitude and for comparing field members. After that all that remains is to define a Common Lisp structure `mring`. The file `mring` has many examples.

  To compute the factorization, the first task is to convert each matrix entry to a member of the indicated field. When conversion isn't possible, the factorization halts with an error message. Members of the field needn't be Maxima expressions. Members of the `complexfield`, for example, are Common Lisp complex numbers. Thus after computing the factorization, the matrix entries must be converted to Maxima expressions.

  See also `get_lu_factors`.

  Examples:

      (%i1) w[i,j] := random (1.0) + %i * random (1.0);
      (%o1)          w     := random(1.) + %i random(1.)
                      i, j
      (%i2) showtime : true$
      Evaluation took 0.00 seconds (0.00 elapsed)
      (%i3) M : genmatrix (w, 100, 100)$
      Evaluation took 7.40 seconds (8.23 elapsed)
      (%i4) lu_factor (M, complexfield)$
      Evaluation took 28.71 seconds (35.00 elapsed)
      (%i5) lu_factor (M, generalring)$
      Evaluation took 109.24 seconds (152.10 elapsed)
      (%i6) showtime : false$

      (%i7) M : matrix ([1 - z, 3], [3, 8 - z]);
                              [ 1 - z    3   ]
      (%o7)                   [              ]
                              [   3    8 - z ]
      (%i8) lu_factor (M, generalring);
                [ 1 - z         3        ]
                [                        ]
      (%o8)    [[   3            9       ], [1, 2], generalring]
                [ -----  - z - ----- + 8 ]
                [ 1 - z        1 - z     ]
      (%i9) get_lu_factors (%);
                        [   1    0 ]  [ 1 - z         3        ]
              [ 1  0 ]  [          ]  [                        ]
      (%o9)  [[      ], [   3      ], [                9       ]]
              [ 0  1 ]  [ -----  1 ]  [   0    - z - ----- + 8 ]
                        [ 1 - z    ]  [              1 - z     ]
      (%i10) %[1] . %[2] . %[3];
                              [ 1 - z    3   ]
      (%o10)                  [              ]
                              [   3    8 - z ]

  Categories: Matrix decompositions · Package linearalgebra

Function: mat_cond
`mat_cond` (M, 1)
`mat_cond` (M, inf)

  Return the p-norm matrix condition number of the matrix m. The allowed values for p are 1 and inf. This function uses the LU factorization to invert the matrix m. Thus the running time for `mat_cond` is proportional to the cube of the matrix size; `lu_factor` determines lower and upper bounds for the infinity norm condition number in time proportional to the square of the matrix size.

  Categories: Package linearalgebra

Function: mat_norm
`mat_norm` (M, 1)
`mat_norm` (M, inf)
`mat_norm` (M, frobenius)

  Return the matrix p-norm of the matrix M. The allowed values for p are 1, `inf`, and `frobenius` (the Frobenius matrix norm). The matrix M should be an unblocked matrix.

  Categories: Package linearalgebra

Function: matrixp
`matrixp` (e, p)
`matrixp` (e)

  Given an optional argument p, return `true` if e is a matrix and p evaluates to `true` for every matrix element. When `matrixp` is not given an optional argument, return `true` if `e` is a matrix. In all other cases, return `false`.

  See also `blockmatrixp`

  Categories: Package linearalgebra · Predicate functions

Function: matrix_size (M)

  Return a two member list that gives the number of rows and columns, respectively of the matrix M.

  Categories: Package linearalgebra

Function: mat_fullunblocker (M)

  If M is a block matrix, unblock the matrix to all levels. If M is a matrix, return M; otherwise, signal an error.

  Categories: Package linearalgebra

Function: mat_trace (M)

  Return the trace of the matrix M. If M isn't a matrix, return a noun form. When M is a block matrix, `mat_trace(M)` returns the same value as does `mat_trace(mat_unblocker(m))`.

  Categories: Package linearalgebra

Function: mat_unblocker (M)

  If M is a block matrix, unblock M one level. If M is a matrix, `mat_unblocker (M)` returns M; otherwise, signal an error.

  Thus if each entry of M is matrix, `mat_unblocker (M)` returns an unblocked matrix, but if each entry of M is a block matrix, `mat_unblocker (M)` returns a block matrix with one less level of blocking.

  If you use block matrices, most likely you'll want to set `matrix_element_mult` to `"."` and `matrix_element_transpose` to `'transpose`. See also `mat_fullunblocker`.

  Example:

      (%i1) A : matrix ([1, 2], [3, 4]);
                                  [ 1  2 ]
      (%o1)                       [      ]
                                  [ 3  4 ]
      (%i2) B : matrix ([7, 8], [9, 10]);
                                  [ 7  8  ]
      (%o2)                       [       ]
                                  [ 9  10 ]
      (%i3) matrix ([A, B]);
                           [ [ 1  2 ]  [ 7  8  ] ]
      (%o3)                [ [      ]  [       ] ]
                           [ [ 3  4 ]  [ 9  10 ] ]
      (%i4) mat_unblocker (%);
                               [ 1  2  7  8  ]
      (%o4)                    [             ]
                               [ 3  4  9  10 ]

  Categories: Package linearalgebra

Function: nullspace (M)

  If M is a matrix, return `span (v_1, ..., v_n)`, where the set `{v_1, ..., v_n}` is a basis for the nullspace of M. The span of the empty set is `{0}`. Thus, when the nullspace has only one member, return `span ()`.

  Categories: Package linearalgebra

Function: nullity (M)

  If M is a matrix, return the dimension of the nullspace of M.

  Categories: Package linearalgebra

Function: orthogonal_complement (v_1, …, v_n)

  Return `span (u_1, ..., u_m)`, where the set `{u_1, ..., u_m}` is a basis for the orthogonal complement of the set `(v_1, ..., v_n)`.

  Each vector v_1 through v_n must be a column vector.

  Categories: Package linearalgebra

Function: polynomialp
`polynomialp` (p, L, coeffp, exponp)
`polynomialp` (p, L, coeffp)
`polynomialp` (p, L)

  Return `true` if p is a polynomial in the variables in the list L. The predicate coeffp must evaluate to `true` for each coefficient, and the predicate exponp must evaluate to `true` for all exponents of the variables in L. If you want to use a non-default value for exponp, you must supply coeffp with a value even if you want to use the default for coeffp.

  The command `polynomialp (p, L, coeffp)` is equivalent to `polynomialp (p, L, coeffp, 'nonnegintegerp)` and the command `polynomialp (p, L)` is equivalent to `polynomialp (p, L, 'constantp, 'nonnegintegerp)`.

  The polynomial needn't be expanded:

      (%i1) polynomialp ((x + 1)*(x + 2), [x]);
      (%o1)                         true
      (%i2) polynomialp ((x + 1)*(x + 2)^a, [x]);
      (%o2)                         false

  An example using non-default values for coeffp and exponp:

      (%i1) polynomialp ((x + 1)*(x + 2)^(3/2), [x], numberp, numberp);
      (%o1)                         true
      (%i2) polynomialp ((x^(1/2) + 1)*(x + 2)^(3/2), [x], numberp,
                                                              numberp);
      (%o2)                         true

  Polynomials with two variables:

      (%i1) polynomialp (x^2 + 5*x*y + y^2, [x]);
      (%o1)                         false
      (%i2) polynomialp (x^2 + 5*x*y + y^2, [x, y]);
      (%o2)                         true

  Categories: Package linearalgebra · Predicate functions

Function: polytocompanion (p, x)

  If p is a polynomial in x, return the companion matrix of p. For a monic polynomial p of degree n, we have `p = (-1)^n charpoly (polytocompanion (p, x))`.

  When p isn't a polynomial in x, signal an error.

  Categories: Package linearalgebra

Function: ptriangularize (M, v)

  If M is a matrix with each entry a polynomial in v, return a matrix M2 such that

  (1) M2 is upper triangular,

  (2) `M2 = E_n ... E_1 M`, where E_1 through E_n are elementary matrices whose entries are polynomials in v,

  (3) `|det (M)| = |det (M2)|`,

  Note: This function doesn't check that every entry is a polynomial in v.

  Categories: Package linearalgebra

Function: rowop (M, i, j, theta)

  If M is a matrix, return the matrix that results from doing the row operation `R_i <- R_i - theta * R_j`. If M doesn't have a row i or j, signal an error.

  Categories: Package linearalgebra

Function: rank (M)

  Return the rank of that matrix M. The rank is the dimension of the column space. Example:

      (%i1) rank(matrix([1,2],[2,4]));
      (%o1)                                  1
      (%i2) rank(matrix([1,b],[c,d]));
      Proviso:  {d - b c # 0}
      (%o2)                                  2

  Categories: Package linearalgebra

Function: rowswap (M, i, j)

  If M is a matrix, swap rows i and j. If M doesn't have a row i or j, signal an error.

  Categories: Package linearalgebra

Function: toeplitz
`toeplitz` (col)
`toeplitz` (col, row)

  Return a Toeplitz matrix T. The first first column of T is col; except for the first entry, the first row of T is row. The default for row is complex conjugate of col. Example:

      (%i1)  toeplitz([1,2,3],[x,y,z]);
                                        [ 1  y  z ]
                                        [         ]
      (%o1)                             [ 2  1  y ]
                                        [         ]
                                        [ 3  2  1 ]
      (%i2)  toeplitz([1,1+%i]);

                                    [   1     1 - %I ]
      (%o2)                         [                ]
                                    [ %I + 1    1    ]

  Categories: Package linearalgebra

Function: vandermonde_matrix ([x_1, ..., x_n])

  Return a n by n matrix whose i-th row is `[1, x_i, x_i^2, ... x_i^(n-1)]`.

  Categories: Package linearalgebra

Function: zerofor
`zerofor` (M)
`zerofor` (M, fld)

  Return a zero matrix that has the same shape as the matrix M. Every entry of the zero matrix is the additive identity of the field fld; the default for fld is generalring.

  The first argument M should be a square matrix or a non-matrix. When M is a matrix, each entry of M can be a square matrix - thus M can be a blocked Maxima matrix. The matrix can be blocked to any (finite) depth.

  See also `identfor`

  Categories: Package linearalgebra

Function: zeromatrixp (M)

  If M is not a block matrix, return `true` if `is (equal (e, 0))` is true for each element e of the matrix M. If M is a block matrix, return `true` if `zeromatrixp` evaluates to `true` for each element of e.

  Categories: Package linearalgebra · Predicate functions

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# 69. lsquares

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## 69.1 Introduction to lsquares

`lsquares` is a collection of functions to implement the method of least squares to estimate parameters for a model from numerical data.

Categories: Statistical estimation · Share packages · Package lsquares

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## 69.2 Functions and Variables for lsquares

Function: lsquares_estimates
`lsquares_estimates` (D, x, e, a)
`lsquares_estimates` (D, x, e, a, initial = L, tol = t)

  Estimate parameters a to best fit the equation e in the variables x and a to the data D, as determined by the method of least squares. `lsquares_estimates` first seeks an exact solution, and if that fails, then seeks an approximate solution.

  The return value is a list of lists of equations of the form `[a = ..., b = ..., c = ...]`. Each element of the list is a distinct, equivalent minimum of the mean square error.

  The data D must be a matrix. Each row is one datum (which may be called a `record' or `case' in some contexts), and each column contains the values of one variable across all data. The list of variables x gives a name for each column of D, even the columns which do not enter the analysis. The list of parameters a gives the names of the parameters for which estimates are sought. The equation e is an expression or equation in the variables x and a; if e is not an equation, it is treated the same as `e = 0`.

  Additional arguments to `lsquares_estimates` are specified as equations and passed on verbatim to the function `lbfgs` which is called to find estimates by a numerical method when an exact result is not found.

  If some exact solution can be found (via `solve`), the data D may contain non-numeric values. However, if no exact solution is found, each element of D must have a numeric value. This includes numeric constants such as `%pi` and `%e` as well as literal numbers (integers, rationals, ordinary floats, and bigfloats). Numerical calculations are carried out with ordinary floating-point arithmetic, so all other kinds of numbers are converted to ordinary floats for calculations.

  If `lsquares_estimates` needs excessive amounts of time or runs out of memory `lsquares_estimates_approximate`, which skips the attempt to find an exact solution, might still succeed.

  `load(lsquares)` loads this function.

  See also `lsquares_estimates_exact`, `lsquares_estimates_approximate`,
  `lsquares_mse`, `lsquares_residuals`, and `lsquares_residual_mse`.

  Examples:

  A problem for which an exact solution is found.

      (%i1) load ("lsquares")$
      (%i2) M : matrix (
              [1,1,1], [3/2,1,2], [9/4,2,1], [3,2,2], [2,2,1]);
                                 [ 1  1  1 ]
                                 [         ]
                                 [ 3       ]
                                 [ -  1  2 ]
                                 [ 2       ]
                                 [         ]
      (%o2)                      [ 9       ]
                                 [ -  2  1 ]
                                 [ 4       ]
                                 [         ]
                                 [ 3  2  2 ]
                                 [         ]
                                 [ 2  2  1 ]
      (%i3) lsquares_estimates (
               M, [z,x,y], (z+D)^2 = A*x+B*y+C, [A,B,C,D]);
                        59        27      10921        107
      (%o3)     [[A = - --, B = - --, C = -----, D = - ---]]
                        16        16      1024         32

  A problem for which no exact solution is found, so `lsquares_estimates` resorts to numerical approximation.

      (%i1) load ("lsquares")$
      (%i2) M : matrix ([1, 1], [2, 7/4], [3, 11/4], [4, 13/4]);
                                  [ 1  1  ]
                                  [       ]
                                  [    7  ]
                                  [ 2  -  ]
                                  [    4  ]
                                  [       ]
      (%o2)                       [    11 ]
                                  [ 3  -- ]
                                  [    4  ]
                                  [       ]
                                  [    13 ]
                                  [ 4  -- ]
                                  [    4  ]
      (%i3) lsquares_estimates (
        M, [x,y], y=a*x^b+c, [a,b,c], initial=[3,3,3], iprint=[-1,0]);
      (%o3) [[a = 1.375751433061394, b = 0.7148891534417651,
                                             c = - 0.4020908910062951]]

  Exponential functions aren't well-conditioned for least min square fitting. In case that fitting to them fails it might be possible to get rid of the exponential function using an logarithm.

      (%i1) load ("lsquares")$
      (%i2) yvalues:[1,3,5,60,200,203,80]$
      (%i3) time:[1,2,4,5,6,8,10]$
      (%i4) f:y=a*exp(b*t);
                                         b t
      (%o4)                      y = a %e
      (%i5) yvalues_log:log(yvalues)$
      (%i6) f_log:log(subst(y=exp(y),f));
                                          b t
      (%o6)                   y = log(a %e   )
      (%i7) lsquares_estimates(
          transpose(matrix(yvalues_log,time)),
          [y,t],
          f_log,
          [a,b]
       );
      *************************************************
        N=    2   NUMBER OF CORRECTIONS=25
             INITIAL VALUES
       F=  6.802906290754687D+00   GNORM=  2.851243373781393D+01
      *************************************************

         I  NFN     FUNC                    GNORM                   STEPLENGTH

         1    3     1.141838765593467D+00   1.067358003667488D-01   1.390943719972406D-02
         2    5     1.141118195694385D+00   1.237977833033414D-01   5.000000000000000D+00
         3    6     1.136945723147959D+00   3.806696991691383D-01   1.000000000000000D+00
         4    7     1.133958243220262D+00   3.865103550379243D-01   1.000000000000000D+00
         5    8     1.131725773805499D+00   2.292258231154026D-02   1.000000000000000D+00
         6    9     1.131625585698168D+00   2.664440547017370D-03   1.000000000000000D+00
         7   10     1.131620564856599D+00   2.519366958715444D-04   1.000000000000000D+00

       THE MINIMIZATION TERMINATED WITHOUT DETECTING ERRORS.
       IFLAG = 0
      (%o7)   [[a = 1.155904145765554, b = 0.5772666876959847]]

  Categories: Package lsquares · Numerical methods

Function: lsquares_estimates_exact (MSE, a)

  Estimate parameters a to minimize the mean square error MSE, by constructing a system of equations and attempting to solve them symbolically via `solve`. The mean square error is an expression in the parameters a, such as that returned by `lsquares_mse`.

  The return value is a list of lists of equations of the form `[a = ..., b = ..., c = ...]`. The return value may contain zero, one, or two or more elements. If two or more elements are returned, each represents a distinct, equivalent minimum of the mean square error.

  See also `lsquares_estimates`, `lsquares_estimates_approximate`, `lsquares_mse`, `lsquares_residuals`, and `lsquares_residual_mse`.

  Example:

      (%i1) load ("lsquares")$
      (%i2) M : matrix (
               [1,1,1], [3/2,1,2], [9/4,2,1], [3,2,2], [2,2,1]);
                                 [ 1  1  1 ]
                                 [         ]
                                 [ 3       ]
                                 [ -  1  2 ]
                                 [ 2       ]
                                 [         ]
      (%o2)                      [ 9       ]
                                 [ -  2  1 ]
                                 [ 4       ]
                                 [         ]
                                 [ 3  2  2 ]
                                 [         ]
                                 [ 2  2  1 ]
      (%i3) mse : lsquares_mse (M, [z, x, y], (z + D)^2 = A*x + B*y + C);
               5
              ====
              \                                         2     2
               >    ((- B M    ) - A M     + (M     + D)  - C)
              /            i, 3       i, 2     i, 1
              ====
              i = 1
      (%o3)   -------------------------------------------------
                                      5
      (%i4) lsquares_estimates_exact (mse, [A, B, C, D]);
                        59        27      10921        107
      (%o4)     [[A = - --, B = - --, C = -----, D = - ---]]
                        16        16      1024         32

  Categories: Package lsquares

Function: lsquares_estimates_approximate (MSE, a, initial = L, tol = t)

  Estimate parameters a to minimize the mean square error MSE, via the numerical minimization function `lbfgs`. The mean square error is an expression in the parameters a, such as that returned by `lsquares_mse`.

  The solution returned by `lsquares_estimates_approximate` is a local (perhaps global) minimum of the mean square error. For consistency with `lsquares_estimates_exact`, the return value is a nested list which contains one element, namely a list of equations of the form `[a = ..., b = ..., c = ...]`.

  Additional arguments to `lsquares_estimates_approximate` are specified as equations and passed on verbatim to the function `lbfgs`.

  MSE must evaluate to a number when the parameters are assigned numeric values. This requires that the data from which MSE was constructed comprise only numeric constants such as `%pi` and `%e` and literal numbers (integers, rationals, ordinary floats, and bigfloats). Numerical calculations are carried out with ordinary floating-point arithmetic, so all other kinds of numbers are converted to ordinary floats for calculations.

  `load(lsquares)` loads this function.

  See also `lsquares_estimates`, `lsquares_estimates_exact`, `lsquares_mse`,
  `lsquares_residuals`, and `lsquares_residual_mse`.

  Example:

      (%i1) load ("lsquares")$
      (%i2) M : matrix (
               [1,1,1], [3/2,1,2], [9/4,2,1], [3,2,2], [2,2,1]);
                                 [ 1  1  1 ]
                                 [         ]
                                 [ 3       ]
                                 [ -  1  2 ]
                                 [ 2       ]
                                 [         ]
      (%o2)                      [ 9       ]
                                 [ -  2  1 ]
                                 [ 4       ]
                                 [         ]
                                 [ 3  2  2 ]
                                 [         ]
                                 [ 2  2  1 ]
      (%i3) mse : lsquares_mse (M, [z, x, y], (z + D)^2 = A*x + B*y + C);
               5
              ====
              \                                         2     2
               >    ((- B M    ) - A M     + (M     + D)  - C)
              /            i, 3       i, 2     i, 1
              ====
              i = 1
      (%o3)   -------------------------------------------------
                                      5
      (%i4) lsquares_estimates_approximate (
              mse, [A, B, C, D], iprint = [-1, 0]);
      (%o4) [[A = - 3.678504947401971, B = - 1.683070351177937,
                       C = 10.63469950148714, D = - 3.340357993175297]]

  Categories: Package lsquares · Numerical methods

Function: lsquares_mse (D, x, e)

  Returns the mean square error (MSE), a summation expression, for the equation e in the variables x, with data D.

  The MSE is defined as:

                          n
                         ====
                     1   \                        2
                     -    >    (lhs(e ) - rhs(e ))
                     n   /           i         i
                         ====
                         i = 1

  where n is the number of data and `e[i]` is the equation e evaluated with the variables in x assigned values from the `i`-th datum, `D[i]`.

  `load(lsquares)` loads this function.

  Example:

      (%i1) load ("lsquares")$
      (%i2) M : matrix (
               [1,1,1], [3/2,1,2], [9/4,2,1], [3,2,2], [2,2,1]);
                                 [ 1  1  1 ]
                                 [         ]
                                 [ 3       ]
                                 [ -  1  2 ]
                                 [ 2       ]
                                 [         ]
      (%o2)                      [ 9       ]
                                 [ -  2  1 ]
                                 [ 4       ]
                                 [         ]
                                 [ 3  2  2 ]
                                 [         ]
                                 [ 2  2  1 ]
      (%i3) mse : lsquares_mse (M, [z, x, y], (z + D)^2 = A*x + B*y + C);
               5
              ====
              \                                         2     2
               >    ((- B M    ) - A M     + (M     + D)  - C)
              /            i, 3       i, 2     i, 1
              ====
              i = 1
      (%o3)   -------------------------------------------------
                                      5
      (%i4) diff (mse, D);
      (%o4)
            5
           ====
           \                                                     2
         4  >    (M     + D) ((- B M    ) - A M     + (M     + D)  - C)
           /       i, 1             i, 3       i, 2     i, 1
           ====
           i = 1
         --------------------------------------------------------------
                                       5
      (%i5) ''mse, nouns;
                     2                 2         9 2               2
      (%o5) (((D + 3)  - C - 2 B - 2 A)  + ((D + -)  - C - B - 2 A)
                                                 4
                 2               2         3 2               2
       + ((D + 2)  - C - B - 2 A)  + ((D + -)  - C - 2 B - A)
                                           2
                 2             2
       + ((D + 1)  - C - B - A) )/5

      (%i3) mse : lsquares_mse (M, [z, x, y], (z + D)^2 = A*x + B*y + C);
                 5
                ====
                \                 2                         2
                 >    ((D + M    )  - C - M     B - M     A)
                /            i, 1          i, 3      i, 2
                ====
                i = 1
      (%o3)     ---------------------------------------------
                                      5

      (%i4) diff (mse, D);
               5
              ====
              \                             2
            4  >    (D + M    ) ((D + M    )  - C - M     B - M     A)
              /           i, 1         i, 1          i, 3      i, 2
              ====
              i = 1
      (%o4) ----------------------------------------------------------
                                        5

      (%i5) ''mse, nouns;
                     2                 2         9 2               2
      (%o5) (((D + 3)  - C - 2 B - 2 A)  + ((D + -)  - C - B - 2 A)
                                                 4
                 2               2         3 2               2
       + ((D + 2)  - C - B - 2 A)  + ((D + -)  - C - 2 B - A)
                                           2
                 2             2
       + ((D + 1)  - C - B - A) )/5

  Categories: Package lsquares

Function: lsquares_residuals (D, x, e, a)

  Returns the residuals for the equation e with specified parameters a and data D.

  D is a matrix, x is a list of variables, e is an equation or general expression; if not an equation, e is treated as if it were `e = 0`. a is a list of equations which specify values for any free parameters in e aside from x.

  The residuals are defined as:

                              lhs(e ) - rhs(e )
                                   i         i

  where `e[i]` is the equation e evaluated with the variables in x assigned values from the `i`-th datum, `D[i]`, and assigning any remaining free variables from a.

  `load(lsquares)` loads this function.

  Example:

      (%i1) load ("lsquares")$
      (%i2) M : matrix (
               [1,1,1], [3/2,1,2], [9/4,2,1], [3,2,2], [2,2,1]);
                                 [ 1  1  1 ]
                                 [         ]
                                 [ 3       ]
                                 [ -  1  2 ]
                                 [ 2       ]
                                 [         ]
      (%o2)                      [ 9       ]
                                 [ -  2  1 ]
                                 [ 4       ]
                                 [         ]
                                 [ 3  2  2 ]
                                 [         ]
                                 [ 2  2  1 ]
      (%i3) a : lsquares_estimates (
                M, [z,x,y], (z+D)^2 = A*x+B*y+C, [A,B,C,D]);
                        59        27      10921        107
      (%o3)     [[A = - --, B = - --, C = -----, D = - ---]]
                        16        16      1024         32
      (%i4) lsquares_residuals (
                M, [z,x,y], (z+D)^2 = A*x+B*y+C, first(a));
                           13    13    13  13  13
      (%o4)               [--, - --, - --, --, --]
                           64    64    32  64  64

  Categories: Package lsquares

Function: lsquares_residual_mse (D, x, e, a)

  Returns the residual mean square error (MSE) for the equation e with specified parameters a and data D.

  The residual MSE is defined as:

                          n
                         ====
                     1   \                        2
                     -    >    (lhs(e ) - rhs(e ))
                     n   /           i         i
                         ====
                         i = 1

  where `e[i]` is the equation e evaluated with the variables in x assigned values from the `i`-th datum, `D[i]`, and assigning any remaining free variables from a.

  `load(lsquares)` loads this function.

  Example:

      (%i1) load ("lsquares")$
      (%i2) M : matrix (
               [1,1,1], [3/2,1,2], [9/4,2,1], [3,2,2], [2,2,1]);
                                 [ 1  1  1 ]
                                 [         ]
                                 [ 3       ]
                                 [ -  1  2 ]
                                 [ 2       ]
                                 [         ]
      (%o2)                      [ 9       ]
                                 [ -  2  1 ]
                                 [ 4       ]
                                 [         ]
                                 [ 3  2  2 ]
                                 [         ]
                                 [ 2  2  1 ]
      (%i3) a : lsquares_estimates (
             M, [z,x,y], (z+D)^2 = A*x+B*y+C, [A,B,C,D]);
                        59        27      10921        107
      (%o3)     [[A = - --, B = - --, C = -----, D = - ---]]
                        16        16      1024         32
      (%i4) lsquares_residual_mse (
             M, [z,x,y], (z + D)^2 = A*x + B*y + C, first (a));
                                    169
      (%o4)                         ----
                                    2560

  Categories: Package lsquares

Function: plsquares
`plsquares` (Mat,VarList,depvars)
`plsquares` (Mat,VarList,depvars,maxexpon)
`plsquares` (Mat,VarList,depvars,maxexpon,maxdegree)

  Multivariable polynomial adjustment of a data table by the "least squares" method. Mat is a matrix containing the data, VarList is a list of variable names (one for each Mat column, but use "-" instead of varnames to ignore Mat columns), depvars is the name of a dependent variable or a list with one or more names of dependent variables (which names should be in VarList), maxexpon is the optional maximum exponent for each independent variable (1 by default), and maxdegree is the optional maximum polynomial degree (maxexpon by default); note that the sum of exponents of each term must be equal or smaller than maxdegree, and if `maxdgree = 0` then no limit is applied.

  If depvars is the name of a dependent variable (not in a list), `plsquares` returns the adjusted polynomial. If depvars is a list of one or more dependent variables, `plsquares` returns a list with the adjusted polynomial(s). The Coefficients of Determination are displayed in order to inform about the goodness of fit, which ranges from 0 (no correlation) to 1 (exact correlation). These values are also stored in the global variable DETCOEF (a list if depvars is a list).

  A simple example of multivariable linear adjustment:

      (%i1) load("plsquares")$

      (%i2) plsquares(matrix([1,2,0],[3,5,4],[4,7,9],[5,8,10]),
                      [x,y,z],z);
           Determination Coefficient for z = .9897039897039897
                             11 y - 9 x - 14
      (%o2)              z = ---------------
                                    3

  The same example without degree restrictions:

      (%i3) plsquares(matrix([1,2,0],[3,5,4],[4,7,9],[5,8,10]),
                      [x,y,z],z,1,0);
           Determination Coefficient for z = 1.0
                          x y + 23 y - 29 x - 19
      (%o3)           z = ----------------------
                                    6

  How many diagonals does a N-sides polygon have? What polynomial degree should be used?

      (%i4) plsquares(matrix([3,0],[4,2],[5,5],[6,9],[7,14],[8,20]),
                      [N,diagonals],diagonals,5);
           Determination Coefficient for diagonals = 1.0
                                      2
                                     N  - 3 N
      (%o4)              diagonals = --------
                                        2
      (%i5) ev(%, N=9);   /* Testing for a 9 sides polygon */
      (%o5)                 diagonals = 27

  How many ways do we have to put two queens without they are threatened into a n x n chessboard?

      (%i6) plsquares(matrix([0,0],[1,0],[2,0],[3,8],[4,44]),
                      [n,positions],[positions],4);
           Determination Coefficient for [positions] = [1.0]
                               4       3      2
                            3 n  - 10 n  + 9 n  - 2 n
      (%o6)    [positions = -------------------------]
                                        6
      (%i7) ev(%[1], n=8); /* Testing for a (8 x 8) chessboard */
      (%o7)                positions = 1288

  An example with six dependent variables:

      (%i8) mtrx:matrix([0,0,0,0,0,1,1,1],[0,1,0,1,1,1,0,0],
                        [1,0,0,1,1,1,0,0],[1,1,1,1,0,0,0,1])$
      (%i8) plsquares(mtrx,[a,b,_And,_Or,_Xor,_Nand,_Nor,_Nxor],
                           [_And,_Or,_Xor,_Nand,_Nor,_Nxor],1,0);
            Determination Coefficient for
      [_And, _Or, _Xor, _Nand, _Nor, _Nxor] =
      [1.0, 1.0, 1.0, 1.0, 1.0, 1.0]
      (%o2) [_And = a b, _Or = - a b + b + a,
      _Xor = - 2 a b + b + a, _Nand = 1 - a b,
      _Nor = a b - b - a + 1, _Nxor = 2 a b - b - a + 1]

  To use this function write first `load("lsquares")`.

  Categories: Package lsquares · Numerical methods

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# 70. minpack

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## 70.1 Introduction to minpack

`Minpack` is a Common Lisp translation (via `f2cl`) of the Fortran library MINPACK, as obtained from Netlib.

Categories: Numerical methods · Optimization · Share packages · Package minpack

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## 70.2 Functions and Variables for minpack

Function: minpack_lsquares (flist, varlist, guess [, tolerance, jacobian])

  Compute the point that minimizes the sum of the squares of the functions in the list flist. The variables are in the list varlist. An initial guess of the optimum point must be provided in guess.

  The optional keyword arguments, tolerance and jacobian provide some control over the algorithm. tolerance is the estimated relative error desired in the sum of squares. jacobian can be used to specify the Jacobian. If jacobian is not given or is `true` (the default), the Jacobian is computed from flist. If jacobian is `false`, a numerical approximation is used.

  `minpack_lsquares` returns a list. The first item is the estimated solution; the second is the sum of squares, and the third indicates the success of the algorithm. The possible values are

  `0`

    improper input parameters.

  `1`

    algorithm estimates that the relative error in the sum of squares is at most `tolerance`.

  `2`

    algorithm estimates that the relative error between x and the solution is at most `tolerance`.

  `3`

    conditions for info = 1 and info = 2 both hold.

  `4`

    fvec is orthogonal to the columns of the jacobian to machine precision.

  `5`

    number of calls to fcn with iflag = 1 has reached 100*(n+1).

  `6`

    tol is too small. no further reduction in the sum of squares is possible.

  `7`

    tol is too small. no further improvement in the approximate solution x is possible.

      /* Problem 6: Powell singular function */
      (%i1) powell(x1,x2,x3,x4) :=
               [x1+10*x2, sqrt(5)*(x3-x4), (x2-2*x3)^2,
                    sqrt(10)*(x1-x4)^2]$
      (%i2) minpack_lsquares(powell(x1,x2,x3,x4), [x1,x2,x3,x4],
                             [3,-1,0,1]);
      (%o2) [[1.652117596168394e-17, - 1.652117596168393e-18,
              2.643388153869468e-18, 2.643388153869468e-18],
             6.109327859207777e-34, 4]

      /* Same problem but use numerical approximation to Jacobian */
      (%i3) minpack_lsquares(powell(x1,x2,x3,x4), [x1,x2,x3,x4],
                             [3,-1,0,1], jacobian = false);
      (%o3) [[5.060282149485331e-11, - 5.060282149491206e-12,
              2.179447843547218e-11, 2.179447843547218e-11],
             3.534491794847031e-21, 5]

Function: minpack_solve (flist, varlist, guess [, tolerance, jacobian])

  Solve a system of `n` equations in `n` unknowns. The `n` equations are given in the list flist, and the unknowns are in varlist. An initial guess of the solution must be provided in guess.

  The optional keyword arguments, tolerance and jacobian provide some control over the algorithm. tolerance is the estimated relative error desired in the sum of squares. jacobian can be used to specify the Jacobian. If jacobian is not given or is `true` (the default), the Jacobian is computed from flist. If jacobian is `false`, a numerical approximation is used.

  `minpack_solve` returns a list. The first item is the estimated solution; the second is the sum of squares, and the third indicates the success of the algorithm. The possible values are

  `0`

    improper input parameters.

  `1`

    algorithm estimates that the relative error in the solution is at most `tolerance`.

  `2`

    number of calls to fcn with iflag = 1 has reached 100*(n+1).

  `3`

    tol is too small. no further reduction in the sum of squares is possible.

  `4`

    Iteration is not making good progress.

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# 71. makeOrders

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## 71.1 Functions and Variables for makeOrders

Function: makeOrders (indvarlist,orderlist)

  Returns a list of all powers for a polynomial up to and including the arguments.

      (%i1) load("makeOrders")$

      (%i2) makeOrders([a,b],[2,3]);
      (%o2) [[0, 0], [0, 1], [0, 2], [0, 3], [1, 0], [1, 1],
                  [1, 2], [1, 3], [2, 0], [2, 1], [2, 2], [2, 3]]
      (%i3) expand((1+a+a^2)*(1+b+b^2+b^3));
             2  3      3    3    2  2      2    2    2
      (%o3) a  b  + a b  + b  + a  b  + a b  + b  + a  b + a b
                                                        2
                                                 + b + a  + a + 1

  where `[0, 1]` is associated with the term b and `[2, 3]` with a^2 b^3.

  To use this function write first `load("makeOrders")`.

  Categories: Polynomials · Share packages · Package makeOrders

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# 72. mnewton

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## 72.1 Introduction to mnewton

`mnewton` is an implementation of Newton's method for solving nonlinear equations in one or more variables.

Categories: Numerical methods · Share packages · Package mnewton

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## 72.2 Functions and Variables for mnewton

Option variable: newtonepsilon

  Default value: `10.0^(-fpprec/2)`

  Precision to determine when the `mnewton` function has converged towards the solution. If `newtonepsilon` is a bigfloat, then `mnewton` computations are done with bigfloats. See also `mnewton`.

  Categories: Package mnewton

Option variable: newtonmaxiter

  Default value: `50`

  Maximum number of iterations to stop the `mnewton` function if it does not converge or if it converges too slowly.

  See also `mnewton`.

  Categories: Package mnewton

Function: mnewton (FuncList,VarList,GuessList)

  Multiple nonlinear functions solution using the Newton method. FuncList is the list of functions to solve, VarList is the list of variable names, and GuessList is the list of initial approximations.

  The solution is returned in the same format that `solve()` returns. If the solution is not found, `[]` is returned.

  This function is controlled by global variables `newtonepsilon` and `newtonmaxiter`.

      (%i1) load("mnewton")$

      (%i2) mnewton([x1+3*log(x1)-x2^2, 2*x1^2-x1*x2-5*x1+1],
                    [x1, x2], [5, 5]);
      (%o2) [[x1 = 3.756834008012769, x2 = 2.779849592817897]]
      (%i3) mnewton([2*a^a-5],[a],[1]);
      (%o3)             [[a = 1.70927556786144]]
      (%i4) mnewton([2*3^u-v/u-5, u+2^v-4], [u, v], [2, 2]);
      (%o4) [[u = 1.066618389595407, v = 1.552564766841786]]

  The variable `newtonepsilon` controls the precision of the approximations. It also controls if computations are performed with floats or bigfloats.

      (%i1) load(mnewton)$

      (%i2) (fpprec : 25, newtonepsilon : bfloat(10^(-fpprec+5)))$

      (%i3) mnewton([2*3^u-v/u-5, u+2^v-4], [u, v], [2, 2]);
      (%o3) [[u = 1.066618389595406772591173b0,
                                     v = 1.552564766841786450100418b0]]

  To use this function write first `load("mnewton")`. See also `newtonepsilon` and `newtonmaxiter`.

  Categories: Package mnewton

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# 73. numericalio

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## 73.1 Introduction to numericalio

`numericalio` is a collection of functions to read and write files and streams. Functions for plain-text input and output can read and write numbers (integer, float, or bigfloat), symbols, and strings. Functions for binary input and output can read and write only floating-point numbers.

If there already exists a list, matrix, or array object to store input data, `numericalio` input functions can write data into that object. Otherwise, `numericalio` can guess, to some degree, the structure of an object to store the data, and return that object.

Categories: File input · File output · Share packages · Package numericalio

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### 73.1.1 Plain-text input and output

In plain-text input and output, it is assumed that each item to read or write is an atom: an integer, float, bigfloat, string, or symbol, and not a rational or complex number or any other kind of nonatomic expression. The `numericalio` functions may attempt to do something sensible faced with nonatomic expressions, but the results are not specified here and subject to change.

Atoms in both input and output files have the same format as in Maxima batch files or the interactive console. In particular, strings are enclosed in double quotes, backslash `\` prevents any special interpretation of the next character, and the question mark `?` is recognized at the beginning of a symbol to mean a Lisp symbol (as opposed to a Maxima symbol). No continuation character (to join broken lines) is recognized.

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### 73.1.2 Separator flag values for input

The functions for plain-text input and output take an optional argument, separator_flag, that tells what character separates data.

For plain-text input, these values of separator_flag are recognized: `comma` for comma separated values, `pipe` for values separated by the vertical bar character `|`, `semicolon` for values separated by semicolon `;`, and `space` for values separated by space or tab characters. If the file name ends in `.csv` and separator_flag is not specified, `comma` is assumed. If the file name ends in something other than `.csv` and `separator_flag` is not specified, `space` is assumed.

In plain-text input, multiple successive space and tab characters count as a single separator. However, multiple comma, pipe, or semicolon characters are significant. Successive comma, pipe, or semicolon characters (with or without intervening spaces or tabs) are considered to have `false` between the separators. For example, `1234,,Foo` is treated the same as `1234,false,Foo`.

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### 73.1.3 Separator flag values for output

For plain-text output, `tab`, for values separated by the tab character, is recognized as a value of separator_flag, as well as `comma`, `pipe`, `semicolon`, and `space`.

In plain-text output, `false` atoms are written as such; a list `[1234, false, Foo]` is written `1234,false,Foo`, and there is no attempt to collapse the output to `1234,,Foo`.

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### 73.1.4 Binary floating-point input and output

`numericalio` functions can read and write 8-byte IEEE 754 floating-point numbers. These numbers can be stored either least significant byte first or most significant byte first, according to the global flag set by `assume_external_byte_order`. If not specified, `numericalio` assumes the external byte order is most-significant byte first.

Other kinds of numbers are coerced to 8-byte floats; `numericalio` cannot read or write binary non-numeric data.

Some Lisp implementations do not recognize IEEE 754 special values (positive and negative infinity, not-a-number values, denormalized values). The effect of reading such values with `numericalio` is undefined.

`numericalio` includes functions to open a stream for reading or writing a stream of bytes.

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## 73.2 Functions and Variables for plain-text input and output

Function: read_matrix
`read_matrix` (S)
`read_matrix` (S, M)
`read_matrix` (S, separator_flag)
`read_matrix` (S, M, separator_flag)

  `read_matrix(S)` reads the source S and returns its entire content as a matrix. The size of the matrix is inferred from the input data; each line of the file becomes one row of the matrix. If some lines have different lengths, `read_matrix` complains.

  `read_matrix(S, M)` read the source S into the matrix M, until M is full or the source is exhausted. Input data are read into the matrix in row-major order; the input need not have the same number of rows and columns as M.

  The source S may be a file name or a stream which for example allows skipping the very first line of a file (that may be useful, if you read CSV data, where the first line often contains the description of the columns):

      s : openr("data.txt");
      readline(s);  /* skip the first line */
      M : read_matrix(s, 'comma);  /* read the following (comma-separated) lines into matrix M */
      close(s);

  The recognized values of separator_flag are `comma`, `pipe`, `semicolon`, and `space`. If separator_flag is not specified, the file is assumed space-delimited.

  Categories: Package numericalio · File input

Function: read_array
`read_array` (S, A)
`read_array` (S, A, separator_flag)

  Reads the source S into the array A, until A is full or the source is exhausted. Input data are read into the array in row-major order; the input need not conform to the dimensions of A.

  The source S may be a file name or a stream.

  The recognized values of separator_flag are `comma`, `pipe`, `semicolon`, and `space`. If separator_flag is not specified, the file is assumed space-delimited.

  Categories: Package numericalio · File input

Function: read_hashed_array
`read_hashed_array` (S, A)
`read_hashed_array` (S, A, separator_flag)

  Reads the source S and returns its entire content as a hashed array. The source S may be a file name or a stream.

  `read_hashed_array` treats the first item on each line as a hash key, and associates the remainder of the line (as a list) with the key. For example, the line `567 12 17 32 55` is equivalent to `A[567]: [12, 17, 32, 55]$`. Lines need not have the same numbers of elements.

  The recognized values of separator_flag are `comma`, `pipe`, `semicolon`, and `space`. If separator_flag is not specified, the file is assumed space-delimited.

  Categories: Package numericalio · File input

Function: read_nested_list
`read_nested_list` (S)
`read_nested_list` (S, separator_flag)

  Reads the source S and returns its entire content as a nested list. The source S may be a file name or a stream.

  `read_nested_list` returns a list which has a sublist for each line of input. Lines need not have the same numbers of elements. Empty lines are not ignored: an empty line yields an empty sublist.

  The recognized values of separator_flag are `comma`, `pipe`, `semicolon`, and `space`. If separator_flag is not specified, the file is assumed space-delimited.

  Categories: Package numericalio · File input

Function: read_list
`read_list` (S)
`read_list` (S, L)
`read_list` (S, separator_flag)
`read_list` (S, L, separator_flag)

  `read_list(S)` reads the source S and returns its entire content as a flat list.

  `read_list(S, L)` reads the source S into the list L, until L is full or the source is exhausted.

  The source S may be a file name or a stream.

  The recognized values of separator_flag are `comma`, `pipe`, `semicolon`, and `space`. If separator_flag is not specified, the file is assumed space-delimited.

  Categories: Package numericalio · File input

Function: write_data
`write_data` (X, D)
`write_data` (X, D, separator_flag)

  Writes the object X to the destination D.

  `write_data` writes a matrix in row-major order, with one line per row.

  `write_data` writes an array created by `array` or `make_array` in row-major order, with a new line at the end of every slab. Higher-dimensional slabs are separated by additional new lines.

  `write_data` writes a hashed array with each key followed by its associated list on one line.

  `write_data` writes a nested list with each sublist on one line.

  `write_data` writes a flat list all on one line.

  The destination D may be a file name or a stream. When the destination is a file name, the global variable `file_output_append` governs whether the output file is appended or truncated. When the destination is a stream, no special action is taken by `write_data` after all the data are written; in particular, the stream remains open.

  The recognized values of separator_flag are `comma`, `pipe`, `semicolon`, `space`, and `tab`. If separator_flag is not specified, the file is assumed space-delimited.

  Categories: Package numericalio · File output

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## 73.3 Functions and Variables for binary input and output

Function: assume_external_byte_order (byte_order_flag)

  Tells `numericalio` the byte order for reading and writing binary data. Two values of byte_order_flag are recognized: `lsb` which indicates least-significant byte first, also called little-endian byte order; and `msb` which indicates most-significant byte first, also called big-endian byte order.

  If not specified, `numericalio` assumes the external byte order is most-significant byte first.

  Categories: Package numericalio

Function: openr_binary (file_name)

  Returns an input stream of 8-bit unsigned bytes to read the file named by file_name.

  Categories: Package numericalio · File input

Function: openw_binary (file_name)

  Returns an output stream of 8-bit unsigned bytes to write the file named by file_name.

  Categories: Package numericalio · File output

Function: opena_binary (file_name)

  Returns an output stream of 8-bit unsigned bytes to append the file named by file_name.

  Categories: Package numericalio · File output

Function: read_binary_matrix (S, M)

  Reads binary 8-byte floating point numbers from the source S into the matrix M until M is full, or the source is exhausted. Elements of M are read in row-major order.

  The source S may be a file name or a stream.

  The byte order in elements of the source is specified by `assume_external_byte_order`.

  Categories: Package numericalio · File input

Function: read_binary_array (S, A)

  Reads binary 8-byte floating point numbers from the source S into the array A until A is full, or the source is exhausted. A must be an array created by `array` or `make_array`. Elements of A are read in row-major order.

  The source S may be a file name or a stream.

  The byte order in elements of the source is specified by `assume_external_byte_order`.

  Categories: Package numericalio · File input

Function: read_binary_list
`read_binary_list` (S)
`read_binary_list` (S, L)

  `read_binary_list(S)` reads the entire content of the source S as a sequence of binary 8-byte floating point numbers, and returns it as a list. The source S may be a file name or a stream.

  `read_binary_list(S, L)` reads 8-byte binary floating point numbers from the source S until the list L is full, or the source is exhausted.

  The byte order in elements of the source is specified by `assume_external_byte_order`.

  Categories: Package numericalio · File input

Function: write_binary_data (X, D)

  Writes the object X, comprising binary 8-byte IEEE 754 floating-point numbers, to the destination D. Other kinds of numbers are coerced to 8-byte floats. `write_binary_data` cannot write non-numeric data.

  The object X may be a list, a nested list, a matrix, or an array created by `array` or `make_array`; X cannot be an undeclared array or any other type of object. `write_binary_data` writes nested lists, matrices, and arrays in row-major order.

  The destination D may be a file name or a stream. When the destination is a file name, the global variable `file_output_append` governs whether the output file is appended or truncated. When the destination is a stream, no special action is taken by `write_binary_data` after all the data are written; in particular, the stream remains open.

  The byte order in elements of the destination is specified by `assume_external_byte_order`.

  Categories: Package numericalio · File output

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# 74. operatingsystem

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## 74.1 Introduction to operatingsystem

Package `operatingsystem` contains functions for operatingsystem-tasks, like file system operations.

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## 74.2 Directory operations

Function: chdir (dir)

  Change to directory dir

Function: mkdir (dir)

  Create directory dir

Function: rmdir (dir)

  remove directory dir

Function: getcurrentdirectory ()

  returns the current working directory.

  See also `directory`.

Examples:

    (%i1) load("operatingsystem")$
    (%i2) mkdir("testdirectory")$
    (%i3) chdir("testdirectory")$
    (%i4) chdir("..")$
    (%i5) rmdir("testdirectory")$

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## 74.3 File operations

Function: copy_file (file1, file2)

  copies file file1 to file2

Function: rename_file (file1, file2)

  renames file file1 to file2

Function: delete_file (file1)

  deletes file file1

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## 74.4 Environment operations

Function: getenv (env)

  Get the value of the environmentvariable env

  Example:

      (%i1) load("operatingsystem")$
      (%i2) getenv("PATH");
      (%o2) /usr/local/sbin:/usr/local/bin:/usr/sbin:/usr/bin:/sbin:/bin

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# 75. opsubst

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## 75.1 Functions and Variables for opsubst

Function: opsubst
`opsubst` (f,g,e)
`opsubst` (g=f,e)
`opsubst` ([g1=f1,g2=f2,..., gn=fn],e)

  The function `opsubst` is similar to the function `subst`, except that `opsubst` only makes substitutions for the operators in an expression. In general, When f is an operator in the expression e, substitute g for f in the expression e.

  To determine the operator, `opsubst` sets `inflag` to true. This means `opsubst` substitutes for the internal, not the displayed, operator in the expression.

  Examples:

      (%i1) load ("opsubst")$

      (%i2) opsubst(f,g,g(g(x)));
      (%o2)                     f(f(x))
      (%i3) opsubst(f,g,g(g));
      (%o3)                       f(g)
      (%i4) opsubst(f,g[x],g[x](z));
      (%o4)                       f(z)
      (%i5) opsubst(g[x],f, f(z));
      (%o5)                      g (z)
                                  x
      (%i6) opsubst(tan, sin, sin(sin));
      (%o6)                     tan(sin)
      (%i7) opsubst([f=g,g=h],f(x));
      (%o7)                       h(x)

  Internally, Maxima does not use the unary negation, division, or the subtraction operators; thus:

      (%i8) opsubst("+","-",a-b);
      (%o8)                     a - b
      (%i9) opsubst("f","-",-a);
      (%o9)                      - a
      (%i10) opsubst("^^","/",a/b);
                                   a
      (%o10)                       -
                                   b

  The internal representation of -a*b is *(-1,a,b); thus

      (%i11) opsubst("[","*", -a*b);
      (%o11)                  [- 1, a, b]

  When either operator isn't a Maxima symbol, generally some other function will signal an error:

      (%i12) opsubst(a+b,f, f(x));

      Improper name or value in functional position:
      b + a
       -- an error.  Quitting.  To debug this try debugmode(true);

  However, subscripted operators are allowed:

      (%i13) opsubst(g[5],f, f(x));
      (%o13)                     g (x)
                                  5

  To use this function write first `load("opsubst")`.

  Categories: Expressions · Share packages · Package opsubst

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# 76. orthopoly

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## 76.1 Introduction to orthogonal polynomials

`orthopoly` is a package for symbolic and numerical evaluation of several kinds of orthogonal polynomials, including Chebyshev, Laguerre, Hermite, Jacobi, Legendre, and ultraspherical (Gegenbauer) polynomials. Additionally, `orthopoly` includes support for the spherical Bessel, spherical Hankel, and spherical harmonic functions.

For the most part, `orthopoly` follows the conventions of Abramowitz and Stegun Handbook of Mathematical Functions, Chapter 22 (10th printing, December 1972); additionally, we use Gradshteyn and Ryzhik, Table of Integrals, Series, and Products (1980 corrected and enlarged edition), and Eugen Merzbacher Quantum Mechanics (2nd edition, 1970).

Barton Willis of the University of Nebraska at Kearney (UNK) wrote the `orthopoly` package and its documentation. The package is released under the GNU General Public License (GPL).

Categories: Orthogonal polynomials · Share packages · Package orthopoly

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### 76.1.1 Getting Started with orthopoly

`load ("orthopoly")` loads the `orthopoly` package.

To find the third-order Legendre polynomial,

    (%i1) legendre_p (3, x);
                          3             2
                 5 (1 - x)    15 (1 - x)
    (%o1)      - ---------- + ----------- - 6 (1 - x) + 1
                     2             2

To express this as a sum of powers of x, apply ratsimp or rat to the result.

    (%i2) [ratsimp (%), rat (%)];
                            3           3
                         5 x  - 3 x  5 x  - 3 x
    (%o2)/R/            [----------, ----------]
                             2           2

Alternatively, make the second argument to `legendre_p` (its "main" variable) a canonical rational expression (CRE).

    (%i1) legendre_p (3, rat (x));
                                  3
                               5 x  - 3 x
    (%o1)/R/                   ----------
                                   2

For floating point evaluation, `orthopoly` uses a running error analysis to estimate an upper bound for the error. For example,

    (%i1) jacobi_p (150, 2, 3, 0.2);
    (%o1) interval(- 0.062017037936715, 1.533267919277521E-11)

Intervals have the form `interval (c, r)`, where c is the center and r is the radius of the interval. Since Maxima does not support arithmetic on intervals, in some situations, such as graphics, you want to suppress the error and output only the center of the interval. To do this, set the option variable `orthopoly_returns_intervals` to `false`.

    (%i1) orthopoly_returns_intervals : false;
    (%o1)                         false
    (%i2) jacobi_p (150, 2, 3, 0.2);
    (%o2)                  - 0.062017037936715

Refer to the section see Floating point Evaluation for more information.

Most functions in `orthopoly` have a `gradef` property; thus

    (%i1) diff (hermite (n, x), x);
    (%o1)                     2 n H     (x)
                                   n - 1
    (%i2) diff (gen_laguerre (n, a, x), x);
                  (a)               (a)
               n L   (x) - (n + a) L     (x) unit_step(n)
                  n                 n - 1
    (%o2)      ------------------------------------------
                                   x

The unit step function in the second example prevents an error that would otherwise arise by evaluating with n equal to 0.

    (%i3) ev (%, n = 0);
    (%o3)                           0

The `gradef` property only applies to the "main" variable; derivatives with respect other arguments usually result in an error message; for example

    (%i1) diff (hermite (n, x), x);
    (%o1)                     2 n H     (x)
                                   n - 1
    (%i2) diff (hermite (n, x), n);

    Maxima doesn't know the derivative of hermite with respect the first
    argument
     -- an error.  Quitting.  To debug this try debugmode(true);

Generally, functions in `orthopoly` map over lists and matrices. For the mapping to fully evaluate, the option variables `doallmxops` and `listarith` must both be `true` (the defaults). To illustrate the mapping over matrices, consider

    (%i1) hermite (2, x);
                                         2
    (%o1)                    - 2 (1 - 2 x )
    (%i2) m : matrix ([0, x], [y, 0]);
                                [ 0  x ]
    (%o2)                       [      ]
                                [ y  0 ]
    (%i3) hermite (2, m);
                   [                             2  ]
                   [      - 2        - 2 (1 - 2 x ) ]
    (%o3)          [                                ]
                   [             2                  ]
                   [ - 2 (1 - 2 y )       - 2       ]

In the second example, the `i, j` element of the value is `hermite (2, m[i,j])`; this is not the same as computing `-2 + 4 m . m`, as seen in the next example.

    (%i4) -2 * matrix ([1, 0], [0, 1]) + 4 * m . m;
                        [ 4 x y - 2      0     ]
    (%o4)               [                      ]
                        [     0      4 x y - 2 ]

If you evaluate a function at a point outside its domain, generally `orthopoly` returns the function unevaluated. For example,

    (%i1) legendre_p (2/3, x);
    (%o1)                        P   (x)
                                  2/3

`orthopoly` supports translation into TeX; it also does two-dimensional output on a terminal.

    (%i1) spherical_harmonic (l, m, theta, phi);
                              m
    (%o1)                    Y (theta, phi)
                              l
    (%i2) tex (%);
    $$Y_{l}^{m}\left(\vartheta,\varphi\right)$$
    (%o2)                         false
    (%i3) jacobi_p (n, a, a - b, x/2);
                              (a, a - b) x
    (%o3)                    P          (-)
                              n          2
    (%i4) tex (%);
    $$P_{n}^{\left(a,a-b\right)}\left({{x}\over{2}}\right)$$
    (%o4)                         false

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### 76.1.2 Limitations

When an expression involves several orthogonal polynomials with symbolic orders, it's possible that the expression actually vanishes, yet Maxima is unable to simplify it to zero. If you divide by such a quantity, you'll be in trouble. For example, the following expression vanishes for integers n greater than 1, yet Maxima is unable to simplify it to zero.

    (%i1) (2*n - 1) * legendre_p (n - 1, x) * x - n * legendre_p (n, x)
          + (1 - n) * legendre_p (n - 2, x);
    (%o1)  (2 n - 1) P     (x) x - n P (x) + (1 - n) P     (x)
                      n - 1           n               n - 2

For a specific n, we can reduce the expression to zero.

    (%i2) ev (% ,n = 10, ratsimp);
    (%o2)                           0

Generally, the polynomial form of an orthogonal polynomial is ill-suited for floating point evaluation. Here's an example.

    (%i1) p : jacobi_p (100, 2, 3, x)$

    (%i2) subst (0.2, x, p);
    (%o2)                3.4442767023833592E+35
    (%i3) jacobi_p (100, 2, 3, 0.2);
    (%o3)  interval(0.18413609135169, 6.8990300925815987E-12)
    (%i4) float(jacobi_p (100, 2, 3, 2/10));
    (%o4)                   0.18413609135169

The true value is about 0.184; this calculation suffers from extreme subtractive cancellation error. Expanding the polynomial and then evaluating, gives a better result.

    (%i5) p : expand(p)$
    (%i6) subst (0.2, x, p);
    (%o6) 0.18413609766122982

This isn't a general rule; expanding the polynomial does not always result in an expression that is better suited for numerical evaluation. By far, the best way to do numerical evaluation is to make one or more of the function arguments floating point numbers. By doing that, specialized floating point algorithms are used for evaluation.

Maxima's `float` function is somewhat indiscriminate; if you apply `float` to an expression involving an orthogonal polynomial with a symbolic degree or order parameter, these parameters may be converted into floats; after that, the expression will not evaluate fully. Consider

    (%i1) assoc_legendre_p (n, 1, x);
                                   1
    (%o1)                         P (x)
                                   n
    (%i2) float (%);
                                  1.0
    (%o2)                        P   (x)
                                  n
    (%i3) ev (%, n=2, x=0.9);
                                 1.0
    (%o3)                       P   (0.9)
                                 2

The expression in (%o3) will not evaluate to a float; `orthopoly` doesn't recognize floating point values where it requires an integer. Similarly, numerical evaluation of the `pochhammer` function for orders that exceed `pochhammer_max_index` can be troublesome; consider

    (%i1) x :  pochhammer (1, 10), pochhammer_max_index : 5;
    (%o1)                         (1)
                                     10

Applying `float` doesn't evaluate x to a float

    (%i2) float (x);
    (%o2)                       (1.0)
                                     10.0

To evaluate x to a float, you'll need to bind `pochhammer_max_index` to 11 or greater and apply `float` to x.

    (%i3) float (x), pochhammer_max_index : 11;
    (%o3)                       3628800.0

The default value of `pochhammer_max_index` is 100; change its value after loading `orthopoly`.

Finally, be aware that reference books vary on the definitions of the orthogonal polynomials; we've generally used the conventions of Abramowitz and Stegun.

Before you suspect a bug in orthopoly, check some special cases to determine if your definitions match those used by `orthopoly`. Definitions often differ by a normalization; occasionally, authors use "shifted" versions of the functions that makes the family orthogonal on an interval other than (-1, 1). To define, for example, a Legendre polynomial that is orthogonal on (0, 1), define

    (%i1) shifted_legendre_p (n, x) := legendre_p (n, 2*x - 1)$

    (%i2) shifted_legendre_p (2, rat (x));
                                2
    (%o2)/R/                 6 x  - 6 x + 1
    (%i3) legendre_p (2, rat (x));
                                   2
                                3 x  - 1
    (%o3)/R/                    --------
                                   2

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### 76.1.3 Floating point Evaluation

Most functions in `orthopoly` use a running error analysis to estimate the error in floating point evaluation; the exceptions are the spherical Bessel functions and the associated Legendre polynomials of the second kind. For numerical evaluation, the spherical Bessel functions call SLATEC functions. No specialized method is used for numerical evaluation of the associated Legendre polynomials of the second kind.

The running error analysis ignores errors that are second or higher order in the machine epsilon (also known as unit roundoff). It also ignores a few other errors. It's possible (although unlikely) that the actual error exceeds the estimate.

Intervals have the form `interval (c, r)`, where c is the center of the interval and r is its radius. The center of an interval can be a complex number, and the radius is always a positive real number.

Here is an example.

    (%i1) fpprec : 50$

    (%i2) y0 : jacobi_p (100, 2, 3, 0.2);
    (%o2) interval(0.1841360913516871, 6.8990300925815987E-12)
    (%i3) y1 : bfloat (jacobi_p (100, 2, 3, 1/5));
    (%o3) 1.8413609135168563091370224958913493690868904463668b-1

Let's test that the actual error is smaller than the error estimate

    (%i4) is (abs (part (y0, 1) - y1) < part (y0, 2));
    (%o4)                         true

Indeed, for this example the error estimate is an upper bound for the true error.

Maxima does not support arithmetic on intervals.

    (%i1) legendre_p (7, 0.1) + legendre_p (8, 0.1);
    (%o1) interval(0.18032072148437508, 3.1477135311021797E-15)
            + interval(- 0.19949294375000004, 3.3769353084291579E-15)

A user could define arithmetic operators that do interval math. To define interval addition, we can define

    (%i1) infix ("@+")$

    (%i2) "@+"(x,y) := interval (part (x, 1) + part (y, 1), part (x, 2)
          + part (y, 2))$

    (%i3) legendre_p (7, 0.1) @+ legendre_p (8, 0.1);
    (%o3) interval(- 0.019172222265624955, 6.5246488395313372E-15)

The special floating point routines get called when the arguments are complex. For example,

    (%i1) legendre_p (10, 2 + 3.0*%i);
    (%o1) interval(- 3.876378825E+7 %i - 6.0787748E+7,
                                               1.2089173052721777E-6)

Let's compare this to the true value.

    (%i1) float (expand (legendre_p (10, 2 + 3*%i)));
    (%o1)          - 3.876378825E+7 %i - 6.0787748E+7

Additionally, when the arguments are big floats, the special floating point routines get called; however, the big floats are converted into double floats and the final result is a double.

    (%i1) ultraspherical (150, 0.5b0, 0.9b0);
    (%o1) interval(- 0.043009481257265, 3.3750051301228864E-14)

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### 76.1.4 Graphics and `orthopoly`

To plot expressions that involve the orthogonal polynomials, you must do two things:

1. Set the option variable `orthopoly_returns_intervals` to `false`,
2. Quote any calls to `orthopoly` functions.

If function calls aren't quoted, Maxima evaluates them to polynomials before plotting; consequently, the specialized floating point code doesn't get called. Here is an example of how to plot an expression that involves a Legendre polynomial.

    (%i1) plot2d ('(legendre_p (5, x)), [x, 0, 1]),
                            orthopoly_returns_intervals : false;
    (%o1)

The entire expression `legendre_p (5, x)` is quoted; this is different than just quoting the function name using `'legendre_p (5, x)`.

Categories: Plotting

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### 76.1.5 Miscellaneous Functions

The `orthopoly` package defines the Pochhammer symbol and a unit step function. `orthopoly` uses the Kronecker delta function and the unit step function in `gradef` statements.

To convert Pochhammer symbols into quotients of gamma functions, use `makegamma`.

    (%i1) makegamma (pochhammer (x, n));
                              gamma(x + n)
    (%o1)                     ------------
                                gamma(x)
    (%i2) makegamma (pochhammer (1/2, 1/2));
                                    1
    (%o2)                       ---------
                                sqrt(%pi)

Derivatives of the Pochhammer symbol are given in terms of the `psi` function.

    (%i1) diff (pochhammer (x, n), x);
    (%o1)             (x)  (psi (x + n) - psi (x))
                         n     0             0
    (%i2) diff (pochhammer (x, n), n);
    (%o2)                   (x)  psi (x + n)
                               n    0

You need to be careful with the expression in (%o1); the difference of the `psi` functions has polynomials when `x = -1, -2, .., -n`. These polynomials cancel with factors in `pochhammer (x, n)` making the derivative a degree `n - 1` polynomial when n is a positive integer.

The Pochhammer symbol is defined for negative orders through its representation as a quotient of gamma functions. Consider

    (%i1) q : makegamma (pochhammer (x, n));
                              gamma(x + n)
    (%o1)                     ------------
                                gamma(x)
    (%i2) sublis ([x=11/3, n= -6], q);
                                   729
    (%o2)                        - ----
                                   2240

Alternatively, we can get this result directly.

    (%i1) pochhammer (11/3, -6);
                                   729
    (%o1)                        - ----
                                   2240

The unit step function is left-continuous; thus

    (%i1) [unit_step (-1/10), unit_step (0), unit_step (1/10)];
    (%o1)                       [0, 0, 1]

If you need a unit step function that is neither left or right continuous at zero, define your own using `signum`; for example,

    (%i1) xunit_step (x) := (1 + signum (x))/2$

    (%i2) [xunit_step (-1/10), xunit_step (0), xunit_step (1/10)];
                                    1
    (%o2)                       [0, -, 1]
                                    2

Do not redefine `unit_step` itself; some code in `orthopoly` requires that the unit step function be left-continuous.

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### 76.1.6 Algorithms

Generally, `orthopoly` does symbolic evaluation by using a hypergeometic representation of the orthogonal polynomials. The hypergeometic functions are evaluated using the (undocumented) functions `hypergeo11` and `hypergeo21`. The exceptions are the half-integer Bessel functions and the associated Legendre function of the second kind. The half-integer Bessel functions are evaluated using an explicit representation, and the associated Legendre function of the second kind is evaluated using recursion.

For floating point evaluation, we again convert most functions into a hypergeometic form; we evaluate the hypergeometic functions using forward recursion. Again, the exceptions are the half-integer Bessel functions and the associated Legendre function of the second kind. Numerically, the half-integer Bessel functions are evaluated using the SLATEC code.

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## 76.2 Functions and Variables for orthogonal polynomials

Function: assoc_legendre_p (n, m, x)

  The associated Legendre function of the first kind of degree n and order m.

  Reference: Abramowitz and Stegun, equations 22.5.37, page 779, 8.6.6 (second equation), page 334, and 8.2.5, page 333.

  Categories: Package orthopoly

Function: assoc_legendre_q (n, m, x)

  The associated Legendre function of the second kind of degree n and order m.

  Reference: Abramowitz and Stegun, equation 8.5.3 and 8.1.8.

  Categories: Package orthopoly

Function: chebyshev_t (n, x)

  The Chebyshev polynomial of the first kind of degree n.

  Reference: Abramowitz and Stegun, equation 22.5.47, page 779.

  Categories: Package orthopoly

Function: chebyshev_u (n, x)

  The Chebyshev polynomial of the second kind of degree n.

  Reference: Abramowitz and Stegun, equation 22.5.48, page 779.

  Categories: Package orthopoly

Function: gen_laguerre (n, a, x)

  The generalized Laguerre polynomial of degree n.

  Reference: Abramowitz and Stegun, equation 22.5.54, page 780.

  Categories: Package orthopoly

Function: hermite (n, x)

  The Hermite polynomial of degree n.

  Reference: Abramowitz and Stegun, equation 22.5.55, page 780.

  Categories: Package orthopoly

Function: intervalp (e)

  Return `true` if the input is an interval and return false if it isn't.

  Categories: Package orthopoly · Predicate functions

Function: jacobi_p (n, a, b, x)

  The Jacobi polynomial.

  The Jacobi polynomials are actually defined for all a and b; however, the Jacobi polynomial weight `(1 - x)^a (1 + x)^b` isn't integrable for `a <= -1` or `b <= -1`.

  Reference: Abramowitz and Stegun, equation 22.5.42, page 779.

  Categories: Package orthopoly

Function: laguerre (n, x)

  The Laguerre polynomial of degree n.

  Reference: Abramowitz and Stegun, equations 22.5.16 and 22.5.54, page 780.

  Categories: Package orthopoly

Function: legendre_p (n, x)

  The Legendre polynomial of the first kind of degree n.

  Reference: Abramowitz and Stegun, equations 22.5.50 and 22.5.51, page 779.

  Categories: Package orthopoly

Function: legendre_q (n, x)

  The Legendre function of the second kind of degree n.

  Reference: Abramowitz and Stegun, equations 8.5.3 and 8.1.8.

  Categories: Package orthopoly

Function: orthopoly_recur (f, args)

  Returns a recursion relation for the orthogonal function family f with arguments args. The recursion is with respect to the polynomial degree.

      (%i1) orthopoly_recur (legendre_p, [n, x]);
                          (2 n + 1) P (x) x - n P     (x)
                                     n           n - 1
      (%o1)   P     (x) = -------------------------------
               n + 1                   n + 1

  The second argument to `orthopoly_recur` must be a list with the correct number of arguments for the function f; if it isn't, Maxima signals an error.

      (%i1) orthopoly_recur (jacobi_p, [n, x]);

      Function jacobi_p needs 4 arguments, instead it received 2
       -- an error.  Quitting.  To debug this try debugmode(true);

  Additionally, when f isn't the name of one of the families of orthogonal polynomials, an error is signalled.

      (%i1) orthopoly_recur (foo, [n, x]);

      A recursion relation for foo isn't known to Maxima
       -- an error.  Quitting.  To debug this try debugmode(true);

  Categories: Package orthopoly

Variable: orthopoly_returns_intervals

  Default value: `true`

  When `orthopoly_returns_intervals` is `true`, floating point results are returned in the form `interval (c, r)`, where c is the center of an interval and r is its radius. The center can be a complex number; in that case, the interval is a disk in the complex plane.

  Categories: Package orthopoly

Function: orthopoly_weight (f, args)

  Returns a three element list; the first element is the formula of the weight for the orthogonal polynomial family f with arguments given by the list args; the second and third elements give the lower and upper endpoints of the interval of orthogonality. For example,

      (%i1) w : orthopoly_weight (hermite, [n, x]);
                                  2
                               - x
      (%o1)                 [%e    , - inf, inf]
      (%i2) integrate(w[1]*hermite(3, x)*hermite(2, x), x, w[2], w[3]);
      (%o2)                           0

  The main variable of f must be a symbol; if it isn't, Maxima signals an error.

  Categories: Package orthopoly

Function: pochhammer (x, n)

  The Pochhammer symbol. For nonnegative integers n with `n <= pochhammer_max_index`, the expression `pochhammer (x, n)` evaluates to the product `x (x + 1) (x + 2) ... (x + n - 1)` when `n > 0` and to 1 when `n = 0`. For negative n, `pochhammer (x, n)` is defined as `(-1)^n / pochhammer (1 - x, -n)`. Thus

      (%i1) pochhammer (x, 3);
      (%o1)                   x (x + 1) (x + 2)
      (%i2) pochhammer (x, -3);
                                       1
      (%o2)               - -----------------------
                            (1 - x) (2 - x) (3 - x)

  To convert a Pochhammer symbol into a quotient of gamma functions, (see Abramowitz and Stegun, equation 6.1.22) use `makegamma`; for example

      (%i1) makegamma (pochhammer (x, n));
                                gamma(x + n)
      (%o1)                     ------------
                                  gamma(x)

  When n exceeds `pochhammer_max_index` or when n is symbolic, `pochhammer` returns a noun form.

      (%i1) pochhammer (x, n);
      (%o1)                         (x)
                                       n

  Categories: Package orthopoly · Gamma and factorial functions

Variable: pochhammer_max_index

  Default value: 100

  `pochhammer (n, x)` expands to a product if and only if `n <= pochhammer_max_index`.

  Examples:

      (%i1) pochhammer (x, 3), pochhammer_max_index : 3;
      (%o1)                   x (x + 1) (x + 2)
      (%i2) pochhammer (x, 4), pochhammer_max_index : 3;
      (%o2)                         (x)
                                       4

  Reference: Abramowitz and Stegun, equation 6.1.16, page 256.

  Categories: Package orthopoly · Gamma and factorial functions

Function: spherical_bessel_j (n, x)

  The spherical Bessel function of the first kind.

  Reference: Abramowitz and Stegun, equations 10.1.8, page 437 and 10.1.15, page 439.

  Categories: Package orthopoly · Bessel functions

Function: spherical_bessel_y (n, x)

  The spherical Bessel function of the second kind.

  Reference: Abramowitz and Stegun, equations 10.1.9, page 437 and 10.1.15, page 439.

  Categories: Package orthopoly · Bessel functions

Function: spherical_hankel1 (n, x)

  The spherical Hankel function of the first kind.

  Reference: Abramowitz and Stegun, equation 10.1.36, page 439.

  Categories: Package orthopoly · Bessel functions

Function: spherical_hankel2 (n, x)

  The spherical Hankel function of the second kind.

  Reference: Abramowitz and Stegun, equation 10.1.17, page 439.

  Categories: Package orthopoly · Bessel functions

Function: spherical_harmonic (n, m, x, y)

  The spherical harmonic function.

  Reference: Merzbacher 9.64.

  Categories: Package orthopoly

Function: unit_step (x)

  The left-continuous unit step function; thus `unit_step (x)` vanishes for `x <= 0` and equals 1 for `x > 0`.

  If you want a unit step function that takes on the value 1/2 at zero, use `(1 + signum (x))/2`.

  Categories: Package orthopoly · Mathematical functions

Function: ultraspherical (n, a, x)

  The ultraspherical polynomial (also known as the Gegenbauer polynomial).

  Reference: Abramowitz and Stegun, equation 22.5.46, page 779.

  Categories: Package orthopoly

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# 77. ratpow

The package `ratpow` provides functions that find the exponents of the denominator in a CRE polynomial. If the exponents in the denominator are needed instead `ratdenom` can be used to extract this denominator first. Returned coefficients are in CRE form except for numbers.

In order to get a list of vars in a CRE polynomial `showratvars` can be used.

For information about CREs see also `rat`, `ratdisrep` and `showratvars`.

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## 77.1 Functions and Variables for ratpow

Function: ratp_hipow (expr, x)

  Finds the highest power of the main variable in `ratnumer(expr)`

      (%i1) load("ratpow")$
      (%i2) ratp_hipow( x^(5/2) + x^2 , x);
      (%o2)                           2
      (%i3) ratp_hipow( x^(5/2) + x^2 , sqrt(x));
      (%o3)                           5

  Categories: Rational expressions Package ratpow

Function: ratp_lopow (expr, x)

  Finds the lowest power of the main variable in `ratnumer(expr)`

      (%i1) load("ratpow")$
      (%i2) ratp_lopow( x^5 + x^2 , x);
      (%o2)                           2

  The following example will return 0 since `1` equals `x^0`:

      (%i1) load("ratpow")$
      (%i2) ratp_lopow( x^5 + x^2 + 1, x);
      (%o2)                           0

  The CRE form of the following equation contains `sqrt(x)` and `x`. Since they are interpreted as independent variables `ratp_lopow` returns `0` in this case:

      (%i1) load("ratpow")$
      (%i2) g:sqrt(x)^5 + sqrt(x)^2;
                                   5/2
      (%o2)                       x    + x
      (%i3) showratvars(g);
                                    1/2
      (%o3)                       [x   , x]
      (%i4) ratp_lopow( g, x);
      (%o4)                           0
      (%i5) ratp_lopow( g, sqrt(x));
      (%o5)                           0

  Categories: Rational expressions Package ratpow

Function: ratp_coeffs (expr, x)

  Generates a list of powers and coefficients of the main variable `ratnumer(expr)`.

      (%i1) load("ratpow")$
      (%i2) ratp_coeffs( 4*x^3 + x + sqrt(x), x);
      (%o2)/R/         [[3, 4], [1, 1], [0, sqrt(x)]]

  Categories: Rational expressions Package ratpow

Function: ratp_dense_coeffs (expr, x)

  Generates a list of coefficients in `ratnumer(expr)`; returned coefficients are in CRE form except for numbers.

      (%i1) load("ratpow")$
      (%i2) ratp_dense_coeffs( 4*x^3 + x + sqrt(x), x);
      (%o2)/R/               [4, 0, 1, sqrt(x)]

  Categories: Rational expressions Package ratpow

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# 78. romberg

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## 78.1 Functions and Variables for romberg

Function: romberg
`romberg` (expr, x, a, b)
`romberg` (F, a, b)

  Computes a numerical integration by Romberg's method.

  `romberg(expr, x, a, b)` returns an estimate of the integral `integrate(expr, x, a, b)`. expr must be an expression which evaluates to a floating point value when x is bound to a floating point value.

  `romberg(F, a, b)` returns an estimate of the integral `integrate(F(x), x, a, b)` where `x` represents the unnamed, sole argument of F; the actual argument is not named `x`. F must be a Maxima or Lisp function which returns a floating point value when the argument is a floating point value. F may name a translated or compiled Maxima function.

  The accuracy of `romberg` is governed by the global variables `rombergabs` and `rombergtol`. `romberg` terminates successfully when the absolute difference between successive approximations is less than `rombergabs`, or the relative difference in successive approximations is less than `rombergtol`. Thus when `rombergabs` is 0.0 (the default) only the relative error test has any effect on `romberg`.

  `romberg` halves the stepsize at most `rombergit` times before it gives up; the maximum number of function evaluations is therefore `2^rombergit`. If the error criterion established by `rombergabs` and `rombergtol` is not satisfied, `romberg` prints an error message. `romberg` always makes at least `rombergmin` iterations; this is a heuristic intended to prevent spurious termination when the integrand is oscillatory.

  `romberg` repeatedly evaluates the integrand after binding the variable of integration to a specific value (and not before). This evaluation policy makes it possible to nest calls to `romberg`, to compute multidimensional integrals. However, the error calculations do not take the errors of nested integrations into account, so errors may be underestimated. Also, methods devised especially for multidimensional problems may yield the same accuracy with fewer function evaluations.

  `load(romberg)` loads this function.

  See also `QUADPACK`, a collection of numerical integration functions.

  Examples:

  A 1-dimensional integration.

      (%i1) load ("romberg");
      (%o1)    /usr/share/maxima/5.11.0/share/numeric/romberg.lisp
      (%i2) f(x) := 1/((x - 1)^2 + 1/100) + 1/((x - 2)^2 + 1/1000)
                    + 1/((x - 3)^2 + 1/200);
                          1                 1                1
      (%o2) f(x) := -------------- + --------------- + --------------
                           2    1           2    1            2    1
                    (x - 1)  + ---   (x - 2)  + ----   (x - 3)  + ---
                               100              1000              200
      (%i3) rombergtol : 1e-6;
      (%o3)                 9.9999999999999995E-7
      (%i4) rombergit : 15;
      (%o4)                          15
      (%i5) estimate : romberg (f(x), x, -5, 5);
      (%o5)                   173.6730736617464
      (%i6) exact : integrate (f(x), x, -5, 5);
      (%o6) 10 sqrt(10) atan(70 sqrt(10))
       + 10 sqrt(10) atan(30 sqrt(10)) + 10 sqrt(2) atan(80 sqrt(2))
       + 10 sqrt(2) atan(20 sqrt(2)) + 10 atan(60) + 10 atan(40)
      (%i7) abs (estimate - exact) / exact, numer;
      (%o7)                7.5527060865060088E-11

  A 2-dimensional integration, implemented by nested calls to `romberg`.

      (%i1) load ("romberg");
      (%o1)    /usr/share/maxima/5.11.0/share/numeric/romberg.lisp
      (%i2) g(x, y) := x*y / (x + y);
                                          x y
      (%o2)                   g(x, y) := -----
                                         x + y
      (%i3) rombergtol : 1e-6;
      (%o3)                 9.9999999999999995E-7
      (%i4) estimate : romberg (romberg (g(x, y), y, 0, x/2), x, 1, 3);
      (%o4)                   0.81930239628356
      (%i5) assume (x > 0);
      (%o5)                        [x > 0]
      (%i6) integrate (integrate (g(x, y), y, 0, x/2), x, 1, 3);
                                                3
                                          2 log(-) - 1
                          9                     2        9
      (%o6)       - 9 log(-) + 9 log(3) + ------------ + -
                          2                    6         2
      (%i7) exact : radcan (%);
                          26 log(3) - 26 log(2) - 13
      (%o7)             - --------------------------
                                      3
      (%i8) abs (estimate - exact) / exact, numer;
      (%o8)                1.3711979871851024E-10

  Categories: Package romberg · Numerical methods

Option variable: rombergabs

  Default value: 0.0

  The accuracy of `romberg` is governed by the global variables `rombergabs` and `rombergtol`. `romberg` terminates successfully when the absolute difference between successive approximations is less than `rombergabs`, or the relative difference in successive approximations is less than `rombergtol`. Thus when `rombergabs` is 0.0 (the default) only the relative error test has any effect on `romberg`.

  See also `rombergit` and `rombergmin`.

  Categories: Package romberg

Option variable: rombergit

  Default value: 11

  `romberg` halves the stepsize at most `rombergit` times before it gives up; the maximum number of function evaluations is therefore `2^rombergit`. `romberg` always makes at least `rombergmin` iterations; this is a heuristic intended to prevent spurious termination when the integrand is oscillatory.

  See also `rombergabs` and `rombergtol`.

  Categories: Package romberg

Option variable: rombergmin

  Default value: 0

  `romberg` always makes at least `rombergmin` iterations; this is a heuristic intended to prevent spurious termination when the integrand is oscillatory.

  See also `rombergit`, `rombergabs`, and `rombergtol`.

  Categories: Package romberg

Option variable: rombergtol

  Default value: 1e-4

  The accuracy of `romberg` is governed by the global variables `rombergabs` and `rombergtol`. `romberg` terminates successfully when the absolute difference between successive approximations is less than `rombergabs`, or the relative difference in successive approximations is less than `rombergtol`. Thus when `rombergabs` is 0.0 (the default) only the relative error test has any effect on `romberg`.

  See also `rombergit` and `rombergmin`.

  Categories: Package romberg

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# 79. simplex

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## 79.1 Introduction to simplex

`simplex` is a package for linear optimization using the simplex algorithm.

Example:

    (%i1) load("simplex")$
    (%i2) minimize_lp(x+y, [3*x+2*y>2, x+4*y>3]);
                      9        7       1
    (%o2)            [--, [y = --, x = -]]
                      10       10      5

Categories: Numerical methods · Optimization · Share packages · Package simplex

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### 79.1.1 Tests for simplex

There are some tests in the directory `share/simplex/Tests`.

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#### 79.1.1.1 klee_minty

The function `klee_minty` produces input for `linear_program`, for which exponential time for solving is required without scaling.

Example:

    load(klee_minty)$
    apply(linear_program, klee_minty(6));

A better approach:

    epsilon_sx : 0$
    scale_sx : true$
    apply(linear_program, klee_minty(10));

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#### 79.1.1.2 NETLIB

Some smaller problems from netlib (http://www.netlib.org/lp/data/) test suite are converted to a format, readable by Maxima. Problems are `adlittle`, `afiro`, `kb2` and `sc50a`. Each problem has three input files in CSV format for matrix A and vectors b and c.

Example:

    A : read_matrix("adlittle_A.csv", 'csv)$
    b : read_list("adlittle_b.csv", 'csv)$
    c : read_list("adlittle_c.csv", 'csv)$
    linear_program(A, b, c)$
    %[2]
    => 225494.963126615

Results:

    PROBLEM        MINIMUM                SCALING
    adlittle       225494.963126615       no
    afiro          - 464.7531428571429    no
    kb2            - 1749.900129055996    yes
    sc50a          - 64.5750770585645     no

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## 79.2 Functions and Variables for simplex

Option variable: epsilon_lp

  Default value: `10^-8`

  Epsilon used for numerical computations in `linear_program`.

  See also: `linear_program`.

  Categories: Package simplex

Function: linear_program (A, b, c)

  `linear_program` is an implementation of the simplex algorithm. `linear_program(A, b, c)` computes a vector x for which `c.x` is minimum possible among vectors for which `A.x = b` and `x >= 0`. Argument A is a matrix and arguments b and c are lists.

  `linear_program` returns a list which contains the minimizing vector x and the minimum value `c.x`. If the problem is not bounded, it returns "Problem not bounded!" and if the problem is not feasible, it returns "Problem not feasible!".

  To use this function first load the `simplex` package with `load(simplex);`.

  Example:

      (%i2) A: matrix([1,1,-1,0], [2,-3,0,-1], [4,-5,0,0])$
      (%i3) b: [1,1,6]$
      (%i4) c: [1,-2,0,0]$
      (%i5) linear_program(A, b, c);
                         13     19        3
      (%o5)            [[--, 4, --, 0], - -]
                         2      2         2

  See also: `minimize_lp`, `scale_lp`, and `epsilon_lp`.

  Categories: Package simplex · Numerical methods

Function: maximize_lp (obj, cond, [pos])

  Maximizes linear objective function obj subject to some linear constraints cond. See `minimize_lp` for detailed description of arguments and return value.

  See also: `minimize_lp`.

  Categories: Package simplex · Numerical methods

Function: minimize_lp (obj, cond, [pos])

  Minimizes a linear objective function obj subject to some linear constraints cond. cond a list of linear equations or inequalities. In strict inequalities `>` is replaced by `>=` and `<` by `<=`. The optional argument pos is a list of decision variables which are assumed to be positive.

  If the minimum exists, `minimize_lp` returns a list which contains the minimum value of the objective function and a list of decision variable values for which the minimum is attained. If the problem is not bounded, `minimize_lp` returns "Problem not bounded!" and if the problem is not feasible, it returns "Problem not feasible!".

  The decision variables are not assumed to be nonegative by default. If all decision variables are nonegative, set `nonegative_lp` to `true`. If only some of decision variables are positive, list them in the optional argument pos (note that this is more efficient than adding constraints).

  `minimize_lp` uses the simplex algorithm which is implemented in maxima `linear_program` function.

  To use this function first load the `simplex` package with `load(simplex);`.

  Examples:

      (%i1) minimize_lp(x+y, [3*x+y=0, x+2*y>2]);
                            4       6        2
      (%o1)                [-, [y = -, x = - -]]
                            5       5        5
      (%i2) minimize_lp(x+y, [3*x+y>0, x+2*y>2]), nonegative_lp=true;
      (%o2)                [1, [y = 1, x = 0]]
      (%i3) minimize_lp(x+y, [3*x+y=0, x+2*y>2]), nonegative_lp=true;
      (%o3)                Problem not feasible!
      (%i4) minimize_lp(x+y, [3*x+y>0]);
      (%o4)                Problem not bounded!

  See also: `maximize_lp`, `nonegative_lp`, `epsilon_lp`.

  Categories: Package simplex · Numerical methods

Option variable: nonegative_lp

  Default value: `false`

  If `nonegative_lp` is true all decision variables to `minimize_lp` and `maximize_lp` are assumed to be positive.

  See also: `minimize_lp`.

  Categories: Package simplex

Option variable: scale_lp

  Default value: `false`

  When `scale_lp` is `true`, `linear_program` scales its input so that the maximum absolute value in each row or column is 1.

  Categories: Package simplex

Variable: pivot_count_sx

  After `linear_program` returns, `pivot_count_sx` is the number of pivots in last computation.

  Categories: Package simplex

Variable: pivot_max_sx

  `pivot_max_sx` is the maximum number of pivots allowed by `linear_program`.

  Categories: Package simplex

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# 80. simplification

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## 80.1 Introduction to simplification

The directory `maxima/share/simplification` contains several scripts which implement simplification rules and functions, and also some functions not related to simplification.

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## 80.2 Package absimp

The `absimp` package contains pattern-matching rules that extend the built-in simplification rules for the `abs` and `signum` functions. `absimp` respects relations established with the built-in `assume` function and by declarations such as `modedeclare (m, even, n, odd)` for even or odd integers.

`absimp` defines `unitramp` and `unitstep` functions in terms of `abs` and `signum`.

`load ("absimp")` loads this package. `demo (absimp)` shows a demonstration of this package.

Examples:

    (%i1) load ("absimp")$
    (%i2) (abs (x))^2;
                                    2
    (%o2)                          x
    (%i3) diff (abs (x), x);
                                   x
    (%o3)                        ------
                                 abs(x)
    (%i4) cosh (abs (x));
    (%o4)                        cosh(x)

Categories: Simplification functions · Rules and patterns · Share packages · Package absimp

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## 80.3 Package facexp

The `facexp` package contains several related functions that provide the user with the ability to structure expressions by controlled expansion. This capability is especially useful when the expression contains variables that have physical meaning, because it is often true that the most economical form of such an expression can be obtained by fully expanding the expression with respect to those variables, and then factoring their coefficients. While it is true that this procedure is not difficult to carry out using standard Maxima functions, additional fine-tuning may also be desirable, and these finishing touches can be more difficult to apply.

The function `facsum` and its related forms provide a convenient means for controlling the structure of expressions in this way. Another function, `collectterms`, can be used to add two or more expressions that have already been simplified to this form, without resimplifying the whole expression again. This function may be useful when the expressions are very large.

`load ("facexp")` loads this package. `demo (facexp)` shows a demonstration of this package.

Categories: Expressions · Share packages · Package facexp

Function: facsum (expr, arg_1, ..., arg_n)

  Returns a form of expr which depends on the arguments arg_1, ..., arg_n. The arguments can be any form suitable for `ratvars`, or they can be lists of such forms. If the arguments are not lists, then the form returned is fully expanded with respect to the arguments, and the coefficients of the arguments are factored. These coefficients are free of the arguments, except perhaps in a non-rational sense.

  If any of the arguments are lists, then all such lists are combined into a single list, and instead of calling `factor` on the coefficients of the arguments, `facsum` calls itself on these coefficients, using this newly constructed single list as the new argument list for this recursive call. This process can be repeated to arbitrary depth by nesting the desired elements in lists.

  It is possible that one may wish to `facsum` with respect to more complicated subexpressions, such as `log (x + y)`. Such arguments are also permissible.

  Occasionally the user may wish to obtain any of the above forms for expressions which are specified only by their leading operators. For example, one may wish to `facsum` with respect to all `log`'s. In this situation, one may include among the arguments either the specific `log`'s which are to be treated in this way, or alternatively, either the expression `operator (log)` or `'operator (log)`. If one wished to `facsum` the expression expr with respect to the operators op_1, ..., op_n, one would evaluate `facsum (expr, operator (op_1, ..., op_n))`. The `operator` form may also appear inside list arguments.

  In addition, the setting of the switches `facsum_combine` and `nextlayerfactor` may affect the result of `facsum`.

  Categories: Package facexp · Expressions

Global variable: nextlayerfactor

  Default value: `false`

  When `nextlayerfactor` is `true`, recursive calls of `facsum` are applied to the factors of the factored form of the coefficients of the arguments.

  When `false`, `facsum` is applied to each coefficient as a whole whenever recusive calls to `facsum` occur.

  Inclusion of the atom `nextlayerfactor` in the argument list of `facsum` has the effect of `nextlayerfactor: true`, but for the next level of the expression only. Since `nextlayerfactor` is always bound to either `true` or `false`, it must be presented single-quoted whenever it appears in the argument list of `facsum`.

  Categories: Package facexp · Expressions

Global variable: facsum_combine

  Default value: `true`

  `facsum_combine` controls the form of the final result returned by `facsum` when its argument is a quotient of polynomials. If `facsum_combine` is `false` then the form will be returned as a fully expanded sum as described above, but if `true`, then the expression returned is a ratio of polynomials, with each polynomial in the form described above.

  The `true` setting of this switch is useful when one wants to `facsum` both the numerator and denominator of a rational expression, but does not want the denominator to be multiplied through the terms of the numerator.

  Categories: Package facexp · Expressions

Function: factorfacsum (expr, arg_1, ... arg_n)

  Returns a form of expr which is obtained by calling `facsum` on the factors of expr with arg_1, ... arg_n as arguments. If any of the factors of expr is raised to a power, both the factor and the exponent will be processed in this way.

  Categories: Package facexp · Expressions

Function: collectterms (expr, arg_1, …, arg_n)

  Collects all terms that contain arg_1 ... arg_n. If several expressions have been simplified with the following functions `facsum`, `factorfacsum`, `factenexpand`, `facexpten` or `factorfacexpten`, and they are to be added together, it may be desirable to combine them using the function `collecterms`. `collecterms` can take as arguments all of the arguments that can be given to these other associated functions with the exception of `nextlayerfactor`, which has no effect on `collectterms`. The advantage of `collectterms` is that it returns a form similar to `facsum`, but since it is adding forms that have already been processed by `facsum`, it does not need to repeat that effort. This capability is especially useful when the expressions to be summed are very large.

  See also `factor`.

  Example:

      (%i1) (exp(x)+2)*x+exp(x);
                                   x          x
      (%o1)                   x (%e  + 2) + %e
      (%i2) collectterms(expand(%),exp(x));
                                        x
      (%o2)                   (x + 1) %e  + 2 x

  Categories: Package facexp · Expressions

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## 80.4 Package functs

Categories: Share packages · Package functs

Function: rempart (expr, n)

  Removes part n from the expression expr.

  If n is a list of the form `[l, m]` then parts l thru m are removed.

  To use this function write first `load(functs)`.

  Categories: Package functs · Expressions

Function: wronskian ([f_1, ..., f_n], x)

  Returns the Wronskian matrix of the list of expressions [f_1, ..., f_n] in the variable x. The determinant of the Wronskian matrix is the Wronskian determinant of the list of expressions.

  To use `wronskian`, first `load(functs)`. Example:

      (%i1) load ("functs")$
      (%i2) wronskian([f(x), g(x)],x);
                          [   f(x)       g(x)    ]
                          [                      ]
      (%o2)               [ d          d         ]
                          [ -- (f(x))  -- (g(x)) ]
                          [ dx         dx        ]

  Categories: Package functs · Differential calculus

Function: tracematrix (M)

  Returns the trace (sum of the diagonal elements) of matrix M.

  To use this function write first `load(functs)`.

  Categories: Package functs · Matrices

Function: rational (z)

  Multiplies numerator and denominator of z by the complex conjugate of denominator, thus rationalizing the denominator. Returns canonical rational expression (CRE) form if given one, else returns general form.

  To use this function write first `load(functs)`.

  Categories: Package functs · Expressions

Function: nonzeroandfreeof (x, expr)

  Returns `true` if expr is nonzero and `freeof (x, expr)` returns `true`. Returns `false` otherwise.

  To use this function write first `load(functs)`.

  Categories: Package functs · Expressions

Function: linear (expr, x)

  When expr is an expression of the form `a*x + b` where a is nonzero, and a and b are free of x, `linear` returns a list of three equations, one for each of the three formal variables b, a, and x. Otherwise, `linear` returns `false`.

  `load(antid)` loads this function.

  Example:

      (%i1) load ("antid");
      (%o1)  /maxima/share/integration/antid.mac
      (%i2) linear ((1 - w)*(1 - x)*z, z);
      (%o2) [bargumentb = 0, aargumenta = (w - 1) x - w + 1,
                                                        xargumentx = z]
      (%i3) linear (cos(u - v) + cos(u + v), u);
      (%o3)                         false

  Categories: Package antid · Expressions

Function: gcdivide (p, q)

  When the option variable `takegcd` is `true` which is the default, `gcdivide` divides the polynomials p and q by their greatest common divisor and returns the ratio of the results. `gcdivde` calls the function `ezgcd` to divide the polynomials by the greatest common divisor.

  When `takegcd` is `false`, `gcdivide` returns the ratio `p/q`.

  To use this function write first `load(functs)`.

  See also `ezgcd`, `gcd`, `gcdex`, and `poly_gcd`.

  Example:

      (%i1) load(functs)$

      (%i2) p1:6*x^3+19*x^2+19*x+6;
                              3       2
      (%o2)                6 x  + 19 x  + 19 x + 6
      (%i3) p2:6*x^5+13*x^4+12*x^3+13*x^2+6*x;
                        5       4       3       2
      (%o3)          6 x  + 13 x  + 12 x  + 13 x  + 6 x
      (%i4) gcdivide(p1, p2);
                                   x + 1
      (%o4)                        ------
                                    3
                                   x  + x
      (%i5) takegcd:false;
      (%o5)                         false
      (%i6) gcdivide(p1, p2);
                             3       2
                          6 x  + 19 x  + 19 x + 6
      (%o6)          ----------------------------------
                        5       4       3       2
                     6 x  + 13 x  + 12 x  + 13 x  + 6 x
      (%i7) ratsimp(%);
                                   x + 1
      (%o7)                        ------
                                    3
                                   x  + x

  Categories: Package functs · Polynomials

Function: arithmetic (a, d, n)

  Returns the n-th term of the arithmetic series `a, a + d, a + 2*d, ..., a + (n - 1)*d`.

  To use this function write first `load(functs)`.

  Categories: Package functs · Sums and products

Function: geometric (a, r, n)

  Returns the n-th term of the geometric series `a, a*r, a*r^2, ..., a*r^(n - 1)`.

  To use this function write first `load(functs)`.

  Categories: Package functs · Sums and products

Function: harmonic (a, b, c, n)

  Returns the n-th term of the harmonic series `a/b, a/(b + c), a/(b + 2*c), ..., a/(b + (n - 1)*c)`.

  To use this function write first `load(functs)`.

  Categories: Package functs · Sums and products

Function: arithsum (a, d, n)

  Returns the sum of the arithmetic series from 1 to n.

  To use this function write first `load(functs)`.

  Categories: Package functs · Sums and products

Function: geosum (a, r, n)

  Returns the sum of the geometric series from 1 to n. If n is infinity (`inf`) then a sum is finite only if the absolute value of r is less than 1.

  To use this function write first `load(functs)`.

  Categories: Package functs · Sums and products

Function: gaussprob (x)

  Returns the Gaussian probability function `%e^(-x^2/2) / sqrt(2*%pi)`.

  To use this function write first `load(functs)`.

  Categories: Package functs · Mathematical functions

Function: gd (x)

  Returns the Gudermannian function `2*atan(%e^x)-%pi/2`.

  To use this function write first `load(functs)`.

  Categories: Package functs · Mathematical functions

Function: agd (x)

  Returns the inverse Gudermannian function `log (tan (%pi/4 + x/2))`.

  To use this function write first `load(functs)`.

  Categories: Package functs · Mathematical functions

Function: vers (x)

  Returns the versed sine `1 - cos (x)`.

  To use this function write first `load(functs)`.

  Categories: Package functs · Mathematical functions

Function: covers (x)

  Returns the coversed sine `1 - sin (x)`.

  To use this function write first `load(functs)`.

  Categories: Package functs · Mathematical functions

Function: exsec (x)

  Returns the exsecant `sec (x) - 1`.

  To use this function write first `load(functs)`.

  Categories: Package functs · Mathematical functions

Function: hav (x)

  Returns the haversine `(1 - cos(x))/2`.

  To use this function write first `load(functs)`.

  Categories: Package functs · Mathematical functions

Function: combination (n, r)

  Returns the number of combinations of n objects taken r at a time.

  To use this function write first `load(functs)`.

  Categories: Package functs · Mathematical functions

Function: permutation (n, r)

  Returns the number of permutations of r objects selected from a set of n objects.

  To use this function write first `load(functs)`.

  Categories: Package functs · Mathematical functions

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## 80.5 Package ineq

The `ineq` package contains simplification rules for inequalities.

Example session:

    (%i1) load(ineq)$
    tellsimp: warning: rule will treat '+
                              ' as noncommutative and nonassociative.
    tellsimp: warning: rule will treat '+
                              ' as noncommutative and nonassociative.
    tellsimp: warning: rule will treat '+
                              ' as noncommutative and nonassociative.
    tellsimp: warning: rule will treat '+
                              ' as noncommutative and nonassociative.
    tellsimp: warning: rule will treat '+
                              ' as noncommutative and nonassociative.
    tellsimp: warning: rule will treat '+
                              ' as noncommutative and nonassociative.
    tellsimp: warning: rule will treat '+
                              ' as noncommutative and nonassociative.
    tellsimp: warning: rule will treat '+
                              ' as noncommutative and nonassociative.
    (%i2) a>=4;  /* a sample inequality */
    (%o2)                        a >= 4
    (%o3)                     b + a > c + 4
    (%o4)                       7 x < 7 y
    (%o5)                    - 2 x <= - 6 z
                                     2
    (%o6)                      1 <= a  + 1
    (%o8)                       2 x < 3 x
    (%o9)                        a >= b
    (%o10)                   a + 3 >= b + 3
    (%o11)                       a >= b
    (%o12)                     a >= c - b
    (%o13)                     b + a >= c
    (%o14)                 (- c) + b + a >= 0
    (%o15)                   c - b - a <= 0
                                   2
    (%o16)                  (z - 1)  > - 2 z
                                2
    (%o17)                     z  + 1 > 0
    (%o18)                        true
    (%i19) (b>c)+%; /* add a second, strict inequality */

Be careful about using parentheses around the inequalities: when the user types in `(A > B) + (C = 5)` the result is `A + C > B + 5`, but `A > B + C = 5` is a syntax error, and `(A > B + C) = 5` is something else entirely.

Do `disprule (all)` to see a complete listing of the rule definitions.

The user will be queried if Maxima is unable to decide the sign of a quantity multiplying an inequality.

The most common mis-feature is illustrated by:

    (%i1) eq: a > b;
    (%o1)                         a > b
    (%i2) 2*eq;
    (%o2)                       2 (a > b)
    (%i3) % - eq;
    (%o3)                         a > b

Another problem is 0 times an inequality; the default to have this turn into 0 has been left alone. However, if you type `X*some_inequality` and Maxima asks about the sign of `X` and you respond `zero` (or `z`), the program returns `X*some_inequality` and not use the information that `X` is 0. You should do `ev (%, x: 0)` in such a case, as the database will only be used for comparison purposes in decisions, and not for the purpose of evaluating `X`.

The user may note a slower response when this package is loaded, as the simplifier is forced to examine more rules than without the package, so you might wish to remove the rules after making use of them. Do `kill (rules)` to eliminate all of the rules (including any that you might have defined); or you may be more selective by killing only some of them; or use `remrule` on a specific rule.

Note that if you load this package after defining your own rules you will clobber your rules that have the same name. The rules in this package are: `*rule1`, ..., `*rule8`, `+rule1`, ..., `+rule18`, and you must enclose the rulename in quotes to refer to it, as in `remrule ("+", "+rule1")` to specifically remove the first rule on `"+"` or `disprule ("*rule2")` to display the definition of the second multiplicative rule.

Categories: Simplification functions · Rules and patterns · Share packages · Package ineq

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## 80.6 Package rducon

Categories: Expressions · Share packages · Package rducon

Function: reduce_consts (expr)

  Replaces constant subexpressions of expr with constructed constant atoms, saving the definition of all these constructed constants in the list of equations `const_eqns`, and returning the modified expr. Those parts of expr are constant which return `true` when operated on by the function `constantp`. Hence, before invoking `reduce_consts`, one should do

      declare ([objects to be given the constant property], constant)$

  to set up a database of the constant quantities occurring in your expressions.

  If you are planning to generate Fortran output after these symbolic calculations, one of the first code sections should be the calculation of all constants. To generate this code segment, do

      map ('fortran, const_eqns)$

  Variables besides `const_eqns` which affect `reduce_consts` are:

  `const_prefix` (default value: `xx`) is the string of characters used to prefix all symbols generated by `reduce_consts` to represent constant subexpressions.

  `const_counter` (default value: 1) is the integer index used to generate unique symbols to represent each constant subexpression found by `reduce_consts`.

  `load ("rducon")` loads this function. `demo (rducon)` shows a demonstration of this function.

  Categories: Package rducon · Expressions

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## 80.7 Package scifac

Categories: Expressions · Share packages · Package scifac

Function: gcfac (expr)

  `gcfac` is a factoring function that attempts to apply the same heuristics which scientists apply in trying to make expressions simpler. `gcfac` is limited to monomial-type factoring. For a sum, `gcfac` does the following:

  1. Factors over the integers.
  2. Factors out the largest powers of terms occurring as coefficients, regardless of the complexity of the terms.
  3. Uses (1) and (2) in factoring adjacent pairs of terms.
  4. Repeatedly and recursively applies these techniques until the expression no longer changes.

  Item (3) does not necessarily do an optimal job of pairwise factoring because of the combinatorially-difficult nature of finding which of all possible rearrangements of the pairs yields the most compact pair-factored result.

  `load ("scifac")` loads this function. `demo (scifac)` shows a demonstration of this function.

  Categories: Package scifac · Expressions

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## 80.8 Package sqdnst

Function: sqrtdenest (expr)

  Denests `sqrt` of simple, numerical, binomial surds, where possible. E.g.

      (%i1) load ("sqdnst")$
      (%i2) sqrt(sqrt(3)/2+1)/sqrt(11*sqrt(2)-12);
                                   sqrt(3)
                              sqrt(------- + 1)
                                      2
      (%o2)                 ---------------------
                            sqrt(11 sqrt(2) - 12)
      (%i3) sqrtdenest(%);
                                 sqrt(3)   1
                                 ------- + -
                                    2      2
      (%o3)                     -------------
                                   1/4    3/4
                                3 2    - 2

  Sometimes it helps to apply `sqrtdenest` more than once, on such as `(19601-13860 sqrt(2))^(7/4)`.

  `load ("sqdnst")` loads this function.

  Categories: Expressions · Share packages · Package sqdnst

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# 81. solve_rec

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## 81.1 Introduction to solve_rec

`solve_rec` is a package for solving linear recurrences with polynomial coefficients.

A demo is available with `demo(solve_rec);`.

Example:

    (%i1) load("solve_rec")$
    (%i2) solve_rec((n+4)*s[n+2] + s[n+1] - (n+1)*s[n], s[n]);
                                        n
                     %k  (2 n + 3) (- 1)          %k
                       1                            2
    (%o2)       s  = -------------------- + ---------------
                 n     (n + 1) (n + 2)      (n + 1) (n + 2)

Categories: Linear recurrences · Share packages · Package solve_rec

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## 81.2 Functions and Variables for solve_rec

Function: reduce_order (rec, sol, var)

  Reduces the order of linear recurrence rec when a particular solution sol is known. The reduced reccurence can be used to get other solutions.

  Example:

      (%i3) rec: x[n+2] = x[n+1] + x[n]/n;
                                            x
                                             n
      (%o3)               x      = x      + --
                           n + 2    n + 1   n
      (%i4) solve_rec(rec, x[n]);
      WARNING: found some hypergeometrical solutions!
      (%o4)                    x  = %k  n
                                n     1
      (%i5) reduce_order(rec, n, x[n]);
      (%t5)                    x  = n %z
                                n       n

                                 n - 1
                                 ====
                                 \
      (%t6)                %z  =  >     %u
                             n   /        %j
                                 ====
                                 %j = 0

      (%o6)             (- n - 2) %u     - %u
                                    n + 1     n
      (%i6) solve_rec((n+2)*%u[n+1] + %u[n], %u[n]);
                                           n
                                  %k  (- 1)
                                    1
      (%o6)                 %u  = ----------
                              n    (n + 1)!

      So the general solution is

                   n - 1
                   ====        j
                   \      (- 1)
             %k  n  >    -------- + %k  n
               2   /     (j + 1)!     1
                   ====
                   j = 0

  Categories: Package solve_rec

Option variable: simplify_products

  Default value: `true`

  If `simplify_products` is `true`, `solve_rec` will try to simplify products in result.

  See also: `solve_rec`.

  Categories: Package solve_rec

Function: simplify_sum (expr)

  Tries to simplify all sums appearing in expr to a closed form.

  To use this function first load the `simplify_sum` package with `load(simplify_sum)`.

  Example:

      (%i1) load("simplify_sum")$
      (%i2) sum(binomial(n+k,k)/2^k,k,1,n)+sum(binomial(2*n,2*k),k,1,n);
              n                          n
             ====                       ====
             \     binomial(n + k, k)   \
      (%o2)   >    ------------------ +  >    binomial(2 n, 2 k)
             /              k           /
             ====          2            ====
             k = 1                      k = 1
      (%i3) simplify_sum(%);

                               2 n - 1    n
      (%o3)                   2        + 2  - 2

  Categories: Package solve_rec · Sums and products · Simplification functions

Function: solve_rec (eqn, var, [init])

  Solves for hypergeometrical solutions to linear recurrence eqn with polynomials coefficient in variable var. Optional arguments init are initial conditions.

  `solve_rec` can solve linear recurrences with constant coefficients, finds hypergeometrical solutions to homogeneous linear recurrences with polynomial coefficients, rational solutions to linear recurrences with polynomial coefficients and can solve Ricatti type recurrences.

  Note that the running time of the algorithm used to find hypergeometrical solutions is exponential in the degree of the leading and trailing coefficient.

  To use this function first load the `solve_rec` package with `load(solve_rec);`.

  Example of linear recurrence with constant coefficients:

      (%i2) solve_rec(a[n]=a[n-1]+a[n-2]+n/2^n, a[n]);
                              n          n
                 (sqrt(5) - 1)  %k  (- 1)
                                  1           n
      (%o2) a  = ------------------------- - ----
             n               n                  n
                            2                5 2
                                                      n
                                         (sqrt(5) + 1)  %k
                                                          2    2
                                       + ------------------ - ----
                                                  n              n
                                                 2            5 2

  Example of linear recurrence with polynomial coefficients:

      (%i7) 2*x*(x+1)*y[x] - (x^2+3*x-2)*y[x+1] + (x-1)*y[x+2];
                               2
      (%o7) (x - 1) y      - (x  + 3 x - 2) y      + 2 x (x + 1) y
                     x + 2                   x + 1                x
      (%i8) solve_rec(%, y[x], y[1]=1, y[3]=3);
                                    x
                                 3 2    x!
      (%o9)                 y  = ---- - --
                             x    4     2

  Example of Ricatti type recurrence:

      (%i2) x*y[x+1]*y[x] - y[x+1]/(x+2) + y[x]/(x-1) = 0;
                                  y         y
                                   x + 1     x
      (%o2)         x y  y      - ------ + ----- = 0
                       x  x + 1   x + 2    x - 1
      (%i3) solve_rec(%, y[x], y[3]=5)$
      (%i4) ratsimp(minfactorial(factcomb(%)));
                                         3
                                     30 x  - 30 x
      (%o4) y  = - -------------------------------------------------
             x        6      5       4       3       2
                   5 x  - 3 x  - 25 x  + 15 x  + 20 x  - 12 x - 1584

  See also: `solve_rec_rat`, `simplify_products` and `product_use_gamma`.

  Categories: Package solve_rec

Function: solve_rec_rat (eqn, var, [init])

  Solves for rational solutions to linear recurrences. See solve_rec for description of arguments.

  To use this function first load the `solve_rec` package with `load(solve_rec);`.

  Example:

      (%i1) (x+4)*a[x+3] + (x+3)*a[x+2] - x*a[x+1] + (x^2-1)*a[x];
      (%o1)  (x + 4) a      + (x + 3) a      - x a
                      x + 3            x + 2      x + 1
                                                         2
                                                     + (x  - 1) a
                                                                  x
      (%i2) solve_rec_rat(% = (x+2)/(x+1), a[x]);
                             1
      (%o2)      a  = ---------------
                  x   (x - 1) (x + 1)

  See also: `solve_rec`.

  Categories: Package solve_rec

Option variable: product_use_gamma

  Default value: `true`

  When simplifying products, `solve_rec` introduces gamma function into the expression if `product_use_gamma` is `true`.

  See also: `simplify_products`, `solve_rec`.

  Categories: Package solve_rec

Function: summand_to_rec
`summand_to_rec` (summand, k, n)
`summand_to_rec` (summand, [k, lo, hi], n)

  Returns the recurrence sattisfied by the sum

           hi
          ====
          \
           >     summand
          /
          ====
        k = lo

  where summand is hypergeometrical in k and n. If lo and hi are omitted, they are assumed to be `lo = -inf` and `hi = inf`.

  To use this function first load the `simplify_sum` package with `load(simplify_sum)`.

  Example:

      (%i1) load("simplify_sum")$
      (%i2) summand: binom(n,k);
      (%o2)                           binomial(n, k)
      (%i3) summand_to_rec(summand,k,n);
      (%o3)                      2 sm  - sm      = 0
                                     n     n + 1
      (%i7) summand: binom(n, k)/(k+1);
                                      binomial(n, k)
      (%o7)                           --------------
                                          k + 1
      (%i8) summand_to_rec(summand, [k, 0, n], n);
      (%o8)               2 (n + 1) sm  - (n + 2) sm      = - 1
                                      n             n + 1

  Categories: Package solve_rec

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# 82. stats

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## 82.1 Introduction to stats

Package `stats` contains a set of classical statistical inference and hypothesis testing procedures.

All these functions return an `inference_result` Maxima object which contains the necessary results for population inferences and decision making.

Global variable `stats_numer` controls whether results are given in floating point or symbolic and rational format; its default value is `true` and results are returned in floating point format.

Package `descriptive` contains some utilities to manipulate data structures (lists and matrices); for example, to extract subsamples. It also contains some examples on how to use package `numericalio` to read data from plain text files. See `descriptive` and `numericalio` for more details.

Package `stats` loads packages `descriptive`, `distrib` and `inference_result`.

For comments, bugs or suggestions, please contact the author at

'mario AT edu DOT xunta DOT es'.

Categories: Statistical inference · Share packages · Package stats

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## 82.2 Functions and Variables for inference_result

Function: inference_result (title, values, numbers)

  Constructs an `inference_result` object of the type returned by the stats functions. Argument title is a string with the name of the procedure; values is a list with elements of the form `symbol = value` and numbers is a list with positive integer numbers ranging from one to `length(values)`, indicating which values will be shown by default.

  Example:

  This is a simple example showing results concerning a rectangle. The title of this object is the string `"Rectangle"`, it stores five results, named `'base`, `'height`, `'diagonal`, `'area`, and `'perimeter`, but only the first, second, fifth, and fourth will be displayed. The `'diagonal` is stored in this object, but it is not displayed; to access its value, make use of function `take_inference`.

      (%i1) load(inference_result)$
      (%i2) b: 3$ h: 2$
      (%i3) inference_result("Rectangle",
                              ['base=b,
                               'height=h,
                               'diagonal=sqrt(b^2+h^2),
                               'area=b*h,
                               'perimeter=2*(b+h)],
                              [1,2,5,4] );
                              |   Rectangle
                              |
                              |    base = 3
                              |
      (%o3)                   |   height = 2
                              |
                              | perimeter = 10
                              |
                              |    area = 6
      (%i4) take_inference('diagonal,%);
      (%o4)                        sqrt(13)

  See also `take_inference`.

  Categories: Package stats

Function: inferencep (obj)

  Returns `true` or `false`, depending on whether obj is an `inference_result` object or not.

  Categories: Package stats

Function: items_inference (obj)

  Returns a list with the names of the items stored in obj, which must be an `inference_result` object.

  Example:

  The `inference_result` object stores two values, named `'pi` and `'e`, but only the second is displayed. The `items_inference` function returns the names of all items, no matter they are displayed or not.

      (%i1) load(inference_result)$
      (%i2) inference_result("Hi", ['pi=%pi,'e=%e],[2]);
                                  |   Hi
      (%o2)                       |
                                  | e = %e
      (%i3) items_inference(%);
      (%o3)                        [pi, e]

  Categories: Package stats

Function: take_inference
`take_inference` (n, obj)
`take_inference` (name, obj)
`take_inference` (list, obj)

  Returns the n-th value stored in obj if n is a positive integer, or the item named name if this is the name of an item. If the first argument is a list of numbers and/or symbols, function `take_inference` returns a list with the corresponding results.

  Example:

  Given an `inference_result` object, function `take_inference` is called in order to extract some information stored in it.

      (%i1) load(inference_result)$
      (%i2) b: 3$ h: 2$
      (%i3) sol: inference_result("Rectangle",
                                  ['base=b,
                                   'height=h,
                                   'diagonal=sqrt(b^2+h^2),
                                   'area=b*h,
                                   'perimeter=2*(b+h)],
                                  [1,2,5,4] );
                              |   Rectangle
                              |
                              |    base = 3
                              |
      (%o3)                   |   height = 2
                              |
                              | perimeter = 10
                              |
                              |    area = 6
      (%i4) take_inference('base,sol);
      (%o4)                           3
      (%i5) take_inference(5,sol);
      (%o5)                          10
      (%i6) take_inference([1,'diagonal],sol);
      (%o6)                     [3, sqrt(13)]
      (%i7) take_inference(items_inference(sol),sol);
      (%o7)                [3, 2, sqrt(13), 6, 10]

  See also `inference_result`, and `take_inference`.

  Categories: Package stats

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## 82.3 Functions and Variables for stats

Option variable: stats_numer

  Default value: `true`

  If `stats_numer` is `true`, inference statistical functions return their results in floating point numbers. If it is `false`, results are given in symbolic and rational format.

  Categories: Package stats · Numerical evaluation

Function: test_mean
`test_mean` (x)
`test_mean` (x, options ...)

  This is the mean t-test. Argument x is a list or a column matrix containing a one dimensional sample. It also performs an asymptotic test based on the Central Limit Theorem if option `'asymptotic` is `true`.

  Options:

  - `'mean`, default `0`, is the mean value to be checked.
  - `'alternative`, default `'twosided`, is the alternative hypothesis; valid values are: `'twosided`, `'greater` and `'less`.
  - `'dev`, default `'unknown`, this is the value of the standard deviation when it is known; valid values are: `'unknown` or a positive expression.
  - `'conflevel`, default `95/100`, confidence level for the confidence interval; it must be an expression which takes a value in (0,1).
  - `'asymptotic`, default `false`, indicates whether it performs an exact t-test or an asymptotic one based on the Central Limit Theorem; valid values are `true` and `false`.

  The output of function `test_mean` is an `inference_result` Maxima object showing the following results:

  1. `'mean_estimate`: the sample mean.
  2. `'conf_level`: confidence level selected by the user.
  3. `'conf_interval`: confidence interval for the population mean.
  4. `'method`: inference procedure.
  5. `'hypotheses`: null and alternative hypotheses to be tested.
  6. `'statistic`: value of the sample statistic used for testing the null hypothesis.
  7. `'distribution`: distribution of the sample statistic, together with its parameter(s).
  8. `'p_value`: p-value of the test.

  Examples:

  Performs an exact t-test with unknown variance. The null hypothesis is H_0: mean=50 against the one sided alternative H_1: mean<50; according to the results, the p-value is too great, there are no evidence for rejecting H_0.

      (%i1) load("stats")$
      (%i2) data: [78,64,35,45,45,75,43,74,42,42]$
      (%i3) test_mean(data,'conflevel=0.9,'alternative='less,'mean=50);
                |                 MEAN TEST
                |
                |            mean_estimate = 54.3
                |
                |              conf_level = 0.9
                |
                | conf_interval = [minf, 61.51314273502712]
                |
      (%o3)     |  method = Exact t-test. Unknown variance.
                |
                | hypotheses = H0: mean = 50 , H1: mean < 50
                |
                |       statistic = .8244705235071678
                |
                |       distribution = [student_t, 9]
                |
                |        p_value = .7845100411786889

  This time Maxima performs an asymptotic test, based on the Central Limit Theorem. The null hypothesis is H_0: equal(mean, 50) against the two sided alternative H_1: not equal(mean, 50); according to the results, the p-value is very small, H_0 should be rejected in favor of the alternative H_1. Note that, as indicated by the `Method` component, this procedure should be applied to large samples.

      (%i1) load("stats")$
      (%i2) test_mean([36,118,52,87,35,256,56,178,57,57,89,34,25,98,35,
                    98,41,45,198,54,79,63,35,45,44,75,42,75,45,45,
                    45,51,123,54,151],
                    'asymptotic=true,'mean=50);
                |                       MEAN TEST
                |
                |           mean_estimate = 74.88571428571429
                |
                |                   conf_level = 0.95
                |
                | conf_interval = [57.72848600856194, 92.04294256286663]
                |
      (%o2)     |    method = Large sample z-test. Unknown variance.
                |
                |       hypotheses = H0: mean = 50 , H1: mean # 50
                |
                |             statistic = 2.842831192874313
                |
                |             distribution = [normal, 0, 1]
                |
                |             p_value = .004471474652002261

  Categories: Package stats

Function: test_means_difference
`test_means_difference` (x1, x2)
`test_means_difference` (x1, x2, options ...)

  This is the difference of means t-test for two samples. Arguments x1 and x2 are lists or column matrices containing two independent samples. In case of different unknown variances (see options `'dev1`, `'dev2` and `'varequal` bellow), the degrees of freedom are computed by means of the Welch approximation. It also performs an asymptotic test based on the Central Limit Theorem if option `'asymptotic` is set to `true`.

  Options:

  - `'alternative`, default `'twosided`, is the alternative hypothesis; valid values are: `'twosided`, `'greater` and `'less`.
  - `'dev1`, default `'unknown`, this is the value of the standard deviation of the x1 sample when it is known; valid values are: `'unknown` or a positive expression.
  - `'dev2`, default `'unknown`, this is the value of the standard deviation of the x2 sample when it is known; valid values are: `'unknown` or a positive expression.
  - `'varequal`, default `false`, whether variances should be considered to be equal or not; this option takes effect only when `'dev1` and/or `'dev2` are `'unknown`.
  - `'conflevel`, default `95/100`, confidence level for the confidence interval; it must be an expression which takes a value in (0,1).
  - `'asymptotic`, default `false`, indicates whether it performs an exact t-test or an asymptotic one based on the Central Limit Theorem; valid values are `true` and `false`.

  The output of function `test_means_difference` is an `inference_result` Maxima object showing the following results:

  1. `'diff_estimate`: the difference of means estimate.
  2. `'conf_level`: confidence level selected by the user.
  3. `'conf_interval`: confidence interval for the difference of means.
  4. `'method`: inference procedure.
  5. `'hypotheses`: null and alternative hypotheses to be tested.
  6. `'statistic`: value of the sample statistic used for testing the null hypothesis.
  7. `'distribution`: distribution of the sample statistic, together with its parameter(s).
  8. `'p_value`: p-value of the test.

  Examples:

  The equality of means is tested with two small samples x and y, against the alternative H_1: m_1>m_2, being m_1 and m_2 the populations means; variances are unknown and supposed to be different.

      (%i1) load("stats")$
      (%i2) x: [20.4,62.5,61.3,44.2,11.1,23.7]$
      (%i3) y: [1.2,6.9,38.7,20.4,17.2]$
      (%i4) test_means_difference(x,y,'alternative='greater);
                  |              DIFFERENCE OF MEANS TEST
                  |
                  |         diff_estimate = 20.31999999999999
                  |
                  |                 conf_level = 0.95
                  |
                  |    conf_interval = [- .04597417812882298, inf]
                  |
      (%o4)       |        method = Exact t-test. Welch approx.
                  |
                  | hypotheses = H0: mean1 = mean2 , H1: mean1 > mean2
                  |
                  |           statistic = 1.838004300728477
                  |
                  |    distribution = [student_t, 8.62758740184604]
                  |
                  |            p_value = .05032746527991905

  The same test as before, but now variances are supposed to be equal.

      (%i1) load("stats")$
      (%i2) x: [20.4,62.5,61.3,44.2,11.1,23.7]$
      (%i3) y: matrix([1.2],[6.9],[38.7],[20.4],[17.2])$
      (%i4) test_means_difference(x,y,'alternative='greater,
                                                       'varequal=true);
                  |              DIFFERENCE OF MEANS TEST
                  |
                  |         diff_estimate = 20.31999999999999
                  |
                  |                 conf_level = 0.95
                  |
                  |     conf_interval = [- .7722627696897568, inf]
                  |
      (%o4)       |   method = Exact t-test. Unknown equal variances
                  |
                  | hypotheses = H0: mean1 = mean2 , H1: mean1 > mean2
                  |
                  |           statistic = 1.765996124515009
                  |
                  |           distribution = [student_t, 9]
                  |
                  |            p_value = .05560320992529344

  Categories: Package stats

Function: test_variance
`test_variance` (x)
`test_variance` (x, options, ...)

  This is the variance chi^2-test. Argument x is a list or a column matrix containing a one dimensional sample taken from a normal population.

  Options:

  - `'mean`, default `'unknown`, is the population's mean, when it is known.
  - `'alternative`, default `'twosided`, is the alternative hypothesis; valid values are: `'twosided`, `'greater` and `'less`.
  - `'variance`, default `1`, this is the variance value (positive) to be checked.
  - `'conflevel`, default `95/100`, confidence level for the confidence interval; it must be an expression which takes a value in (0,1).

  The output of function `test_variance` is an `inference_result` Maxima object showing the following results:

  1. `'var_estimate`: the sample variance.
  2. `'conf_level`: confidence level selected by the user.
  3. `'conf_interval`: confidence interval for the population variance.
  4. `'method`: inference procedure.
  5. `'hypotheses`: null and alternative hypotheses to be tested.
  6. `'statistic`: value of the sample statistic used for testing the null hypothesis.
  7. `'distribution`: distribution of the sample statistic, together with its parameter.
  8. `'p_value`: p-value of the test.

  Examples:

  It is tested whether the variance of a population with unknown mean is equal to or greater than 200.

      (%i1) load("stats")$
      (%i2) x: [203,229,215,220,223,233,208,228,209]$
      (%i3) test_variance(x,'alternative='greater,'variance=200);
                   |                  VARIANCE TEST
                   |
                   |              var_estimate = 110.75
                   |
                   |                conf_level = 0.95
                   |
                   |     conf_interval = [57.13433376937479, inf]
                   |
      (%o3)        | method = Variance Chi-square test. Unknown mean.
                   |
                   |    hypotheses = H0: var = 200 , H1: var > 200
                   |
                   |                 statistic = 4.43
                   |
                   |             distribution = [chi2, 8]
                   |
                   |           p_value = .8163948512777689

  Categories: Package stats

Function: test_variance_ratio
`test_variance_ratio` (x1, x2)
`test_variance_ratio` (x1, x2, options ...)

  This is the variance ratio F-test for two normal populations. Arguments x1 and x2 are lists or column matrices containing two independent samples.

  Options:

  - `'alternative`, default `'twosided`, is the alternative hypothesis; valid values are: `'twosided`, `'greater` and `'less`.
  - `'mean1`, default `'unknown`, when it is known, this is the mean of the population from which x1 was taken.
  - `'mean2`, default `'unknown`, when it is known, this is the mean of the population from which x2 was taken.
  - `'conflevel`, default `95/100`, confidence level for the confidence interval of the ratio; it must be an expression which takes a value in (0,1).

  The output of function `test_variance_ratio` is an `inference_result` Maxima object showing the following results:

  1. `'ratio_estimate`: the sample variance ratio.
  2. `'conf_level`: confidence level selected by the user.
  3. `'conf_interval`: confidence interval for the variance ratio.
  4. `'method`: inference procedure.
  5. `'hypotheses`: null and alternative hypotheses to be tested.
  6. `'statistic`: value of the sample statistic used for testing the null hypothesis.
  7. `'distribution`: distribution of the sample statistic, together with its parameters.
  8. `'p_value`: p-value of the test.

  Examples:

  The equality of the variances of two normal populations is checked against the alternative that the first is greater than the second.

      (%i1) load("stats")$
      (%i2) x: [20.4,62.5,61.3,44.2,11.1,23.7]$
      (%i3) y: [1.2,6.9,38.7,20.4,17.2]$
      (%i4) test_variance_ratio(x,y,'alternative='greater);
                    |              VARIANCE RATIO TEST
                    |
                    |       ratio_estimate = 2.316933391522034
                    |
                    |               conf_level = 0.95
                    |
                    |    conf_interval = [.3703504689507268, inf]
                    |
      (%o4)         | method = Variance ratio F-test. Unknown means.
                    |
                    | hypotheses = H0: var1 = var2 , H1: var1 > var2
                    |
                    |         statistic = 2.316933391522034
                    |
                    |            distribution = [f, 5, 4]
                    |
                    |          p_value = .2179269692254457

  Categories: Package stats

Function: test_proportion
`test_proportion` (x, n)
`test_proportion` (x, n, options ...)

  Inferences on a proportion. Argument x is the number of successes in n trials in a Bernoulli experiment with unknown probability.

  Options:

  - `'proportion`, default `1/2`, is the value of the proportion to be checked.
  - `'alternative`, default `'twosided`, is the alternative hypothesis; valid values are: `'twosided`, `'greater` and `'less`.
  - `'conflevel`, default `95/100`, confidence level for the confidence interval; it must be an expression which takes a value in (0,1).
  - `'asymptotic`, default `false`, indicates whether it performs an exact test based on the binomial distribution, or an asymptotic one based on the Central Limit Theorem; valid values are `true` and `false`.
  - `'correct`, default `true`, indicates whether Yates correction is applied or not.

  The output of function `test_proportion` is an `inference_result` Maxima object showing the following results:

  1. `'sample_proportion`: the sample proportion.
  2. `'conf_level`: confidence level selected by the user.
  3. `'conf_interval`: Wilson confidence interval for the proportion.
  4. `'method`: inference procedure.
  5. `'hypotheses`: null and alternative hypotheses to be tested.
  6. `'statistic`: value of the sample statistic used for testing the null hypothesis.
  7. `'distribution`: distribution of the sample statistic, together with its parameters.
  8. `'p_value`: p-value of the test.

  Examples:

  Performs an exact test. The null hypothesis is H_0: p=1/2 against the one sided alternative H_1: p<1/2.

      (%i1) load("stats")$
      (%i2) test_proportion(45, 103, alternative = less);
               |            PROPORTION TEST
               |
               | sample_proportion = .4368932038834951
               |
               |           conf_level = 0.95
               |
               | conf_interval = [0, 0.522714149150231]
               |
      (%o2)    |     method = Exact binomial test.
               |
               | hypotheses = H0: p = 0.5 , H1: p < 0.5
               |
               |             statistic = 45
               |
               |  distribution = [binomial, 103, 0.5]
               |
               |      p_value = .1184509388901454

  A two sided asymptotic test. Confidence level is 99/100.

      (%i1) load("stats")$
      (%i2) fpprintprec:7$
      (%i3) test_proportion(45, 103,
                        conflevel = 99/100, asymptotic=true);
            |                 PROPORTION TEST
            |
            |           sample_proportion = .43689
            |
            |                conf_level = 0.99
            |
            |        conf_interval = [.31422, .56749]
            |
      (%o3) | method = Asympthotic test with Yates correction.
            |
            |     hypotheses = H0: p = 0.5 , H1: p # 0.5
            |
            |               statistic = .43689
            |
            |      distribution = [normal, 0.5, .048872]
            |
            |                p_value = .19662

  Categories: Package stats

Function: test_proportions_difference
`test_proportions_difference` (x1, n1, x2, n2)
`test_proportions_difference` (x1, n1, x2, n2, options …)

  Inferences on the difference of two proportions. Argument x1 is the number of successes in n1 trials in a Bernoulli experiment in the first population, and x2 and n2 are the corresponding values in the second population. Samples are independent and the test is asymptotic.

  Options:

  - `'alternative`, default `'twosided`, is the alternative hypothesis; valid values are: `'twosided` (`p1 # p2`), `'greater` (`p1 > p2`) and `'less` (`p1 < p2`).
  - `'conflevel`, default `95/100`, confidence level for the confidence interval; it must be an expression which takes a value in (0,1).
  - `'correct`, default `true`, indicates whether Yates correction is applied or not.

  The output of function `test_proportions_difference` is an `inference_result` Maxima object showing the following results:

  1. `'proportions`: list with the two sample proportions.
  2. `'conf_level`: confidence level selected by the user.
  3. `'conf_interval`: Confidence interval for the difference of proportions `p1 - p2`.
  4. `'method`: inference procedure and warning message in case of any of the samples sizes is less than 10.
  5. `'hypotheses`: null and alternative hypotheses to be tested.
  6. `'statistic`: value of the sample statistic used for testing the null hypothesis.
  7. `'distribution`: distribution of the sample statistic, together with its parameters.
  8. `'p_value`: p-value of the test.

  Examples:

  A machine produced 10 defective articles in a batch of 250. After some maintenance work, it produces 4 defective in a batch of 150. In order to know if the machine has improved, we test the null hypothesis `H0:p1=p2`, against the alternative `H0:p1>p2`, where `p1` and `p2` are the probabilities for one produced article to be defective before and after maintenance. According to the p value, there is not enough evidence to accept the alternative.

      (%i1) load("stats")$
      (%i2) fpprintprec:7$
      (%i3) test_proportions_difference(10, 250, 4, 150,
                                      alternative = greater);
            |       DIFFERENCE OF PROPORTIONS TEST
            |
            |       proportions = [0.04, .02666667]
            |
            |              conf_level = 0.95
            |
            |      conf_interval = [- .02172761, 1]
            |
      (%o3) | method = Asymptotic test. Yates correction.
            |
            |   hypotheses = H0: p1 = p2 , H1: p1 > p2
            |
            |            statistic = .01333333
            |
            |    distribution = [normal, 0, .01898069]
            |
            |             p_value = .2411936

  Exact standard deviation of the asymptotic normal distribution when the data are unknown.

      (%i1) load("stats")$
      (%i2) stats_numer: false$
      (%i3) sol: test_proportions_difference(x1,n1,x2,n2)$
      (%i4) last(take_inference('distribution,sol));
                     1    1                  x2 + x1
                    (-- + --) (x2 + x1) (1 - -------)
                     n2   n1                 n2 + n1
      (%o4)    sqrt(---------------------------------)
                                 n2 + n1

  Categories: Package stats

Function: test_sign
`test_sign` (x)
`test_sign` (x, options …)

  This is the non parametric sign test for the median of a continuous population. Argument x is a list or a column matrix containing a one dimensional sample.

  Options:

  - `'alternative`, default `'twosided`, is the alternative hypothesis; valid values are: `'twosided`, `'greater` and `'less`.
  - `'median`, default `0`, is the median value to be checked.

  The output of function `test_sign` is an `inference_result` Maxima object showing the following results:

  1. `'med_estimate`: the sample median.
  2. `'method`: inference procedure.
  3. `'hypotheses`: null and alternative hypotheses to be tested.
  4. `'statistic`: value of the sample statistic used for testing the null hypothesis.
  5. `'distribution`: distribution of the sample statistic, together with its parameter(s).
  6. `'p_value`: p-value of the test.

  Examples:

  Checks whether the population from which the sample was taken has median 6, against the alternative H_1: median > 6.

      (%i1) load("stats")$
      (%i2) x: [2,0.1,7,1.8,4,2.3,5.6,7.4,5.1,6.1,6]$
      (%i3) test_sign(x,'median=6,'alternative='greater);
                     |                  SIGN TEST
                     |
                     |              med_estimate = 5.1
                     |
                     |      method = Non parametric sign test.
                     |
      (%o3)          | hypotheses = H0: median = 6 , H1: median > 6
                     |
                     |                statistic = 7
                     |
                     |      distribution = [binomial, 10, 0.5]
                     |
                     |         p_value = .05468749999999989

  Categories: Package stats

Function: test_signed_rank
`test_signed_rank` (x)
`test_signed_rank` (x, options …)

  This is the Wilcoxon signed rank test to make inferences about the median of a continuous population. Argument x is a list or a column matrix containing a one dimensional sample. Performs normal approximation if the sample size is greater than 20, or if there are zeroes or ties.

  See also `pdf_rank_test` and `cdf_rank_test`

  Options:

  - `'median`, default `0`, is the median value to be checked.
  - `'alternative`, default `'twosided`, is the alternative hypothesis; valid values are: `'twosided`, `'greater` and `'less`.

  The output of function `test_signed_rank` is an `inference_result` Maxima object with the following results:

  1. `'med_estimate`: the sample median.
  2. `'method`: inference procedure.
  3. `'hypotheses`: null and alternative hypotheses to be tested.
  4. `'statistic`: value of the sample statistic used for testing the null hypothesis.
  5. `'distribution`: distribution of the sample statistic, together with its parameter(s).
  6. `'p_value`: p-value of the test.

  Examples:

  Checks the null hypothesis H_0: median = 15 against the alternative H_1: median > 15. This is an exact test, since there are no ties.

      (%i1) load("stats")$
      (%i2) x: [17.1,15.9,13.7,13.4,15.5,17.6]$
      (%i3) test_signed_rank(x,median=15,alternative=greater);
                       |             SIGNED RANK TEST
                       |
                       |           med_estimate = 15.7
                       |
                       |           method = Exact test
                       |
      (%o3)            | hypotheses = H0: med = 15 , H1: med > 15
                       |
                       |              statistic = 14
                       |
                       |     distribution = [signed_rank, 6]
                       |
                       |            p_value = 0.28125

  Checks the null hypothesis H_0: equal(median, 2.5) against the alternative H_1: not equal(median, 2.5). This is an approximated test, since there are ties.

      (%i1) load("stats")$
      (%i2) y:[1.9,2.3,2.6,1.9,1.6,3.3,4.2,4,2.4,2.9,1.5,3,2.9,4.2,3.1]$
      (%i3) test_signed_rank(y,median=2.5);
                   |                 SIGNED RANK TEST
                   |
                   |                med_estimate = 2.9
                   |
                   |          method = Asymptotic test. Ties
                   |
      (%o3)        |    hypotheses = H0: med = 2.5 , H1: med # 2.5
                   |
                   |                 statistic = 76.5
                   |
                   | distribution = [normal, 60.5, 17.58195097251724]
                   |
                   |           p_value = .3628097734643669

  Categories: Package stats

Function: test_rank_sum
`test_rank_sum` (x1, x2)
`test_rank_sum` (x1, x2, option)

  This is the Wilcoxon-Mann-Whitney test for comparing the medians of two continuous populations. The first two arguments x1 and x2 are lists or column matrices with the data of two independent samples. Performs normal approximation if any of the sample sizes is greater than 10, or if there are ties.

  Option:

  - `'alternative`, default `'twosided`, is the alternative hypothesis; valid values are: `'twosided`, `'greater` and `'less`.

  The output of function `test_rank_sum` is an `inference_result` Maxima object with the following results:

  1. `'method`: inference procedure.
  2. `'hypotheses`: null and alternative hypotheses to be tested.
  3. `'statistic`: value of the sample statistic used for testing the null hypothesis.
  4. `'distribution`: distribution of the sample statistic, together with its parameters.
  5. `'p_value`: p-value of the test.

  Examples:

  Checks whether populations have similar medians. Samples sizes are small and an exact test is made.

      (%i1) load("stats")$
      (%i2) x:[12,15,17,38,42,10,23,35,28]$
      (%i3) y:[21,18,25,14,52,65,40,43]$
      (%i4) test_rank_sum(x,y);
                    |                 RANK SUM TEST
                    |
                    |              method = Exact test
                    |
                    | hypotheses = H0: med1 = med2 , H1: med1 # med2
      (%o4)         |
                    |                 statistic = 22
                    |
                    |        distribution = [rank_sum, 9, 8]
                    |
                    |          p_value = .1995886466474702

  Now, with greater samples and ties, the procedure makes normal approximation. The alternative hypothesis is H_1: median1 < median2.

      (%i1) load("stats")$
      (%i2) x: [39,42,35,13,10,23,15,20,17,27]$
      (%i3) y: [20,52,66,19,41,32,44,25,14,39,43,35,19,56,27,15]$
      (%i4) test_rank_sum(x,y,'alternative='less);
                   |                  RANK SUM TEST
                   |
                   |          method = Asymptotic test. Ties
                   |
                   |  hypotheses = H0: med1 = med2 , H1: med1 < med2
      (%o4)        |
                   |                 statistic = 48.5
                   |
                   | distribution = [normal, 79.5, 18.95419580097078]
                   |
                   |           p_value = .05096985666598441

  Categories: Package stats

Function: test_normality (x)

  Shapiro-Wilk test for normality. Argument x is a list of numbers, and sample size must be greater than 2 and less or equal than 5000, otherwise, function `test_normality` signals an error message.

  Reference:

  [1] Algorithm AS R94, Applied Statistics (1995), vol.44, no.4, 547-551

  The output of function `test_normality` is an `inference_result` Maxima object with the following results:

  1. `'statistic`: value of the W statistic.
  2. `'p_value`: p-value under normal assumption.

  Examples:

  Checks for the normality of a population, based on a sample of size 9.

      (%i1) load("stats")$
      (%i2) x:[12,15,17,38,42,10,23,35,28]$
      (%i3) test_normality(x);
                             |      SHAPIRO - WILK TEST
                             |
      (%o3)                  | statistic = .9251055695162436
                             |
                             |  p_value = .4361763918860381

  Categories: Package stats

Function: linear_regression
`linear_regression` (x)
`linear_regression` (x option)

  Multivariate linear regression, y_i = b0 + b1*x_1i + b2*x_2i + ... + bk*x_ki + u_i, where u_i are N(0,sigma) independent random variables. Argument x must be a matrix with more than one column. The last column is considered as the responses (y_i).

  Option:

  - `'conflevel`, default `95/100`, confidence level for the confidence intervals; it must be an expression which takes a value in (0,1).

  The output of function `linear_regression` is an `inference_result` Maxima object with the following results:

  1. `'b_estimation`: regression coefficients estimates.
  2. `'b_covariances`: covariance matrix of the regression coefficients estimates.
  3. `b_conf_int`: confidence intervals of the regression coefficients.
  4. `b_statistics`: statistics for testing coefficient.
  5. `b_p_values`: p-values for coefficient tests.
  6. `b_distribution`: probability distribution for coefficient tests.
  7. `v_estimation`: unbiased variance estimator.
  8. `v_conf_int`: variance confidence interval.
  9. `v_distribution`: probability distribution for variance test.
  10. `residuals`: residuals.
  11. `adc`: adjusted determination coefficient.
  12. `aic`: Akaike's information criterion.
  13. `bic`: Bayes's information criterion.

  Only items 1, 4, 5, 6, 7, 8, 9 and 11 above, in this order, are shown by default. The rest remain hidden until the user makes use of functions `items_inference` and `take_inference`.

  Example:

  Fitting a linear model to a trivariate sample. The last column is considered as the responses (y_i).

      (%i2) load("stats")$
      (%i3) X:matrix(
          [58,111,64],[84,131,78],[78,158,83],
          [81,147,88],[82,121,89],[102,165,99],
          [85,174,101],[102,169,102])$
      (%i4) fpprintprec: 4$
      (%i5) res: linear_regression(X);
                   |       LINEAR REGRESSION MODEL
                   |
                   | b_estimation = [9.054, .5203, .2397]
                   |
                   | b_statistics = [.6051, 2.246, 1.74]
                   |
                   | b_p_values = [.5715, .07466, .1423]
                   |
      (%o5)        |   b_distribution = [student_t, 5]
                   |
                   |         v_estimation = 35.27
                   |
                   |     v_conf_int = [13.74, 212.2]
                   |
                   |      v_distribution = [chi2, 5]
                   |
                   |             adc = .7922
      (%i6) items_inference(res);
      (%o6) [b_estimation, b_covariances, b_conf_int, b_statistics,
      b_p_values, b_distribution, v_estimation, v_conf_int,
      v_distribution, residuals, adc, aic, bic]
      (%i7) take_inference('b_covariances, res);
                        [  223.9    - 1.12   - .8532  ]
                        [                             ]
      (%o7)             [ - 1.12    .05367   - .02305 ]
                        [                             ]
                        [ - .8532  - .02305   .01898  ]
      (%i8) take_inference('bic, res);
      (%o8)                          30.98
      (%i9) load("draw")$
      (%i10) draw2d(
          points_joined = true,
          grid = true,
          points(take_inference('residuals, res)) )$

  Categories: Package stats · Statistical estimation

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## 82.4 Functions and Variables for special distributions

Function: pdf_signed_rank (x, n)

  Probability density function of the exact distribution of the signed rank statistic. Argument x is a real number and n a positive integer.

  See also `test_signed_rank`.

  Categories: Package stats

Function: cdf_signed_rank (x, n)

  Cumulative density function of the exact distribution of the signed rank statistic. Argument x is a real number and n a positive integer.

  See also `test_signed_rank`.

  Categories: Package stats

Function: pdf_rank_sum (x, n, m)

  Probability density function of the exact distribution of the rank sum statistic. Argument x is a real number and n and m are both positive integers.

  See also `test_rank_sum`.

  Categories: Package stats

Function: cdf_rank_sum (x, n, m)

  Cumulative density function of the exact distribution of the rank sum statistic. Argument x is a real number and n and m are both positive integers.

  See also `test_rank_sum`.

  Categories: Package stats

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# 83. stirling

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## 83.1 Functions and Variables for stirling

Function: stirling
`stirling` (z,n)
`stirling` (z,n,pred)

  Replace `gamma(x)` with the O(1/x^(2n-1)) Stirling formula. when n isn't a nonnegative integer, signal an error. With the optional third argument `pred`, the Stirling formula is applied only when `pred` is true.

  Reference: Abramowitz & Stegun, " Handbook of mathematical functions", 6.1.40.

  Examples:

      (%i1) load ("stirling")$

      (%i2) stirling(gamma(%alpha+x)/gamma(x),1);
             1/2 - x             x + %alpha - 1/2
      (%o2) x        (x + %alpha)
                                         1           1
                                  --------------- - ---- - %alpha
                                  12 (x + %alpha)   12 x
                                %e
      (%i3) taylor(%,x,inf,1);
                          %alpha       2    %alpha
                %alpha   x       %alpha  - x       %alpha
      (%o3)/T/ x       + -------------------------------- + . . .
                                       2 x
      (%i4) map('factor,%);
                                             %alpha - 1
               %alpha   (%alpha - 1) %alpha x
      (%o4)   x       + -------------------------------
                                        2

  The function `stirling` knows the difference between the variable 'gamma' and the function gamma:

      (%i5) stirling(gamma + gamma(x),0);
                                          x - 1/2   - x
      (%o5)    gamma + sqrt(2) sqrt(%pi) x        %e
      (%i6) stirling(gamma(y) + gamma(x),0);
                               y - 1/2   - y
      (%o6) sqrt(2) sqrt(%pi) y        %e
                                                    x - 1/2   - x
                               + sqrt(2) sqrt(%pi) x        %e

  To apply the Stirling formula only to terms that involve the variable `k`, use an optional third argument; for example

      (%i7) makegamma(pochhammer(a,k)/pochhammer(b,k));
      (%o7) (gamma(b)*gamma(k+a))/(gamma(a)*gamma(k+b))
      (%i8) stirling(%,1, lambda([s], not(freeof(k,s))));
      (%o8) (%e^(b-a)*gamma(b)*(k+a)^(k+a-1/2)*(k+b)^(-k-b+1/2))/gamma(a)

  The terms `gamma(a)` and `gamma(b)` are free of `k`, so the Stirling formula was not applied to these two terms.

  To use this function write first `load("stirling")`.

  Categories: Gamma and factorial functions · Share packages · Package stirling

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# 84. stringproc

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## 84.1 Introduction to String Processing

The package `stringproc` contains functions for processing strings and characters including formatting, encoding and data streams. This package is completed by some tools for cryptography, e.g. base64 and hash functions.

It can be directly loaded via `load(stringproc)` or automatically by using one of its functions.

For questions and bug reports please contact the author. The following command prints his e-mail-address.

`printf(true, "~{~a~}@gmail.com", split(sdowncase("Volker van Nek")))$`

A string is constructed by typing e.g. `"Text"`. When the option variable `stringdisp` is set to `false`, which is the default, the double quotes won't be printed. stringp is a test, if an object is a string.

    (%i1) str: "Text";
    (%o1)                         Text
    (%i2) stringp(str);
    (%o2)                         true

Characters are represented by a string of length 1. charp is the corresponding test.

    (%i1) char: "e";
    (%o1)                           e
    (%i2) charp(char);
    (%o2)                         true

In Maxima position indices in strings are like in list 1-indexed which results to the following consistency.

    (%i1) is(charat("Lisp",1) = charlist("Lisp")[1]);
    (%o1)                         true

A string may contain Maxima expressions. These can be parsed with parse_string.

    (%i1) map(parse_string, ["42" ,"sqrt(2)", "%pi"]);
    (%o1)                   [42, sqrt(2), %pi]
    (%i2) map('float, %);
    (%o2)        [42.0, 1.414213562373095, 3.141592653589793]

Strings can be processed as characters or in binary form as octets. Functions for conversions are string_to_octets and octets_to_string. Usable encodings depend on the platform, the application and the underlying Lisp. (The following shows Maxima in GNU/Linux, compiled with SBCL.)

    (%i1) obase: 16.$
    (%i2) string_to_octets("$£€", "cp1252");
    (%o2)                     [24, 0A3, 80]
    (%i3) string_to_octets("$£€", "utf-8");
    (%o3)               [24, 0C2, 0A3, 0E2, 82, 0AC]

Strings may be written to character streams or as octets to binary streams. The following example demonstrates file in and output of characters.

openw returns an output stream to a file, printf writes formatted to that file and by e.g. close all characters contained in the stream are written to the file.

    (%i1) s: openw("file.txt");
    (%o1)                #<output stream file.txt>
    (%i2) printf(s, "~%~d ~f ~a ~a ~f ~e ~a~%",
    42, 1.234, sqrt(2), %pi, 1.0e-2, 1.0e-2, 1.0b-2)$
    (%i3) close(s)$

openr then returns an input stream from the previously used file and readline returns the line read as a string. The string may be tokenized by e.g. split or tokens and finally parsed by parse_string.

    (%i4) s: openr("file.txt");
    (%o4)                 #<input stream file.txt>
    (%i5) readline(s);
    (%o5)          42 1.234 sqrt(2) %pi 0.01 1.0E-2 1.0b-2
    (%i6) map(parse_string, split(%));
    (%o6)       [42, 1.234, sqrt(2), %pi, 0.01, 0.01, 1.0b-2]
    (%i7) close(s)$

Categories: Strings · Share packages · Package stringproc

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## 84.2 Input and Output

Example: Formatted printing to a file.

    (%i1) s: openw("file.txt");
    (%o1)                      #<output stream file.txt>
    (%i2) control:
    "~2tAn atom: ~20t~a~%~2tand a list: ~20t~{~r ~}~%~2t\
    and an integer: ~20t~d~%"$
    (%i3) printf( s,control, 'true,[1,2,3],42 )$
    (%o3)                                false
    (%i4) close(s);
    (%o4)                                true
    (%i5) s: openr("file.txt");
    (%o5)                      #<input stream file.txt>
    (%i6) while stringp( tmp:readline(s) ) do print(tmp)$
      An atom:          true
      and a list:       one two three
      and an integer:   42
    (%i7) close(s)$

Function: close (stream)

  Closes stream and returns `true` if stream had been open.

  Categories: File input · File output · Package stringproc

Function: flength (stream)

  stream has to be an open stream from or to a file. `flength` then returns the number of bytes which are currently present in this file.

  Example: See writebyte .

  Categories: File input · File output · Package stringproc

Function: flush_output (stream)

  Flushes stream where stream has to be an output stream to a file.

  Example: See writebyte .

  Categories: File output · Package stringproc

Function: fposition
`fposition` (stream)
`fposition` (stream, pos)

  Returns the current position in stream, if pos is not used. If pos is used, `fposition` sets the position in stream. stream has to be a stream from or to a file and pos has to be a positive number.

  Positions in data streams are like in strings or lists 1-indexed, i.e. the first element in stream is in position 1.

  Categories: File input · File output · Package stringproc

Function: freshline
`freshline` ()
`freshline` (stream)

  Writes a new line to the standard output stream if the position is not at the beginning of a line und returns `true`. Using the optional argument stream the new line is written to that stream. There are some cases, where `freshline()` does not work as expected.

  See also newline.

  Categories: File output · Package stringproc

Function: get_output_stream_string (stream)

  Returns a string containing all the characters currently present in stream which must be an open string-output stream. The returned characters are removed from stream.

  Example: See make_string_output_stream .

  Categories: Package stringproc

Function: make_string_input_stream
`make_string_input_stream` (string)
`make_string_input_stream` (string, start)
`make_string_input_stream` (string, start, end)

  Returns an input stream which contains parts of string and an end of file. Without optional arguments the stream contains the entire string and is positioned in front of the first character. start and end define the substring contained in the stream. The first character is available at position 1.

      (%i1) istream : make_string_input_stream("text", 1, 4);
      (%o1)              #<string-input stream from "text">
      (%i2) (while (c : readchar(istream)) # false do sprint(c), newline())$
      t e x
      (%i3) close(istream)$

  Categories: Package stringproc

Function: make_string_output_stream ()

  Returns an output stream that accepts characters. Characters currently present in this stream can be retrieved by get_output_stream_string.

      (%i1) ostream : make_string_output_stream();
      (%o1)               #<string-output stream 09622ea0>
      (%i2) printf(ostream, "foo")$

      (%i3) printf(ostream, "bar")$

      (%i4) string : get_output_stream_string(ostream);
      (%o4)                            foobar
      (%i5) printf(ostream, "baz")$

      (%i6) string : get_output_stream_string(ostream);
      (%o6)                              baz
      (%i7) close(ostream)$

  Categories: Package stringproc

Function: newline
`newline` ()
`newline` (stream)

  Writes a new line to the standard output stream. Using the optional argument stream the new line is written to that stream. There are some cases, where `newline()` does not work as expected.

  See sprint for an example of using `newline()`.

  Categories: File output · Package stringproc

Function: opena (file)

  Returns a character output stream to file. If an existing file is opened, `opena` appends elements at the end of file.

  For binary output see opena_binary .

  Categories: File output · Package stringproc

Function: openr
`openr` (file)
`openr` (file, encoding)

  Returns a character input stream to file. `openr` assumes that file already exists. If reading the file results in a lisp error about its encoding passing the correct string as the argument encoding might help. The available encodings and their names depend on the lisp being used. For sbcl a list of suitable strings can be found at http://www.sbcl.org/manual/#External-Formats.

  For binary input see openr_binary . See also `close` and `openw`.

      (%i1) istream : openr("data.txt","EUC-JP");
      (%o1)     #<FD-STREAM for "file /home/gunter/data.txt" {10099A3AE3}>
      (%i2) close(istream);
      (%o2)                                true

  Categories: File input · Package stringproc

Function: openw (file)

  Returns a character output stream to file. If file does not exist, it will be created. If an existing file is opened, `openw` destructively modifies file.

  For binary output see openw_binary .

  See also `close` and `openr`.

  Categories: File output · Package stringproc

Function: printf
`printf` (dest, string)
`printf` (dest, string, expr_1, ..., expr_n)

  Produces formatted output by outputting the characters of control-string string and observing that a tilde introduces a directive. The character after the tilde, possibly preceded by prefix parameters and modifiers, specifies what kind of formatting is desired. Most directives use one or more elements of the arguments expr_1, ..., expr_n to create their output.

  If dest is a stream or `true`, then `printf` returns `false`. Otherwise, `printf` returns a string containing the output. By default the streams stdin, stdout and stderr are defined. If maxima is running as a server (which is the normal case if maxima communicating with a graphical user interface) `setup-client` will define old_stdout and old_stderr, too.

  `printf` provides the Common Lisp function `format` in Maxima. The following example illustrates the general relation between these two functions.

      (%i1) printf(true, "R~dD~d~%", 2, 2);
      R2D2
      (%o1)                                false
      (%i2) :lisp (format t "R~dD~d~%" 2 2)
      R2D2
      NIL

  The following description is limited to a rough sketch of the possibilities of `printf`. The Lisp function `format` is described in detail in many reference books. Of good help is e.g. the free available online-manual "Common Lisp the Language" by Guy L. Steele. See chapter 22.3.3 there.

         ~%       new line
         ~&       fresh line
         ~t       tab
         ~$       monetary
         ~d       decimal integer
         ~b       binary integer
         ~o       octal integer
         ~x       hexadecimal integer
         ~br      base-b integer
         ~r       spell an integer
         ~p       plural
         ~f       floating point
         ~e       scientific notation
         ~g       ~f or ~e, depending upon magnitude
         ~h       bigfloat
         ~a       uses Maxima function string
         ~s       like ~a, but output enclosed in "double quotes"
         ~~       ~
         ~<       justification, ~> terminates
         ~(       case conversion, ~) terminates
         ~[       selection, ~] terminates
         ~{       iteration, ~} terminates

  The directive ~h for bigfloat is no Lisp-standard and is therefore illustrated below.

  Note that the directive ~* is not supported.

  If dest is a stream or `true`, then `printf` returns `false`. Otherwise, `printf` returns a string containing the output.

      (%i1) printf( false, "~a ~a ~4f ~a ~@r",
                    "String",sym,bound,sqrt(12),144), bound = 1.234;
      (%o1)                 String sym 1.23 2*sqrt(3) CXLIV
      (%i2) printf( false,"~{~a ~}",["one",2,"THREE"] );
      (%o2)                          one 2 THREE
      (%i3) printf(true,"~{~{~9,1f ~}~%~}",mat ),
                mat = args(matrix([1.1,2,3.33],[4,5,6],[7,8.88,9]))$
            1.1       2.0       3.3
            4.0       5.0       6.0
            7.0       8.9       9.0
      (%i4) control: "~:(~r~) bird~p ~[is~;are~] singing."$
      (%i5) printf( false,control, n,n,if n=1 then 1 else 2 ), n=2;
      (%o5)                    Two birds are singing.

  The directive ~h has been introduced to handle bigfloats.

      ~w,d,e,x,o,p@H
       w : width
       d : decimal digits behind floating point
       e : minimal exponent digits
       x : preferred exponent
       o : overflow character
       p : padding character
       @ : display sign for positive numbers

      (%i1) fpprec : 1000$
      (%i2) printf(true, "|~h|~%", 2.b0^-64)$
      |0.0000000000000000000542101086242752217003726400434970855712890625|
      (%i3) fpprec : 26$
      (%i4) printf(true, "|~h|~%", sqrt(2))$
      |1.4142135623730950488016887|
      (%i5) fpprec : 24$
      (%i6) printf(true, "|~h|~%", sqrt(2))$
      |1.41421356237309504880169|
      (%i7) printf(true, "|~28h|~%", sqrt(2))$
      |   1.41421356237309504880169|
      (%i8) printf(true, "|~28,,,,,'*h|~%", sqrt(2))$
      |***1.41421356237309504880169|
      (%i9) printf(true, "|~,18h|~%", sqrt(2))$
      |1.414213562373095049|
      (%i10) printf(true, "|~,,,-3h|~%", sqrt(2))$
      |1414.21356237309504880169b-3|
      (%i11) printf(true, "|~,,2,-3h|~%", sqrt(2))$
      |1414.21356237309504880169b-03|
      (%i12) printf(true, "|~20h|~%", sqrt(2))$
      |1.41421356237309504880169|
      (%i13) printf(true, "|~20,,,,'+h|~%", sqrt(2))$
      |++++++++++++++++++++|

  Categories: File output · Package stringproc

Function: readbyte (stream)

  Removes and returns the first byte in stream which must be a binary input stream. If the end of file is encountered `readbyte` returns `false`.

  Example: Read the first 16 bytes from a file encrypted with AES in OpenSSL.

      (%i1) ibase: obase: 16.$

      (%i2) in: openr_binary("msg.bin");
      (%o2)                       #<input stream msg.bin>
      (%i3) (L:[],  thru 16. do push(readbyte(in), L),  L:reverse(L));
      (%o3) [53, 61, 6C, 74, 65, 64, 5F, 5F, 88, 56, 0DE, 8A, 74, 0FD, 0AD, 0F0]
      (%i4) close(in);
      (%o4)                                true
      (%i5) map(ascii, rest(L,-8));
      (%o5)                      [S, a, l, t, e, d, _, _]
      (%i6) salt: octets_to_number(rest(L,8));
      (%o6)                          8856de8a74fdadf0

  Categories: File input · Package stringproc

Function: readchar (stream)

  Removes and returns the first character in stream. If the end of file is encountered `readchar` returns `false`.

  Example: See make_string_input_stream.

  Categories: File input · Package stringproc

Function: readline (stream)

  Returns a string containing all characters starting at the current position in stream up to the end of the line or `false` if the end of the file is encountered.

  Categories: File input · Package stringproc

Function: sprint (expr_1, …, expr_n)

  Evaluates and displays its arguments one after the other `on a line' starting at the leftmost position. The expressions are printed with a space character right next to the number, and it disregards line length. `newline()` might be used for line breaking.

  Example: Sequential printing with `sprint`. Creating a new line with `newline()`.

      (%i1) for n:0 thru 19 do sprint(fib(n))$
      0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181
      (%i2) for n:0 thru 22 do (
               sprint(fib(n)),
               if mod(n,10) = 9 then newline() )$
      0 1 1 2 3 5 8 13 21 34
      55 89 144 233 377 610 987 1597 2584 4181
      6765 10946 17711

  Categories: Package stringproc

Function: writebyte (byte, stream)

  Writes byte to stream which must be a binary output stream. `writebyte` returns `byte`.

  Example: Write some bytes to a binary file output stream. In this example all bytes correspond to printable characters and are printed by `printfile`. The bytes remain in the stream until `flush_output` or `close` have been called.

      (%i1) ibase: obase: 16.$

      (%i2) bytes: map(cint, charlist("GNU/Linux"));
      (%o2)                [47, 4E, 55, 2F, 4C, 69, 6E, 75, 78]
      (%i3) out: openw_binary("test.bin");
      (%o3)                      #<output stream test.bin>
      (%i4) for i thru 3 do writebyte(bytes[i], out);
      (%o4)                                done
      (%i5) printfile("test.bin")$

      (%i6) flength(out);
      (%o6)                                  0
      (%i7) flush_output(out);
      (%o7)                                true
      (%i8) flength(out);
      (%o8)                                  3
      (%i9) printfile("test.bin")$
      GNU
      (%i0A) for b in rest(bytes,3) do writebyte(b, out);
      (%o0A)                               done
      (%i0B) close(out);
      (%o0B)                               true
      (%i0C) printfile("test.bin")$
      GNU/Linux

  Categories: File output · Package stringproc

[ < ] | [ > ] |  | [ << ] | [ Up ] | [ >> ] |  |  |  |  | [Top] | [Contents] | [Index] | [ ? ]

## 84.3 Characters

Characters are strings of length 1.

Function: adjust_external_format ()

  Prints information about the current external format of the Lisp reader and in case the external format encoding differs from the encoding of the application which runs Maxima `adjust_external_format` tries to adjust the encoding or prints some help or instruction. `adjust_external_format` returns `true` when the external format has been changed and `false` otherwise.

  Functions like cint, unicode, octets_to_string and string_to_octets need UTF-8 as the external format of the Lisp reader to work properly over the full range of Unicode characters.

  Examples (Maxima on Windows, March 2016): Using `adjust_external_format` when the default external format is not equal to the encoding provided by the application.

  1. Command line Maxima

  In case a terminal session is preferred it is recommended to use Maxima compiled with SBCL. Here Unicode support is provided by default and calls to `adjust_external_format` are unnecessary.

  If Maxima is compiled with CLISP or GCL it is recommended to change the terminal encoding from CP850 to CP1252. `adjust_external_format` prints some help.

  CCL reads UTF-8 while the terminal input is CP850 by default. CP1252 is not supported by CCL. `adjust_external_format` prints instructions for changing the terminal encoding and external format both to iso-8859-1.

  2. wxMaxima

  In wxMaxima SBCL reads CP1252 by default but the input from the application is UTF-8 encoded. Adjustment is needed.

  Calling `adjust_external_format` and restarting Maxima permanently changes the default external format to UTF-8.

      (%i1)adjust_external_format();
      The line
      (setf sb-impl::*default-external-format* :utf-8)
      has been appended to the init file
      C:/Users/Username/.sbclrc
      Please restart Maxima to set the external format to UTF-8.
      (%i1) false

  Restarting Maxima.

      (%i1) adjust_external_format();
      The external format is currently UTF-8
      and has not been changed.
      (%i1) false

  Categories: Package stringproc

Function: alphacharp (char)

  Returns `true` if char is an alphabetic character.

  To identify a non-US-ASCII character as an alphabetic character the underlying Lisp must provide full Unicode support. E.g. a German umlaut is detected as an alphabetic character with SBCL in GNU/Linux but not with GCL. (In Windows Maxima, when compiled with SBCL, must be set to UTF-8. See adjust_external_format for more.)

  Example: Examination of non-US-ASCII characters.

  The underlying Lisp (SBCL, GNU/Linux) is able to convert the typed character into a Lisp character and to examine.

      (%i1) alphacharp("ü");
      (%o1)                          true

  In GCL this is not possible. An error break occurs.

      (%i1) alphacharp("u");
      (%o1)                          true
      (%i2) alphacharp("ü");

      package stringproc: ü cannot be converted into a Lisp character.
       -- an error.

  Categories: Predicate functions · Package stringproc

Function: alphanumericp (char)

  Returns `true` if char is an alphabetic character or a digit (only corresponding US-ASCII characters are regarded as digits).

  Note: See remarks on alphacharp.

  Categories: Predicate functions · Package stringproc

Function: ascii (int)

  Returns the US-ASCII character corresponding to the integer int which has to be less than `128`.

  See unicode for converting code points larger than `127`.

  Examples:

      (%i1) for n from 0 thru 127 do (
              ch: ascii(n),
              if alphacharp(ch) then sprint(ch),
              if n = 96 then newline() )$
      A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
      a b c d e f g h i j k l m n o p q r s t u v w x y z

  Categories: Package stringproc

Function: cequal (char_1, char_2)

  Returns `true` if char_1 and char_2 are the same character.

  Categories: Predicate functions · Package stringproc

Function: cequalignore (char_1, char_2)

  Like `cequal` but ignores case which is only possible for non-US-ASCII characters when the underlying Lisp is able to recognize a character as an alphabetic character. See remarks on alphacharp.

  Categories: Predicate functions · Package stringproc

Function: cgreaterp (char_1, char_2)

  Returns `true` if the code point of char_1 is greater than the code point of char_2.

  Categories: Predicate functions · Package stringproc

Function: cgreaterpignore (char_1, char_2)

  Like `cgreaterp` but ignores case which is only possible for non-US-ASCII characters when the underlying Lisp is able to recognize a character as an alphabetic character. See remarks on alphacharp.

  Categories: Predicate functions · Package stringproc

Function: charp (obj)

  Returns `true` if obj is a Maxima-character. See introduction for example.

  Categories: Predicate functions · Package stringproc

Function: cint (char)

  Returns the Unicode code point of char which must be a Maxima character, i.e. a string of length `1`.

  Examples: The hexadecimal code point of some characters (Maxima with SBCL on GNU/Linux).

      (%i1) obase: 16.$
      (%i2) map(cint, ["$","£","€"]);
      (%o2)                           [24, 0A3, 20AC]

  Warning: It is not possible to enter characters corresponding to code points larger than 16 bit in wxMaxima with SBCL on Windows when the external format has not been set to UTF-8. See adjust_external_format.

  CMUCL doesn't process these characters as one character. `cint` then returns `false`. Converting a character to a code point via UTF-8-octets may serve as a workaround:

  `utf8_to_unicode(string_to_octets(character));`

  See utf8_to_unicode, string_to_octets.

  Categories: Package stringproc

Function: clessp (char_1, char_2)

  Returns `true` if the code point of char_1 is less than the code point of char_2.

  Categories: Predicate functions · Package stringproc

Function: clesspignore (char_1, char_2)

  Like `clessp` but ignores case which is only possible for non-US-ASCII characters when the underlying Lisp is able to recognize a character as an alphabetic character. See remarks on alphacharp.

  Categories: Predicate functions · Package stringproc

Function: constituent (char)

  Returns `true` if char is a graphic character but not a space character. A graphic character is a character one can see, plus the space character. (`constituent` is defined by Paul Graham. See Paul Graham, ANSI Common Lisp, 1996, page 67.)

      (%i1) for n from 0 thru 255 do (
      tmp: ascii(n), if constituent(tmp) then sprint(tmp) )$
      ! " #  %  ' ( ) * + , - . / 0 1 2 3 4 5 6 7 8 9 : ; < = > ? @ A B
      C D E F G H I J K L M N O P Q R S T U V W X Y Z [ \ ] ^ _ ` a b c
      d e f g h i j k l m n o p q r s t u v w x y z { | } ~

  Categories: Predicate functions · Package stringproc

Function: digitcharp (char)

  Returns `true` if char is a digit where only the corresponding US-ASCII-character is regarded as a digit.

  Categories: Predicate functions · Package stringproc

Function: lowercasep (char)

  Returns `true` if char is a lowercase character.

  Note: See remarks on alphacharp.

  Categories: Predicate functions · Package stringproc

Variable: newline

  The newline character (ASCII-character 10).

  Categories: Global variables · Package stringproc

Variable: space

  The space character.

  Categories: Global variables · Package stringproc

Variable: tab

  The tab character.

  Categories: Global variables · Package stringproc

Function: unicode (arg)

  Returns the character defined by arg which might be a Unicode code point or a name string if the underlying Lisp provides full Unicode support.

  Example: Characters defined by hexadecimal code points (Maxima with SBCL on GNU/Linux).

      (%i1) ibase: 16.$
      (%i2) map(unicode, [24, 0A3, 20AC]);
      (%o2)                            [$, £, €]

  Warning: In wxMaxima with SBCL on Windows it is not possible to convert code points larger than 16 bit to characters when the external format has not been set to UTF-8. See adjust_external_format for more information.

  CMUCL doesn't process code points larger than 16 bit. In these cases `unicode` returns `false`. Converting a code point to a character via UTF-8 octets may serve as a workaround:

  `octets_to_string(unicode_to_utf8(code_point));`

  See octets_to_string, unicode_to_utf8.

  In case the underlying Lisp provides full Unicode support the character might be specified by its name. The following is possible in ECL, CLISP and SBCL, where in SBCL on Windows the external format has to be set to UTF-8. `unicode(name)` is supported by CMUCL too but again limited to 16 bit characters.

  The string argument to `unicode` is basically the same string returned by `printf` using the "~@c" specifier. But as shown below the prefix "#\" must be omitted. Underlines might be replaced by spaces and uppercase letters by lowercase ones.

  Example (continued): Characters defined by names (Maxima with SBCL on GNU/Linux).

      (%i3) printf(false, "~@c", unicode(0DF));
      (%o3)                    #\LATIN_SMALL_LETTER_SHARP_S
      (%i4) unicode("LATIN_SMALL_LETTER_SHARP_S");
      (%o4)                                  ß
      (%i5) unicode("Latin small letter sharp s");
      (%o5)                                  ß

  Categories: Package stringproc

Function: unicode_to_utf8 (code_point)

  Returns a list containing the UTF-8 code corresponding to the Unicode code_point.

  Examples: Converting Unicode code points to UTF-8 and vice versa.

      (%i1) ibase: obase: 16.$
      (%i2) map(cint, ["$","£","€"]);
      (%o2)                           [24, 0A3, 20AC]
      (%i3) map(unicode_to_utf8, %);
      (%o3)                 [[24], [0C2, 0A3], [0E2, 82, 0AC]]
      (%i4) map(utf8_to_unicode, %);
      (%o4)                           [24, 0A3, 20AC]

  Categories: Package stringproc

Function: uppercasep (char)

  Returns `true` if char is an uppercase character.

  Note: See remarks on alphacharp.

  Categories: Predicate functions · Package stringproc

Variable: us_ascii_only

  This option variable affects Maxima when the character encoding provided by the application which runs Maxima is UTF-8 but the external format of the Lisp reader is not equal to UTF-8.

  On GNU/Linux this is true when Maxima is built with GCL and on Windows in wxMaxima with GCL- and SBCL-builds. With SBCL it is recommended to change the external format to UTF-8. Setting `us_ascii_only` is unnecessary then. See adjust_external_format for details.

  `us_ascii_only` is `false` by default. Maxima itself then (i.e. in the above described situation) parses the UTF-8 encoding.

  When `us_ascii_only` is set to `true` it is assumed that all strings used as arguments to string processing functions do not contain Non-US-ASCII characters. Given that promise, Maxima avoids parsing UTF-8 and strings can be processed more efficiently.

  Categories: Global variables · Package stringproc

Function: utf8_to_unicode (list)

  Returns a Unicode code point corresponding to the list which must contain the UTF-8 encoding of a single character.

  Examples: See unicode_to_utf8.

  Categories: Package stringproc

[ < ] | [ > ] |  | [ << ] | [ Up ] | [ >> ] |  |  |  |  | [Top] | [Contents] | [Index] | [ ? ]

## 84.4 String Processing

Position indices in strings are 1-indexed like in Maxima lists. See example in charat.

Function: charat (string, n)

  Returns the n-th character of string. The first character in string is returned with n = 1.

      (%i1) charat("Lisp",1);
      (%o1)                           L
      (%i2) charlist("Lisp")[1];
      (%o2)                           L

  Categories: Package stringproc

Function: charlist (string)

  Returns the list of all characters in string.

      (%i1) charlist("Lisp");
      (%o1)                     [L, i, s, p]

  Categories: Package stringproc

Function: eval_string (str)

  Parse the string str as a Maxima expression and evaluate it. The string str may or may not have a terminator (dollar sign `$` or semicolon `;`). Only the first expression is parsed and evaluated, if there is more than one.

  Complain if str is not a string.

  Examples:

      (%i1) eval_string ("foo: 42; bar: foo^2 + baz");
      (%o1)                       42
      (%i2) eval_string ("(foo: 42, bar: foo^2 + baz)");
      (%o2)                   baz + 1764

  See also parse_string.

  Categories: Package stringproc

Function: parse_string (str)

  Parse the string str as a Maxima expression (do not evaluate it). The string str may or may not have a terminator (dollar sign `$` or semicolon `;`). Only the first expression is parsed, if there is more than one.

  Complain if str is not a string.

  Examples:

      (%i1) parse_string ("foo: 42; bar: foo^2 + baz");
      (%o1)                    foo : 42
      (%i2) parse_string ("(foo: 42, bar: foo^2 + baz)");
                                         2
      (%o2)          (foo : 42, bar : foo  + baz)

  See also eval_string.

  Categories: Package stringproc

Function: scopy (string)

  Returns a copy of string as a new string.

  Categories: Package stringproc

Function: sdowncase
`sdowncase` (string)
`sdowncase` (string, start)
`sdowncase` (string, start, end)

  Like supcase but uppercase characters are converted to lowercase.

  Categories: Package stringproc

Function: sequal (string_1, string_2)

  Returns `true` if string_1 and string_2 contain the same sequence of characters.

  Categories: Predicate functions · Package stringproc

Function: sequalignore (string_1, string_2)

  Like `sequal` but ignores case which is only possible for non-US-ASCII characters when the underlying Lisp is able to recognize a character as an alphabetic character. See remarks on alphacharp.

  Categories: Predicate functions · Package stringproc

Function: sexplode (string)

  `sexplode` is an alias for function `charlist`.

  Categories: Package stringproc

Function: simplode
`simplode` (list)
`simplode` (list, delim)

  `simplode` takes a list of expressions and concatenates them into a string. If no delimiter delim is specified, `simplode` uses no delimiter. delim can be any string.

  Examples:

      (%i1) simplode(["xx[",3,"]:",expand((x+y)^3)]);
      (%o1)             xx[3]:y^3+3*x*y^2+3*x^2*y+x^3
      (%i2) simplode( sexplode("stars")," * " );
      (%o2)                   s * t * a * r * s
      (%i3) simplode( ["One","more","coffee."]," " );
      (%o3)                   One more coffee.

  Categories: Package stringproc

Function: sinsert (seq, string, pos)

  Returns a string that is a concatenation of `substring(string, 1, pos-1)`, the string seq and `substring (string, pos)`. Note that the first character in string is in position 1.

  Examples:

      (%i1) s: "A submarine."$
      (%i2) concat( substring(s,1,3),"yellow ",substring(s,3) );
      (%o2)                  A yellow submarine.
      (%i3) sinsert("hollow ",s,3);
      (%o3)                  A hollow submarine.

  Categories: Package stringproc

Function: sinvertcase
`sinvertcase` (string)
`sinvertcase` (string, start)
`sinvertcase` (string, start, end)

  Returns string except that each character from position start to end is inverted. If end is not given, all characters from start to the end of string are replaced.

  Examples:

      (%i1) sinvertcase("sInvertCase");
      (%o1)                      SiNVERTcASE

  Categories: Package stringproc

Function: slength (string)

  Returns the number of characters in string.

  Categories: Package stringproc

Function: smake (num, char)

  Returns a new string with a number of num characters char.

  Example:

      (%i1) smake(3,"w");
      (%o1)                          www

  Categories: Package stringproc

Function: smismatch
`smismatch` (string_1, string_2)
`smismatch` (string_1, string_2, test)

  Returns the position of the first character of string_1 at which string_1 and string_2 differ or `false`. Default test function for matching is `sequal`. If `smismatch` should ignore case, use `sequalignore` as test.

  Example:

      (%i1) smismatch("seven","seventh");
      (%o1)                           6

  Categories: Package stringproc

Function: split
`split` (string)
`split` (string, delim)
`split` (string, delim, multiple)

  Returns the list of all tokens in string. Each token is an unparsed string. `split` uses delim as delimiter. If delim is not given, the space character is the default delimiter. multiple is a boolean variable with `true` by default. Multiple delimiters are read as one. This is useful if tabs are saved as multiple space characters. If multiple is set to `false`, each delimiter is noted.

  Examples:

      (%i1) split("1.2   2.3   3.4   4.5");
      (%o1)                 [1.2, 2.3, 3.4, 4.5]
      (%i2) split("first;;third;fourth",";",false);
      (%o2)               [first, , third, fourth]

  Categories: Package stringproc

Function: sposition (char, string)

  Returns the position of the first character in string which matches char. The first character in string is in position 1. For matching characters ignoring case see ssearch.

  Categories: Package stringproc

Function: sremove
`sremove` (seq, string)
`sremove` (seq, string, test)
`sremove` (seq, string, test, start)
`sremove` (seq, string, test, start, end)

  Returns a string like string but without all substrings matching seq. Default test function for matching is `sequal`. If `sremove` should ignore case while searching for seq, use `sequalignore` as test. Use start and end to limit searching. Note that the first character in string is in position 1.

  Examples:

      (%i1) sremove("n't","I don't like coffee.");
      (%o1)                   I do like coffee.
      (%i2) sremove ("DO ",%,'sequalignore);
      (%o2)                    I like coffee.

  Categories: Package stringproc

Function: sremovefirst
`sremovefirst` (seq, string)
`sremovefirst` (seq, string, test)
`sremovefirst` (seq, string, test, start)
`sremovefirst` (seq, string, test, start, end)

  Like `sremove` except that only the first substring that matches seq is removed.

  Categories: Package stringproc

Function: sreverse (string)

  Returns a string with all the characters of string in reverse order.

  Categories: Package stringproc

Function: ssearch
`ssearch` (seq, string)
`ssearch` (seq, string, test)
`ssearch` (seq, string, test, start)
`ssearch` (seq, string, test, start, end)

  Returns the position of the first substring of string that matches the string seq. Default test function for matching is `sequal`. If `ssearch` should ignore case, use `sequalignore` as test. Use start and end to limit searching. Note that the first character in string is in position 1.

  Example:

      (%i1) ssearch("~s","~{~S ~}~%",'sequalignore);
      (%o1)                                  4

  Categories: Package stringproc

Function: ssort
`ssort` (string)
`ssort` (string, test)

  Returns a string that contains all characters from string in an order such there are no two successive characters c and d such that `test (c, d)` is `false` and `test (d, c)` is `true`. Default test function for sorting is clessp. The set of test functions is `{clessp, clesspignore, cgreaterp, cgreaterpignore, cequal, cequalignore}`.

  Examples:

      (%i1) ssort("I don't like Mondays.");
      (%o1)                    '.IMaddeiklnnoosty
      (%i2) ssort("I don't like Mondays.",'cgreaterpignore);
      (%o2)                 ytsoonnMlkIiedda.'

  Categories: Package stringproc

Function: ssubst
`ssubst` (new, old, string)
`ssubst` (new, old, string, test)
`ssubst` (new, old, string, test, start)
`ssubst` (new, old, string, test, start, end)

  Returns a string like string except that all substrings matching old are replaced by new. old and new need not to be of the same length. Default test function for matching is `sequal`. If `ssubst` should ignore case while searching for old, use `sequalignore` as test. Use start and end to limit searching. Note that the first character in string is in position 1.

  Examples:

      (%i1) ssubst("like","hate","I hate Thai food. I hate green tea.");
      (%o1)          I like Thai food. I like green tea.
      (%i2) ssubst("Indian","thai",%,'sequalignore,8,12);
      (%o2)         I like Indian food. I like green tea.

  Categories: Package stringproc

Function: ssubstfirst
`ssubstfirst` (new, old, string)
`ssubstfirst` (new, old, string, test)
`ssubstfirst` (new, old, string, test, start)
`ssubstfirst` (new, old, string, test, start, end)

  Like `subst` except that only the first substring that matches old is replaced.

  Categories: Package stringproc

Function: strim (seq,string)

  Returns a string like string, but with all characters that appear in seq removed from both ends.

  Examples:

      (%i1) "/* comment */"$
      (%i2) strim(" /*",%);
      (%o2)                        comment
      (%i3) slength(%);
      (%o3)                           7

  Categories: Package stringproc

Function: striml (seq, string)

  Like `strim` except that only the left end of string is trimmed.

  Categories: Package stringproc

Function: strimr (seq, string)

  Like `strim` except that only the right end of string is trimmed.

  Categories: Package stringproc

Function: stringp (obj)

  Returns `true` if obj is a string. See introduction for example.

  Categories: Predicate functions · Package stringproc

Function: substring
`substring` (string, start)
`substring` (string, start, end)

  Returns the substring of string beginning at position start and ending at position end. The character at position end is not included. If end is not given, the substring contains the rest of the string. Note that the first character in string is in position 1.

  Examples:

      (%i1) substring("substring",4);
      (%o1)                        string
      (%i2) substring(%,4,6);
      (%o2)                          in

  Categories: Package stringproc

Function: supcase
`supcase` (string)
`supcase` (string, start)
`supcase` (string, start, end)

  Returns string except that lowercase characters from position start to end are replaced by the corresponding uppercase ones. If end is not given, all lowercase characters from start to the end of string are replaced.

  Example:

      (%i1) supcase("english",1,2);
      (%o1)                        English

  Categories: Package stringproc

Function: tokens
`tokens` (string)
`tokens` (string, test)

  Returns a list of tokens, which have been extracted from string. The tokens are substrings whose characters satisfy a certain test function. If test is not given, constituent is used as the default test. `{constituent, alphacharp, digitcharp, lowercasep, uppercasep, charp, characterp, alphanumericp}` is the set of test functions. (The Lisp-version of `tokens` is written by Paul Graham. ANSI Common Lisp, 1996, page 67.)

  Examples:

      (%i1) tokens("24 October 2005");
      (%o1)                  [24, October, 2005]
      (%i2) tokens("05-10-24",'digitcharp);
      (%o2)                     [05, 10, 24]
      (%i3) map(parse_string,%);
      (%o3)                      [5, 10, 24]

  Categories: Package stringproc

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## 84.5 Octets and Utilities for Cryptography

Function: base64 (arg)

  Returns the base64-representation of arg as a string. The argument arg may be a string, a non-negative integer or a list of octets.

  Examples:

      (%i1) base64: base64("foo bar baz");
      (%o1)                          Zm9vIGJhciBiYXo=
      (%i2) string: base64_decode(base64);
      (%o2)                            foo bar baz
      (%i3) obase: 16.$
      (%i4) integer: base64_decode(base64, 'number);
      (%o4)                       666f6f206261722062617a
      (%i5) octets: base64_decode(base64, 'list);
      (%o5)            [66, 6F, 6F, 20, 62, 61, 72, 20, 62, 61, 7A]
      (%i6) ibase: 16.$
      (%i7) base64(octets);
      (%o7)                          Zm9vIGJhciBiYXo=

  Note that if arg contains umlauts (resp. octets larger than 127) the resulting base64-string is platform dependend. However the decoded string will be equal to the original.

  Categories: Package stringproc

Function: base64_decode
`base64_decode` (base64-string)
`base64_decode` (base64-string, return-type)

  By default `base64_decode` decodes the base64-string back to the original string.

  The optional argument return-type allows `base64_decode` to alternatively return the corresponding number or list of octets. return-type may be `string`, `number` or `list`.

  Example: See base64.

  Categories: Package stringproc

Function: crc24sum
`crc24sum` (octets)
`crc24sum` (octets, return-type)

  By default `crc24sum` returns the `CRC24` checksum of an octet-list as a string.

  The optional argument return-type allows `crc24sum` to alternatively return the corresponding number or list of octets. return-type may be `string`, `number` or `list`.

  Example:

      -----BEGIN PGP SIGNATURE-----
      Version: GnuPG v2.0.22 (GNU/Linux)

      iQEcBAEBAgAGBQJVdCTzAAoJEG/1Mgf2DWAqCSYH/AhVFwhu1D89C3/QFcgVvZTM
      wnOYzBUURJAL/cT+IngkLEpp3hEbREcugWp+Tm6aw3R4CdJ7G3FLxExBH/5KnDHi
      rBQu+I7+3ySK2hpryQ6Wx5J9uZSa4YmfsNteR8up0zGkaulJeWkS4pjiRM+auWVe
      vajlKZCIK52P080DG7Q2dpshh4fgTeNwqCuCiBhQ73t8g1IaLdhDN6EzJVjGIzam
      /spqT/sTo6sw8yDOJjvU+Qvn6/mSMjC/YxjhRMaQt9EMrR1AZ4ukBF5uG1S7mXOH
      WdiwkSPZ3gnIBhM9SuC076gLWZUNs6NqTeE3UzMjDAFhH3jYk1T7mysCvdtIkms=
      =WmeC
      -----END PGP SIGNATURE-----

      (%i1) ibase : obase : 16.$
      (%i2) sig64 : sconcat(
       "iQEcBAEBAgAGBQJVdCTzAAoJEG/1Mgf2DWAqCSYH/AhVFwhu1D89C3/QFcgVvZTM",
       "wnOYzBUURJAL/cT+IngkLEpp3hEbREcugWp+Tm6aw3R4CdJ7G3FLxExBH/5KnDHi",
       "rBQu+I7+3ySK2hpryQ6Wx5J9uZSa4YmfsNteR8up0zGkaulJeWkS4pjiRM+auWVe",
       "vajlKZCIK52P080DG7Q2dpshh4fgTeNwqCuCiBhQ73t8g1IaLdhDN6EzJVjGIzam",
       "/spqT/sTo6sw8yDOJjvU+Qvn6/mSMjC/YxjhRMaQt9EMrR1AZ4ukBF5uG1S7mXOH",
       "WdiwkSPZ3gnIBhM9SuC076gLWZUNs6NqTeE3UzMjDAFhH3jYk1T7mysCvdtIkms=" )$
      (%i3) octets: base64_decode(sig64, 'list)$
      (%i4) crc24: crc24sum(octets, 'list);
      (%o4)                          [5A, 67, 82]
      (%i5) base64(crc24);
      (%o5)                              WmeC

  Categories: Package stringproc

Function: md5sum
`md5sum` (arg)
`md5sum` (arg, return-type)

  Returns the `MD5` checksum of a string, a non-negative integer or a list of octets. The default return value is a string containing 32 hex characters.

  The optional argument return-type allows `md5sum` to alternatively return the corresponding number or list of octets. return-type may be `string`, `number` or `list`.

  Examples:

      (%i1) ibase: obase: 16.$
      (%i2) msg: "foo bar baz"$
      (%i3) string: md5sum(msg);
      (%o3)                  ab07acbb1e496801937adfa772424bf7
      (%i4) integer: md5sum(msg, 'number);
      (%o4)                 0ab07acbb1e496801937adfa772424bf7
      (%i5) octets: md5sum(msg, 'list);
      (%o5)        [0AB,7,0AC,0BB,1E,49,68,1,93,7A,0DF,0A7,72,42,4B,0F7]
      (%i6) sdowncase( printf(false, "~{~2,'0x~^:~}", octets) );
      (%o6)           ab:07:ac:bb:1e:49:68:01:93:7a:df:a7:72:42:4b:f7

  Note that in case arg contains German umlauts or other non-ASCII characters (resp. octets larger than 127) the `MD5` checksum is platform dependend.

  Categories: Package stringproc

Function: mgf1_sha1
`mgf1_sha1` (seed, len)
`mgf1_sha1` (seed, len, return-type)

  Returns a pseudo random number of variable length. By default the returned value is a number with a length of len octets.

  The optional argument return-type allows `mgf1_sha1` to alternatively return the corresponding list of len octets. return-type may be `number` or `list`.

  The computation of the returned value is described in `RFC 3447`, appendix `B.2.1 MGF1`. `SHA1` ist used as hash function, i.e. the randomness of the computed number relies on the randomness of `SHA1` hashes.

  Example:

      (%i1) ibase: obase: 16.$
      (%i2) number: mgf1_sha1(4711., 8);
      (%o2)                        0e0252e5a2a42fea1
      (%i3) octets: mgf1_sha1(4711., 8, 'list);
      (%o3)                  [0E0,25,2E,5A,2A,42,0FE,0A1]

  Categories: Package stringproc

Function: number_to_octets (number)

  Returns an octet-representation of number as a list of octets. The number must be a non-negative integer.

  Example:

      (%i1) ibase : obase : 16.$
      (%i2) octets: [0ca,0fe,0ba,0be]$
      (%i3) number: octets_to_number(octets);
      (%o3)                            0cafebabe
      (%i4) number_to_octets(number);
      (%o4)                      [0CA, 0FE, 0BA, 0BE]

  Categories: Package stringproc

Function: octets_to_number (octets)

  Returns a number by concatenating the octets in the list of octets.

  Example: See number_to_octets.

  Categories: Package stringproc

Function: octets_to_oid (octets)

  Computes an object identifier (OID) from the list of octets.

  Example: RSA encryption OID

      (%i1) ibase : obase : 16.$
      (%i2) oid: octets_to_oid([2A,86,48,86,0F7,0D,1,1,1]);
      (%o2)                      1.2.840.113549.1.1.1
      (%i3) oid_to_octets(oid);
      (%o3)               [2A, 86, 48, 86, 0F7, 0D, 1, 1, 1]

  Categories: Package stringproc

Function: octets_to_string
`octets_to_string` (octets)
`octets_to_string` (octets, encoding)

  Decodes the list of octets into a string according to current system defaults. When decoding octets corresponding to Non-US-ASCII characters the result depends on the platform, application and underlying Lisp.

  Example: Using system defaults (Maxima compiled with GCL, which uses no format definition and simply passes through the UTF-8-octets encoded by the GNU/Linux terminal).

      (%i1) octets: string_to_octets("abc");
      (%o1)                            [61, 62, 63]
      (%i2) octets_to_string(octets);
      (%o2)                                 abc
      (%i3) ibase: obase: 16.$
      (%i4) unicode(20AC);
      (%o4)                                  €
      (%i5) octets: string_to_octets(%);
      (%o5)                           [0E2, 82, 0AC]
      (%i6) octets_to_string(octets);
      (%o6)                                  €
      (%i7) utf8_to_unicode(octets);
      (%o7)                                20AC

  In case the external format of the Lisp reader is equal to UTF-8 the optional argument encoding allows to set the encoding for the octet to string conversion. If necessary see adjust_external_format for changing the external format.

  Some names of supported encodings (see corresponding Lisp manual for more):
  CCL, CLISP, SBCL: `utf-8, ucs-2be, ucs-4be, iso-8859-1, cp1252, cp850`
  CMUCL: `utf-8, utf-16-be, utf-32-be, iso8859-1, cp1252`
  ECL: `utf-8, ucs-2be, ucs-4be, iso-8859-1, windows-cp1252, dos-cp850`

  Example (continued): Using the optional encoding argument (Maxima compiled with SBCL, GNU/Linux terminal).

      (%i8) string_to_octets("€", "ucs-2be");
      (%o8)                              [20, 0AC]

  Categories: Package stringproc

Function: oid_to_octets (oid-string)

  Convertes an object identifier (OID) to a list of octets.

  Example: See octets_to_oid.

  Categories: Package stringproc

Function: sha1sum
`sha1sum` (arg)
`sha1sum` (arg, return-type)

  Returns the `SHA1` fingerprint of a string, a non-negative integer or a list of octets. The default return value is a string containing 40 hex characters.

  The optional argument return-type allows `sha1sum` to alternatively return the corresponding number or list of octets. return-type may be `string`, `number` or `list`.

  Example:

      (%i1) ibase: obase: 16.$
      (%i2) msg: "foo bar baz"$
      (%i3) string: sha1sum(msg);
      (%o3)              c7567e8b39e2428e38bf9c9226ac68de4c67dc39
      (%i4) integer: sha1sum(msg, 'number);
      (%o4)             0c7567e8b39e2428e38bf9c9226ac68de4c67dc39
      (%i5) octets: sha1sum(msg, 'list);
      (%o5)  [0C7,56,7E,8B,39,0E2,42,8E,38,0BF,9C,92,26,0AC,68,0DE,4C,67,0DC,39]
      (%i6) sdowncase( printf(false, "~{~2,'0x~^:~}", octets) );
      (%o6)     c7:56:7e:8b:39:e2:42:8e:38:bf:9c:92:26:ac:68:de:4c:67:dc:39

  Note that in case arg contains German umlauts or other non-ASCII characters (resp. octets larger than 127) the `SHA1` fingerprint is platform dependend.

  Categories: Package stringproc

Function: sha256sum
`sha256sum` (arg)
`sha256sum` (arg, return-type)

  Returns the `SHA256` fingerprint of a string, a non-negative integer or a list of octets. The default return value is a string containing 64 hex characters.

  The optional argument return-type allows `sha256sum` to alternatively return the corresponding number or list of octets (see sha1sum).

  Example:

      (%i1) string: sha256sum("foo bar baz");
      (%o1)  dbd318c1c462aee872f41109a4dfd3048871a03dedd0fe0e757ced57dad6f2d7

  Note that in case arg contains German umlauts or other non-ASCII characters (resp. octets larger than 127) the `SHA256` fingerprint is platform dependend.

  Categories: Package stringproc

Function: string_to_octets
`string_to_octets` (string)
`string_to_octets` (string, encoding)

  Encodes a string into a list of octets according to current system defaults. When encoding strings containing Non-US-ASCII characters the result depends on the platform, application and underlying Lisp.

  In case the external format of the Lisp reader is equal to UTF-8 the optional argument encoding allows to set the encoding for the string to octet conversion. If necessary see adjust_external_format for changing the external format.

  See octets_to_string for examples and some more information.

  Categories: Package stringproc

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# 85. to_poly_solve

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## 85.1 Functions and Variables for to_poly_solve

The packages `to_poly` and `to_poly_solve` are experimental; the specifications of the functions in these packages might change or the some of the functions in these packages might be merged into other Maxima functions.

Barton Willis (Professor of Mathematics, University of Nebraska at Kearney) wrote the `to_poly` and `to_poly_solve` packages and the English language user documentation for these packages.

Operator: %and

  The operator `%and` is a simplifying nonshort-circuited logical conjunction. Maxima simplifies an `%and` expression to either true, false, or a logically equivalent, but simplified, expression. The operator `%and` is associative, commutative, and idempotent. Thus when `%and` returns a noun form, the arguments of `%and` form a non-redundant sorted list; for example

      (%i1) a %and (a %and b);
      (%o1)                       a %and b

  If one argument to a conjunction is the explicit the negation of another argument, `%and` returns false:

      (%i2) a %and (not a);
      (%o2)                         false

  If any member of the conjunction is false, the conjunction simplifies to false even if other members are manifestly non-boolean; for example

      (%i3) 42 %and false;
      (%o3)                         false

  Any argument of an `%and` expression that is an inequation (that is, an inequality or equation), is simplified using the Fourier elimination package. The Fourier elimination simplifier has a pre-processor that converts some, but not all, nonlinear inequations into linear inequations; for example the Fourier elimination code simplifies `abs(x) + 1 > 0` to true, so

      (%i4) (x < 1) %and (abs(x) + 1 > 0);
      (%o4)                         x < 1

  Notes

  - The option variable `prederror` does not alter the simplification `%and` expressions.
  - To avoid operator precedence errors, compound expressions involving the operators `%and, %or`, and `not` should be fully parenthesized.
  - The Maxima operators `and` and `or` are both short-circuited. Thus `and` isn't associative or commutative.

  Limitations The conjunction `%and` simplifies inequations locally, not globally. This means that conjunctions such as

      (%i5) (x < 1) %and (x > 1);
      (%o5)                 (x > 1) %and (x < 1)

  do not simplify to false. Also, the Fourier elimination code ignores the fact database;

      (%i6) assume(x > 5);
      (%o6)                        [x > 5]
      (%i7) (x > 1) %and (x > 2);
      (%o7)                 (x > 1) %and (x > 2)

  Finally, nonlinear inequations that aren't easily converted into an equivalent linear inequation aren't simplified.

  There is no support for distributing `%and` over `%or`; neither is there support for distributing a logical negation over `%and`.

  To use ``load(to_poly_solve)'`

  Related functions `%or, %if, and, or, not`

  Status The operator `%and` is experimental; the specifications of this function might change and its functionality might be merged into other Maxima functions.

Operator: %if (bool, a, b)

  The operator `%if` is a simplifying conditional. The conditional bool should be boolean-valued. When the conditional is true, return the second argument; when the conditional is false, return the third; in all other cases, return a noun form.

  Maxima inequations (either an inequality or an equality) are not boolean-valued; for example, Maxima does not simplify 5 < 6 to true, and it does not simplify 5 = 6 to false; however, in the context of a conditional to an `%if` statement, Maxima automatically attempts to determine the truth value of an inequation. Examples:

      (%i1) f : %if(x # 1, 2, 8);
      (%o1)                 %if(x - 1 # 0, 2, 8)
      (%i2) [subst(x = -1,f), subst(x=1,f)];
      (%o2)                        [2, 8]

  If the conditional involves an inequation, Maxima simplifies it using the Fourier elimination package.

  Notes

  - If the conditional is manifestly non-boolean, Maxima returns a noun form:

      (%i3) %if(42,1,2);
      (%o3)                     %if(42, 1, 2)

  - The Maxima operator `if` is nary, the operator `%if` isn't nary.

  Limitations The Fourier elimination code only simplifies nonlinear inequations that are readily convertible to an equivalent linear inequation.

  To use: ``load(to_poly_solve)'`

  Status: The operator `%if` is experimental; its specifications might change and its functionality might be merged into other Maxima functions.

Operator: %or

  The operator `%or` is a simplifying nonshort-circuited logical disjunction. Maxima simplifies an `%or` expression to either true, false, or a logically equivalent, but simplified, expression. The operator `%or` is associative, commutative, and idempotent. Thus when `%or` returns a noun form, the arguments of `%or` form a non-redundant sorted list; for example

      (%i1) a %or (a %or b);
      (%o1)                        a %or b

  If one member of the disjunction is the explicit the negation of another member, `%or` returns true:

      (%i2) a %or (not a);
      (%o2)                         true

  If any member of the disjunction is true, the disjunction simplifies to true even if other members of the disjunction are manifestly non-boolean; for example

      (%i3) 42 %or true;
      (%o3)                         true

  Any argument of an `%or` expression that is an inequation (that is, an inequality or equation), is simplified using the Fourier elimination package. The Fourier elimination code simplifies `abs(x) + 1 > 0` to true, so we have

      (%i4) (x < 1) %or (abs(x) + 1 > 0);
      (%o4)                         true

  Notes

  - The option variable `prederror` does not alter the simplification of `%or` expressions.
  - You should parenthesize compound expressions involving the operators `%and, %or`, and `not`; the binding powers of these operators might not match your expectations.
  - The Maxima operators `and` and `or` are both short-circuited. Thus `or` isn't associative or commutative.

  Limitations The conjunction `%or` simplifies inequations locally, not globally. This means that conjunctions such as

      (%i1) (x < 1) %or (x >= 1);
      (%o1) (x > 1) %or (x >= 1)

  do not simplify to true. Further, the Fourier elimination code ignores the fact database;

      (%i2) assume(x > 5);
      (%o2)                        [x > 5]
      (%i3) (x > 1) %and (x > 2);
      (%o3)                 (x > 1) %and (x > 2)

  Finally, nonlinear inequations that aren't easily converted into an equivalent linear inequation aren't simplified.

  The algorithm that looks for terms that cannot both be false is weak; also there is no support for distributing `%or` over `%and`; neither is there support for distributing a logical negation over `%or`.

  To use ``load(to_poly_solve)'`

  Related functions `%or, %if, and, or, not`

  Status The operator `%or` is experimental; the specifications of this function might change and its functionality might be merged into other Maxima functions.

Function: complex_number_p (x)

  The predicate `complex_number_p` returns true if its argument is either `a + %i * b`, `a`, `%i b`, or `%i`, where `a` and `b` are either rational or floating point numbers (including big floating point); for all other inputs, `complex_number_p` returns false; for example

      (%i1) map('complex_number_p,[2/3, 2 + 1.5 * %i, %i]);
      (%o1)                  [true, true, true]
      (%i2) complex_number_p((2+%i)/(5-%i));
      (%o2)                         false
      (%i3) complex_number_p(cos(5 - 2 * %i));
      (%o3)                         false

  Related functions `isreal_p`

  To use ``load(to_poly_solve)'`

  Status The operator `complex_number_p` is experimental; its specifications might change and its functionality might be merged into other Maxima functions.

Function: compose_functions (l)

  The function call `compose_functions(l)` returns a lambda form that is the composition of the functions in the list l. The functions are applied from right to left; for example

      (%i1) compose_functions([cos, exp]);
                                              %g151
      (%o1)             lambda([%g151], cos(%e     ))
      (%i2) %(x);
                                        x
      (%o2)                       cos(%e )

  When the function list is empty, return the identity function:

      (%i3) compose_functions([]);
      (%o3)                lambda([%g152], %g152)
      (%i4)  %(x);
      (%o4)                           x

  Notes

  - When Maxima determines that a list member isn't a symbol or a lambda form, `funmake` (not `compose_functions`) signals an error:

      (%i5) compose_functions([a < b]);

      funmake: first argument must be a symbol, subscripted symbol,
      string, or lambda expression; found: a < b
      #0: compose_functions(l=[a < b])(to_poly_solve.mac line 40)
       -- an error. To debug this try: debugmode(true);

  - To avoid name conflicts, the independent variable is determined by the function `new_variable`.

        (%i6) compose_functions([%g0]);
        (%o6)              lambda([%g154], %g0(%g154))
        (%i7) compose_functions([%g0]);
        (%o7)              lambda([%g155], %g0(%g155))

    Although the independent variables are different, Maxima is able to to deduce that these lambda forms are semantically equal:

        (%i8) is(equal(%o6,%o7));
        (%o8)                         true

  To use ``load(to_poly_solve)'`

  Status The function `compose_functions` is experimental; its specifications might change and its functionality might be merged into other Maxima functions.

Function: dfloat (x)

  The function `dfloat` is a similar to `float`, but the function `dfloat` applies `rectform` when `float` fails to evaluate to an IEEE double floating point number; thus

      (%i1) float(4.5^(1 + %i));
                                     %i + 1
      (%o1)                       4.5
      (%i2) dfloat(4.5^(1 + %i));
      (%o2)        4.48998802962884 %i + .3000124893895671

  Notes

  - The rectangular form of an expression might be poorly suited for numerical evaluation-for example, the rectangular form might needlessly involve the difference of floating point numbers (subtractive cancellation).
  - The identifier `float` is both an option variable (default value false) and a function name.

  Related functions `float, bfloat`

  To use ``load(to_poly_solve)'`

  Status The function `dfloat` is experimental; its specifications might change and its functionality might be merged into other Maxima functions.

Function: elim (l, x)

  The function `elim` eliminates the variables in the set or list `x` from the equations in the set or list `l`. Each member of `x` must be a symbol; the members of `l` can either be equations, or expressions that are assumed to equal zero.

  The function `elim` returns a list of two lists; the first is the list of expressions with the variables eliminated; the second is the list of pivots; thus, the second list is a list of expressions that `elim` used to eliminate the variables.

  Here is a example of eliminating between linear equations:

      (%i1) elim(set(x + y + z = 1, x - y  - z = 8, x - z = 1),
                 set(x,y));
      (%o1)            [[2 z - 7], [y + 7, z - x + 1]]

  Eliminating `x` and `y` yields the single equation `2 z - 7 = 0`; the equations `y + 7 = 0` and `z - z + 1 = 1` were used as pivots. Eliminating all three variables from these equations, triangularizes the linear system:

      (%i2) elim(set(x + y + z = 1, x - y  - z = 8, x - z = 1),
                 set(x,y,z));
      (%o2)           [[], [2 z - 7, y + 7, z - x + 1]]

  Of course, the equations needn't be linear:

      (%i3) elim(set(x^2 - 2 * y^3 = 1,  x - y = 5), [x,y]);
                           3    2
      (%o3)       [[], [2 y  - y  - 10 y - 24, y - x + 5]]

  The user doesn't control the order the variables are eliminated. Instead, the algorithm uses a heuristic to attempt to choose the best pivot and the best elimination order.

  Notes

  - Unlike the related function `eliminate`, the function `elim` does not invoke `solve` when the number of equations equals the number of variables.
  - The function `elim` works by applying resultants; the option variable `resultant` determines which algorithm Maxima uses. Using `sqfr`, Maxima factors each resultant and suppresses multiple zeros.
  - The `elim` will triangularize a nonlinear set of polynomial equations; the solution set of the triangularized set can be larger than that solution set of the untriangularized set. Thus, the triangularized equations can have spurious solutions.

  Related functions elim_allbut, eliminate_using, eliminate

  Option variables resultant

  To use ``load(to_poly)'`

  Status The function `elim` is experimental; its specifications might change and its functionality might be merged into other Maxima functions.

Function: elim_allbut (l, x)

  This function is similar to `elim`, except that it eliminates all the variables in the list of equations `l` except for those variables that in in the list `x`

      (%i1) elim_allbut([x+y = 1, x - 5*y = 1],[]);
      (%o1)                 [[], [y, y + x - 1]]
      (%i2) elim_allbut([x+y = 1, x - 5*y = 1],[x]);
      (%o2)                [[x - 1], [y + x - 1]]

  To use ``load(to_poly)'`

  Option variables resultant

  Related functions elim, eliminate_using, eliminate

  Status The function `elim_allbut` is experimental; its specifications might change and its functionality might be merged into other Maxima functions.

Function: eliminate_using (l, e, x)

  Using `e` as the pivot, eliminate the symbol `x` from the list or set of equations in `l`. The function `eliminate_using` returns a set.

      (%i1) eq : [x^2 - y^2 - z^3 , x*y - z^2 - 5, x - y + z];
                     3    2    2     2
      (%o1)      [- z  - y  + x , - z  + x y - 5, z - y + x]
      (%i2) eliminate_using(eq,first(eq),z);
              3              2      2      3    2
      (%o2) {y  + (1 - 3 x) y  + 3 x  y - x  - x ,
                              4    3  3       2  2             4
                             y  - x  y  + 13 x  y  - 75 x y + x  + 125}
      (%i3) eliminate_using(eq,second(eq),z);
              2            2       4    3  3       2  2             4
      (%o3) {y  - 3 x y + x  + 5, y  - x  y  + 13 x  y  - 75 x y + x
                                                                 + 125}
      (%i4) eliminate_using(eq, third(eq),z);
              2            2       3              2      2      3    2
      (%o4) {y  - 3 x y + x  + 5, y  + (1 - 3 x) y  + 3 x  y - x  - x }

  Option variables resultant

  Related functions elim, eliminate, elim_allbut

  To use ``load(to_poly)'`

  Status The function `eliminate_using` is experimental; its specifications might change and its functionality might be merged into other Maxima functions.

Function: fourier_elim ([eq1, eq2, …], [var1, var, …])

  Fourier elimination is the analog of Gauss elimination for linear inequations (equations or inequalities). The function call `fourier_elim([eq1, eq2, ...], [var1, var2, ...])` does Fourier elimination on a list of linear inequations `[eq1, eq2, ...]` with respect to the variables `[var1, var2, ...]`; for example

      (%i1) fourier_elim([y-x < 5, x - y < 7, 10 < y],[x,y]);
      (%o1)            [y - 5 < x, x < y + 7, 10 < y]
      (%i2) fourier_elim([y-x < 5, x - y < 7, 10 < y],[y,x]);
      (%o2)        [max(10, x - 7) < y, y < x + 5, 5 < x]

  Eliminating first with respect to x and second with respect to y yields lower and upper bounds for x that depend on y, and lower and upper bounds for y that are numbers. Eliminating in the other order gives x dependent lower and upper bounds for y, and numerical lower and upper bounds for x.

  When necessary, `fourier_elim` returns a disjunction of lists of inequations:

      (%i3) fourier_elim([x # 6],[x]);
      (%o3)                  [x < 6] or [6 < x]

  When the solution set is empty, `fourier_elim` returns `emptyset`, and when the solution set is all reals, `fourier_elim` returns `universalset`; for example

      (%i4) fourier_elim([x < 1, x > 1],[x]);
      (%o4)                       emptyset
      (%i5) fourier_elim([minf < x, x < inf],[x]);
      (%o5)                     universalset

  For nonlinear inequations, `fourier_elim` returns a (somewhat) simplified list of inequations:

      (%i6) fourier_elim([x^3 - 1 > 0],[x]);
                     2                             2
      (%o6) [1 < x, x  + x + 1 > 0] or [x < 1, - (x  + x + 1) > 0]
      (%i7) fourier_elim([cos(x) < 1/2],[x]);
      (%o7)                  [1 - 2 cos(x) > 0]

  Instead of a list of inequations, the first argument to `fourier_elim` may be a logical disjunction or conjunction:

      (%i8) fourier_elim((x + y < 5) and (x - y >8),[x,y]);
                                                    3
      (%o8)            [y + 8 < x, x < 5 - y, y < - -]
                                                    2
      (%i9) fourier_elim(((x + y < 5) and x < 1) or  (x - y >8),[x,y]);
      (%o9)          [y + 8 < x] or [x < min(1, 5 - y)]

  The function `fourier_elim` supports the inequation operators `<, <=, >, >=, #`, and `=`.

  The Fourier elimination code has a preprocessor that converts some nonlinear inequations that involve the absolute value, minimum, and maximum functions into linear in equations. Additionally, the preprocessor handles some expressions that are the product or quotient of linear terms:

      (%i10) fourier_elim([max(x,y) > 6, x # 8, abs(y-1) > 12],[x,y]);
      (%o10) [6 < x, x < 8, y < - 11] or [8 < x, y < - 11]
       or [x < 8, 13 < y] or [x = y, 13 < y] or [8 < x, x < y, 13 < y]
       or [y < x, 13 < y]
      (%i11) fourier_elim([(x+6)/(x-9) <= 6],[x]);
      (%o11)           [x = 12] or [12 < x] or [x < 9]
      (%i12) fourier_elim([x^2 - 1 # 0],[x]);
      (%o12)      [- 1 < x, x < 1] or [1 < x] or [x < - 1]

  To use ``load(fourier_elim)'`

Function: isreal_p (e)

  The predicate `isreal_p` returns true when Maxima is able to determine that `e` is real-valued on the entire real line; it returns false when Maxima is able to determine that `e` isn't real-valued on some nonempty subset of the real line; and it returns a noun form for all other cases.

      (%i1) map('isreal_p, [-1, 0, %i, %pi]);
      (%o1)               [true, true, false, true]

  Maxima variables are assumed to be real; thus

      (%i2) isreal_p(x);
      (%o2)                         true

  The function `isreal_p` examines the fact database:

      (%i3) declare(z,complex)$

      (%i4) isreal_p(z);
      (%o4)                      isreal_p(z)

  Limitations Too often, `isreal_p` returns a noun form when it should be able to return false; a simple example: the logarithm function isn't real-valued on the entire real line, so `isreal_p(log(x))` should return false; however

      (%i5) isreal_p(log(x));
      (%o5)                   isreal_p(log(x))

  To use ``load(to_poly_solve)'`

  Related functions complex_number_p

  Status The function `isreal_p` is experimental; its specifications might change and its functionality might be merged into other Maxima functions.

Function: new_variable (type)

  Return a unique symbol of the form `%[z,n,r,c,g]k`, where `k` is an integer. The allowed values for type are integer, natural_number, real, natural_number, and general. (By natural number, we mean the nonnegative integers; thus zero is a natural number. Some, but not all,definitions of natural number exclude zero.)

  When type isn't one of the allowed values, type defaults to general. For integers, natural numbers, and complex numbers, Maxima automatically appends this information to the fact database.

      (%i1) map('new_variable,
                ['integer, 'natural_number, 'real, 'complex, 'general]);
      (%o1)          [%z144, %n145, %r146, %c147, %g148]
      (%i2) nicedummies(%);
      (%o2)               [%z0, %n0, %r0, %c0, %g0]
      (%i3) featurep(%z0, 'integer);
      (%o3)                         true
      (%i4) featurep(%n0, 'integer);
      (%o4)                         true
      (%i5) is(%n0 >= 0);
      (%o5)                         true
      (%i6) featurep(%c0, 'complex);
      (%o6)                         true

  Note Generally, the argument to `new_variable` should be quoted. The quote will protect against errors similar to

      (%i7) integer : 12$

      (%i8) new_variable(integer);
      (%o8)                         %g149
      (%i9) new_variable('integer);
      (%o9)                         %z150

  Related functions nicedummies

  To use ``load(to_poly_solve)'`

  Status The function `new_variable` is experimental; its specifications might change and its functionality might be merged into other Maxima functions.

Function: nicedummies

  Starting with zero, the function `nicedummies` re-indexes the variables in an expression that were introduced by `new_variable`;

      (%i1) new_variable('integer) + 52 * new_variable('integer);
      (%o1)                   52 %z136 + %z135
      (%i2) new_variable('integer) - new_variable('integer);
      (%o2)                     %z137 - %z138
      (%i3) nicedummies(%);
      (%o3)                       %z0 - %z1

  Related functions new_variable

  To use ``load(to_poly_solve)'`

  Status The function `nicedummies` is experimental; its specifications might change and its functionality might be merged into other Maxima functions.

Function: parg (x)

  The function `parg` is a simplifying version of the complex argument function `carg`; thus

      (%i1) map('parg,[1,1+%i,%i, -1 + %i, -1]);
                              %pi  %pi  3 %pi
      (%o1)               [0, ---, ---, -----, %pi]
                               4    2     4

  Generally, for a non-constant input, `parg` returns a noun form; thus

      (%i2) parg(x + %i * sqrt(x));
      (%o2)                 parg(x + %i sqrt(x))

  When `sign` can determine that the input is a positive or negative real number, `parg` will return a non-noun form for a non-constant input. Here are two examples:

      (%i3) parg(abs(x));
      (%o3) 0
      (%i4) parg(-x^2-1);
      (%o4)                          %pi

  Note The `sign` function mostly ignores the variables that are declared to be complex (`declare(x,complex)`); for variables that are declared to be complex, the `parg` can return incorrect values; for example

      (%i1) declare(x,complex)$

      (%i2) parg(x^2 + 1);
      (%o2) 0

  Related function carg, isreal_p

  To use ``load(to_poly_solve)'`

  Status The function `parg` is experimental; its specifications might change and its functionality might be merged into other Maxima functions.

Function: real_imagpart_to_conjugate (e)

  The function `real_imagpart_to_conjugate` replaces all occurrences of `realpart` and `imagpart` to algebraically equivalent expressions involving the `conjugate`.

      (%i1) declare(x, complex)$

      (%i2) real_imagpart_to_conjugate(realpart(x) +  imagpart(x) = 3);
                conjugate(x) + x   %i (x - conjugate(x))
      (%o2)     ---------------- - --------------------- = 3
                       2                     2

  To use ``load(to_poly_solve)'`

  Status The function `real_imagpart_to_conjugate` is experimental; its specifications might change and its functionality might be merged into other Maxima functions.

Function: rectform_log_if_constant (e)

  The function `rectform_log_if_constant` converts all terms of the form `log(c)` to `rectform(log(c))`, where `c` is either a declared constant expression or explicitly declared constant

      (%i1) rectform_log_if_constant(log(1-%i) - log(x - %i));
                                       log(2)   %i %pi
      (%o1)            - log(x - %i) + ------ - ------
                                         2        4
      (%i2) declare(a,constant, b,constant)$

      (%i3) rectform_log_if_constant(log(a + %i*b));
                             2    2
                        log(b  + a )
      (%o3)             ------------ + %i atan2(b, a)
                             2

  To use ``load(to_poly_solve)'`

  Status The function `rectform_log_if_constant` is experimental; the specifications of this function might change might change and its functionality might be merged into other Maxima functions.

Function: simp_inequality (e)

  The function `simp_inequality` applies some simplifications to conjunctions and disjunctions of inequations.

  Limitations The function `simp_inequality` is limited in at least two ways; first, the simplifications are local; thus

      (%i1) simp_inequality((x > minf) %and (x < 0));
      (%o1) (x>1) %and (x<1)

  And second, `simp_inequality` doesn't consult the fact database:

      (%i2) assume(x > 0)$

      (%i3) simp_inequality(x > 0);
      (%o3)                         x > 0

  To use ``load(fourier_elim)'`

  Status The function `simp_inequality` is experimental; its specifications might change and its functionality might be merged into other Maxima functions.

Function: standardize_inverse_trig (e)

  This function applies the identities `cot(x) = atan(1/x), acsc(x) = asin(1/x),` and similarly for `asec, acoth, acsch` and `asech` to an expression. See Abramowitz and Stegun, Eqs. 4.4.6 through 4.4.8 and 4.6.4 through 4.6.6.

  To use ``load(to_poly_solve)'`

  Status The function `standardize_inverse_trig` is experimental; its specifications might change and its functionality might be merged into other Maxima functions.

Function: subst_parallel (l, e)

  When `l` is a single equation or a list of equations, substitute the right hand side of each equation for the left hand side. The substitutions are made in parallel; for example

      (%i1) load(to_poly_solve)$

      (%i2) subst_parallel([x=y,y=x], [x,y]);
      (%o2)                        [y, x]

  Compare this to substitutions made serially:

      (%i3) subst([x=y,y=x],[x,y]);
      (%o3)                        [x, x]

  The function `subst_parallel` is similar to `sublis` except that `subst_parallel` allows for substitution of nonatoms; for example

      (%i4) subst_parallel([x^2 = a, y = b], x^2 * y);
      (%o4)                          a b
      (%i5) sublis([x^2 = a, y = b], x^2 * y);

                                                                   2
      sublis: left-hand side of equation must be a symbol; found: x
       -- an error. To debug this try: debugmode(true);

  The substitutions made by `subst_parallel` are literal, not semantic; thus `subst_parallel` does not recognize that x * y is a subexpression of x^2 * y

      (%i6) subst_parallel([x * y = a], x^2 * y);
                                     2
      (%o6)                         x  y

  The function `subst_parallel` completes all substitutions before simplifications. This allows for substitutions into conditional expressions where errors might occur if the simplifications were made earlier:

      (%i7) subst_parallel([x = 0], %if(x < 1, 5, log(x)));
      (%o7)                           5
      (%i8) subst([x = 0], %if(x < 1, 5, log(x)));

      log: encountered log(0).
       -- an error. To debug this try: debugmode(true);

  Related functions subst, sublis, ratsubst

  To use ``load(to_poly_solve_extra.lisp)'`

  Status The function `subst_parallel` is experimental; the specifications of this function might change might change and its functionality might be merged into other Maxima functions.

Function: to_poly (e, l)

  The function `to_poly` attempts to convert the equation `e` into a polynomial system along with inequality constraints; the solutions to the polynomial system that satisfy the constraints are solutions to the equation `e`. Informally, `to_poly` attempts to polynomialize the equation e; an example might clarify:

      (%i1) load(to_poly_solve)$

      (%i2) to_poly(sqrt(x) = 3, [x]);
                                  2
      (%o2) [[%g130 - 3, x = %g130 ],
                            %pi                               %pi
                         [- --- < parg(%g130), parg(%g130) <= ---], []]
                             2                                 2

  The conditions `-%pi/2<parg(%g130),parg(%g130)<=%pi/2` tell us that `%g130` is in the range of the square root function. When this is true, the solution set to `sqrt(x) = 3` is the same as the solution set to `%g130-3,x=%g130^2`.

  To polynomialize trigonometric expressions, it is necessary to introduce a non algebraic substitution; these non algebraic substitutions are returned in the third list returned by `to_poly`; for example

      (%i3) to_poly(cos(x),[x]);
                      2                                 %i x
      (%o3)    [[%g131  + 1], [2 %g131 # 0], [%g131 = %e    ]]

  Constant terms aren't polynomializied unless the number one is a member of the variable list; for example

      (%i4) to_poly(x = sqrt(5),[x]);
      (%o4)                [[x - sqrt(5)], [], []]
      (%i5) to_poly(x = sqrt(5),[1,x]);
                                  2
      (%o5) [[x - %g132, 5 = %g132 ],
                            %pi                               %pi
                         [- --- < parg(%g132), parg(%g132) <= ---], []]
                             2                                 2

  To generate a polynomial with sqrt(5) + sqrt(7) as one of its roots, use the commands

      (%i6) first(elim_allbut(first(to_poly(x = sqrt(5) + sqrt(7),
                                            [1,x])), [x]));
                                4       2
      (%o6)                   [x  - 24 x  + 4]

  Related functions to_poly_solve

  To use ``load(to_poly)'`

  Status: The function `to_poly` is experimental; its specifications might change and its functionality might be merged into other Maxima functions.

Function: to_poly_solve (e, l, [options])

  The function `to_poly_solve` tries to solve the equations e for the variables l. The equation(s) e can either be a single expression or a set or list of expressions; similarly, l can either be a single symbol or a list of set of symbols. When a member of e isn't explicitly an equation, for example x^2 -1, the solver asummes that the expression vanishes.

  The basic strategy of `to_poly_solve` is to convert the input into a polynomial form and to call `algsys` on the polynomial system. Internally `to_poly_solve` defaults `algexact` to true. To change the default for `algexact`, append 'algexact=false to the `to_poly_solve` argument list.

  When `to_poly_solve` is able to determine the solution set, each member of the solution set is a list in a `%union` object:

      (%i1) load(to_poly_solve)$

      (%i2) to_poly_solve(x*(x-1) = 0, x);
      (%o2)               %union([x = 0], [x = 1])

  When `to_poly_solve` is unable to determine the solution set, a `%solve` nounform is returned (in this case, a warning is printed)

      (%i3) to_poly_solve(x^k + 2* x + 1 = 0, x);

      Nonalgebraic argument given to 'to_poly'
      unable to solve
                                k
      (%o3)            %solve([x  + 2 x + 1 = 0], [x])

  Subsitution into a `%solve` nounform can sometimes result in the solution

      (%i4) subst(k = 2, %);
      (%o4)                   %union([x = - 1])

  Especially for trigonometric equations, the solver sometimes needs to introduce an arbitrary integer. These arbitrary integers have the form `%zXXX`, where `XXX` is an integer; for example

      (%i5) to_poly_solve(sin(x) = 0, x);
      (%o5)   %union([x = 2 %pi %z33 + %pi], [x = 2 %pi %z35])

  To re-index these variables to zero, use `nicedummies`:

      (%i6) nicedummies(%);
      (%o6)    %union([x = 2 %pi %z0 + %pi], [x = 2 %pi %z1])

  Occasionally, the solver introduces an arbitrary complex number of the form `%cXXX` or an arbitrary real number of the form `%rXXX`. The function `nicedummies` will re-index these identifiers to zero.

  The solution set sometimes involves simplifing versions of various of logical operators including `%and`, `%or`, or `%if` for conjunction, disjuntion, and implication, respectively; for example

      (%i7) sol : to_poly_solve(abs(x) = a, x);
      (%o7) %union(%if(isnonnegative_p(a), [x = - a], %union()),
                            %if(isnonnegative_p(a), [x = a], %union()))
      (%i8) subst(a = 42, sol);
      (%o8)             %union([x = - 42], [x = 42])
      (%i9) subst(a = -42, sol);
      (%o9)                       %union()

  The empty set is represented by `%union()`.

  The function `to_poly_solve` is able to solve some, but not all, equations involving rational powers, some nonrational powers, absolute values, trigonometric functions, and minimum and maximum. Also, some it can solve some equations that are solvable in in terms of the Lambert W function; some examples:

      (%i1) load(to_poly_solve)$

      (%i2) to_poly_solve(set(max(x,y) = 5, x+y = 2), set(x,y));
      (%o2)      %union([x = - 3, y = 5], [x = 5, y = - 3])
      (%i3) to_poly_solve(abs(1-abs(1-x)) = 10,x);
      (%o3)             %union([x = - 10], [x = 12])
      (%i4) to_poly_solve(set(sqrt(x) + sqrt(y) = 5, x + y = 10),
                          set(x,y));
                           3/2               3/2
                          5    %i - 10      5    %i + 10
      (%o4) %union([x = - ------------, y = ------------],
                               2                 2
                                      3/2                 3/2
                                     5    %i + 10        5    %i - 10
                                [x = ------------, y = - ------------])
                                          2                   2
      (%i5) to_poly_solve(cos(x) * sin(x) = 1/2,x,
                          'simpfuncs = ['expand, 'nicedummies]);
                                               %pi
      (%o5)              %union([x = %pi %z0 + ---])
                                                4
      (%i6) to_poly_solve(x^(2*a) + x^a + 1,x);
                                              2 %i %pi %z81
                                              -------------
                                        1/a         a
                        (sqrt(3) %i - 1)    %e
      (%o6) %union([x = -----------------------------------],
                                        1/a
                                       2
                                                        2 %i %pi %z83
                                                        -------------
                                                  1/a         a
                                (- sqrt(3) %i - 1)    %e
                           [x = -------------------------------------])
                                                 1/a
                                                2
      (%i7) to_poly_solve(x * exp(x) = a, x);
      (%o7)              %union([x = lambert_w(a)])

  For linear inequalities, `to_poly_solve` automatically does Fourier elimination:

      (%i8) to_poly_solve([x + y < 1, x - y >= 8], [x,y]);
                                     7
      (%o8) %union([x = y + 8, y < - -],
                                     2
                                                                    7
                                       [y + 8 < x, x < 1 - y, y < - -])
                                                                    2

  Each optional argument to `to_poly_solve` must be an equation; generally, the order of these options does not matter.

  - `simpfuncs = l`, where `l` is a list of functions. Apply the composition of the members of l to each solution.

        (%i1) to_poly_solve(x^2=%i,x);
                                       1/4             1/4
        (%o1)       %union([x = - (- 1)   ], [x = (- 1)   ])
        (%i2) to_poly_solve(x^2= %i,x, 'simpfuncs = ['rectform]);
                              %i         1             %i         1
        (%o2) %union([x = - ------- - -------], [x = ------- + -------])
                            sqrt(2)   sqrt(2)        sqrt(2)   sqrt(2)

    Sometimes additional simplification can revert a simplification; for example

        (%i3) to_poly_solve(x^2=1,x);
        (%o3)              %union([x = - 1], [x = 1])
        (%i4) to_poly_solve(x^2= 1,x, 'simpfuncs = [polarform]);
                                                %i %pi
        (%o4)            %union([x = 1], [x = %e      ]

    Maxima doesn't try to check that each member of the function list `l` is purely a simplification; thus

        (%i5) to_poly_solve(x^2 = %i,x, 'simpfuncs = [lambda([s],s^2)]);
        (%o5)                   %union([x = %i])

    To convert each solution to a double float, use `simpfunc = ['dfloat]`:

        (%i6) to_poly_solve(x^3 +x + 1 = 0,x,
                            'simpfuncs = ['dfloat]), algexact : true;
        (%o6) %union([x = - .6823278038280178],
        [x = .3411639019140089 - 1.161541399997251 %i],
        [x = 1.161541399997251 %i + .3411639019140089])

  - `use_grobner = true` With this option, the function `poly_reduced_grobner` is applied to the equations before attempting their solution. Primarily, this option provides a workaround for weakness in the function `algsys`. Here is an example of such a workaround:

        (%i7) to_poly_solve([x^2+y^2=2^2,(x-1)^2+(y-1)^2=2^2],[x,y],
                            'use_grobner = true);
                            sqrt(7) - 1      sqrt(7) + 1
        (%o7) %union([x = - -----------, y = -----------],
                                 2                2
                                         sqrt(7) + 1        sqrt(7) - 1
                                    [x = -----------, y = - -----------])
                                              2                  2
        (%i8) to_poly_solve([x^2+y^2=2^2,(x-1)^2+(y-1)^2=2^2],[x,y]);
        (%o8)                       %union()

  - `maxdepth = k`, where `k` is a positive integer. This function controls the maximum recursion depth for the solver. The default value for `maxdepth` is five. When the recursions depth is exceeded, the solver signals an error:

        (%i9) to_poly_solve(cos(x) = x,x, 'maxdepth = 2);

        Unable to solve
        Unable to solve
        (%o9)        %solve([cos(x) = x], [x], maxdepth = 2)

  - `parameters = l`, where `l` is a list of symbols. The solver attempts to return a solution that is valid for all members of the list `l`; for example:

        (%i10) to_poly_solve(a * x = x, x);
        (%o10)                   %union([x = 0])
        (%i11) to_poly_solve(a * x = x, x, 'parameters = [a]);
        (%o11) %union(%if(a - 1 = 0, [x = %c111], %union()),
                                       %if(a - 1 # 0, [x = 0], %union()))

    In `(%o2)`, the solver introduced a dummy variable; to re-index the these dummy variables, use the function `nicedummies`:

        (%i12) nicedummies(%);
        (%o12) %union(%if(a - 1 = 0, [x = %c0], %union()),
                                       %if(a - 1 # 0, [x = 0], %union()))

  The `to_poly_solve` uses data stored in the hashed array `one_to_one_reduce` to solve equations of the form f(a) = f(b). The assignment `one_to_one_reduce['f,'f] : lambda([a,b], a=b)` tells `to_poly_solve` that the solution set of f(a) = f(b) equals the solution set of a=b; for example

      (%i13) one_to_one_reduce['f,'f] : lambda([a,b], a=b)$

      (%i14) to_poly_solve(f(x^2-1) = f(0),x);
      (%o14)             %union([x = - 1], [x = 1])

  More generally, the assignment `one_to_one_reduce['f,'g] : lambda([a,b], w(a, b) = 0` tells `to_poly_solve` that the solution set of f(a) = f(b) equals the solution set of w(a,b) = 0; for example

      (%i15) one_to_one_reduce['f,'g] : lambda([a,b], a = 1 + b/2)$

      (%i16) to_poly_solve(f(x) - g(x),x);
      (%o16)                   %union([x = 2])

  Additionally, the function `to_poly_solve` uses data stored in the hashed array `function_inverse` to solve equations of the form f(a) = b. The assignment `function_inverse['f] : lambda([s], g(s))` informs `to_poly_solve` that the solution set to `f(x) = b` equals the solution set to `x = g(b)`; two examples:

      (%i17) function_inverse['Q] : lambda([s], P(s))$

      (%i18) to_poly_solve(Q(x-1) = 2009,x);
      (%o18)              %union([x = P(2009) + 1])
      (%i19) function_inverse['G] : lambda([s], s+new_variable(integer));
      (%o19)       lambda([s], s + new_variable(integer))
      (%i20) to_poly_solve(G(x - a) = b,x);
      (%o20)             %union([x = b + a + %z125])

  Notes

      (%i1) to_poly_solve([x^2 + y^2 + x * y = 5, x * y = 8],
                          [x^2 + y^2, x * y]);
                                        2    2
      (%o1)           %union([x y = 8, y  + x  = - 3])

      (%i1) declare(x,complex)$

      (%i2) to_poly_solve(x + (5 + %i) * conjugate(x) = 1, x);
                                         %i + 21
      (%o2)              %union([x = - -----------])
                                       25 %i - 125
      (%i3) declare(y,complex)$

      (%i4) to_poly_solve(set(conjugate(x) - y = 42 + %i,
                              x + conjugate(y) = 0), set(x,y));
                                 %i - 42        %i + 42
      (%o4)        %union([x = - -------, y = - -------])
                                    2              2

  Relevant option variables algexact, resultant, algebraic

  Related functions to_poly

  To use ``load(to_poly_solve)'`

  Status: The function `to_poly_solve` is experimental; its specifications might change and its functionality might be merged into other Maxima functions.

Operator: %union (soln_1, soln_2, soln_3, ...)
Operator: %union ()

  `%union(soln_1, soln_2, soln_3, ...)` represents the union of its arguments, each of which represents a solution set, as determined by `to_poly_solve`. `%union()` represents the empty set.

  In many cases, a solution is a list of equations `[x = ..., y = ..., z = ...]` where x, y, and z are one or more unknowns. In such cases, `to_poly_solve` returns a `%union` expression containing one or more such lists.

  The solution set sometimes involves simplifing versions of various of logical operators including `%and`, `%or`, or `%if` for conjunction, disjuntion, and implication, respectively.

  Examples:

  `%union(...)` represents the union of its arguments, each of which represents a solution set, as determined by `to_poly_solve`. In many cases, a solution is a list of equations.

      (%i1) load ("to_poly_solve") $
      (%i2) to_poly_solve ([sqrt(x^2 - y^2), x + y], [x, y]);
      (%o2)    %union([x = 0, y = 0], [x = %c13, y = - %c13])

  `%union()` represents the empty set.

      (%i1) load ("to_poly_solve") $
      (%i2) to_poly_solve (abs(x) = -1, x);
      (%o2)                       %union()

  The solution set sometimes involves simplifing versions of various of logical operators.

      (%i1) load ("to_poly_solve") $
      (%i2) sol : to_poly_solve (abs(x) = a, x);
      (%o2) %union(%if(isnonnegative_p(a), [x = - a], %union()),
                            %if(isnonnegative_p(a), [x = a], %union()))
      (%i3) subst (a = 42, sol);
      (%o3)             %union([x = - 42], [x = 42])
      (%i4) subst (a = -42, sol);
      (%o4)                       %union()

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# 86. unit

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## 86.1 Introduction to Units

The unit package enables the user to convert between arbitrary units and work with dimensions in equations. The functioning of this package is radically different from the original Maxima units package - whereas the original was a basic list of definitions, this package uses rulesets to allow the user to chose, on a per dimension basis, what unit final answers should be rendered in. It will separate units instead of intermixing them in the display, allowing the user to readily identify the units associated with a particular answer. It will allow a user to simplify an expression to its fundamental Base Units, as well as providing fine control over simplifying to derived units. Dimensional analysis is possible, and a variety of tools are available to manage conversion and simplification options. In addition to customizable automatic conversion, units also provides a traditional manual conversion option.

Note - when unit conversions are inexact Maxima will make approximations resulting in fractions. This is a consequence of the techniques used to simplify units. The messages warning of this type of substitution are disabled by default in the case of units (normally they are on) since this situation occurs frequently and the warnings clutter the output. (The existing state of ratprint is restored after unit conversions, so user changes to that setting will be preserved otherwise.) If the user needs this information for units, they can set unitverbose:on to reactivate the printing of warnings from the unit conversion process.

unit is included in Maxima in the share/contrib/unit directory. It obeys normal Maxima package loading conventions:

    (%i1) load("unit")$
    *******************************************************************
    *                       Units version 0.50                        *
    *          Definitions based on the NIST Reference on             *
    *              Constants, Units, and Uncertainty                  *
    *       Conversion factors from various sources including         *
    *                   NIST and the GNU units package                *
    *******************************************************************

    Redefining necessary functions...
    WARNING: DEFUN/DEFMACRO: redefining function TOPLEVEL-MACSYMA-EVAL ...
    WARNING: DEFUN/DEFMACRO: redefining function MSETCHK ...
    WARNING: DEFUN/DEFMACRO: redefining function KILL1 ...
    WARNING: DEFUN/DEFMACRO: redefining function NFORMAT ...
    Initializing unit arrays...
    Done.

The WARNING messages are expected and not a cause for concern - they indicate the unit package is redefining functions already defined in Maxima proper. This is necessary in order to properly handle units. The user should be aware that if other changes have been made to these functions by other packages those changes will be overwritten by this loading process.

The unit.mac file also loads a lisp file unit-functions.lisp which contains the lisp functions needed for the package.

Clifford Yapp is the primary author. He has received valuable assistance from Barton Willis of the University of Nebraska at Kearney (UNK), Robert Dodier, and other intrepid folk of the Maxima mailing list.

There are probably lots of bugs. Let me know. `float` and `numer` don't do what is expected.

TODO : dimension functionality, handling of temperature, showabbr and friends. Show examples with addition of quantities containing units.

Categories: Physical units · Share packages · Package unit

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## 86.2 Functions and Variables for Units

Function: setunits (list)

  By default, the unit package does not use any derived dimensions, but will convert all units to the seven fundamental dimensions using MKS units.

      (%i2) N;
                                           kg m
      (%o2)                                ----
                                             2
                                            s
      (%i3) dyn;
                                         1      kg m
      (%o3)                           (------) (----)
                                       100000     2
                                                 s
      (%i4) g;
                                          1
      (%o4)                             (----) (kg)
                                         1000
      (%i5) centigram*inch/minutes^2;
                                        127        kg m
      (%o5)                       (-------------) (----)
                                   1800000000000     2
                                                    s

  In some cases this is the desired behavior. If the user wishes to use other units, this is achieved with the `setunits` command:

      (%i6) setunits([centigram,inch,minute]);
      (%o6)                                done
      (%i7) N;
                                  1800000000000   %in cg
      (%o7)                      (-------------) (------)
                                       127            2
                                                  %min
      (%i8) dyn;
                                     18000000   %in cg
      (%o8)                         (--------) (------)
                                       127          2
                                                %min
      (%i9) g;
      (%o9)                             (100) (cg)
      (%i10) centigram*inch/minutes^2;
                                          %in cg
      (%o10)                              ------
                                              2
                                          %min

  The setting of units is quite flexible. For example, if we want to get back to kilograms, meters, and seconds as defaults for those dimensions we can do:

      (%i11) setunits([kg,m,s]);
      (%o11)                               done
      (%i12) centigram*inch/minutes^2;
                                        127        kg m
      (%o12)                      (-------------) (----)
                                   1800000000000     2
                                                    s

  Derived units are also handled by this command:

      (%i17) setunits(N);
      (%o17)                               done
      (%i18) N;
      (%o18)                                 N
      (%i19) dyn;
                                          1
      (%o19)                           (------) (N)
                                        100000
      (%i20) kg*m/s^2;
      (%o20)                                 N
      (%i21) centigram*inch/minutes^2;
                                          127
      (%o21)                        (-------------) (N)
                                     1800000000000

  Notice that the unit package recognized the non MKS combination of mass, length, and inverse time squared as a force, and converted it to Newtons. This is how Maxima works in general. If, for example, we prefer dyne to Newtons, we simply do the following:

      (%i22) setunits(dyn);
      (%o22)                               done
      (%i23) kg*m/s^2;
      (%o23)                          (100000) (dyn)
      (%i24) centigram*inch/minutes^2;
                                        127
      (%o24)                         (--------) (dyn)
                                      18000000

  To discontinue simplifying to any force, we use the uforget command:

      (%i26) uforget(dyn);
      (%o26)                               false
      (%i27) kg*m/s^2;
                                           kg m
      (%o27)                               ----
                                             2
                                            s
      (%i28) centigram*inch/minutes^2;
                                        127        kg m
      (%o28)                      (-------------) (----)
                                   1800000000000     2
                                                    s

  This would have worked equally well with `uforget(N)` or `uforget(%force)`.

  See also `uforget`. To use this function write first `load("unit")`.

  Categories: Package unit

Function: uforget (list)

  By default, the unit package converts all units to the seven fundamental dimensions using MKS units. This behavior can be changed with the `setunits` command. After that, the user can restore the default behavior for a particular dimension by means of the `uforget` command:

      (%i13) setunits([centigram,inch,minute]);
      (%o13)                               done
      (%i14) centigram*inch/minutes^2;
                                          %in cg
      (%o14)                              ------
                                              2
                                          %min
      (%i15) uforget([cg,%in,%min]);
      (%o15)                      [false, false, false]
      (%i16) centigram*inch/minutes^2;
                                        127        kg m
      (%o16)                      (-------------) (----)
                                   1800000000000     2
                                                    s

  `uforget` operates on dimensions, not units, so any unit of a particular dimension will work. The dimension itself is also a legal argument.

  See also `setunits`. To use this function write first `load("unit")`.

  Categories: Package unit

Function: convert (expr, list)

  When resetting the global environment is overkill, there is the `convert` command, which allows one time conversions. It can accept either a single argument or a list of units to use in conversion. When a convert operation is done, the normal global evaluation system is bypassed, in order to avoid the desired result being converted again. As a consequence, for inexact calculations "rat" warnings will be visible if the global environment controlling this behavior (`ratprint`) is true. This is also useful for spot-checking the accuracy of a global conversion. Another feature is `convert` will allow a user to do Base Dimension conversions even if the global environment is set to simplify to a Derived Dimension.

      (%i2) kg*m/s^2;
                                           kg m
      (%o2)                                ----
                                             2
                                            s
      (%i3) convert(kg*m/s^2,[g,km,s]);
                                           g km
      (%o3)                                ----
                                             2
                                            s
      (%i4) convert(kg*m/s^2,[g,inch,minute]);

      `rat' replaced 39.37007874015748 by 5000/127 = 39.37007874015748
                                    18000000000   %in g
      (%o4)                        (-----------) (-----)
                                        127           2
                                                  %min
      (%i5) convert(kg*m/s^2,[N]);
      (%o5)                                  N
      (%i6) convert(kg*m^2/s^2,[N]);
      (%o6)                                 m N
      (%i7) setunits([N,J]);
      (%o7)                                done
      (%i8) convert(kg*m^2/s^2,[N]);
      (%o8)                                 m N
      (%i9) convert(kg*m^2/s^2,[N,inch]);

      `rat' replaced 39.37007874015748 by 5000/127 = 39.37007874015748
                                       5000
      (%o9)                           (----) (%in N)
                                       127
      (%i10) convert(kg*m^2/s^2,[J]);
      (%o10)                                 J
      (%i11) kg*m^2/s^2;
      (%o11)                                 J
      (%i12) setunits([g,inch,s]);
      (%o12)                               done
      (%i13) kg*m/s^2;
      (%o13)                                 N
      (%i14) uforget(N);
      (%o14)                               false
      (%i15) kg*m/s^2;
                                      5000000   %in g
      (%o15)                         (-------) (-----)
                                        127       2
                                                 s
      (%i16) convert(kg*m/s^2,[g,inch,s]);

      `rat' replaced 39.37007874015748 by 5000/127 = 39.37007874015748
                                      5000000   %in g
      (%o16)                         (-------) (-----)
                                        127       2
                                                 s

  See also `setunits` and `uforget`. To use this function write first `load("unit")`.

  Categories: Package unit

Optional variable: usersetunits

  Default value: none

  If a user wishes to have a default unit behavior other than that described, they can make use of maxima-init.mac and the usersetunits variable. The unit package will check on startup to see if this variable has been assigned a list. If it has, it will use setunits on that list and take the units from that list to be defaults. `uforget` will revert to the behavior defined by usersetunits over its own defaults. For example, if we have a maxima-init.mac file containing:

      usersetunits : [N,J];

  we would see the following behavior:

      (%i1) load("unit")$
      *******************************************************************
      *                       Units version 0.50                        *
      *          Definitions based on the NIST Reference on             *
      *              Constants, Units, and Uncertainty                  *
      *       Conversion factors from various sources including         *
      *                   NIST and the GNU units package                *
      *******************************************************************

      Redefining necessary functions...
      WARNING: DEFUN/DEFMACRO: redefining function
       TOPLEVEL-MACSYMA-EVAL ...
      WARNING: DEFUN/DEFMACRO: redefining function MSETCHK ...
      WARNING: DEFUN/DEFMACRO: redefining function KILL1 ...
      WARNING: DEFUN/DEFMACRO: redefining function NFORMAT ...
      Initializing unit arrays...
      Done.
      User defaults found...
      User defaults initialized.
      (%i2) kg*m/s^2;
      (%o2)                                  N
      (%i3) kg*m^2/s^2;
      (%o3)                                  J
      (%i4) kg*m^3/s^2;
      (%o4)                                 J m
      (%i5) kg*m*km/s^2;
      (%o5)                             (1000) (J)
      (%i6) setunits([dyn,eV]);
      (%o6)                                done
      (%i7) kg*m/s^2;
      (%o7)                           (100000) (dyn)
      (%i8) kg*m^2/s^2;
      (%o8)                     (6241509596477042688) (eV)
      (%i9) kg*m^3/s^2;
      (%o9)                    (6241509596477042688) (eV m)
      (%i10) kg*m*km/s^2;
      (%o10)                   (6241509596477042688000) (eV)
      (%i11) uforget([dyn,eV]);
      (%o11)                           [false, false]
      (%i12) kg*m/s^2;
      (%o12)                                 N
      (%i13) kg*m^2/s^2;
      (%o13)                                 J
      (%i14) kg*m^3/s^2;
      (%o14)                                J m
      (%i15) kg*m*km/s^2;
      (%o15)                            (1000) (J)

  Without `usersetunits`, the initial inputs would have been converted to MKS, and uforget would have resulted in a return to MKS rules. Instead, the user preferences are respected in both cases. Notice these can still be overridden if desired. To completely eliminate this simplification - i.e. to have the user defaults reset to factory defaults - the `dontusedimension` command can be used. `uforget` can restore user settings again, but only if `usedimension` frees it for use. Alternately, `kill(usersetunits)` will completely remove all knowledge of the user defaults from the session. Here are some examples of how these various options work.

      (%i2) kg*m/s^2;
      (%o2)                                  N
      (%i3) kg*m^2/s^2;
      (%o3)                                  J
      (%i4) setunits([dyn,eV]);
      (%o4)                                done
      (%i5) kg*m/s^2;
      (%o5)                           (100000) (dyn)
      (%i6) kg*m^2/s^2;
      (%o6)                     (6241509596477042688) (eV)
      (%i7) uforget([dyn,eV]);
      (%o7)                          [false, false]
      (%i8) kg*m/s^2;
      (%o8)                                  N
      (%i9) kg*m^2/s^2;
      (%o9)                                  J
      (%i10) dontusedimension(N);
      (%o10)                             [%force]
      (%i11) dontusedimension(J);
      (%o11)                         [%energy, %force]
      (%i12) kg*m/s^2;
                                           kg m
      (%o12)                               ----
                                             2
                                            s
      (%i13) kg*m^2/s^2;
                                               2
                                           kg m
      (%o13)                               -----
                                             2
                                            s
      (%i14) setunits([dyn,eV]);
      (%o14)                               done
      (%i15) kg*m/s^2;
                                           kg m
      (%o15)                               ----
                                             2
                                            s
      (%i16) kg*m^2/s^2;
                                               2
                                           kg m
      (%o16)                               -----
                                             2
                                            s
      (%i17) uforget([dyn,eV]);
      (%o17)                         [false, false]
      (%i18) kg*m/s^2;
                                           kg m
      (%o18)                               ----
                                             2
                                            s
      (%i19) kg*m^2/s^2;
                                               2
                                           kg m
      (%o19)                               -----
                                             2
                                            s
      (%i20) usedimension(N);
      Done.  To have Maxima simplify to this dimension, use
      setunits([unit]) to select a unit.
      (%o20)                               true
      (%i21) usedimension(J);
      Done.  To have Maxima simplify to this dimension, use
      setunits([unit]) to select a unit.
      (%o21)                               true
      (%i22) kg*m/s^2;
                                           kg m
      (%o22)                               ----
                                             2
                                            s
      (%i23) kg*m^2/s^2;
                                               2
                                           kg m
      (%o23)                               -----
                                             2
                                            s
      (%i24) setunits([dyn,eV]);
      (%o24)                               done
      (%i25) kg*m/s^2;
      (%o25)                          (100000) (dyn)
      (%i26) kg*m^2/s^2;
      (%o26)                    (6241509596477042688) (eV)
      (%i27) uforget([dyn,eV]);
      (%o27)                           [false, false]
      (%i28) kg*m/s^2;
      (%o28)                                 N
      (%i29) kg*m^2/s^2;
      (%o29)                                 J
      (%i30) kill(usersetunits);
      (%o30)                               done
      (%i31) uforget([dyn,eV]);
      (%o31)                          [false, false]
      (%i32) kg*m/s^2;
                                           kg m
      (%o32)                               ----
                                             2
                                            s
      (%i33) kg*m^2/s^2;
                                               2
                                           kg m
      (%o33)                               -----
                                             2
                                            s

  Unfortunately this wide variety of options is a little confusing at first, but once the user grows used to them they should find they have very full control over their working environment.

  Categories: Package unit

Function: metricexpandall (x)

  Rebuilds global unit lists automatically creating all desired metric units. x is a numerical argument which is used to specify how many metric prefixes the user wishes defined. The arguments are as follows, with each higher number defining all lower numbers' units:

                 0 - none. Only base units
                 1 - kilo, centi, milli
      (default)  2 - giga, mega, kilo, hecto, deka, deci, centi, milli,
                     micro, nano
                 3 - peta, tera, giga, mega, kilo, hecto, deka, deci,
                     centi, milli, micro, nano, pico, femto
                 4 - all

  Normally, Maxima will not define the full expansion since this results in a very large number of units, but `metricexpandall` can be used to rebuild the list in a more or less complete fashion. The relevant variable in the unit.mac file is %unitexpand.

  Categories: Package unit

Variable: %unitexpand

  Default value: `2`

  This is the value supplied to `metricexpandall` during the initial loading of unit.

  Categories: Package unit

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# 87. zeilberger

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## 87.1 Introduction to zeilberger

`zeilberger` is a implementation of Zeilberger's algorithm for definite hypergeometric summation, and also Gosper's algorithm for indefinite hypergeometric summation.

`zeilberger` makes use of the "filtering" optimization method developed by Axel Riese.

`zeilberger` was developed by Fabrizio Caruso.

`load ("zeilberger")` loads this package.

Categories: Sums and products · Share packages · Package zeilberger

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### 87.1.1 The indefinite summation problem

`zeilberger` implements Gosper's algorithm for indefinite hypergeometric summation. Given a hypergeometric term F_k in k we want to find its hypergeometric anti-difference, that is, a hypergeometric term f_k such that

F_k = f_(k+1) - f_k.

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### 87.1.2 The definite summation problem

`zeilberger` implements Zeilberger's algorithm for definite hypergeometric summation. Given a proper hypergeometric term (in n and k) F_(n,k) and a positive integer d we want to find a d-th order linear recurrence with polynomial coefficients (in n) for F_(n,k) and a rational function R in n and k such that

a_0 F_(n,k) + ... + a_d F_(n+d),k = Delta_k(R(n,k) F_(n,k)),

where Delta_k is the k-forward difference operator, i.e., Delta_k(t_k) := t_(k+1) - t_k.

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### 87.1.3 Verbosity levels

There are also verbose versions of the commands which are called by adding one of the following prefixes:

`Summary`

  Just a summary at the end is shown

`Verbose`

  Some information in the intermidiate steps

`VeryVerbose`

  More information

`Extra`

  Even more information including information on the linear system in Zeilberger's algorithm

For example:
`GosperVerbose`, `parGosperVeryVerbose`, `ZeilbergerExtra`, `AntiDifferenceSummary`.

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## 87.2 Functions and Variables for zeilberger

Function: AntiDifference (F_k, k)

  Returns the hypergeometric anti-difference of F_k, if it exists.
  Otherwise `AntiDifference` returns `no_hyp_antidifference`.

  Categories: Package zeilberger

Function: Gosper (F_k, k)

  Returns the rational certificate R(k) for F_k, that is, a rational function such that F_k = R(k+1) F_(k+1) - R(k) F_k, if it exists. Otherwise, `Gosper` returns `no_hyp_sol`.

  Categories: Package zeilberger

Function: GosperSum (F_k, k, a, b)

  Returns the summmation of F_k from k = a to k = b if F_k has a hypergeometric anti-difference. Otherwise, `GosperSum` returns `nongosper_summable`.

  Examples:

      (%i1) load ("zeilberger")$
      (%i2) GosperSum ((-1)^k*k / (4*k^2 - 1), k, 1, n);
      Dependent equations eliminated:  (1)
                                 3       n + 1
                            (n + -) (- 1)
                                 2               1
      (%o2)               - ------------------ - -
                                        2        4
                            2 (4 (n + 1)  - 1)
      (%i3) GosperSum (1 / (4*k^2 - 1), k, 1, n);
                                      3
                                - n - -
                                      2       1
      (%o3)                  -------------- + -
                                      2       2
                             4 (n + 1)  - 1
      (%i4) GosperSum (x^k, k, 1, n);
                                n + 1
                               x          x
      (%o4)                    ------ - -----
                               x - 1    x - 1
      (%i5) GosperSum ((-1)^k*a! / (k!*(a - k)!), k, 1, n);
                                      n + 1
                      a! (n + 1) (- 1)              a!
      (%o5)       - ------------------------- - ----------
                    a (- n + a - 1)! (n + 1)!   a (a - 1)!
      (%i6) GosperSum (k*k!, k, 1, n);
      Dependent equations eliminated:  (1)
      (%o6)                     (n + 1)! - 1
      (%i7) GosperSum ((k + 1)*k! / (k + 1)!, k, 1, n);
                        (n + 1) (n + 2) (n + 1)!
      (%o7)             ------------------------ - 1
                                (n + 2)!
      (%i8) GosperSum (1 / ((a - k)!*k!), k, 1, n);
      (%o8)                  NON_GOSPER_SUMMABLE

  Categories: Package zeilberger

Function: parGosper (F_(n,k), k, n, d)

  Attempts to find a d-th order recurrence for F_(n,k).

  The algorithm yields a sequence [s_1, s_2, ..., s_m] of solutions. Each solution has the form

  [R(n, k), [a_0, a_1, ..., a_d]].

  `parGosper` returns `[]` if it fails to find a recurrence.

  Categories: Package zeilberger

Function: Zeilberger (F_(n,k), k, n)

  Attempts to compute the indefinite hypergeometric summation of F_(n,k).

  `Zeilberger` first invokes `Gosper`, and if that fails to find a solution, then invokes `parGosper` with order 1, 2, 3, ..., up to `MAX_ORD`. If Zeilberger finds a solution before reaching `MAX_ORD`, it stops and returns the solution.

  The algorithms yields a sequence [s_1, s_2, ..., s_m] of solutions. Each solution has the form

  [R(n,k), [a_0, a_1, ..., a_d]].

  `Zeilberger` returns `[]` if it fails to find a solution.

  `Zeilberger` invokes `Gosper` only if `Gosper_in_Zeilberger` is `true`.

  Categories: Package zeilberger

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## 87.3 General global variables

Global variable: MAX_ORD

  Default value: 5

  `MAX_ORD` is the maximum recurrence order attempted by `Zeilberger`.

  Categories: Package zeilberger

Global variable: simplified_output

  Default value: `false`

  When `simplified_output` is `true`, functions in the `zeilberger` package attempt further simplification of the solution.

  Categories: Package zeilberger

Global variable: linear_solver

  Default value: `linsolve`

  `linear_solver` names the solver which is used to solve the system of equations in Zeilberger's algorithm.

  Categories: Package zeilberger

Global variable: warnings

  Default value: `true`

  When `warnings` is `true`, functions in the `zeilberger` package print warning messages during execution.

  Categories: Package zeilberger

Global variable: Gosper_in_Zeilberger

  Default value: `true`

  When `Gosper_in_Zeilberger` is `true`, the `Zeilberger` function calls `Gosper` before calling `parGosper`. Otherwise, `Zeilberger` goes immediately to `parGosper`.

  Categories: Package zeilberger

Global variable: trivial_solutions

  Default value: `true`

  When `trivial_solutions` is `true`, `Zeilberger` returns solutions which have certificate equal to zero, or all coefficients equal to zero.

  Categories: Package zeilberger

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## 87.4 Variables related to the modular test

Global variable: mod_test

  Default value: `false`

  When `mod_test` is `true`, `parGosper` executes a modular test for discarding systems with no solutions.

  Categories: Package zeilberger

Global variable: modular_linear_solver

  Default value: `linsolve`

  `modular_linear_solver` names the linear solver used by the modular test in `parGosper`.

  Categories: Package zeilberger

Global variable: ev_point

  Default value: `big_primes[10]`

  `ev_point` is the value at which the variable n is evaluated when executing the modular test in `parGosper`.

  Categories: Package zeilberger

Global variable: mod_big_prime

  Default value: `big_primes[1]`

  `mod_big_prime` is the modulus used by the modular test in `parGosper`.

  Categories: Package zeilberger

Global variable: mod_threshold

  Default value: 4

  `mod_threshold` is the greatest order for which the modular test in `parGosper` is attempted.

  Categories: Package zeilberger

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# 88. Error and warning messages

This chapter provides detailed information about the meaning of some error messages or on how to recover from errors.

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## 88.1 Error messages

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### 88.1.1 apply: no such "list" element

One common cause for this error message is that square brackets operator (`[ ]`) was used trying to access a list element that whose element number was `< 1` or `> length(list)`.

Error messages

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### 88.1.2 argument must be a non-atomic expression

This normally means that a list, a set or something else that consists of more than one element was expected. One possible cause for this error message is a construct of the following type:

    (%i1) l:[1,2,3];
    (%o1)                       [1, 2, 3]
    (%i2) append(l,4);
    append: argument must be a non-atomic expression; found 4
     -- an error. To debug this try: debugmode(true);

The correct way to append variables or numbers to a list is to wrap them in a single-element list first:

    (%i1) l:[1,2,3];
    (%o1)                       [1, 2, 3]
    (%i2) append(l,[4]);
    (%o2)                     [1, 2, 3, 4]

Categories: Error messages

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### 88.1.3 assignment: cannot assign to <function name>

Maxima supports several assignment operators. When trying to define a function `:=` has to be used.

Error messages

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### 88.1.4 expt: undefined: 0 to a negative exponent.

This message notifies about a classical division by zero error.

Error messages

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### 88.1.5 incorrect syntax: , is not a prefix operator

This might be caused by a command starting with a comma (`,`) or by one comma being directly followed by another one..

Error messages

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### 88.1.6 incorrect syntax: Illegal use of delimiter )

Common reasons for this error appearing are a closing parenthesis without an opening one or a closing parenthesis directly preceded by a comma.

Error messages

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### 88.1.7 loadfile: failed to load <filename>

This error message normally indicates that the file exists, but can not be read. If the file is present and readable there is another possible for this error message: Maxima can compile packages to native binary files in order to make them run faster. If after compiling the file something in the system has changed in a way that makes it incompatible with the binary the binary the file cannot be loaded any more. Maxima normally puts binary files it creates from its own packages in a folder named `binary` within the folder whose name it is printed after typing:

    (%i1) maxima_userdir;
    (%o1)                 /home/gunter/.maxima

If this directory is missing maxima will recreate it again as soon as it has to compile a package.

Error messages

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### 88.1.8 makelist: second argument must evaluate to a number

`makelist` expects the second argument to be the name of the variable whose value is to be stepped. This time instead of the name of a still-undefined variable maxima has found something else, possibly a list or the name of a list.

Categories: Error messages

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### 88.1.9 Only symbols can be bound

The most probable cause for this error is that there was an attempt to either use a number or a variable whose numerical value is known as a loop counter.

Error messages

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### 88.1.10 operators of arguments must all be the same

One possible reason for this error message to appear is a try to use `append` in order to add an equation to a list:

    (%i1) l:[a=1,b=2,c=3];
    (%o1)                 [a = 1, b = 2, c = 3]
    (%i2) append(l,d=5);
    append: operators of arguments must all be the same.
     -- an error. To debug this try: debugmode(true);

In order to add an equation to a list it has to be wrapped in a single-element list first:

    (%i1) l:[a=1,b=2,c=3];
    (%o1)                 [a = 1, b = 2, c = 3]
    (%i2) append(l,[d=5]);
    (%o2)             [a = 1, b = 2, c = 3, d = 5]

Categories: Error messages

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### 88.1.11 Out of memory

Lisp typically handles several types of memory containing at least one stack and a heap that contains user objects. To avoid running out of memory several approaches might be useful:

- If possible, the best solution normally is to use an algorithm that is more memory-efficient.
- Compiling a function might drastically reduce the amount of memory it needs.
- Arrays of a fixed type might be more memory-efficient than lists.
- If maxima is run by sbcl sbcl's memory limit might be set to a value that is too low to solve the current problem. In this case the command-line option `--dynamic-space-size <n>` allows to tell sbcl to reserve `n` megabytes for the heap. It is to note, though, that sbcl has to handle several distinct types of memory and therefore might be able to only reserve about half of the available physical memory. Also note that 32-bit processes might only be able to access 2GB of physical memory.

Categories: Error messages

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### 88.1.12 part: fell off the end

`part()` was used to access the `n`th item in something that has less than `n` items.

Error messages

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### 88.1.13 undefined variable (draw or plot)

A function could not be plotted since it still contained a variable maxima doesn't know the value of.

In order to find out which variable this could be it is sometimes helpful to temporarily replace the name of the drawing command (`draw2d`, `plot2d` or similar) by a random name (for example `ddraw2d`) that doesn't coincide with the name of an existing function to make maxima print out what parameters the drawing command sees.

    (%i1) load("draw")$
    (%i2) f(x):=sin(omega*t);
    (%o2) f(x) := sin(omega t)
    (%i3) draw2d(
            explicit(
              f(x),
              x,1,10
            )
          );
    draw2d (explicit): non defined variable
     -- an error. To debug this try: debugmode(true);
    (%i4) ddraw2d(
            explicit(
              f(x),
              x,1,10
            )
          );
    (%o4) ddraw2d(explicit(sin(omega t), x, 1, 10))

Categories: Error messages

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### 88.1.14 VTK is not installed, which is required for Scene

This might either mean that VTK is actually not installed - or cannot be found by maxima - or that maxima has no write access to the directory whose name is output if the following maxima command is entered:

    (%i1) maxima_tempdir;
    (%o1)                         /tmp

Categories: Error messages

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## 88.2 Warning messages

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### 88.2.1 Encountered undefined variable <x> in translation

A function was compiled but the type of the variable `x` was not known. This means that the compiled command contains additional code that makes it retain all the flexibility maxima provides in respect to this variable. If `x` isn't meant as a variable name but just a named option to a command prepending the named option by a single quote (`'`) should resolve this issue.

Error messages

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### 88.2.2 Rat: replaced <x> by <y> = <z>

Floating-point numbers provide a maximum number of digits that is typically high, but still limited. Good examples that this limitation might be too low even for harmless-looking examples include Wilkinson's Polynomial, The Rump polynomial and the fact that an exact 1/10 cannot be expressed as a binary floating-point number. In places where the floating-point error might add up or hinder terms from cancelling each other out maxima therefore by default replaces them with exact fractions. See also `ratprint`, `ratepsilon`, `bftorat`, `fpprintprec` and `rationalize`.

Warning messages

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# A. Function and Variable Index

Jump to: | ! " # $ % ' * + - . / : ; < = > ? @ [ ] ^ _ ` | ~ A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

| Index Entry | Section
! |  |
| `!` | 10.3 Combinatorial Functions
| `!!` | 10.3 Combinatorial Functions
" |  |
| `''` | 8.1 Functions and Variables for Evaluation
# |  |
| `#` | 7.5 Operators for Equations
$ |  |
| `$` | 4.2 Functions and Variables for Command Line
% |  |
| % | 4.2 Functions and Variables for Command Line
| %% | 4.2 Functions and Variables for Command Line
| `%and` | 85.1 Functions and Variables for to_poly_solve
| %c | 48.2 Functions and Variables for contrib_ode
| %e | 5.3.1 Functions and Variables for Constants
| %e_to_numlog | 10.4 Root, Exponential and Logarithmic Functions
| %edispflag | 4.3 Functions and Variables for Display
| %emode | 10.4 Root, Exponential and Logarithmic Functions
| %enumer | 10.4 Root, Exponential and Logarithmic Functions
| `%f` | 15.8 Hypergeometric Functions
| %gamma | 5.3.1 Functions and Variables for Constants
| %i | 5.3.1 Functions and Variables for Constants
| %iargs | 10.5.2 Functions and Variables for Trigonometric
| `%if` | 85.1 Functions and Variables for to_poly_solve
| %k1 | 48.2 Functions and Variables for contrib_ode
| %k2 | 48.2 Functions and Variables for contrib_ode
| `%m` | 15.8 Hypergeometric Functions
| `%or` | 85.1 Functions and Variables for to_poly_solve
| %phi | 5.3.1 Functions and Variables for Constants
| %pi | 5.3.1 Functions and Variables for Constants
| %piargs | 10.5.2 Functions and Variables for Trigonometric
| %rnum | 20.1 Functions and Variables for Equations
| %rnum_list | 20.1 Functions and Variables for Equations
| `%s` | 15.2 Bessel Functions
| `%th` | 4.2 Functions and Variables for Command Line
| `%union` | 85.1 Functions and Variables for to_poly_solve
| `%union` | 85.1 Functions and Variables for to_poly_solve
| %unitexpand | 86.2 Functions and Variables for Units
| `%w` | 15.8 Hypergeometric Functions
' |  |
| `'` | 8.1 Functions and Variables for Evaluation
* |  |
| `*` | 7.2 Arithmetic operators
| `**` | 7.2 Arithmetic operators
+ |  |
| `+` | 7.2 Arithmetic operators
- |  |
| `-` | 7.2 Arithmetic operators
. |  |
| `.` | 7.2 Arithmetic operators
/ |  |
| `/` | 7.2 Arithmetic operators
: |  |
| `:` | 7.6 Assignment operators
| `::` | 7.6 Assignment operators
| `::=` | 7.6 Assignment operators
| `:=` | 7.6 Assignment operators
; |  |
| `;` | 4.2 Functions and Variables for Command Line
< |  |
| `<` | 7.3 Relational operators
| `<=` | 7.3 Relational operators
= |  |
| `=` | 7.5 Operators for Equations
> |  |
| `>` | 7.3 Relational operators
| `>=` | 7.3 Relational operators
? |  |
| `?` | 4.2 Functions and Variables for Command Line
| `??` | 4.2 Functions and Variables for Command Line
@ |  |
| `@` | 5.6.2 Functions and Variables for Structures
[ |  |
| [ | 5.4.2 Functions and Variables for Lists
] |  |
| ] | 5.4.2 Functions and Variables for Lists
^ |  |
| `^` | 7.2 Arithmetic operators
| `^^` | 7.2 Arithmetic operators
_ |  |
| _ | 4.2 Functions and Variables for Command Line
| __ | 4.2 Functions and Variables for Command Line
` |  |
| ``` | 56.3 Functions and Variables for ezunits
| ``` | 56.3 Functions and Variables for ezunits
| | | 25.2.7 Exterior algebra
~ |  |
| ~ | 25.2.7 Exterior algebra
A |  |
| `abasep` | 27.2 Functions and Variables for atensor
| `abs` | 10.1 Functions for Numbers
| absboxchar | 4.3 Functions and Variables for Display
| `absint` | 28.5 Functions and Variables for Fourier series
| `absolute_real_time` | 32.3 Functions and Variables for Runtime Environment
| `acos` | 10.5.2 Functions and Variables for Trigonometric
| `acosh` | 10.5.2 Functions and Variables for Trigonometric
| `acot` | 10.5.2 Functions and Variables for Trigonometric
| `acoth` | 10.5.2 Functions and Variables for Trigonometric
| `acsc` | 10.5.2 Functions and Variables for Trigonometric
| `acsch` | 10.5.2 Functions and Variables for Trigonometric
| `activate` | 11.3 Functions and Variables for Facts
| activecontexts | 11.3 Functions and Variables for Facts
| adapt_depth | 12.4 Plotting Options
| adapt_depth | 52.2.3 Graphics options
| `add_edge` | 61.2.3 Modifying graphs
| `add_edges` | 61.2.3 Modifying graphs
| `add_vertex` | 61.2.3 Modifying graphs
| `add_vertices` | 61.2.3 Modifying graphs
| `addcol` | 23.2 Functions and Variables for Matrices and Linear Algebra
| additive | 9.2 Functions and Variables for Simplification
| `addmatrices` | 68.2 Functions and Variables for linearalgebra
| `addrow` | 23.2 Functions and Variables for Matrices and Linear Algebra
| adim | 27.2 Functions and Variables for atensor
| `adjacency_matrix` | 61.2.2 Graph properties
| `adjoin` | 35.2 Functions and Variables for Sets
| `adjoint` | 23.2 Functions and Variables for Matrices and Linear Algebra
| `adjust_external_format` | 84.3 Characters
| `af` | 27.2 Functions and Variables for atensor
| aform | 27.2 Functions and Variables for atensor
| `agd` | 80.4 Package functs
| `airy_ai` | 15.3 Airy Functions
| `airy_bi` | 15.3 Airy Functions
| `airy_dai` | 15.3 Airy Functions
| `airy_dbi` | 15.3 Airy Functions
| `alg_type` | 27.2 Functions and Variables for atensor
| algebraic | 14.2 Functions and Variables for Polynomials
| algepsilon | 20.1 Functions and Variables for Equations
| algexact | 20.1 Functions and Variables for Equations
| `algsys` | 20.1 Functions and Variables for Equations
| `alias` | 6.5 Functions and Variables for Expressions
| aliases | 6.5 Functions and Variables for Expressions
| all_dotsimp_denoms | 24.2 Functions and Variables for Affine
| allbut | 6.5 Functions and Variables for Expressions
| allocation | 52.2.3 Graphics options
| `allroots` | 20.1 Functions and Variables for Equations
| allsym | 25.2.2 Tensor symmetries
| alphabetic | 11.2 Functions and Variables for Properties
| `alphacharp` | 84.3 Characters
| `alphanumericp` | 84.3 Characters
| `amortization` | 58.2 Functions and Variables for finance
| `and` | 7.4 Logical operators
| animation | 54.3.3 Scene object's options
| `annuity_fv` | 58.2 Functions and Variables for finance
| `annuity_pv` | 58.2 Functions and Variables for finance
| `antid` | 18.1 Functions and Variables for Differentiation
| `antidiff` | 18.1 Functions and Variables for Differentiation
| `AntiDifference` | 87.2 Functions and Variables for zeilberger
| antisymmetric | 9.2 Functions and Variables for Simplification
| `append` | 5.4.2 Functions and Variables for Lists
| `appendfile` | 13.3 Functions and Variables for File Input and Output
| `apply` | 36.4 Functions and Variables for Function Definition
| `apply1` | 34.2 Functions and Variables for Rules and Patterns
| `apply2` | 34.2 Functions and Variables for Rules and Patterns
| `applyb1` | 34.2 Functions and Variables for Rules and Patterns
| `apropos` | 3.2 Functions and Variables for Help
| `args` | 6.5 Functions and Variables for Expressions
| `arit_amortization` | 58.2 Functions and Variables for finance
| `arithmetic` | 80.4 Package functs
| `arithsum` | 80.4 Package functs
| `array` | 5.5.1 Functions and Variables for Arrays
| `arrayapply` | 5.5.1 Functions and Variables for Arrays
| `arrayinfo` | 5.5.1 Functions and Variables for Arrays
| `arraymake` | 5.5.1 Functions and Variables for Arrays
| arrays | 5.5.1 Functions and Variables for Arrays
| `arraysetapply` | 5.5.1 Functions and Variables for Arrays
| `ascii` | 84.3 Characters
| `asec` | 10.5.2 Functions and Variables for Trigonometric
| `asech` | 10.5.2 Functions and Variables for Trigonometric
| `asin` | 10.5.2 Functions and Variables for Trigonometric
| `asinh` | 10.5.2 Functions and Variables for Trigonometric
| askexp | 33.3 Functions and Variables for Miscellaneous Options
| `askinteger` | 11.3 Functions and Variables for Facts
| `asksign` | 11.3 Functions and Variables for Facts
| `assoc` | 5.4.2 Functions and Variables for Lists
| `assoc_legendre_p` | 76.2 Functions and Variables for orthogonal polynomials
| `assoc_legendre_q` | 76.2 Functions and Variables for orthogonal polynomials
| `assume` | 11.3 Functions and Variables for Facts
| `assume_external_byte_order` | 73.3 Functions and Variables for binary input and output
| assume_pos | 11.3 Functions and Variables for Facts
| assume_pos_pred | 11.3 Functions and Variables for Facts
| assumescalar | 11.3 Functions and Variables for Facts
| asymbol | 27.2 Functions and Variables for atensor
| `asympa` | 40.1 Introduction to asympa
| `at` | 18.1 Functions and Variables for Differentiation
| `atan` | 10.5.2 Functions and Variables for Trigonometric
| `atan2` | 10.5.2 Functions and Variables for Trigonometric
| `atanh` | 10.5.2 Functions and Variables for Trigonometric
| `atensimp` | 27.2 Functions and Variables for atensor
| `atom` | 6.5 Functions and Variables for Expressions
| atomgrad | 18.1 Functions and Variables for Differentiation
| atrig1 | 10.5.2 Functions and Variables for Trigonometric
| `atvalue` | 18.1 Functions and Variables for Differentiation
| `augcoefmatrix` | 23.2 Functions and Variables for Matrices and Linear Algebra
| `augmented_lagrangian_method` | 41.1 Functions and Variables for augmented_lagrangian
| `av` | 27.2 Functions and Variables for atensor
| `average_degree` | 61.2.2 Graph properties
| axes | 12.4 Plotting Options
| axis_3d | 52.2.3 Graphics options
| axis_bottom | 52.2.3 Graphics options
| axis_left | 52.2.3 Graphics options
| axis_right | 52.2.3 Graphics options
| axis_top | 52.2.3 Graphics options
| azimuth | 12.4 Plotting Options
| azimuth | 54.3.1 Scene options
B |  |
| background | 54.3.1 Scene options
| background_color | 52.2.3 Graphics options
| backslash | 5.2.1 Introduction to Strings
| backsubst | 20.1 Functions and Variables for Equations
| `backtrace` | 37.4 Functions and Variables for Program Flow
| `bars` | 52.2.4 Graphics objects
| `barsplot` | 49.4 Functions and Variables for statistical graphs
| `barsplot_description` | 49.4 Functions and Variables for statistical graphs
| `base64` | 84.5 Octets and Utilities for Cryptography
| `base64_decode` | 84.5 Octets and Utilities for Cryptography
| `bashindices` | 28.1 Functions and Variables for Sums and Products
| `batch` | 13.3 Functions and Variables for File Input and Output
| `batchload` | 13.3 Functions and Variables for File Input and Output
| `bc2` | 21.2 Functions and Variables for Differential Equations
| `bdvac` | 26.2.7 Miscellaneous features
| `belln` | 35.2 Functions and Variables for Sets
| `benefit_cost` | 58.2 Functions and Variables for finance
| berlefact | 14.2 Functions and Variables for Polynomials
| `bern` | 29.1 Functions and Variables for Number Theory
| `bernpoly` | 29.1 Functions and Variables for Number Theory
| `bernstein_approx` | 42.1 Functions and Variables for Bernstein
| `bernstein_expand` | 42.1 Functions and Variables for Bernstein
| bernstein_explicit | 42.1 Functions and Variables for Bernstein
| `bernstein_poly` | 42.1 Functions and Variables for Bernstein
| `bessel_i` | 15.2 Bessel Functions
| `bessel_j` | 15.2 Bessel Functions
| `bessel_k` | 15.2 Bessel Functions
| `bessel_simplify` | 48.2 Functions and Variables for contrib_ode
| `bessel_y` | 15.2 Bessel Functions
| besselexpand | 15.2 Bessel Functions
| `beta` | 15.4 Gamma and factorial Functions
| beta_args_sum_to_integer | 15.4 Gamma and factorial Functions
| beta_expand | 15.4 Gamma and factorial Functions
| `beta_incomplete` | 15.4 Gamma and factorial Functions
| `beta_incomplete_generalized` | 15.4 Gamma and factorial Functions
| `beta_incomplete_regularized` | 15.4 Gamma and factorial Functions
| `bezout` | 14.2 Functions and Variables for Polynomials
| `bf_fft` | 22.2 Functions and Variables for fast Fourier transform
| `bf_find_root` | 22.3 Functions for numerical solution of equations
| `bf_find_root` | 22.3 Functions for numerical solution of equations
| `bf_fmin_cobyla` | 47.2 Functions and Variables for cobyla
| `bf_inverse_fft` | 22.2 Functions and Variables for fast Fourier transform
| `bf_inverse_real_fft` | 22.2 Functions and Variables for fast Fourier transform
| `bf_real_fft` | 22.2 Functions and Variables for fast Fourier transform
| `bfallroots` | 20.1 Functions and Variables for Equations
| `bffac` | 15.4 Gamma and factorial Functions
| `bfhzeta` | 29.1 Functions and Variables for Number Theory
| `bfloat` | 5.1.2 Functions and Variables for Numbers
| `bfloatp` | 5.1.2 Functions and Variables for Numbers
| `bfpsi` | 15.4 Gamma and factorial Functions
| `bfpsi0` | 15.4 Gamma and factorial Functions
| bftorat | 5.1.2 Functions and Variables for Numbers
| bftrunc | 5.1.2 Functions and Variables for Numbers
| `bfzeta` | 29.1 Functions and Variables for Number Theory
| `biconnected_components` | 61.2.2 Graph properties
| `bimetric` | 26.2.7 Miscellaneous features
| bindtest | 11.2 Functions and Variables for Properties
| `binomial` | 10.3 Combinatorial Functions
| `bipartition` | 61.2.2 Graph properties
| `bit_and` | 43.1 Functions and Variables for bitwise
| `bit_length` | 43.1 Functions and Variables for bitwise
| `bit_lsh` | 43.1 Functions and Variables for bitwise
| `bit_not` | 43.1 Functions and Variables for bitwise
| `bit_onep` | 43.1 Functions and Variables for bitwise
| `bit_or` | 43.1 Functions and Variables for bitwise
| `bit_rsh` | 43.1 Functions and Variables for bitwise
| `bit_xor` | 43.1 Functions and Variables for bitwise
| `block` | 36.4 Functions and Variables for Function Definition
| `blockmatrixp` | 68.2 Functions and Variables for linearalgebra
| `bode_gain` | 44.1 Functions and Variables for bode
| `bode_phase` | 44.1 Functions and Variables for bode
| border | 52.2.3 Graphics options
| `bothcoef` | 14.2 Functions and Variables for Polynomials
| boundaries_array | 52.4.1 Variables and Functions
| `box` | 6.5 Functions and Variables for Expressions
| boxchar | 6.5 Functions and Variables for Expressions
| `boxplot` | 49.4 Functions and Variables for statistical graphs
| `boxplot_description` | 49.4 Functions and Variables for statistical graphs
| `break` | 36.4 Functions and Variables for Function Definition
| breakup | 20.1 Functions and Variables for Equations
| `bug_report` | 2.1 Functions and Variables for Bug Detection and Reporting
| `build_info` | 2.1 Functions and Variables for Bug Detection and Reporting
| `build_sample` | 49.2 Functions and Variables for data manipulation
| `buildq` | 36.3 Macros
| `burn` | 29.1 Functions and Variables for Number Theory
C |  |
| `cabs` | 10.2 Functions for Complex Numbers
| `canform` | 25.2.2 Tensor symmetries
| `canten` | 25.2.1 Managing indexed objects
| capping | 52.2.3 Graphics options
| capping | 54.3.3 Scene object's options
| `cardinality` | 35.2 Functions and Variables for Sets
| `carg` | 10.2 Functions for Complex Numbers
| `cartan` | 18.1 Functions and Variables for Differentiation
| `cartesian_product` | 35.2 Functions and Variables for Sets
| `catch` | 36.4 Functions and Variables for Function Definition
| `cauchy_matrix` | 23.2 Functions and Variables for Matrices and Linear Algebra
| cauchysum | 28.3 Functions and Variables for Series
| `cbffac` | 15.4 Gamma and factorial Functions
| cbrange | 52.2.3 Graphics options
| cbtics | 52.2.3 Graphics options
| `cdf_bernoulli` | 51.3 Functions and Variables for discrete distributions
| `cdf_beta` | 51.2 Functions and Variables for continuous distributions
| `cdf_binomial` | 51.3 Functions and Variables for discrete distributions
| `cdf_cauchy` | 51.2 Functions and Variables for continuous distributions
| `cdf_chi2` | 51.2 Functions and Variables for continuous distributions
| `cdf_continuous_uniform` | 51.2 Functions and Variables for continuous distributions
| `cdf_discrete_uniform` | 51.3 Functions and Variables for discrete distributions
| `cdf_empirical` | 49.3 Functions and Variables for descriptive statistics
| `cdf_exp` | 51.2 Functions and Variables for continuous distributions
| `cdf_f` | 51.2 Functions and Variables for continuous distributions
| `cdf_gamma` | 51.2 Functions and Variables for continuous distributions
| `cdf_general_finite_discrete` | 51.3 Functions and Variables for discrete distributions
| `cdf_geometric` | 51.3 Functions and Variables for discrete distributions
| `cdf_gumbel` | 51.2 Functions and Variables for continuous distributions
| `cdf_hypergeometric` | 51.3 Functions and Variables for discrete distributions
| `cdf_laplace` | 51.2 Functions and Variables for continuous distributions
| `cdf_logistic` | 51.2 Functions and Variables for continuous distributions
| `cdf_lognormal` | 51.2 Functions and Variables for continuous distributions
| `cdf_negative_binomial` | 51.3 Functions and Variables for discrete distributions
| `cdf_noncentral_chi2` | 51.2 Functions and Variables for continuous distributions
| `cdf_noncentral_student_t` | 51.2 Functions and Variables for continuous distributions
| `cdf_normal` | 51.2 Functions and Variables for continuous distributions
| `cdf_pareto` | 51.2 Functions and Variables for continuous distributions
| `cdf_poisson` | 51.3 Functions and Variables for discrete distributions
| `cdf_rank_sum` | 82.4 Functions and Variables for special distributions
| `cdf_rayleigh` | 51.2 Functions and Variables for continuous distributions
| `cdf_signed_rank` | 82.4 Functions and Variables for special distributions
| `cdf_student_t` | 51.2 Functions and Variables for continuous distributions
| `cdf_weibull` | 51.2 Functions and Variables for continuous distributions
| `cdisplay` | 26.2.8 Utility functions
| `ceiling` | 10.1 Functions for Numbers
| center | 54.3.3 Scene object's options
| `central_moment` | 49.3 Functions and Variables for descriptive statistics
| `cequal` | 84.3 Characters
| `cequalignore` | 84.3 Characters
| `cf` | 29.1 Functions and Variables for Number Theory
| `cfdisrep` | 29.1 Functions and Variables for Number Theory
| `cfexpand` | 29.1 Functions and Variables for Number Theory
| cflength | 29.1 Functions and Variables for Number Theory
| cframe_flag | 26.2.9 Variables used by `ctensor`
| `cgeodesic` | 26.2.7 Miscellaneous features
| `cgreaterp` | 84.3 Characters
| `cgreaterpignore` | 84.3 Characters
| `changename` | 25.2.1 Managing indexed objects
| `changevar` | 19.2 Functions and Variables for Integration
| `chaosgame` | 54.2 Graphical analysis of discrete dynamical systems
| `charat` | 84.4 String Processing
| `charfun` | 11.4 Functions and Variables for Predicates
| `charfun2` | 64.2 Functions and Variables for interpol
| `charlist` | 84.4 String Processing
| `charp` | 84.3 Characters
| `charpoly` | 23.2 Functions and Variables for Matrices and Linear Algebra
| `chdir` | 74.2 Directory operations
| `chebyshev_t` | 76.2 Functions and Variables for orthogonal polynomials
| `chebyshev_u` | 76.2 Functions and Variables for orthogonal polynomials
| `check_overlaps` | 24.2 Functions and Variables for Affine
| `checkdiv` | 26.2.7 Miscellaneous features
| `chinese` | 29.1 Functions and Variables for Number Theory
| `cholesky` | 68.2 Functions and Variables for linearalgebra
| `christof` | 26.2.2 The tensors of curved space
| `chromatic_index` | 61.2.2 Graph properties
| `chromatic_number` | 61.2.2 Graph properties
| `cint` | 84.3 Characters
| `circulant_graph` | 61.2.1 Building graphs
| `clear_edge_weight` | 61.2.2 Graph properties
| `clear_rules` | 34.2 Functions and Variables for Rules and Patterns
| `clear_vertex_label` | 61.2.2 Graph properties
| `clebsch_gordan` | 46.1 Functions and Variables for clebsch_gordan
| `clebsch_graph` | 61.2.1 Building graphs
| `clessp` | 84.3 Characters
| `clesspignore` | 84.3 Characters
| `close` | 84.2 Input and Output
| `closefile` | 13.3 Functions and Variables for File Input and Output
| `cmetric` | 26.2.1 Initialization and setup
| cnonmet_flag | 26.2.9 Variables used by `ctensor`
| `coeff` | 14.2 Functions and Variables for Polynomials
| `coefmatrix` | 23.2 Functions and Variables for Matrices and Linear Algebra
| `cograd` | 26.2.7 Miscellaneous features
| `col` | 23.2 Functions and Variables for Matrices and Linear Algebra
| `collapse` | 6.5 Functions and Variables for Expressions
| `collectterms` | 80.3 Package facexp
| color | 12.4 Plotting Options
| color | 52.2.3 Graphics options
| color | 54.3.3 Scene object's options
| color_bar | 12.4 Plotting Options
| color_bar_tics | 12.4 Plotting Options
| colorbox | 52.2.3 Graphics options
| `columnop` | 68.2 Functions and Variables for linearalgebra
| columns | 52.2.3 Graphics options
| `columnspace` | 68.2 Functions and Variables for linearalgebra
| `columnswap` | 68.2 Functions and Variables for linearalgebra
| `columnvector` | 23.2 Functions and Variables for Matrices and Linear Algebra
| `combination` | 80.4 Package functs
| `combine` | 9.2 Functions and Variables for Simplification
| commutative | 9.2 Functions and Variables for Simplification
| `comp2pui` | 30.2.1 Changing bases
| `compare` | 11.4 Functions and Variables for Predicates
| `compfile` | 36.4 Functions and Variables for Function Definition
| `compile` | 36.4 Functions and Variables for Function Definition
| `compile_file` | 36.4 Functions and Variables for Function Definition
| `complement_graph` | 61.2.1 Building graphs
| `complete_bipartite_graph` | 61.2.1 Building graphs
| `complete_graph` | 61.2.1 Building graphs
| complex | 11.2 Functions and Variables for Properties
| `complex_number_p` | 85.1 Functions and Variables for to_poly_solve
| `components` | 25.2.1 Managing indexed objects
| `compose_functions` | 85.1 Functions and Variables for to_poly_solve
| `concan` | 25.2.1 Managing indexed objects
| `concat` | 5.2.2 Functions and Variables for Strings
| cone | 54.3.2 Scene objects
| `conjugate` | 10.2 Functions for Complex Numbers
| `conmetderiv` | 25.2.3 Indicial tensor calculus
| `connect_vertices` | 61.2.3 Modifying graphs
| `connected_components` | 61.2.2 Graph properties
| `cons` | 5.4.2 Functions and Variables for Lists
| `constant` | 11.2 Functions and Variables for Properties
| `constantp` | 11.2 Functions and Variables for Properties
| `constituent` | 84.3 Characters
| `constvalue` | 56.3 Functions and Variables for ezunits
| `cont2part` | 30.2.2 Changing representations
| `content` | 14.2 Functions and Variables for Polynomials
| context | 11.3 Functions and Variables for Facts
| contexts | 11.3 Functions and Variables for Facts
| `continuous_freq` | 49.2 Functions and Variables for data manipulation
| `contortion` | 26.2.6 Torsion and nonmetricity
| contour | 52.2.3 Graphics options
| contour_levels | 52.2.3 Graphics options
| `contour_plot` | 12.3 Functions and Variables for Plotting
| `contract` | 25.2.1 Managing indexed objects
| `contract` | 30.2.2 Changing representations
| `contract_edge` | 61.2.3 Modifying graphs
| `contragrad` | 26.2.7 Miscellaneous features
| `contrib_ode` | 48.2 Functions and Variables for contrib_ode
| `convert` | 86.2 Functions and Variables for Units
| `coord` | 25.2.3 Indicial tensor calculus
| `copy` | 6.5 Functions and Variables for Expressions
| `copy_file` | 74.3 File operations
| `copy_graph` | 61.2.1 Building graphs
| `copylist` | 5.4.2 Functions and Variables for Lists
| `copymatrix` | 23.2 Functions and Variables for Matrices and Linear Algebra
| `cor` | 49.3 Functions and Variables for descriptive statistics
| `cos` | 10.5.2 Functions and Variables for Trigonometric
| `cosh` | 10.5.2 Functions and Variables for Trigonometric
| cosnpiflag | 28.5 Functions and Variables for Fourier series
| `cot` | 10.5.2 Functions and Variables for Trigonometric
| `coth` | 10.5.2 Functions and Variables for Trigonometric
| `cov` | 49.3 Functions and Variables for descriptive statistics
| `cov1` | 49.3 Functions and Variables for descriptive statistics
| `covdiff` | 25.2.4 Tensors in curved spaces
| `covect` | 23.2 Functions and Variables for Matrices and Linear Algebra
| `covers` | 80.4 Package functs
| `crc24sum` | 84.5 Octets and Utilities for Cryptography
| `create_graph` | 61.2.1 Building graphs
| `create_list` | 5.4.2 Functions and Variables for Lists
| `csc` | 10.5.2 Functions and Variables for Trigonometric
| `csch` | 10.5.2 Functions and Variables for Trigonometric
| `csetup` | 26.2.1 Initialization and setup
| `cspline` | 64.2 Functions and Variables for interpol
| ct_coords | 26.2.9 Variables used by `ctensor`
| `ct_coordsys` | 26.2.1 Initialization and setup
| `ctaylor` | 26.2.3 Taylor series expansion
| ctaypov | 26.2.9 Variables used by `ctensor`
| ctaypt | 26.2.9 Variables used by `ctensor`
| ctayswitch | 26.2.9 Variables used by `ctensor`
| ctayvar | 26.2.9 Variables used by `ctensor`
| ctorsion_flag | 26.2.9 Variables used by `ctensor`
| `ctransform` | 26.2.7 Miscellaneous features
| `ctranspose` | 68.2 Functions and Variables for linearalgebra
| ctrgsimp | 26.2.9 Variables used by `ctensor`
| cube | 54.3.2 Scene objects
| `cube_graph` | 61.2.1 Building graphs
| `cuboctahedron_graph` | 61.2.1 Building graphs
| current_let_rule_package | 34.2 Functions and Variables for Rules and Patterns
| `cv` | 49.3 Functions and Variables for descriptive statistics
| `cycle_digraph` | 61.2.1 Building graphs
| `cycle_graph` | 61.2.1 Building graphs
| cylinder | 54.3.2 Scene objects
| `cylindrical` | 52.2.4 Graphics objects
D |  |
| data_file_name | 52.2.3 Graphics options
| `days360` | 58.2 Functions and Variables for finance
| `dblint` | 19.2 Functions and Variables for Integration
| `deactivate` | 11.3 Functions and Variables for Facts
| debugmode | 38.3 Functions and Variables for Debugging
| `declare` | 11.2 Functions and Variables for Properties
| `declare_constvalue` | 56.3 Functions and Variables for ezunits
| `declare_dimensions` | 56.3 Functions and Variables for ezunits
| `declare_fundamental_dimensions` | 56.3 Functions and Variables for ezunits
| `declare_fundamental_units` | 56.3 Functions and Variables for ezunits
| `declare_qty` | 56.3 Functions and Variables for ezunits
| `declare_translated` | 36.4 Functions and Variables for Function Definition
| `declare_unit_conversion` | 56.3 Functions and Variables for ezunits
| `declare_units` | 56.3 Functions and Variables for ezunits
| `declare_weights` | 24.2 Functions and Variables for Affine
| `decode_time` | 32.3 Functions and Variables for Runtime Environment
| decreasing | 11.2 Functions and Variables for Properties
| `decsym` | 25.2.2 Tensor symmetries
| default_let_rule_package | 34.2 Functions and Variables for Rules and Patterns
| `defcon` | 25.2.1 Managing indexed objects
| `define` | 36.4 Functions and Variables for Function Definition
| `define_alt_display` | 39.2 Functions and Variables for alt-display
| `define_variable` | 36.4 Functions and Variables for Function Definition
| `defint` | 19.2 Functions and Variables for Integration
| `defmatch` | 34.2 Functions and Variables for Rules and Patterns
| `defrule` | 34.2 Functions and Variables for Rules and Patterns
| `defstruct` | 5.6.2 Functions and Variables for Structures
| `deftaylor` | 28.3 Functions and Variables for Series
| `degree_sequence` | 61.2.2 Graph properties
| `del` | 18.1 Functions and Variables for Differentiation
| delay | 52.2.3 Graphics options
| `delete` | 5.4.2 Functions and Variables for Lists
| `delete_file` | 74.3 File operations
| `deleten` | 26.2.8 Utility functions
| `delta` | 18.1 Functions and Variables for Differentiation
| `demo` | 3.2 Functions and Variables for Help
| `demoivre` | 9.2 Functions and Variables for Simplification
| `demoivre` | 9.2 Functions and Variables for Simplification
| `denom` | 14.2 Functions and Variables for Polynomials
| dependencies | 18.1 Functions and Variables for Differentiation
| dependencies | 18.1 Functions and Variables for Differentiation
| `depends` | 18.1 Functions and Variables for Differentiation
| derivabbrev | 18.1 Functions and Variables for Differentiation
| `derivdegree` | 18.1 Functions and Variables for Differentiation
| `derivlist` | 18.1 Functions and Variables for Differentiation
| derivsubst | 18.1 Functions and Variables for Differentiation
| `describe` | 3.2 Functions and Variables for Help
| `desolve` | 21.2 Functions and Variables for Differential Equations
| `determinant` | 23.2 Functions and Variables for Matrices and Linear Algebra
| detout | 23.2 Functions and Variables for Matrices and Linear Algebra
| `dfloat` | 85.1 Functions and Variables for to_poly_solve
| `dgauss_a` | 48.2 Functions and Variables for contrib_ode
| `dgauss_b` | 48.2 Functions and Variables for contrib_ode
| `dgeev` | 65.2 Functions and Variables for lapack
| `dgemm` | 65.2 Functions and Variables for lapack
| `dgeqrf` | 65.2 Functions and Variables for lapack
| `dgesv` | 65.2 Functions and Variables for lapack
| `dgesvd` | 65.2 Functions and Variables for lapack
| `diag` | 50.1 Functions and Variables for diag
| `diag_matrix` | 68.2 Functions and Variables for linearalgebra
| `diagmatrix` | 23.2 Functions and Variables for Matrices and Linear Algebra
| `diagmatrixp` | 26.2.8 Utility functions
| diagmetric | 26.2.9 Variables used by `ctensor`
| `diameter` | 61.2.2 Graph properties
| `diff` | 18.1 Functions and Variables for Differentiation
| `diff` | 25.2.3 Indicial tensor calculus
| `digitcharp` | 84.3 Characters
| dim | 26.2.9 Variables used by `ctensor`
| `dimacs_export` | 61.2.4 Reading and writing to files
| `dimacs_import` | 61.2.4 Reading and writing to files
| `dimension` | 20.1 Functions and Variables for Equations
| `dimensionless` | 56.3 Functions and Variables for ezunits
| `dimensions` | 56.3 Functions and Variables for ezunits
| `dimensions_as_list` | 56.3 Functions and Variables for ezunits
| `direct` | 30.2.3 Groups and orbits
| `directory` | 13.3 Functions and Variables for File Input and Output
| `discrete_freq` | 49.2 Functions and Variables for data manipulation
| `disjoin` | 35.2 Functions and Variables for Sets
| `disjointp` | 35.2 Functions and Variables for Sets
| `disolate` | 6.5 Functions and Variables for Expressions
| `disp` | 4.3 Functions and Variables for Display
| `dispcon` | 25.2.1 Managing indexed objects
| dispflag | 20.1 Functions and Variables for Equations
| `dispform` | 6.5 Functions and Variables for Expressions
| `dispfun` | 36.4 Functions and Variables for Function Definition
| `dispJordan` | 50.1 Functions and Variables for diag
| `display` | 4.3 Functions and Variables for Display
| display2d | 4.3 Functions and Variables for Display
| display_format_internal | 4.3 Functions and Variables for Display
| `disprule` | 34.2 Functions and Variables for Rules and Patterns
| `dispterms` | 4.3 Functions and Variables for Display
| `distrib` | 9.2 Functions and Variables for Simplification
| distribute_over | 9.2 Functions and Variables for Simplification
| `divide` | 14.2 Functions and Variables for Polynomials
| `divisors` | 35.2 Functions and Variables for Sets
| `divsum` | 29.1 Functions and Variables for Number Theory
| `dkummer_m` | 48.2 Functions and Variables for contrib_ode
| `dkummer_u` | 48.2 Functions and Variables for contrib_ode
| `dlange` | 65.2 Functions and Variables for lapack
| `do` | 37.4 Functions and Variables for Program Flow
| doallmxops | 23.2 Functions and Variables for Matrices and Linear Algebra
| `dodecahedron_graph` | 61.2.1 Building graphs
| domain | 9.2 Functions and Variables for Simplification
| domxexpt | 23.2 Functions and Variables for Matrices and Linear Algebra
| domxmxops | 23.2 Functions and Variables for Matrices and Linear Algebra
| domxnctimes | 23.2 Functions and Variables for Matrices and Linear Algebra
| dontfactor | 23.2 Functions and Variables for Matrices and Linear Algebra
| doscmxops | 23.2 Functions and Variables for Matrices and Linear Algebra
| doscmxplus | 23.2 Functions and Variables for Matrices and Linear Algebra
| dot0nscsimp | 23.2 Functions and Variables for Matrices and Linear Algebra
| dot0simp | 23.2 Functions and Variables for Matrices and Linear Algebra
| dot1simp | 23.2 Functions and Variables for Matrices and Linear Algebra
| dotassoc | 23.2 Functions and Variables for Matrices and Linear Algebra
| dotconstrules | 23.2 Functions and Variables for Matrices and Linear Algebra
| dotdistrib | 23.2 Functions and Variables for Matrices and Linear Algebra
| dotexptsimp | 23.2 Functions and Variables for Matrices and Linear Algebra
| dotident | 23.2 Functions and Variables for Matrices and Linear Algebra
| `dotproduct` | 68.2 Functions and Variables for linearalgebra
| dotscrules | 23.2 Functions and Variables for Matrices and Linear Algebra
| `dotsimp` | 24.2 Functions and Variables for Affine
| `dpart` | 6.5 Functions and Variables for Expressions
| `draw` | 52.2.2 Functions
| `draw2d` | 52.2.2 Functions
| `draw3d` | 52.2.2 Functions
| `draw_file` | 52.2.2 Functions
| `draw_graph` | 61.2.5 Visualization
| draw_graph_program | 61.2.5 Visualization
| draw_realpart | 52.2.3 Graphics options
| `drawdf` | 53.2.1 Functions
| `dscalar` | 18.1 Functions and Variables for Differentiation
| `dscalar` | 26.2.7 Miscellaneous features
E |  |
| `echelon` | 23.2 Functions and Variables for Matrices and Linear Algebra
| edge_color | 61.2.5 Visualization
| `edge_coloring` | 61.2.2 Graph properties
| `edge_connectivity` | 61.2.2 Graph properties
| edge_partition | 61.2.5 Visualization
| edge_type | 61.2.5 Visualization
| edge_width | 61.2.5 Visualization
| `edges` | 61.2.2 Graph properties
| `eigens_by_jacobi` | 68.2 Functions and Variables for linearalgebra
| `eigenvalues` | 23.2 Functions and Variables for Matrices and Linear Algebra
| `eigenvectors` | 23.2 Functions and Variables for Matrices and Linear Algebra
| `eighth` | 5.4.2 Functions and Variables for Lists
| `einstein` | 26.2.2 The tensors of curved space
| `eivals` | 23.2 Functions and Variables for Matrices and Linear Algebra
| `eivects` | 23.2 Functions and Variables for Matrices and Linear Algebra
| `elapsed_real_time` | 32.3 Functions and Variables for Runtime Environment
| `elapsed_run_time` | 32.3 Functions and Variables for Runtime Environment
| `ele2comp` | 30.2.1 Changing bases
| `ele2polynome` | 30.2.5 Polynomials and their roots
| `ele2pui` | 30.2.1 Changing bases
| `elem` | 30.2.1 Changing bases
| `elementp` | 35.2 Functions and Variables for Sets
| elevation | 12.4 Plotting Options
| elevation | 54.3.1 Scene options
| `elevation_grid` | 52.2.4 Graphics objects
| `elim` | 85.1 Functions and Variables for to_poly_solve
| `elim_allbut` | 85.1 Functions and Variables for to_poly_solve
| `eliminate` | 14.2 Functions and Variables for Polynomials
| `eliminate_using` | 85.1 Functions and Variables for to_poly_solve
| `ellipse` | 52.2.4 Graphics objects
| `elliptic_e` | 16.3 Functions and Variables for Elliptic Integrals
| `elliptic_ec` | 16.3 Functions and Variables for Elliptic Integrals
| `elliptic_eu` | 16.3 Functions and Variables for Elliptic Integrals
| `elliptic_f` | 16.3 Functions and Variables for Elliptic Integrals
| `elliptic_kc` | 16.3 Functions and Variables for Elliptic Integrals
| `elliptic_pi` | 16.3 Functions and Variables for Elliptic Integrals
| `ematrix` | 23.2 Functions and Variables for Matrices and Linear Algebra
| `empty_graph` | 61.2.1 Building graphs
| `emptyp` | 35.2 Functions and Variables for Sets
| `encode_time` | 32.3 Functions and Variables for Runtime Environment
| `endcons` | 5.4.2 Functions and Variables for Lists
| endphi | 54.3.3 Scene object's options
| endtheta | 54.3.3 Scene object's options
| engineering_format_floats | 55.1 Functions and Variables for engineering-format
| enhanced3d | 52.2.3 Graphics options
| `entermatrix` | 23.2 Functions and Variables for Matrices and Linear Algebra
| `entertensor` | 25.2.1 Managing indexed objects
| `entier` | 10.1 Functions for Numbers
| epsilon_lp | 79.2 Functions and Variables for simplex
| `equal` | 11.4 Functions and Variables for Predicates
| `equalp` | 28.5 Functions and Variables for Fourier series
| `equiv_classes` | 35.2 Functions and Variables for Sets
| `erf` | 15.6 Error Function
| `erf_generalized` | 15.6 Error Function
| erf_representation | 15.6 Error Function
| `erfc` | 15.6 Error Function
| erfflag | 19.2 Functions and Variables for Integration
| `erfi` | 15.6 Error Function
| `errcatch` | 37.4 Functions and Variables for Program Flow
| `error` | 37.4 Functions and Variables for Program Flow
| `error` | 37.4 Functions and Variables for Program Flow
| error_size | 37.4 Functions and Variables for Program Flow
| error_syms | 37.4 Functions and Variables for Program Flow
| error_type | 52.2.3 Graphics options
| `errormsg` | 37.4 Functions and Variables for Program Flow
| `errors` | 52.2.4 Graphics objects
| `euler` | 29.1 Functions and Variables for Number Theory
| `ev` | 8.1 Functions and Variables for Evaluation
| ev_point | 87.4 Variables related to the modular test
| eval | 8.1 Functions and Variables for Evaluation
| `eval_string` | 84.4 String Processing
| even | 11.2 Functions and Variables for Properties
| evenfun | 9.2 Functions and Variables for Simplification
| `evenp` | 5.1.2 Functions and Variables for Numbers
| `every` | 35.2 Functions and Variables for Sets
| evflag | 8.1 Functions and Variables for Evaluation
| evfun | 8.1 Functions and Variables for Evaluation
| `evolution` | 54.2 Graphical analysis of discrete dynamical systems
| `evolution2d` | 54.2 Graphical analysis of discrete dynamical systems
| `evundiff` | 25.2.3 Indicial tensor calculus
| `example` | 3.2 Functions and Variables for Help
| `exp` | 10.4 Root, Exponential and Logarithmic Functions
| `expand` | 9.2 Functions and Variables for Simplification
| `expandwrt` | 9.2 Functions and Variables for Simplification
| expandwrt_denom | 9.2 Functions and Variables for Simplification
| `expandwrt_factored` | 9.2 Functions and Variables for Simplification
| `expintegral_chi` | 15.5 Exponential Integrals
| `expintegral_ci` | 15.5 Exponential Integrals
| `expintegral_e` | 15.5 Exponential Integrals
| `expintegral_e1` | 15.5 Exponential Integrals
| `expintegral_e_simplify` | 48.2 Functions and Variables for contrib_ode
| `expintegral_ei` | 15.5 Exponential Integrals
| `expintegral_li` | 15.5 Exponential Integrals
| `expintegral_shi` | 15.5 Exponential Integrals
| `expintegral_si` | 15.5 Exponential Integrals
| expintexpand | 15.5 Exponential Integrals
| expintrep | 15.5 Exponential Integrals
| `explicit` | 52.2.4 Graphics objects
| `explose` | 30.2.2 Changing representations
| expon | 9.2 Functions and Variables for Simplification
| `exponentialize` | 9.2 Functions and Variables for Simplification
| `exponentialize` | 9.2 Functions and Variables for Simplification
| expop | 9.2 Functions and Variables for Simplification
| `express` | 18.1 Functions and Variables for Differentiation
| `expt` | 4.3 Functions and Variables for Display
| exptdispflag | 4.3 Functions and Variables for Display
| exptisolate | 6.5 Functions and Variables for Expressions
| exptsubst | 6.5 Functions and Variables for Expressions
| `exsec` | 80.4 Package functs
| `extdiff` | 25.2.7 Exterior algebra
| `extract_linear_equations` | 24.2 Functions and Variables for Affine
| `extremal_subset` | 35.2 Functions and Variables for Sets
| `ezgcd` | 14.2 Functions and Variables for Polynomials
F |  |
| `f90` | 57.1 Functions and Variables for f90
| facexpand | 14.2 Functions and Variables for Polynomials
| `facsum` | 80.3 Package facexp
| facsum_combine | 80.3 Package facexp
| `factcomb` | 10.3 Combinatorial Functions
| factlim | 10.3 Combinatorial Functions
| `factor` | 14.2 Functions and Variables for Polynomials
| `factorfacsum` | 80.3 Package facexp
| factorflag | 14.2 Functions and Variables for Polynomials
| `factorial` | 10.3 Combinatorial Functions
| factorial_expand | 10.3 Combinatorial Functions
| `factorout` | 14.2 Functions and Variables for Polynomials
| factors_only | 29.1 Functions and Variables for Number Theory
| `factorsum` | 14.2 Functions and Variables for Polynomials
| `facts` | 11.3 Functions and Variables for Facts
| false | 5.3.1 Functions and Variables for Constants
| `fast_central_elements` | 24.2 Functions and Variables for Affine
| `fast_linsolve` | 24.2 Functions and Variables for Affine
| `fasttimes` | 14.2 Functions and Variables for Polynomials
| fb | 26.2.9 Variables used by `ctensor`
| feature | 11.2 Functions and Variables for Properties
| `featurep` | 11.2 Functions and Variables for Properties
| features | 11.2 Functions and Variables for Properties
| `fernfale` | 59.2 Definitions for IFS fractals
| `fft` | 22.2 Functions and Variables for fast Fourier transform
| `fib` | 29.1 Functions and Variables for Number Theory
| `fibtophi` | 29.1 Functions and Variables for Number Theory
| `fifth` | 5.4.2 Functions and Variables for Lists
| file_name | 52.2.3 Graphics options
| file_name | 61.2.5 Visualization
| file_output_append | 13.3 Functions and Variables for File Input and Output
| `file_search` | 13.3 Functions and Variables for File Input and Output
| file_search_demo | 13.3 Functions and Variables for File Input and Output
| file_search_lisp | 13.3 Functions and Variables for File Input and Output
| file_search_maxima | 13.3 Functions and Variables for File Input and Output
| file_search_tests | 13.3 Functions and Variables for File Input and Output
| file_search_usage | 13.3 Functions and Variables for File Input and Output
| `file_type` | 13.3 Functions and Variables for File Input and Output
| file_type_lisp | 13.3 Functions and Variables for File Input and Output
| file_type_maxima | 13.3 Functions and Variables for File Input and Output
| `filename_merge` | 13.3 Functions and Variables for File Input and Output
| fill_color | 52.2.3 Graphics options
| fill_density | 52.2.3 Graphics options
| `fillarray` | 5.5.1 Functions and Variables for Arrays
| filled_func | 52.2.3 Graphics options
| `find_root` | 22.3 Functions for numerical solution of equations
| `find_root` | 22.3 Functions for numerical solution of equations
| `find_root_abs` | 22.3 Functions for numerical solution of equations
| `find_root_error` | 22.3 Functions for numerical solution of equations
| `find_root_rel` | 22.3 Functions for numerical solution of equations
| `findde` | 26.2.7 Miscellaneous features
| `first` | 5.4.2 Functions and Variables for Lists
| `firstn` | 5.4.2 Functions and Variables for Lists
| `fix` | 10.1 Functions for Numbers
| fixed_vertices | 61.2.5 Visualization
| `flatten` | 35.2 Functions and Variables for Sets
| `flength` | 84.2 Input and Output
| flipflag | 25.2.1 Managing indexed objects
| `float` | 5.1.2 Functions and Variables for Numbers
| float2bf | 5.1.2 Functions and Variables for Numbers
| `floatnump` | 5.1.2 Functions and Variables for Numbers
| `floor` | 10.1 Functions for Numbers
| `flower_snark` | 61.2.1 Building graphs
| `flush` | 25.2.3 Indicial tensor calculus
| `flush1deriv` | 25.2.3 Indicial tensor calculus
| `flush_output` | 84.2 Input and Output
| `flushd` | 25.2.3 Indicial tensor calculus
| `flushnd` | 25.2.3 Indicial tensor calculus
| `fmin_cobyla` | 47.2 Functions and Variables for cobyla
| font | 52.2.3 Graphics options
| font_size | 52.2.3 Graphics options
| `for` | 37.4 Functions and Variables for Program Flow
| `forget` | 11.3 Functions and Variables for Facts
| fortindent | 13.5 Functions and Variables for Fortran Output
| `fortran` | 13.5 Functions and Variables for Fortran Output
| fortspaces | 13.5 Functions and Variables for Fortran Output
| `fourcos` | 28.5 Functions and Variables for Fourier series
| `fourexpand` | 28.5 Functions and Variables for Fourier series
| `fourier` | 28.5 Functions and Variables for Fourier series
| `fourier_elim` | 85.1 Functions and Variables for to_poly_solve
| `fourint` | 28.5 Functions and Variables for Fourier series
| `fourintcos` | 28.5 Functions and Variables for Fourier series
| `fourintsin` | 28.5 Functions and Variables for Fourier series
| `foursimp` | 28.5 Functions and Variables for Fourier series
| `foursin` | 28.5 Functions and Variables for Fourier series
| `fourth` | 5.4.2 Functions and Variables for Lists
| `fposition` | 84.2 Input and Output
| fpprec | 5.1.2 Functions and Variables for Numbers
| fpprintprec | 5.1.2 Functions and Variables for Numbers
| `frame_bracket` | 26.2.4 Frame fields
| `freeof` | 6.5 Functions and Variables for Expressions
| `freshline` | 84.2 Input and Output
| `fresnel_c` | 15.6 Error Function
| `fresnel_s` | 15.6 Error Function
| `from` | 37.4 Functions and Variables for Program Flow
| `from_adjacency_matrix` | 61.2.1 Building graphs
| `frucht_graph` | 61.2.1 Building graphs
| `full_listify` | 35.2 Functions and Variables for Sets
| `fullmap` | 36.4 Functions and Variables for Function Definition
| `fullmapl` | 36.4 Functions and Variables for Function Definition
| `fullratsimp` | 14.2 Functions and Variables for Polynomials
| `fullratsubst` | 14.2 Functions and Variables for Polynomials
| `fullsetify` | 35.2 Functions and Variables for Sets
| `funcsolve` | 20.1 Functions and Variables for Equations
| functions | 36.4 Functions and Variables for Function Definition
| `fundamental_dimensions` | 56.3 Functions and Variables for ezunits
| `fundamental_units` | 56.3 Functions and Variables for ezunits
| `fundef` | 36.4 Functions and Variables for Function Definition
| `funmake` | 36.4 Functions and Variables for Function Definition
| `funp` | 28.5 Functions and Variables for Fourier series
| `fv` | 58.2 Functions and Variables for finance
G |  |
| `gamma` | 15.4 Gamma and factorial Functions
| gamma_expand | 15.4 Gamma and factorial Functions
| `gamma_incomplete` | 15.4 Gamma and factorial Functions
| `gamma_incomplete_generalized` | 15.4 Gamma and factorial Functions
| `gamma_incomplete_lower` | 15.4 Gamma and factorial Functions
| `gamma_incomplete_regularized` | 15.4 Gamma and factorial Functions
| gammalim | 15.4 Gamma and factorial Functions
| `gauss_a` | 48.2 Functions and Variables for contrib_ode
| `gauss_b` | 48.2 Functions and Variables for contrib_ode
| `gaussprob` | 80.4 Package functs
| `gcd` | 14.2 Functions and Variables for Polynomials
| `gcdex` | 14.2 Functions and Variables for Polynomials
| `gcdivide` | 80.4 Package functs
| `gcfac` | 80.7 Package scifac
| `gcfactor` | 14.2 Functions and Variables for Polynomials
| `gd` | 80.4 Package functs
| gdet | 26.2.9 Variables used by `ctensor`
| `gen_laguerre` | 76.2 Functions and Variables for orthogonal polynomials
| `generalized_lambert_w` | 15.10 Functions and Variables for Special Functions
| `genfact` | 10.3 Combinatorial Functions
| genindex | 33.3 Functions and Variables for Miscellaneous Options
| `genmatrix` | 23.2 Functions and Variables for Matrices and Linear Algebra
| gensumnum | 33.3 Functions and Variables for Miscellaneous Options
| `gensym` | 33.3 Functions and Variables for Miscellaneous Options
| `geo_amortization` | 58.2 Functions and Variables for finance
| `geo_annuity_fv` | 58.2 Functions and Variables for finance
| `geo_annuity_pv` | 58.2 Functions and Variables for finance
| `geomap` | 52.4.2 Graphic objects
| `geometric` | 80.4 Package functs
| `geometric_mean` | 49.3 Functions and Variables for descriptive statistics
| geomview_command | 12.3 Functions and Variables for Plotting
| `geosum` | 80.4 Package functs
| `get` | 11.2 Functions and Variables for Properties
| `get_edge_weight` | 61.2.2 Graph properties
| `get_lu_factors` | 68.2 Functions and Variables for linearalgebra
| `get_output_stream_string` | 84.2 Input and Output
| `get_pixel` | 52.3 Functions and Variables for pictures
| `get_plot_option` | 12.3 Functions and Variables for Plotting
| `get_tex_environment` | 13.4 Functions and Variables for TeX Output
| `get_tex_environment_default` | 13.4 Functions and Variables for TeX Output
| `get_vertex_label` | 61.2.2 Graph properties
| `getcurrentdirectory` | 74.2 Directory operations
| `getenv` | 74.4 Environment operations
| `gfactor` | 14.2 Functions and Variables for Polynomials
| `gfactorsum` | 14.2 Functions and Variables for Polynomials
| `ggf` | 60.1 Functions and Variables for ggf
| GGFCFMAX | 60.1 Functions and Variables for ggf
| GGFINFINITY | 60.1 Functions and Variables for ggf
| `girth` | 61.2.2 Graph properties
| `global_variances` | 49.3 Functions and Variables for descriptive statistics
| globalsolve | 20.1 Functions and Variables for Equations
| `gnuplot_close` | 12.6 Gnuplot_pipes Format Functions
| gnuplot_command | 12.3 Functions and Variables for Plotting
| gnuplot_curve_styles | 12.5 Gnuplot Options
| gnuplot_curve_titles | 12.5 Gnuplot Options
| gnuplot_default_term_command | 12.5 Gnuplot Options
| gnuplot_dumb_term_command | 12.5 Gnuplot Options
| gnuplot_file_args | 12.3 Functions and Variables for Plotting
| gnuplot_file_name | 52.2.3 Graphics options
| gnuplot_out_file | 12.5 Gnuplot Options
| gnuplot_pdf_term_command | 12.5 Gnuplot Options
| gnuplot_pm3d | 12.5 Gnuplot Options
| gnuplot_png_term_command | 12.5 Gnuplot Options
| gnuplot_postamble | 12.5 Gnuplot Options
| gnuplot_preamble | 12.5 Gnuplot Options
| gnuplot_ps_term_command | 12.5 Gnuplot Options
| `gnuplot_replot` | 12.6 Gnuplot_pipes Format Functions
| `gnuplot_reset` | 12.6 Gnuplot_pipes Format Functions
| `gnuplot_restart` | 12.6 Gnuplot_pipes Format Functions
| `gnuplot_start` | 12.6 Gnuplot_pipes Format Functions
| gnuplot_svg_term_command | 12.5 Gnuplot Options
| gnuplot_term | 12.5 Gnuplot Options
| gnuplot_view_args | 12.3 Functions and Variables for Plotting
| `go` | 37.4 Functions and Variables for Program Flow
| `Gosper` | 87.2 Functions and Variables for zeilberger
| Gosper_in_Zeilberger | 87.3 General global variables
| `GosperSum` | 87.2 Functions and Variables for zeilberger
| `gr2d` | 52.2.1 Scenes
| `gr3d` | 52.2.1 Scenes
| `gradef` | 18.1 Functions and Variables for Differentiation
| gradefs | 18.1 Functions and Variables for Differentiation
| `gramschmidt` | 23.2 Functions and Variables for Matrices and Linear Algebra
| `graph6_decode` | 61.2.4 Reading and writing to files
| `graph6_encode` | 61.2.4 Reading and writing to files
| `graph6_export` | 61.2.4 Reading and writing to files
| `graph6_import` | 61.2.4 Reading and writing to files
| `graph_center` | 61.2.2 Graph properties
| `graph_charpoly` | 61.2.2 Graph properties
| `graph_eigenvalues` | 61.2.2 Graph properties
| `graph_flow` | 58.2 Functions and Variables for finance
| `graph_order` | 61.2.2 Graph properties
| `graph_periphery` | 61.2.2 Graph properties
| `graph_product` | 61.2.1 Building graphs
| `graph_size` | 61.2.2 Graph properties
| `graph_union` | 61.2.1 Building graphs
| `great_rhombicosidodecahedron_graph` | 61.2.1 Building graphs
| `great_rhombicuboctahedron_graph` | 61.2.1 Building graphs
| grid | 12.4 Plotting Options
| grid | 52.2.3 Graphics options
| grid2d | 12.4 Plotting Options
| `grid_graph` | 61.2.1 Building graphs
| `grind` | 4.3 Functions and Variables for Display
| `grobner_basis` | 24.2 Functions and Variables for Affine
| `grotzch_graph` | 61.2.1 Building graphs
H |  |
| halfangles | 10.5.2 Functions and Variables for Trigonometric
| `hamilton_cycle` | 61.2.2 Graph properties
| `hamilton_path` | 61.2.2 Graph properties
| `hankel` | 68.2 Functions and Variables for linearalgebra
| `hankel_1` | 15.2 Bessel Functions
| `hankel_2` | 15.2 Bessel Functions
| `harmonic` | 80.4 Package functs
| `harmonic_mean` | 49.3 Functions and Variables for descriptive statistics
| `hav` | 80.4 Package functs
| head_angle | 52.2.3 Graphics options
| head_angle | 61.2.5 Visualization
| head_both | 52.2.3 Graphics options
| head_length | 52.2.3 Graphics options
| head_length | 61.2.5 Visualization
| head_type | 52.2.3 Graphics options
| `heawood_graph` | 61.2.1 Building graphs
| height | 54.3.1 Scene options
| height | 54.3.3 Scene object's options
| `hermite` | 76.2 Functions and Variables for orthogonal polynomials
| `hessian` | 68.2 Functions and Variables for linearalgebra
| `hgfred` | 15.10 Functions and Variables for Special Functions
| `hilbert_matrix` | 68.2 Functions and Variables for linearalgebra
| `hilbertmap` | 59.5 Definitions for Peano maps
| `hipow` | 14.2 Functions and Variables for Polynomials
| `histogram` | 49.4 Functions and Variables for statistical graphs
| `histogram_description` | 49.4 Functions and Variables for statistical graphs
| `hodge` | 25.2.7 Exterior algebra
| `horner` | 22.3 Functions for numerical solution of equations
| `hypergeometric` | 15.8 Hypergeometric Functions
| hypergeometric_representation | 15.6 Error Function
| `hypergeometric_simp` | 15.10 Functions and Variables for Special Functions
I |  |
| ibase | 4.3 Functions and Variables for Display
| `ic1` | 21.2 Functions and Variables for Differential Equations
| `ic2` | 21.2 Functions and Variables for Differential Equations
| `ic_convert` | 25.2.9 Interfacing with ctensor
| icc1 | 25.2.5 Moving frames
| icc2 | 25.2.5 Moving frames
| `ichr1` | 25.2.4 Tensors in curved spaces
| `ichr2` | 25.2.4 Tensors in curved spaces
| `icosahedron_graph` | 61.2.1 Building graphs
| `icosidodecahedron_graph` | 61.2.1 Building graphs
| icounter | 25.2.1 Managing indexed objects
| `icurvature` | 25.2.4 Tensors in curved spaces
| `ident` | 23.2 Functions and Variables for Matrices and Linear Algebra
| `identfor` | 68.2 Functions and Variables for linearalgebra
| `identity` | 35.2 Functions and Variables for Sets
| `idiff` | 25.2.3 Indicial tensor calculus
| `idim` | 25.2.4 Tensors in curved spaces
| `idummy` | 25.2.1 Managing indexed objects
| idummyx | 25.2.1 Managing indexed objects
| `ieqn` | 20.1 Functions and Variables for Equations
| ieqnprint | 20.1 Functions and Variables for Equations
| `if` | 37.4 Functions and Variables for Program Flow
| `ifactors` | 29.1 Functions and Variables for Number Theory
| ifb | 25.2.5 Moving frames
| ifc1 | 25.2.5 Moving frames
| ifc2 | 25.2.5 Moving frames
| ifg | 25.2.5 Moving frames
| ifgi | 25.2.5 Moving frames
| ifr | 25.2.5 Moving frames
| iframe_bracket_form | 25.2.5 Moving frames
| `iframes` | 25.2.5 Moving frames
| ifri | 25.2.5 Moving frames
| `ifs` | 54.2 Graphical analysis of discrete dynamical systems
| `igcdex` | 29.1 Functions and Variables for Number Theory
| `igeodesic_coords` | 25.2.4 Tensors in curved spaces
| igeowedge_flag | 25.2.7 Exterior algebra
| ikt1 | 25.2.6 Torsion and nonmetricity
| ikt2 | 25.2.6 Torsion and nonmetricity
| `ilt` | 19.2 Functions and Variables for Integration
| `image` | 52.2.4 Graphics objects
| imaginary | 11.2 Functions and Variables for Properties
| `imagpart` | 10.2 Functions for Complex Numbers
| `imetric` | 25.2.4 Tensors in curved spaces
| `imetric` | 25.2.4 Tensors in curved spaces
| `implicit` | 52.2.4 Graphics objects
| `implicit_derivative` | 63.1 Functions and Variables for impdiff
| `implicit_plot` | 12.3 Functions and Variables for Plotting
| `in` | 37.4 Functions and Variables for Program Flow
| `in_neighbors` | 61.2.2 Graph properties
| inchar | 4.2 Functions and Variables for Command Line
| increasing | 11.2 Functions and Variables for Properties
| ind | 5.3.1 Functions and Variables for Constants
| `indexed_tensor` | 25.2.1 Managing indexed objects
| `indices` | 25.2.1 Managing indexed objects
| `induced_subgraph` | 61.2.1 Building graphs
| inf | 5.3.1 Functions and Variables for Constants
| `inference_result` | 82.2 Functions and Variables for inference_result
| `inferencep` | 82.2 Functions and Variables for inference_result
| infeval | 8.1 Functions and Variables for Evaluation
| infinity | 5.3.1 Functions and Variables for Constants
| `infix` | 7.7 User defined operators
| inflag | 6.5 Functions and Variables for Expressions
| `info_display` | 39.2 Functions and Variables for alt-display
| infolists | 4.2 Functions and Variables for Command Line
| `init_atensor` | 27.2 Functions and Variables for atensor
| `init_ctensor` | 26.2.1 Initialization and setup
| inm | 25.2.6 Torsion and nonmetricity
| inmc1 | 25.2.6 Torsion and nonmetricity
| inmc2 | 25.2.6 Torsion and nonmetricity
| `innerproduct` | 23.2 Functions and Variables for Matrices and Linear Algebra
| `inpart` | 6.5 Functions and Variables for Expressions
| `inprod` | 23.2 Functions and Variables for Matrices and Linear Algebra
| `inrt` | 29.1 Functions and Variables for Number Theory
| intanalysis | 19.2 Functions and Variables for Integration
| integer | 11.2 Functions and Variables for Properties
| `integer_partitions` | 35.2 Functions and Variables for Sets
| `integerp` | 5.1.2 Functions and Variables for Numbers
| integervalued | 11.2 Functions and Variables for Properties
| `integrate` | 19.2 Functions and Variables for Integration
| integrate_use_rootsof | 19.2 Functions and Variables for Integration
| integration_constant | 19.2 Functions and Variables for Integration
| integration_constant_counter | 19.2 Functions and Variables for Integration
| interpolate_color | 52.2.3 Graphics options
| `intersect` | 35.2 Functions and Variables for Sets
| `intersection` | 35.2 Functions and Variables for Sets
| `intervalp` | 76.2 Functions and Variables for orthogonal polynomials
| intfaclim | 14.2 Functions and Variables for Polynomials
| `intopois` | 28.6 Functions and Variables for Poisson series
| `intosum` | 28.1 Functions and Variables for Sums and Products
| `inv_mod` | 29.1 Functions and Variables for Number Theory
| `invariant1` | 26.2.7 Miscellaneous features
| `invariant2` | 26.2.7 Miscellaneous features
| `inverse_fft` | 22.2 Functions and Variables for fast Fourier transform
| `inverse_jacobi_cd` | 16.2 Functions and Variables for Elliptic Functions
| `inverse_jacobi_cn` | 16.2 Functions and Variables for Elliptic Functions
| `inverse_jacobi_cs` | 16.2 Functions and Variables for Elliptic Functions
| `inverse_jacobi_dc` | 16.2 Functions and Variables for Elliptic Functions
| `inverse_jacobi_dn` | 16.2 Functions and Variables for Elliptic Functions
| `inverse_jacobi_ds` | 16.2 Functions and Variables for Elliptic Functions
| `inverse_jacobi_nc` | 16.2 Functions and Variables for Elliptic Functions
| `inverse_jacobi_nd` | 16.2 Functions and Variables for Elliptic Functions
| `inverse_jacobi_ns` | 16.2 Functions and Variables for Elliptic Functions
| `inverse_jacobi_sc` | 16.2 Functions and Variables for Elliptic Functions
| `inverse_jacobi_sd` | 16.2 Functions and Variables for Elliptic Functions
| `inverse_jacobi_sn` | 16.2 Functions and Variables for Elliptic Functions
| `inverse_real_fft` | 22.2 Functions and Variables for fast Fourier transform
| `invert` | 23.2 Functions and Variables for Matrices and Linear Algebra
| `invert_by_adjoint` | 23.2 Functions and Variables for Matrices and Linear Algebra
| `invert_by_lu` | 68.2 Functions and Variables for linearalgebra
| ip_grid | 52.2.3 Graphics options
| ip_grid_in | 52.2.3 Graphics options
| `irr` | 58.2 Functions and Variables for finance
| irrational | 11.2 Functions and Variables for Properties
| `is` | 11.3 Functions and Variables for Facts
| `is_biconnected` | 61.2.2 Graph properties
| `is_bipartite` | 61.2.2 Graph properties
| `is_connected` | 61.2.2 Graph properties
| `is_digraph` | 61.2.2 Graph properties
| `is_edge_in_graph` | 61.2.2 Graph properties
| `is_graph` | 61.2.2 Graph properties
| `is_graph_or_digraph` | 61.2.2 Graph properties
| `is_isomorphic` | 61.2.2 Graph properties
| `is_planar` | 61.2.2 Graph properties
| `is_sconnected` | 61.2.2 Graph properties
| `is_tree` | 61.2.2 Graph properties
| `is_vertex_in_graph` | 61.2.2 Graph properties
| `ishow` | 25.2.1 Managing indexed objects
| `isolate` | 6.5 Functions and Variables for Expressions
| isolate_wrt_times | 6.5 Functions and Variables for Expressions
| `isomorphism` | 61.2.2 Graph properties
| `isqrt` | 29.1 Functions and Variables for Number Theory
| `isreal_p` | 85.1 Functions and Variables for to_poly_solve
| `items_inference` | 82.2 Functions and Variables for inference_result
| iterations | 12.4 Plotting Options
| itr | 25.2.6 Torsion and nonmetricity
J |  |
| `jacobi` | 29.1 Functions and Variables for Number Theory
| `jacobi_cd` | 16.2 Functions and Variables for Elliptic Functions
| `jacobi_cn` | 16.2 Functions and Variables for Elliptic Functions
| `jacobi_cs` | 16.2 Functions and Variables for Elliptic Functions
| `jacobi_dc` | 16.2 Functions and Variables for Elliptic Functions
| `jacobi_dn` | 16.2 Functions and Variables for Elliptic Functions
| `jacobi_ds` | 16.2 Functions and Variables for Elliptic Functions
| `jacobi_nc` | 16.2 Functions and Variables for Elliptic Functions
| `jacobi_nd` | 16.2 Functions and Variables for Elliptic Functions
| `jacobi_ns` | 16.2 Functions and Variables for Elliptic Functions
| `jacobi_p` | 76.2 Functions and Variables for orthogonal polynomials
| `jacobi_sc` | 16.2 Functions and Variables for Elliptic Functions
| `jacobi_sd` | 16.2 Functions and Variables for Elliptic Functions
| `jacobi_sn` | 16.2 Functions and Variables for Elliptic Functions
| `jacobian` | 68.2 Functions and Variables for linearalgebra
| `JF` | 50.1 Functions and Variables for diag
| `join` | 5.4.2 Functions and Variables for Lists
| `jordan` | 50.1 Functions and Variables for diag
| `julia` | 12.3 Functions and Variables for Plotting
| julia_parameter | 59.3 Definitions for complex fractals
| `julia_set` | 59.3 Definitions for complex fractals
| `julia_sin` | 59.3 Definitions for complex fractals
K |  |
| `kdels` | 25.2.1 Managing indexed objects
| `kdelta` | 25.2.1 Managing indexed objects
| keepfloat | 14.2 Functions and Variables for Polynomials
| key | 52.2.3 Graphics options
| key_pos | 52.2.3 Graphics options
| `kill` | 4.2 Functions and Variables for Command Line
| `killcontext` | 11.3 Functions and Variables for Facts
| kinvariant | 26.2.9 Variables used by `ctensor`
| `km` | 49.3 Functions and Variables for descriptive statistics
| `kostka` | 30.2.4 Partitions
| `kron_delta` | 35.2 Functions and Variables for Sets
| `kronecker_product` | 68.2 Functions and Variables for linearalgebra
| kt | 26.2.9 Variables used by `ctensor`
| `kummer_m` | 48.2 Functions and Variables for contrib_ode
| `kummer_u` | 48.2 Functions and Variables for contrib_ode
| `kurtosis` | 49.3 Functions and Variables for descriptive statistics
| `kurtosis_bernoulli` | 51.3 Functions and Variables for discrete distributions
| `kurtosis_beta` | 51.2 Functions and Variables for continuous distributions
| `kurtosis_binomial` | 51.3 Functions and Variables for discrete distributions
| `kurtosis_chi2` | 51.2 Functions and Variables for continuous distributions
| `kurtosis_continuous_uniform` | 51.2 Functions and Variables for continuous distributions
| `kurtosis_discrete_uniform` | 51.3 Functions and Variables for discrete distributions
| `kurtosis_exp` | 51.2 Functions and Variables for continuous distributions
| `kurtosis_f` | 51.2 Functions and Variables for continuous distributions
| `kurtosis_gamma` | 51.2 Functions and Variables for continuous distributions
| `kurtosis_general_finite_discrete` | 51.3 Functions and Variables for discrete distributions
| `kurtosis_geometric` | 51.3 Functions and Variables for discrete distributions
| `kurtosis_gumbel` | 51.2 Functions and Variables for continuous distributions
| `kurtosis_hypergeometric` | 51.3 Functions and Variables for discrete distributions
| `kurtosis_laplace` | 51.2 Functions and Variables for continuous distributions
| `kurtosis_logistic` | 51.2 Functions and Variables for continuous distributions
| `kurtosis_lognormal` | 51.2 Functions and Variables for continuous distributions
| `kurtosis_negative_binomial` | 51.3 Functions and Variables for discrete distributions
| `kurtosis_noncentral_chi2` | 51.2 Functions and Variables for continuous distributions
| `kurtosis_noncentral_student_t` | 51.2 Functions and Variables for continuous distributions
| `kurtosis_normal` | 51.2 Functions and Variables for continuous distributions
| `kurtosis_pareto` | 51.2 Functions and Variables for continuous distributions
| `kurtosis_poisson` | 51.3 Functions and Variables for discrete distributions
| `kurtosis_rayleigh` | 51.2 Functions and Variables for continuous distributions
| `kurtosis_student_t` | 51.2 Functions and Variables for continuous distributions
| `kurtosis_weibull` | 51.2 Functions and Variables for continuous distributions
L |  |
| `label` | 52.2.4 Graphics objects
| label_alignment | 52.2.3 Graphics options
| label_alignment | 61.2.5 Visualization
| label_orientation | 52.2.3 Graphics options
| `labels` | 4.2 Functions and Variables for Command Line
| `lagrange` | 64.2 Functions and Variables for interpol
| `laguerre` | 76.2 Functions and Variables for orthogonal polynomials
| `lambda` | 36.4 Functions and Variables for Function Definition
| `lambert_w` | 15.10 Functions and Variables for Special Functions
| `laplace` | 18.1 Functions and Variables for Differentiation
| `laplacian_matrix` | 61.2.2 Graph properties
| lassociative | 9.2 Functions and Variables for Simplification
| `last` | 5.4.2 Functions and Variables for Lists
| `lastn` | 5.4.2 Functions and Variables for Lists
| `lbfgs` | 66.2 Functions and Variables for lbfgs
| lbfgs_ncorrections | 66.2 Functions and Variables for lbfgs
| lbfgs_nfeval_max | 66.2 Functions and Variables for lbfgs
| `lc2kdt` | 25.2.1 Managing indexed objects
| `lc_l` | 25.2.1 Managing indexed objects
| `lc_u` | 25.2.1 Managing indexed objects
| `lcm` | 29.1 Functions and Variables for Number Theory
| `ldefint` | 19.2 Functions and Variables for Integration
| `ldisp` | 4.3 Functions and Variables for Display
| `ldisplay` | 4.3 Functions and Variables for Display
| leftjust | 4.3 Functions and Variables for Display
| legend | 12.4 Plotting Options
| `legendre_p` | 76.2 Functions and Variables for orthogonal polynomials
| `legendre_q` | 76.2 Functions and Variables for orthogonal polynomials
| `leinstein` | 26.2.2 The tensors of curved space
| `length` | 5.4.2 Functions and Variables for Lists
| `let` | 34.2 Functions and Variables for Rules and Patterns
| let_rule_packages | 34.2 Functions and Variables for Rules and Patterns
| letrat | 34.2 Functions and Variables for Rules and Patterns
| `letrules` | 34.2 Functions and Variables for Rules and Patterns
| `letsimp` | 34.2 Functions and Variables for Rules and Patterns
| `levi_civita` | 25.2.1 Managing indexed objects
| lfg | 26.2.9 Variables used by `ctensor`
| `lfreeof` | 6.5 Functions and Variables for Expressions
| lg | 26.2.9 Variables used by `ctensor`
| `lgtreillis` | 30.2.4 Partitions
| lhospitallim | 17.1 Functions and Variables for Limits
| `lhs` | 20.1 Functions and Variables for Equations
| `li` | 10.4 Root, Exponential and Logarithmic Functions
| `liediff` | 25.2.3 Indicial tensor calculus
| `limit` | 17.1 Functions and Variables for Limits
| limsubst | 17.1 Functions and Variables for Limits
| `Lindstedt` | 67.1 Functions and Variables for lindstedt
| `line_graph` | 61.2.1 Building graphs
| line_type | 52.2.3 Graphics options
| line_width | 52.2.3 Graphics options
| `linear` | 80.4 Package functs
| `linear_program` | 79.2 Functions and Variables for simplex
| `linear_regression` | 82.3 Functions and Variables for stats
| linear_solver | 87.3 General global variables
| `linearinterpol` | 64.2 Functions and Variables for interpol
| linechar | 4.2 Functions and Variables for Command Line
| linel | 4.3 Functions and Variables for Display
| linenum | 4.2 Functions and Variables for Command Line
| linewidth | 54.3.3 Scene object's options
| `linsolve` | 20.1 Functions and Variables for Equations
| linsolve_params | 20.1 Functions and Variables for Equations
| linsolvewarn | 20.1 Functions and Variables for Equations
| lispdisp | 4.3 Functions and Variables for Display
| `list_correlations` | 49.3 Functions and Variables for descriptive statistics
| `list_matrix_entries` | 23.2 Functions and Variables for Matrices and Linear Algebra
| `list_nc_monomials` | 24.2 Functions and Variables for Affine
| listarith | 5.4.2 Functions and Variables for Lists
| `listarray` | 5.5.1 Functions and Variables for Arrays
| listconstvars | 6.5 Functions and Variables for Expressions
| listdummyvars | 6.5 Functions and Variables for Expressions
| `listify` | 35.2 Functions and Variables for Sets
| `listoftens` | 25.2.1 Managing indexed objects
| `listofvars` | 6.5 Functions and Variables for Expressions
| `listp` | 5.4.2 Functions and Variables for Lists
| `listp` | 68.2 Functions and Variables for linearalgebra
| `lmax` | 10.1 Functions for Numbers
| `lmin` | 10.1 Functions for Numbers
| lmxchar | 23.2 Functions and Variables for Matrices and Linear Algebra
| `load` | 13.3 Functions and Variables for File Input and Output
| load_pathname | 13.3 Functions and Variables for File Input and Output
| `loadfile` | 13.3 Functions and Variables for File Input and Output
| loadprint | 13.3 Functions and Variables for File Input and Output
| `local` | 36.4 Functions and Variables for Function Definition
| `locate_matrix_entry` | 68.2 Functions and Variables for linearalgebra
| `log` | 10.4 Root, Exponential and Logarithmic Functions
| `log_gamma` | 15.4 Gamma and factorial Functions
| logabs | 10.4 Root, Exponential and Logarithmic Functions
| logarc | 10.4 Root, Exponential and Logarithmic Functions
| logarc | 10.4 Root, Exponential and Logarithmic Functions
| logcb | 52.2.3 Graphics options
| logconcoeffp | 10.4 Root, Exponential and Logarithmic Functions
| `logcontract` | 10.4 Root, Exponential and Logarithmic Functions
| logexpand | 10.4 Root, Exponential and Logarithmic Functions
| lognegint | 10.4 Root, Exponential and Logarithmic Functions
| logsimp | 10.4 Root, Exponential and Logarithmic Functions
| logx | 12.4 Plotting Options
| logx | 52.2.3 Graphics options
| logx_secondary | 52.2.3 Graphics options
| logy | 12.4 Plotting Options
| logy | 52.2.3 Graphics options
| logy_secondary | 52.2.3 Graphics options
| logz | 52.2.3 Graphics options
| `lopow` | 14.2 Functions and Variables for Polynomials
| `lorentz_gauge` | 25.2.4 Tensors in curved spaces
| `lowercasep` | 84.3 Characters
| `lpart` | 6.5 Functions and Variables for Expressions
| `lratsubst` | 14.2 Functions and Variables for Polynomials
| `lreduce` | 5.4.2 Functions and Variables for Lists
| lriem | 26.2.9 Variables used by `ctensor`
| `lriemann` | 26.2.2 The tensors of curved space
| `lsquares_estimates` | 69.2 Functions and Variables for lsquares
| `lsquares_estimates_approximate` | 69.2 Functions and Variables for lsquares
| `lsquares_estimates_exact` | 69.2 Functions and Variables for lsquares
| `lsquares_mse` | 69.2 Functions and Variables for lsquares
| `lsquares_residual_mse` | 69.2 Functions and Variables for lsquares
| `lsquares_residuals` | 69.2 Functions and Variables for lsquares
| `lsum` | 28.1 Functions and Variables for Sums and Products
| `ltreillis` | 30.2.4 Partitions
| `lu_backsub` | 68.2 Functions and Variables for linearalgebra
| `lu_factor` | 68.2 Functions and Variables for linearalgebra
| `lucas` | 29.1 Functions and Variables for Number Theory
M |  |
| m1pbranch | 5.1.2 Functions and Variables for Numbers
| `macroexpand` | 36.3 Macros
| `macroexpand1` | 36.3 Macros
| macroexpansion | 36.4 Functions and Variables for Function Definition
| macros | 36.3 Macros
| mainvar | 6.5 Functions and Variables for Expressions
| `make_array` | 5.5.1 Functions and Variables for Arrays
| `make_graph` | 61.2.1 Building graphs
| `make_level_picture` | 52.3 Functions and Variables for pictures
| `make_poly_continent` | 52.4.1 Variables and Functions
| `make_poly_country` | 52.4.1 Variables and Functions
| `make_polygon` | 52.4.1 Variables and Functions
| `make_random_state` | 10.6 Random Numbers
| `make_rgb_picture` | 52.3 Functions and Variables for pictures
| `make_string_input_stream` | 84.2 Input and Output
| `make_string_output_stream` | 84.2 Input and Output
| `make_transform` | 12.3 Functions and Variables for Plotting
| `makebox` | 25.2.3 Indicial tensor calculus
| `makefact` | 15.4 Gamma and factorial Functions
| `makegamma` | 15.4 Gamma and factorial Functions
| `makelist` | 5.4.2 Functions and Variables for Lists
| `makeOrders` | 71.1 Functions and Variables for makeOrders
| `makeset` | 35.2 Functions and Variables for Sets
| `mandelbrot` | 12.3 Functions and Variables for Plotting
| `mandelbrot_set` | 59.3 Definitions for complex fractals
| manual_demo | 3.2 Functions and Variables for Help
| `map` | 37.4 Functions and Variables for Program Flow
| `mapatom` | 37.4 Functions and Variables for Program Flow
| maperror | 37.4 Functions and Variables for Program Flow
| `maplist` | 37.4 Functions and Variables for Program Flow
| mapprint | 37.4 Functions and Variables for Program Flow
| `mat_cond` | 68.2 Functions and Variables for linearalgebra
| `mat_fullunblocker` | 68.2 Functions and Variables for linearalgebra
| `mat_function` | 50.1 Functions and Variables for diag
| `mat_norm` | 68.2 Functions and Variables for linearalgebra
| `mat_trace` | 68.2 Functions and Variables for linearalgebra
| `mat_unblocker` | 68.2 Functions and Variables for linearalgebra
| `matchdeclare` | 34.2 Functions and Variables for Rules and Patterns
| `matchfix` | 7.7 User defined operators
| `mathml_display` | 39.2 Functions and Variables for alt-display
| `matrix` | 23.2 Functions and Variables for Matrices and Linear Algebra
| matrix_element_add | 23.2 Functions and Variables for Matrices and Linear Algebra
| matrix_element_mult | 23.2 Functions and Variables for Matrices and Linear Algebra
| matrix_element_transpose | 23.2 Functions and Variables for Matrices and Linear Algebra
| `matrix_size` | 68.2 Functions and Variables for linearalgebra
| `matrixexp` | 23.2 Functions and Variables for Matrices and Linear Algebra
| `matrixmap` | 23.2 Functions and Variables for Matrices and Linear Algebra
| `matrixp` | 23.2 Functions and Variables for Matrices and Linear Algebra
| `matrixp` | 68.2 Functions and Variables for linearalgebra
| `mattrace` | 23.2 Functions and Variables for Matrices and Linear Algebra
| `max` | 10.1 Functions for Numbers
| `max_clique` | 61.2.2 Graph properties
| `max_degree` | 61.2.2 Graph properties
| `max_flow` | 61.2.2 Graph properties
| `max_independent_set` | 61.2.2 Graph properties
| `max_matching` | 61.2.2 Graph properties
| MAX_ORD | 87.3 General global variables
| maxapplydepth | 34.2 Functions and Variables for Rules and Patterns
| maxapplyheight | 34.2 Functions and Variables for Rules and Patterns
| maxima_tempdir | 32.3 Functions and Variables for Runtime Environment
| maxima_userdir | 32.3 Functions and Variables for Runtime Environment
| `maximize_lp` | 79.2 Functions and Variables for simplex
| maxnegex | 9.2 Functions and Variables for Simplification
| maxposex | 9.2 Functions and Variables for Simplification
| maxpsifracdenom | 15.4 Gamma and factorial Functions
| maxpsifracnum | 15.4 Gamma and factorial Functions
| maxpsinegint | 15.4 Gamma and factorial Functions
| maxpsiposint | 15.4 Gamma and factorial Functions
| maxtayorder | 28.3 Functions and Variables for Series
| `maybe` | 11.3 Functions and Variables for Facts
| `md5sum` | 84.5 Octets and Utilities for Cryptography
| `mean` | 49.3 Functions and Variables for descriptive statistics
| `mean_bernoulli` | 51.3 Functions and Variables for discrete distributions
| `mean_beta` | 51.2 Functions and Variables for continuous distributions
| `mean_binomial` | 51.3 Functions and Variables for discrete distributions
| `mean_chi2` | 51.2 Functions and Variables for continuous distributions
| `mean_continuous_uniform` | 51.2 Functions and Variables for continuous distributions
| `mean_deviation` | 49.3 Functions and Variables for descriptive statistics
| `mean_discrete_uniform` | 51.3 Functions and Variables for discrete distributions
| `mean_exp` | 51.2 Functions and Variables for continuous distributions
| `mean_f` | 51.2 Functions and Variables for continuous distributions
| `mean_gamma` | 51.2 Functions and Variables for continuous distributions
| `mean_general_finite_discrete` | 51.3 Functions and Variables for discrete distributions
| `mean_geometric` | 51.3 Functions and Variables for discrete distributions
| `mean_gumbel` | 51.2 Functions and Variables for continuous distributions
| `mean_hypergeometric` | 51.3 Functions and Variables for discrete distributions
| `mean_laplace` | 51.2 Functions and Variables for continuous distributions
| `mean_logistic` | 51.2 Functions and Variables for continuous distributions
| `mean_lognormal` | 51.2 Functions and Variables for continuous distributions
| `mean_negative_binomial` | 51.3 Functions and Variables for discrete distributions
| `mean_noncentral_chi2` | 51.2 Functions and Variables for continuous distributions
| `mean_noncentral_student_t` | 51.2 Functions and Variables for continuous distributions
| `mean_normal` | 51.2 Functions and Variables for continuous distributions
| `mean_pareto` | 51.2 Functions and Variables for continuous distributions
| `mean_poisson` | 51.3 Functions and Variables for discrete distributions
| `mean_rayleigh` | 51.2 Functions and Variables for continuous distributions
| `mean_student_t` | 51.2 Functions and Variables for continuous distributions
| `mean_weibull` | 51.2 Functions and Variables for continuous distributions
| `median` | 49.3 Functions and Variables for descriptive statistics
| `median_deviation` | 49.3 Functions and Variables for descriptive statistics
| `member` | 5.4.2 Functions and Variables for Lists
| `mesh` | 52.2.4 Graphics objects
| mesh_lines_color | 12.4 Plotting Options
| method | 48.2 Functions and Variables for contrib_ode
| `metricexpandall` | 86.2 Functions and Variables for Units
| `mgf1_sha1` | 84.5 Octets and Utilities for Cryptography
| `min` | 10.1 Functions for Numbers
| `min_degree` | 61.2.2 Graph properties
| `min_edge_cut` | 61.2.2 Graph properties
| `min_vertex_cover` | 61.2.2 Graph properties
| `min_vertex_cut` | 61.2.2 Graph properties
| minf | 5.3.1 Functions and Variables for Constants
| `minfactorial` | 10.3 Combinatorial Functions
| `minimalPoly` | 50.1 Functions and Variables for diag
| `minimize_lp` | 79.2 Functions and Variables for simplex
| `minimum_spanning_tree` | 61.2.2 Graph properties
| `minor` | 23.2 Functions and Variables for Matrices and Linear Algebra
| `minpack_lsquares` | 70.2 Functions and Variables for minpack
| `minpack_solve` | 70.2 Functions and Variables for minpack
| `mkdir` | 74.2 Directory operations
| `mnewton` | 72.2 Functions and Variables for mnewton
| `mod` | 29.1 Functions and Variables for Number Theory
| mod_big_prime | 87.4 Variables related to the modular test
| mod_test | 87.4 Variables related to the modular test
| mod_threshold | 87.4 Variables related to the modular test
| mode_check_errorp | 36.4 Functions and Variables for Function Definition
| mode_check_warnp | 36.4 Functions and Variables for Function Definition
| mode_checkp | 36.4 Functions and Variables for Function Definition
| `mode_declare` | 36.4 Functions and Variables for Function Definition
| `mode_identity` | 36.4 Functions and Variables for Function Definition
| `ModeMatrix` | 50.1 Functions and Variables for diag
| modular_linear_solver | 87.4 Variables related to the modular test
| modulus | 14.2 Functions and Variables for Polynomials
| `moebius` | 35.2 Functions and Variables for Sets
| `mon2schur` | 30.2.1 Changing bases
| `mono` | 24.2 Functions and Variables for Affine
| `monomial_dimensions` | 24.2 Functions and Variables for Affine
| `multi_display_for_texinfo` | 39.2 Functions and Variables for alt-display
| `multi_elem` | 30.2.1 Changing bases
| `multi_orbit` | 30.2.3 Groups and orbits
| `multi_pui` | 30.2.1 Changing bases
| `multibernstein_poly` | 42.1 Functions and Variables for Bernstein
| `multinomial` | 30.2.7 Miscellaneous
| `multinomial_coeff` | 35.2 Functions and Variables for Sets
| multiplicative | 9.2 Functions and Variables for Simplification
| multiplicities | 20.1 Functions and Variables for Equations
| `multiplot_mode` | 52.2.2 Functions
| `multsym` | 30.2.3 Groups and orbits
| `multthru` | 9.2 Functions and Variables for Simplification
| `mycielski_graph` | 61.2.1 Building graphs
| myoptions | 4.2 Functions and Variables for Command Line
N |  |
| `nary` | 7.7 User defined operators
| `natural_unit` | 56.3 Functions and Variables for ezunits
| `nc_degree` | 24.2 Functions and Variables for Affine
| `ncexpt` | 4.3 Functions and Variables for Display
| `ncharpoly` | 23.2 Functions and Variables for Matrices and Linear Algebra
| `negative_picture` | 52.3 Functions and Variables for pictures
| negdistrib | 9.2 Functions and Variables for Simplification
| negsumdispflag | 4.3 Functions and Variables for Display
| `neighbors` | 61.2.2 Graph properties
| `new` | 5.6.2 Functions and Variables for Structures
| `new_graph` | 61.2.1 Building graphs
| `new_variable` | 85.1 Functions and Variables for to_poly_solve
| `newcontext` | 11.3 Functions and Variables for Facts
| `newdet` | 23.2 Functions and Variables for Matrices and Linear Algebra
| `newline` | 84.2 Input and Output
| `newton` | 22.3 Functions for numerical solution of equations
| newtonepsilon | 72.2 Functions and Variables for mnewton
| newtonmaxiter | 72.2 Functions and Variables for mnewton
| `next` | 37.4 Functions and Variables for Program Flow
| `next_prime` | 29.1 Functions and Variables for Number Theory
| nextlayerfactor | 80.3 Package facexp
| `nicedummies` | 85.1 Functions and Variables for to_poly_solve
| `niceindices` | 28.3 Functions and Variables for Series
| niceindicespref | 28.3 Functions and Variables for Series
| `ninth` | 5.4.2 Functions and Variables for Lists
| nm | 26.2.9 Variables used by `ctensor`
| nmc | 26.2.9 Variables used by `ctensor`
| noeval | 8.1 Functions and Variables for Evaluation
| `nofix` | 7.7 User defined operators
| nolabels | 4.2 Functions and Variables for Command Line
| `nonarray` | 11.2 Functions and Variables for Properties
| `noncentral_moment` | 49.3 Functions and Variables for descriptive statistics
| nonegative_lp | 79.2 Functions and Variables for simplex
| noninteger | 11.2 Functions and Variables for Properties
| `nonmetricity` | 26.2.6 Torsion and nonmetricity
| `nonnegintegerp` | 5.1.2 Functions and Variables for Numbers
| nonscalar | 11.2 Functions and Variables for Properties
| `nonscalarp` | 11.2 Functions and Variables for Properties
| `nonzeroandfreeof` | 80.4 Package functs
| `not` | 7.4 Logical operators
| `notequal` | 11.4 Functions and Variables for Predicates
| noun | 6.5 Functions and Variables for Expressions
| noundisp | 6.5 Functions and Variables for Expressions
| `nounify` | 6.5 Functions and Variables for Expressions
| nouns | 8.1 Functions and Variables for Evaluation
| np | 26.2.9 Variables used by `ctensor`
| npi | 26.2.9 Variables used by `ctensor`
| `nptetrad` | 26.2.5 Algebraic classification
| `npv` | 58.2 Functions and Variables for finance
| `nroots` | 20.1 Functions and Variables for Equations
| `nterms` | 6.5 Functions and Variables for Expressions
| `ntermst` | 26.2.8 Utility functions
| `nthroot` | 20.1 Functions and Variables for Equations
| nticks | 12.4 Plotting Options
| nticks | 52.2.3 Graphics options
| ntrig | 10.5.2 Functions and Variables for Trigonometric
| `nullity` | 68.2 Functions and Variables for linearalgebra
| `nullspace` | 68.2 Functions and Variables for linearalgebra
| `num` | 14.2 Functions and Variables for Polynomials
| `num_distinct_partitions` | 35.2 Functions and Variables for Sets
| `num_partitions` | 35.2 Functions and Variables for Sets
| `number_to_octets` | 84.5 Octets and Utilities for Cryptography
| `numbered_boundaries` | 52.4.1 Variables and Functions
| `numberp` | 5.1.2 Functions and Variables for Numbers
| numer | 5.1.2 Functions and Variables for Numbers
| numer_pbranch | 5.1.2 Functions and Variables for Numbers
| `numerval` | 5.1.2 Functions and Variables for Numbers
| `numfactor` | 15.4 Gamma and factorial Functions
| `nusum` | 28.3 Functions and Variables for Series
| `nzeta` | 15.10 Functions and Variables for Special Functions
| `nzetai` | 15.10 Functions and Variables for Special Functions
| `nzetar` | 15.10 Functions and Variables for Special Functions
O |  |
| obase | 4.3 Functions and Variables for Display
| `octets_to_number` | 84.5 Octets and Utilities for Cryptography
| `octets_to_oid` | 84.5 Octets and Utilities for Cryptography
| `octets_to_string` | 84.5 Octets and Utilities for Cryptography
| odd | 11.2 Functions and Variables for Properties
| `odd_girth` | 61.2.2 Graph properties
| oddfun | 9.2 Functions and Variables for Simplification
| `oddp` | 5.1.2 Functions and Variables for Numbers
| `ode2` | 21.2 Functions and Variables for Differential Equations
| `ode_check` | 48.2 Functions and Variables for contrib_ode
| `odelin` | 48.2 Functions and Variables for contrib_ode
| `oid_to_octets` | 84.5 Octets and Utilities for Cryptography
| `op` | 6.5 Functions and Variables for Expressions
| opacity | 54.3.3 Scene object's options
| `opena` | 84.2 Input and Output
| `opena_binary` | 73.3 Functions and Variables for binary input and output
| `openr` | 84.2 Input and Output
| `openr_binary` | 73.3 Functions and Variables for binary input and output
| `openw` | 84.2 Input and Output
| `openw_binary` | 73.3 Functions and Variables for binary input and output
| `operatorp` | 6.5 Functions and Variables for Expressions
| opproperties | 9.2 Functions and Variables for Simplification
| `opsubst` | 75.1 Functions and Variables for opsubst
| `optimize` | 6.5 Functions and Variables for Expressions
| optimprefix | 6.5 Functions and Variables for Expressions
| optionset | 4.2 Functions and Variables for Command Line
| `or` | 7.4 Logical operators
| `orbit` | 30.2.3 Groups and orbits
| `orbits` | 54.2 Graphical analysis of discrete dynamical systems
| `ordergreat` | 6.5 Functions and Variables for Expressions
| `ordergreatp` | 6.5 Functions and Variables for Expressions
| `orderless` | 6.5 Functions and Variables for Expressions
| `orderlessp` | 6.5 Functions and Variables for Expressions
| orientation | 54.3.3 Scene object's options
| origin | 54.3.3 Scene object's options
| `orthogonal_complement` | 68.2 Functions and Variables for linearalgebra
| `orthopoly_recur` | 76.2 Functions and Variables for orthogonal polynomials
| orthopoly_returns_intervals | 76.2 Functions and Variables for orthogonal polynomials
| `orthopoly_weight` | 76.2 Functions and Variables for orthogonal polynomials
| `out_neighbors` | 61.2.2 Graph properties
| outative | 9.2 Functions and Variables for Simplification
| outchar | 4.2 Functions and Variables for Command Line
| `outermap` | 37.4 Functions and Variables for Program Flow
| `outofpois` | 28.6 Functions and Variables for Poisson series
P |  |
| packagefile | 33.3 Functions and Variables for Miscellaneous Options
| `pade` | 28.3 Functions and Variables for Series
| palette | 12.4 Plotting Options
| palette | 52.2.3 Graphics options
| `parabolic_cylinder_d` | 15.9 Parabolic Cylinder Functions
| `parametric` | 52.2.4 Graphics objects
| `parametric_surface` | 52.2.4 Graphics objects
| `parg` | 85.1 Functions and Variables for to_poly_solve
| `parGosper` | 87.2 Functions and Variables for zeilberger
| `parse_string` | 84.4 String Processing
| `parse_timedate` | 32.3 Functions and Variables for Runtime Environment
| `part` | 6.5 Functions and Variables for Expressions
| `part2cont` | 30.2.2 Changing representations
| `partfrac` | 29.1 Functions and Variables for Number Theory
| `partition` | 6.5 Functions and Variables for Expressions
| `partition_set` | 35.2 Functions and Variables for Sets
| `partpol` | 30.2.2 Changing representations
| partswitch | 6.5 Functions and Variables for Expressions
| `path_digraph` | 61.2.1 Building graphs
| `path_graph` | 61.2.1 Building graphs
| `pathname_directory` | 13.3 Functions and Variables for File Input and Output
| `pathname_name` | 13.3 Functions and Variables for File Input and Output
| `pathname_type` | 13.3 Functions and Variables for File Input and Output
| `pdf_bernoulli` | 51.3 Functions and Variables for discrete distributions
| `pdf_beta` | 51.2 Functions and Variables for continuous distributions
| `pdf_binomial` | 51.3 Functions and Variables for discrete distributions
| `pdf_cauchy` | 51.2 Functions and Variables for continuous distributions
| `pdf_chi2` | 51.2 Functions and Variables for continuous distributions
| `pdf_continuous_uniform` | 51.2 Functions and Variables for continuous distributions
| `pdf_discrete_uniform` | 51.3 Functions and Variables for discrete distributions
| `pdf_exp` | 51.2 Functions and Variables for continuous distributions
| `pdf_f` | 51.2 Functions and Variables for continuous distributions
| pdf_file | 12.4 Plotting Options
| `pdf_gamma` | 51.2 Functions and Variables for continuous distributions
| `pdf_general_finite_discrete` | 51.3 Functions and Variables for discrete distributions
| `pdf_geometric` | 51.3 Functions and Variables for discrete distributions
| `pdf_gumbel` | 51.2 Functions and Variables for continuous distributions
| `pdf_hypergeometric` | 51.3 Functions and Variables for discrete distributions
| `pdf_laplace` | 51.2 Functions and Variables for continuous distributions
| `pdf_logistic` | 51.2 Functions and Variables for continuous distributions
| `pdf_lognormal` | 51.2 Functions and Variables for continuous distributions
| `pdf_negative_binomial` | 51.3 Functions and Variables for discrete distributions
| `pdf_noncentral_chi2` | 51.2 Functions and Variables for continuous distributions
| `pdf_noncentral_student_t` | 51.2 Functions and Variables for continuous distributions
| `pdf_normal` | 51.2 Functions and Variables for continuous distributions
| `pdf_pareto` | 51.2 Functions and Variables for continuous distributions
| `pdf_poisson` | 51.3 Functions and Variables for discrete distributions
| `pdf_rank_sum` | 82.4 Functions and Variables for special distributions
| `pdf_rayleigh` | 51.2 Functions and Variables for continuous distributions
| `pdf_signed_rank` | 82.4 Functions and Variables for special distributions
| `pdf_student_t` | 51.2 Functions and Variables for continuous distributions
| `pdf_weibull` | 51.2 Functions and Variables for continuous distributions
| `pearson_skewness` | 49.3 Functions and Variables for descriptive statistics
| `permanent` | 23.2 Functions and Variables for Matrices and Linear Algebra
| `permut` | 30.2.7 Miscellaneous
| `permutation` | 80.4 Package functs
| `permutations` | 35.2 Functions and Variables for Sets
| `petersen_graph` | 61.2.1 Building graphs
| `petrov` | 26.2.5 Algebraic classification
| pfeformat | 4.3 Functions and Variables for Display
| phiresolution | 54.3.3 Scene object's options
| `pickapart` | 6.5 Functions and Variables for Expressions
| `picture_equalp` | 52.3 Functions and Variables for pictures
| `picturep` | 52.3 Functions and Variables for pictures
| piece | 6.5 Functions and Variables for Expressions
| `piechart` | 49.4 Functions and Variables for statistical graphs
| `piechart_description` | 49.4 Functions and Variables for statistical graphs
| pivot_count_sx | 79.2 Functions and Variables for simplex
| pivot_max_sx | 79.2 Functions and Variables for simplex
| `planar_embedding` | 61.2.2 Graph properties
| `playback` | 4.2 Functions and Variables for Command Line
| `plog` | 10.4 Root, Exponential and Logarithmic Functions
| `plot2d` | 12.3 Functions and Variables for Plotting
| `plot3d` | 12.3 Functions and Variables for Plotting
| plot_format | 12.4 Plotting Options
| plot_options | 12.3 Functions and Variables for Plotting
| plot_realpart | 12.4 Plotting Options
| `plotdf` | 22.5 Functions for numerical solution of differential equations
| `ploteq` | 22.5 Functions for numerical solution of differential equations
| `plsquares` | 69.2 Functions and Variables for lsquares
| png_file | 12.4 Plotting Options
| `pochhammer` | 76.2 Functions and Variables for orthogonal polynomials
| pochhammer_max_index | 76.2 Functions and Variables for orthogonal polynomials
| point_size | 52.2.3 Graphics options
| point_type | 12.4 Plotting Options
| point_type | 52.2.3 Graphics options
| `points` | 52.2.4 Graphics objects
| points_joined | 52.2.3 Graphics options
| pointsize | 54.3.3 Scene object's options
| `poisdiff` | 28.6 Functions and Variables for Poisson series
| `poisexpt` | 28.6 Functions and Variables for Poisson series
| `poisint` | 28.6 Functions and Variables for Poisson series
| poislim | 28.6 Functions and Variables for Poisson series
| `poismap` | 28.6 Functions and Variables for Poisson series
| `poisplus` | 28.6 Functions and Variables for Poisson series
| `poissimp` | 28.6 Functions and Variables for Poisson series
| poisson | 28.6 Functions and Variables for Poisson series
| `poissubst` | 28.6 Functions and Variables for Poisson series
| `poistimes` | 28.6 Functions and Variables for Poisson series
| `poistrim` | 28.6 Functions and Variables for Poisson series
| `polar` | 52.2.4 Graphics objects
| `polar_to_xy` | 12.3 Functions and Variables for Plotting
| `polarform` | 10.2 Functions for Complex Numbers
| `polartorect` | 22.2 Functions and Variables for fast Fourier transform
| `poly_add` | 62.2.2 Simple operators in grobner
| `poly_buchberger` | 62.2.3 Other functions in grobner
| `poly_buchberger_criterion` | 62.2.3 Other functions in grobner
| poly_coefficient_ring | 62.2.1 Global switches for grobner
| `poly_colon_ideal` | 62.2.4 Standard postprocessing of Groebner Bases
| `poly_content` | 62.2.3 Other functions in grobner
| `poly_depends_p` | 62.2.4 Standard postprocessing of Groebner Bases
| `poly_elimination_ideal` | 62.2.4 Standard postprocessing of Groebner Bases
| poly_elimination_order | 62.2.1 Global switches for grobner
| `poly_exact_divide` | 62.2.3 Other functions in grobner
| `poly_expand` | 62.2.3 Other functions in grobner
| `poly_expt` | 62.2.3 Other functions in grobner
| `poly_gcd` | 62.2.4 Standard postprocessing of Groebner Bases
| `poly_grobner` | 62.2.4 Standard postprocessing of Groebner Bases
| poly_grobner_algorithm | 62.2.1 Global switches for grobner
| poly_grobner_debug | 62.2.1 Global switches for grobner
| `poly_grobner_equal` | 62.2.4 Standard postprocessing of Groebner Bases
| `poly_grobner_member` | 62.2.4 Standard postprocessing of Groebner Bases
| `poly_grobner_subsetp` | 62.2.4 Standard postprocessing of Groebner Bases
| `poly_ideal_intersection` | 62.2.4 Standard postprocessing of Groebner Bases
| `poly_ideal_polysaturation` | 62.2.4 Standard postprocessing of Groebner Bases
| `poly_ideal_polysaturation1` | 62.2.4 Standard postprocessing of Groebner Bases
| `poly_ideal_saturation` | 62.2.4 Standard postprocessing of Groebner Bases
| `poly_ideal_saturation1` | 62.2.4 Standard postprocessing of Groebner Bases
| `poly_lcm` | 62.2.4 Standard postprocessing of Groebner Bases
| `poly_minimization` | 62.2.4 Standard postprocessing of Groebner Bases
| poly_monomial_order | 62.2.1 Global switches for grobner
| `poly_multiply` | 62.2.2 Simple operators in grobner
| `poly_normal_form` | 62.2.3 Other functions in grobner
| `poly_normalize` | 62.2.2 Simple operators in grobner
| `poly_normalize_list` | 62.2.4 Standard postprocessing of Groebner Bases
| `poly_polysaturation_extension` | 62.2.4 Standard postprocessing of Groebner Bases
| poly_primary_elimination_order | 62.2.1 Global switches for grobner
| `poly_primitive_part` | 62.2.2 Simple operators in grobner
| `poly_pseudo_divide` | 62.2.3 Other functions in grobner
| `poly_reduced_grobner` | 62.2.4 Standard postprocessing of Groebner Bases
| `poly_reduction` | 62.2.4 Standard postprocessing of Groebner Bases
| poly_return_term_list | 62.2.1 Global switches for grobner
| `poly_s_polynomial` | 62.2.2 Simple operators in grobner
| `poly_saturation_extension` | 62.2.4 Standard postprocessing of Groebner Bases
| poly_secondary_elimination_order | 62.2.1 Global switches for grobner
| `poly_subtract` | 62.2.2 Simple operators in grobner
| poly_top_reduction_only | 62.2.1 Global switches for grobner
| `polydecomp` | 14.2 Functions and Variables for Polynomials
| polyfactor | 20.1 Functions and Variables for Equations
| `polygon` | 52.2.4 Graphics objects
| `polymod` | 14.2 Functions and Variables for Polynomials
| `polynome2ele` | 30.2.5 Polynomials and their roots
| `polynomialp` | 68.2 Functions and Variables for linearalgebra
| `polytocompanion` | 68.2 Functions and Variables for linearalgebra
| `pop` | 5.4.2 Functions and Variables for Lists
| posfun | 11.2 Functions and Variables for Properties
| position | 54.3.3 Scene object's options
| `postfix` | 7.7 User defined operators
| `potential` | 19.2 Functions and Variables for Integration
| `power_mod` | 29.1 Functions and Variables for Number Theory
| powerdisp | 4.3 Functions and Variables for Display
| `powerseries` | 28.3 Functions and Variables for Series
| `powerset` | 35.2 Functions and Variables for Sets
| pred | 8.1 Functions and Variables for Evaluation
| prederror | 37.4 Functions and Variables for Program Flow
| `prefix` | 7.7 User defined operators
| `prev_prime` | 29.1 Functions and Variables for Number Theory
| `primep` | 29.1 Functions and Variables for Number Theory
| primep_number_of_tests | 29.1 Functions and Variables for Number Theory
| `primes` | 29.1 Functions and Variables for Number Theory
| `principal_components` | 49.3 Functions and Variables for descriptive statistics
| `print` | 4.3 Functions and Variables for Display
| `print_graph` | 61.2.2 Graph properties
| `printf` | 84.2 Input and Output
| `printfile` | 13.3 Functions and Variables for File Input and Output
| `printpois` | 28.6 Functions and Variables for Poisson series
| `printprops` | 11.2 Functions and Variables for Properties
| `prodrac` | 30.2.5 Polynomials and their roots
| `product` | 28.1 Functions and Variables for Sums and Products
| product_use_gamma | 81.2 Functions and Variables for solve_rec
| program | 61.2.5 Visualization
| programmode | 20.1 Functions and Variables for Equations
| prompt | 4.2 Functions and Variables for Command Line
| `properties` | 11.2 Functions and Variables for Properties
| proportional_axes | 52.2.3 Graphics options
| props | 11.2 Functions and Variables for Properties
| `propvars` | 11.2 Functions and Variables for Properties
| ps_file | 12.4 Plotting Options
| psexpand | 28.3 Functions and Variables for Series
| `psi` | 15.4 Gamma and factorial Functions
| `psi` | 26.2.5 Algebraic classification
| `psubst` | 6.5 Functions and Variables for Expressions
| `ptriangularize` | 68.2 Functions and Variables for linearalgebra
| `pui` | 30.2.1 Changing bases
| `pui2comp` | 30.2.1 Changing bases
| `pui2ele` | 30.2.1 Changing bases
| `pui2polynome` | 30.2.5 Polynomials and their roots
| `pui_direct` | 30.2.3 Groups and orbits
| `puireduc` | 30.2.1 Changing bases
| `push` | 5.4.2 Functions and Variables for Lists
| `put` | 11.2 Functions and Variables for Properties
| `pv` | 58.2 Functions and Variables for finance
Q |  |
| `qput` | 11.2 Functions and Variables for Properties
| `qrange` | 49.3 Functions and Variables for descriptive statistics
| `qty` | 56.3 Functions and Variables for ezunits
| `quad_control` | 19.4 Functions and Variables for QUADPACK
| `quad_qag` | 19.4 Functions and Variables for QUADPACK
| `quad_qagi` | 19.4 Functions and Variables for QUADPACK
| `quad_qagp` | 19.4 Functions and Variables for QUADPACK
| `quad_qags` | 19.4 Functions and Variables for QUADPACK
| `quad_qawc` | 19.4 Functions and Variables for QUADPACK
| `quad_qawf` | 19.4 Functions and Variables for QUADPACK
| `quad_qawo` | 19.4 Functions and Variables for QUADPACK
| `quad_qaws` | 19.4 Functions and Variables for QUADPACK
| `quadrilateral` | 52.2.4 Graphics objects
| `quantile` | 49.3 Functions and Variables for descriptive statistics
| `quantile_bernoulli` | 51.3 Functions and Variables for discrete distributions
| `quantile_beta` | 51.2 Functions and Variables for continuous distributions
| `quantile_binomial` | 51.3 Functions and Variables for discrete distributions
| `quantile_cauchy` | 51.2 Functions and Variables for continuous distributions
| `quantile_chi2` | 51.2 Functions and Variables for continuous distributions
| `quantile_continuous_uniform` | 51.2 Functions and Variables for continuous distributions
| `quantile_discrete_uniform` | 51.3 Functions and Variables for discrete distributions
| `quantile_exp` | 51.2 Functions and Variables for continuous distributions
| `quantile_f` | 51.2 Functions and Variables for continuous distributions
| `quantile_gamma` | 51.2 Functions and Variables for continuous distributions
| `quantile_general_finite_discrete` | 51.3 Functions and Variables for discrete distributions
| `quantile_geometric` | 51.3 Functions and Variables for discrete distributions
| `quantile_gumbel` | 51.2 Functions and Variables for continuous distributions
| `quantile_hypergeometric` | 51.3 Functions and Variables for discrete distributions
| `quantile_laplace` | 51.2 Functions and Variables for continuous distributions
| `quantile_logistic` | 51.2 Functions and Variables for continuous distributions
| `quantile_lognormal` | 51.2 Functions and Variables for continuous distributions
| `quantile_negative_binomial` | 51.3 Functions and Variables for discrete distributions
| `quantile_noncentral_chi2` | 51.2 Functions and Variables for continuous distributions
| `quantile_noncentral_student_t` | 51.2 Functions and Variables for continuous distributions
| `quantile_normal` | 51.2 Functions and Variables for continuous distributions
| `quantile_pareto` | 51.2 Functions and Variables for continuous distributions
| `quantile_poisson` | 51.3 Functions and Variables for discrete distributions
| `quantile_rayleigh` | 51.2 Functions and Variables for continuous distributions
| `quantile_student_t` | 51.2 Functions and Variables for continuous distributions
| `quantile_weibull` | 51.2 Functions and Variables for continuous distributions
| `quartile_skewness` | 49.3 Functions and Variables for descriptive statistics
| `quit` | 4.2 Functions and Variables for Command Line
| `qunit` | 29.1 Functions and Variables for Number Theory
| `quotient` | 14.2 Functions and Variables for Polynomials
R |  |
| `racah_v` | 46.1 Functions and Variables for clebsch_gordan
| `racah_w` | 46.1 Functions and Variables for clebsch_gordan
| `radcan` | 9.2 Functions and Variables for Simplification
| radexpand | 9.2 Functions and Variables for Simplification
| `radius` | 61.2.2 Graph properties
| radsubstflag | 14.2 Functions and Variables for Polynomials
| `random` | 10.6 Random Numbers
| `random_bernoulli` | 51.3 Functions and Variables for discrete distributions
| `random_beta` | 51.2 Functions and Variables for continuous distributions
| `random_binomial` | 51.3 Functions and Variables for discrete distributions
| `random_bipartite_graph` | 61.2.1 Building graphs
| `random_cauchy` | 51.2 Functions and Variables for continuous distributions
| `random_chi2` | 51.2 Functions and Variables for continuous distributions
| `random_continuous_uniform` | 51.2 Functions and Variables for continuous distributions
| `random_digraph` | 61.2.1 Building graphs
| `random_discrete_uniform` | 51.3 Functions and Variables for discrete distributions
| `random_exp` | 51.2 Functions and Variables for continuous distributions
| `random_f` | 51.2 Functions and Variables for continuous distributions
| `random_gamma` | 51.2 Functions and Variables for continuous distributions
| `random_general_finite_discrete` | 51.3 Functions and Variables for discrete distributions
| `random_geometric` | 51.3 Functions and Variables for discrete distributions
| `random_graph` | 61.2.1 Building graphs
| `random_graph1` | 61.2.1 Building graphs
| `random_gumbel` | 51.2 Functions and Variables for continuous distributions
| `random_hypergeometric` | 51.3 Functions and Variables for discrete distributions
| `random_laplace` | 51.2 Functions and Variables for continuous distributions
| `random_logistic` | 51.2 Functions and Variables for continuous distributions
| `random_lognormal` | 51.2 Functions and Variables for continuous distributions
| `random_negative_binomial` | 51.3 Functions and Variables for discrete distributions
| `random_network` | 61.2.1 Building graphs
| `random_noncentral_chi2` | 51.2 Functions and Variables for continuous distributions
| `random_noncentral_student_t` | 51.2 Functions and Variables for continuous distributions
| `random_normal` | 51.2 Functions and Variables for continuous distributions
| `random_pareto` | 51.2 Functions and Variables for continuous distributions
| `random_permutation` | 35.2 Functions and Variables for Sets
| `random_poisson` | 51.3 Functions and Variables for discrete distributions
| `random_rayleigh` | 51.2 Functions and Variables for continuous distributions
| `random_regular_graph` | 61.2.1 Building graphs
| `random_student_t` | 51.2 Functions and Variables for continuous distributions
| `random_tournament` | 61.2.1 Building graphs
| `random_tree` | 61.2.1 Building graphs
| `random_weibull` | 51.2 Functions and Variables for continuous distributions
| `range` | 49.3 Functions and Variables for descriptive statistics
| `rank` | 23.2 Functions and Variables for Matrices and Linear Algebra
| `rank` | 68.2 Functions and Variables for linearalgebra
| rassociative | 9.2 Functions and Variables for Simplification
| `rat` | 14.2 Functions and Variables for Polynomials
| ratalgdenom | 14.2 Functions and Variables for Polynomials
| ratchristof | 26.2.9 Variables used by `ctensor`
| `ratcoef` | 14.2 Functions and Variables for Polynomials
| `ratdenom` | 14.2 Functions and Variables for Polynomials
| ratdenomdivide | 14.2 Functions and Variables for Polynomials
| `ratdiff` | 14.2 Functions and Variables for Polynomials
| `ratdisrep` | 14.2 Functions and Variables for Polynomials
| rateinstein | 26.2.9 Variables used by `ctensor`
| ratepsilon | 5.1.2 Functions and Variables for Numbers
| `ratexpand` | 14.2 Functions and Variables for Polynomials
| `ratexpand` | 14.2 Functions and Variables for Polynomials
| ratfac | 14.2 Functions and Variables for Polynomials
| `ratinterpol` | 64.2 Functions and Variables for interpol
| `rational` | 80.4 Package functs
| `rationalize` | 5.1.2 Functions and Variables for Numbers
| ratmx | 23.2 Functions and Variables for Matrices and Linear Algebra
| `ratnumer` | 14.2 Functions and Variables for Polynomials
| `ratnump` | 5.1.2 Functions and Variables for Numbers
| `ratp` | 14.2 Functions and Variables for Polynomials
| `ratp_coeffs` | 77.1 Functions and Variables for ratpow
| `ratp_dense_coeffs` | 77.1 Functions and Variables for ratpow
| `ratp_hipow` | 77.1 Functions and Variables for ratpow
| `ratp_lopow` | 77.1 Functions and Variables for ratpow
| ratprint | 14.2 Functions and Variables for Polynomials
| ratriemann | 26.2.9 Variables used by `ctensor`
| `ratsimp` | 14.2 Functions and Variables for Polynomials
| `ratsimp` | 14.2 Functions and Variables for Polynomials
| ratsimpexpons | 14.2 Functions and Variables for Polynomials
| `ratsubst` | 14.2 Functions and Variables for Polynomials
| `ratvars` | 14.2 Functions and Variables for Polynomials
| `ratvars` | 14.2 Functions and Variables for Polynomials
| `ratvars` | 14.2 Functions and Variables for Polynomials
| ratvarswitch | 14.2 Functions and Variables for Polynomials
| `ratweight` | 14.2 Functions and Variables for Polynomials
| ratweights | 14.2 Functions and Variables for Polynomials
| ratweyl | 26.2.9 Variables used by `ctensor`
| ratwtlvl | 14.2 Functions and Variables for Polynomials
| `read` | 4.2 Functions and Variables for Command Line
| `read_array` | 73.2 Functions and Variables for plain-text input and output
| `read_binary_array` | 73.3 Functions and Variables for binary input and output
| `read_binary_list` | 73.3 Functions and Variables for binary input and output
| `read_binary_matrix` | 73.3 Functions and Variables for binary input and output
| `read_hashed_array` | 73.2 Functions and Variables for plain-text input and output
| `read_list` | 73.2 Functions and Variables for plain-text input and output
| `read_matrix` | 73.2 Functions and Variables for plain-text input and output
| `read_nested_list` | 73.2 Functions and Variables for plain-text input and output
| `read_xpm` | 52.3 Functions and Variables for pictures
| `readbyte` | 84.2 Input and Output
| `readchar` | 84.2 Input and Output
| `readline` | 84.2 Input and Output
| `readonly` | 4.2 Functions and Variables for Command Line
| real | 11.2 Functions and Variables for Properties
| `real_fft` | 22.2 Functions and Variables for fast Fourier transform
| `real_imagpart_to_conjugate` | 85.1 Functions and Variables for to_poly_solve
| realonly | 20.1 Functions and Variables for Equations
| `realpart` | 10.2 Functions for Complex Numbers
| `realroots` | 20.1 Functions and Variables for Equations
| `rearray` | 5.5.1 Functions and Variables for Arrays
| `rectangle` | 52.2.4 Graphics objects
| `rectform` | 10.2 Functions for Complex Numbers
| `rectform_log_if_constant` | 85.1 Functions and Variables for to_poly_solve
| `recttopolar` | 22.2 Functions and Variables for fast Fourier transform
| `rediff` | 25.2.3 Indicial tensor calculus
| redraw | 61.2.5 Visualization
| `reduce_consts` | 80.6 Package rducon
| `reduce_order` | 81.2 Functions and Variables for solve_rec
| refcheck | 38.3 Functions and Variables for Debugging
| `region` | 52.2.4 Graphics objects
| `region_boundaries` | 52.4.1 Variables and Functions
| `region_boundaries_plus` | 52.4.1 Variables and Functions
| `rem` | 11.2 Functions and Variables for Properties
| `remainder` | 14.2 Functions and Variables for Polynomials
| `remarray` | 5.5.1 Functions and Variables for Arrays
| `rembox` | 6.5 Functions and Variables for Expressions
| `remcomps` | 25.2.1 Managing indexed objects
| `remcon` | 25.2.1 Managing indexed objects
| `remcoord` | 25.2.3 Indicial tensor calculus
| `remfun` | 28.5 Functions and Variables for Fourier series
| `remfunction` | 36.4 Functions and Variables for Function Definition
| `remlet` | 34.2 Functions and Variables for Rules and Patterns
| `remove` | 11.2 Functions and Variables for Properties
| `remove_constvalue` | 56.3 Functions and Variables for ezunits
| `remove_dimensions` | 56.3 Functions and Variables for ezunits
| `remove_edge` | 61.2.3 Modifying graphs
| `remove_fundamental_dimensions` | 56.3 Functions and Variables for ezunits
| `remove_fundamental_units` | 56.3 Functions and Variables for ezunits
| `remove_plot_option` | 12.3 Functions and Variables for Plotting
| `remove_vertex` | 61.2.3 Modifying graphs
| `rempart` | 80.4 Package functs
| `remrule` | 34.2 Functions and Variables for Rules and Patterns
| `remsym` | 25.2.2 Tensor symmetries
| `remvalue` | 33.3 Functions and Variables for Miscellaneous Options
| `rename` | 25.2.1 Managing indexed objects
| `rename_file` | 74.3 File operations
| `reset` | 4.2 Functions and Variables for Command Line
| `reset_displays` | 39.2 Functions and Variables for alt-display
| `residue` | 19.2 Functions and Variables for Integration
| resolution | 54.3.3 Scene object's options
| `resolvante` | 30.2.6 Resolvents
| `resolvante_alternee1` | 30.2.6 Resolvents
| `resolvante_bipartite` | 30.2.6 Resolvents
| `resolvante_diedrale` | 30.2.6 Resolvents
| `resolvante_klein` | 30.2.6 Resolvents
| `resolvante_klein3` | 30.2.6 Resolvents
| `resolvante_produit_sym` | 30.2.6 Resolvents
| `resolvante_unitaire` | 30.2.6 Resolvents
| `resolvante_vierer` | 30.2.6 Resolvents
| `rest` | 5.4.2 Functions and Variables for Lists
| restart | 54.3.1 Scene options
| `resultant` | 14.2 Functions and Variables for Polynomials
| `return` | 37.4 Functions and Variables for Program Flow
| `reveal` | 6.5 Functions and Variables for Expressions
| `reverse` | 5.4.2 Functions and Variables for Lists
| `revert` | 28.3 Functions and Variables for Series
| `revert2` | 28.3 Functions and Variables for Series
| `rgb2level` | 52.3 Functions and Variables for pictures
| `rhs` | 20.1 Functions and Variables for Equations
| ric | 26.2.9 Variables used by `ctensor`
| `ricci` | 26.2.2 The tensors of curved space
| riem | 26.2.9 Variables used by `ctensor`
| `riemann` | 26.2.2 The tensors of curved space
| `rinvariant` | 26.2.2 The tensors of curved space
| `risch` | 19.2 Functions and Variables for Integration
| `rk` | 22.5 Functions for numerical solution of differential equations
| `rmdir` | 74.2 Directory operations
| rmxchar | 23.2 Functions and Variables for Matrices and Linear Algebra
| `rncombine` | 33.3 Functions and Variables for Miscellaneous Options
| `romberg` | 78.1 Functions and Variables for romberg
| rombergabs | 78.1 Functions and Variables for romberg
| rombergit | 78.1 Functions and Variables for romberg
| rombergmin | 78.1 Functions and Variables for romberg
| rombergtol | 78.1 Functions and Variables for romberg
| `room` | 32.3 Functions and Variables for Runtime Environment
| rootsconmode | 20.1 Functions and Variables for Equations
| `rootscontract` | 20.1 Functions and Variables for Equations
| rootsepsilon | 20.1 Functions and Variables for Equations
| `round` | 10.1 Functions for Numbers
| `row` | 23.2 Functions and Variables for Matrices and Linear Algebra
| `rowop` | 68.2 Functions and Variables for linearalgebra
| `rowswap` | 68.2 Functions and Variables for linearalgebra
| `rreduce` | 5.4.2 Functions and Variables for Lists
| `run_testsuite` | 2.1 Functions and Variables for Bug Detection and Reporting
| run_viewer | 12.4 Plotting Options
S |  |
| same_xy | 12.4 Plotting Options
| same_xyz | 12.4 Plotting Options
| `save` | 13.3 Functions and Variables for File Input and Output
| savedef | 36.4 Functions and Variables for Function Definition
| savefactors | 14.2 Functions and Variables for Polynomials
| `saving` | 58.2 Functions and Variables for finance
| scalar | 11.2 Functions and Variables for Properties
| scalarmatrixp | 23.2 Functions and Variables for Matrices and Linear Algebra
| `scalarp` | 11.2 Functions and Variables for Properties
| scale | 54.3.3 Scene object's options
| scale_lp | 79.2 Functions and Variables for simplex
| `scaled_bessel_i` | 15.2 Bessel Functions
| `scaled_bessel_i0` | 15.2 Bessel Functions
| `scaled_bessel_i1` | 15.2 Bessel Functions
| `scalefactors` | 23.2 Functions and Variables for Matrices and Linear Algebra
| `scanmap` | 37.4 Functions and Variables for Program Flow
| `scatterplot` | 49.4 Functions and Variables for statistical graphs
| `scatterplot_description` | 49.4 Functions and Variables for statistical graphs
| `scene` | 54.3 Visualization with VTK
| `schur2comp` | 30.2.1 Changing bases
| `sconcat` | 5.2.2 Functions and Variables for Strings
| `scopy` | 84.4 String Processing
| `scsimp` | 9.2 Functions and Variables for Simplification
| `scurvature` | 26.2.2 The tensors of curved space
| `sdowncase` | 84.4 String Processing
| `sec` | 10.5.2 Functions and Variables for Trigonometric
| `sech` | 10.5.2 Functions and Variables for Trigonometric
| `second` | 5.4.2 Functions and Variables for Lists
| `sequal` | 84.4 String Processing
| `sequalignore` | 84.4 String Processing
| `set_alt_display` | 39.2 Functions and Variables for alt-display
| `set_draw_defaults` | 52.2.2 Functions
| `set_edge_weight` | 61.2.2 Graph properties
| `set_partitions` | 35.2 Functions and Variables for Sets
| `set_plot_option` | 12.3 Functions and Variables for Plotting
| `set_prompt` | 39.2 Functions and Variables for alt-display
| `set_random_state` | 10.6 Random Numbers
| `set_tex_environment` | 13.4 Functions and Variables for TeX Output
| `set_tex_environment_default` | 13.4 Functions and Variables for TeX Output
| `set_up_dot_simplifications` | 24.2 Functions and Variables for Affine
| `set_vertex_label` | 61.2.2 Graph properties
| setcheck | 38.3 Functions and Variables for Debugging
| setcheckbreak | 38.3 Functions and Variables for Debugging
| `setdifference` | 35.2 Functions and Variables for Sets
| `setelmx` | 23.2 Functions and Variables for Matrices and Linear Algebra
| `setequalp` | 35.2 Functions and Variables for Sets
| `setify` | 35.2 Functions and Variables for Sets
| `setp` | 35.2 Functions and Variables for Sets
| `setunits` | 86.2 Functions and Variables for Units
| `setup_autoload` | 33.3 Functions and Variables for Miscellaneous Options
| setval | 38.3 Functions and Variables for Debugging
| `seventh` | 5.4.2 Functions and Variables for Lists
| `sexplode` | 84.4 String Processing
| `sf` | 27.2 Functions and Variables for atensor
| `sha1sum` | 84.5 Octets and Utilities for Cryptography
| `sha256sum` | 84.5 Octets and Utilities for Cryptography
| share_testsuite_files | 2.1 Functions and Variables for Bug Detection and Reporting
| `shortest_path` | 61.2.2 Graph properties
| `shortest_weighted_path` | 61.2.2 Graph properties
| `show` | 25.2.1 Managing indexed objects
| show_edge_color | 61.2.5 Visualization
| show_edge_type | 61.2.5 Visualization
| show_edge_width | 61.2.5 Visualization
| show_edges | 61.2.5 Visualization
| show_id | 61.2.5 Visualization
| show_label | 61.2.5 Visualization
| show_vertex_color | 61.2.5 Visualization
| show_vertex_size | 61.2.5 Visualization
| show_vertex_type | 61.2.5 Visualization
| show_vertices | 61.2.5 Visualization
| show_weight | 61.2.5 Visualization
| `showcomps` | 25.2.1 Managing indexed objects
| `showratvars` | 14.2 Functions and Variables for Polynomials
| showtime | 4.2 Functions and Variables for Command Line
| `sierpinskiale` | 59.2 Definitions for IFS fractals
| `sierpinskimap` | 59.5 Definitions for Peano maps
| `sign` | 11.3 Functions and Variables for Facts
| `signum` | 10.1 Functions for Numbers
| `similaritytransform` | 23.2 Functions and Variables for Matrices and Linear Algebra
| simp | 9.2 Functions and Variables for Simplification
| `simp_inequality` | 85.1 Functions and Variables for to_poly_solve
| simplified_output | 87.3 General global variables
| simplify_products | 81.2 Functions and Variables for solve_rec
| `simplify_sum` | 81.2 Functions and Variables for solve_rec
| `simplode` | 84.4 String Processing
| `simpmetderiv` | 25.2.3 Indicial tensor calculus
| simpproduct | 28.1 Functions and Variables for Sums and Products
| simpsum | 28.1 Functions and Variables for Sums and Products
| `simtran` | 23.2 Functions and Variables for Matrices and Linear Algebra
| `sin` | 10.5.2 Functions and Variables for Trigonometric
| `sinh` | 10.5.2 Functions and Variables for Trigonometric
| sinnpiflag | 28.5 Functions and Variables for Fourier series
| `sinsert` | 84.4 String Processing
| `sinvertcase` | 84.4 String Processing
| `sixth` | 5.4.2 Functions and Variables for Lists
| `skewness` | 49.3 Functions and Variables for descriptive statistics
| `skewness_bernoulli` | 51.3 Functions and Variables for discrete distributions
| `skewness_beta` | 51.2 Functions and Variables for continuous distributions
| `skewness_binomial` | 51.3 Functions and Variables for discrete distributions
| `skewness_chi2` | 51.2 Functions and Variables for continuous distributions
| `skewness_continuous_uniform` | 51.2 Functions and Variables for continuous distributions
| `skewness_discrete_uniform` | 51.3 Functions and Variables for discrete distributions
| `skewness_exp` | 51.2 Functions and Variables for continuous distributions
| `skewness_f` | 51.2 Functions and Variables for continuous distributions
| `skewness_gamma` | 51.2 Functions and Variables for continuous distributions
| `skewness_general_finite_discrete` | 51.3 Functions and Variables for discrete distributions
| `skewness_geometric` | 51.3 Functions and Variables for discrete distributions
| `skewness_gumbel` | 51.2 Functions and Variables for continuous distributions
| `skewness_hypergeometric` | 51.3 Functions and Variables for discrete distributions
| `skewness_laplace` | 51.2 Functions and Variables for continuous distributions
| `skewness_logistic` | 51.2 Functions and Variables for continuous distributions
| `skewness_lognormal` | 51.2 Functions and Variables for continuous distributions
| `skewness_negative_binomial` | 51.3 Functions and Variables for discrete distributions
| `skewness_noncentral_chi2` | 51.2 Functions and Variables for continuous distributions
| `skewness_noncentral_student_t` | 51.2 Functions and Variables for continuous distributions
| `skewness_normal` | 51.2 Functions and Variables for continuous distributions
| `skewness_pareto` | 51.2 Functions and Variables for continuous distributions
| `skewness_poisson` | 51.3 Functions and Variables for discrete distributions
| `skewness_rayleigh` | 51.2 Functions and Variables for continuous distributions
| `skewness_student_t` | 51.2 Functions and Variables for continuous distributions
| `skewness_weibull` | 51.2 Functions and Variables for continuous distributions
| `slength` | 84.4 String Processing
| `smake` | 84.4 String Processing
| `small_rhombicosidodecahedron_graph` | 61.2.1 Building graphs
| `small_rhombicuboctahedron_graph` | 61.2.1 Building graphs
| `smax` | 49.3 Functions and Variables for descriptive statistics
| `smin` | 49.3 Functions and Variables for descriptive statistics
| `smismatch` | 84.4 String Processing
| `snowmap` | 59.4 Definitions for Koch snowflakes
| `snub_cube_graph` | 61.2.1 Building graphs
| `snub_dodecahedron_graph` | 61.2.1 Building graphs
| `solve` | 20.1 Functions and Variables for Equations
| `solve_rec` | 81.2 Functions and Variables for solve_rec
| `solve_rec_rat` | 81.2 Functions and Variables for solve_rec
| solvedecomposes | 20.1 Functions and Variables for Equations
| solveexplicit | 20.1 Functions and Variables for Equations
| solvefactors | 20.1 Functions and Variables for Equations
| solvenullwarn | 20.1 Functions and Variables for Equations
| solveradcan | 20.1 Functions and Variables for Equations
| solvetrigwarn | 20.1 Functions and Variables for Equations
| `some` | 35.2 Functions and Variables for Sets
| `somrac` | 30.2.5 Polynomials and their roots
| `sort` | 5.4.2 Functions and Variables for Lists
| space | 84.3 Characters
| sparse | 23.2 Functions and Variables for Matrices and Linear Algebra
| `sparse6_decode` | 61.2.4 Reading and writing to files
| `sparse6_encode` | 61.2.4 Reading and writing to files
| `sparse6_export` | 61.2.4 Reading and writing to files
| `sparse6_import` | 61.2.4 Reading and writing to files
| `specint` | 15.10 Functions and Variables for Special Functions
| sphere | 54.3.2 Scene objects
| `spherical` | 52.2.4 Graphics objects
| `spherical_bessel_j` | 76.2 Functions and Variables for orthogonal polynomials
| `spherical_bessel_y` | 76.2 Functions and Variables for orthogonal polynomials
| `spherical_hankel1` | 76.2 Functions and Variables for orthogonal polynomials
| `spherical_hankel2` | 76.2 Functions and Variables for orthogonal polynomials
| `spherical_harmonic` | 76.2 Functions and Variables for orthogonal polynomials
| `spherical_to_xyz` | 12.3 Functions and Variables for Plotting
| `splice` | 36.3 Macros
| `split` | 84.4 String Processing
| `sposition` | 84.4 String Processing
| spring_embedding_depth | 61.2.5 Visualization
| `sprint` | 84.2 Input and Output
| `sqfr` | 14.2 Functions and Variables for Polynomials
| `sqrt` | 10.4 Root, Exponential and Logarithmic Functions
| `sqrtdenest` | 80.8 Package sqdnst
| sqrtdispflag | 4.3 Functions and Variables for Display
| `sremove` | 84.4 String Processing
| `sremovefirst` | 84.4 String Processing
| `sreverse` | 84.4 String Processing
| `ssearch` | 84.4 String Processing
| `ssort` | 84.4 String Processing
| `sstatus` | 32.3 Functions and Variables for Runtime Environment
| `ssubst` | 84.4 String Processing
| `ssubstfirst` | 84.4 String Processing
| `staircase` | 54.2 Graphical analysis of discrete dynamical systems
| `standardize` | 49.2 Functions and Variables for data manipulation
| `standardize_inverse_trig` | 85.1 Functions and Variables for to_poly_solve
| stardisp | 4.3 Functions and Variables for Display
| `starplot` | 49.4 Functions and Variables for statistical graphs
| `starplot_description` | 49.4 Functions and Variables for statistical graphs
| startphi | 54.3.3 Scene object's options
| starttheta | 54.3.3 Scene object's options
| stats_numer | 82.3 Functions and Variables for stats
| `status` | 32.3 Functions and Variables for Runtime Environment
| `std` | 49.3 Functions and Variables for descriptive statistics
| `std1` | 49.3 Functions and Variables for descriptive statistics
| `std_bernoulli` | 51.3 Functions and Variables for discrete distributions
| `std_beta` | 51.2 Functions and Variables for continuous distributions
| `std_binomial` | 51.3 Functions and Variables for discrete distributions
| `std_chi2` | 51.2 Functions and Variables for continuous distributions
| `std_continuous_uniform` | 51.2 Functions and Variables for continuous distributions
| `std_discrete_uniform` | 51.3 Functions and Variables for discrete distributions
| `std_exp` | 51.2 Functions and Variables for continuous distributions
| `std_f` | 51.2 Functions and Variables for continuous distributions
| `std_gamma` | 51.2 Functions and Variables for continuous distributions
| `std_general_finite_discrete` | 51.3 Functions and Variables for discrete distributions
| `std_geometric` | 51.3 Functions and Variables for discrete distributions
| `std_gumbel` | 51.2 Functions and Variables for continuous distributions
| `std_hypergeometric` | 51.3 Functions and Variables for discrete distributions
| `std_laplace` | 51.2 Functions and Variables for continuous distributions
| `std_logistic` | 51.2 Functions and Variables for continuous distributions
| `std_lognormal` | 51.2 Functions and Variables for continuous distributions
| `std_negative_binomial` | 51.3 Functions and Variables for discrete distributions
| `std_noncentral_chi2` | 51.2 Functions and Variables for continuous distributions
| `std_noncentral_student_t` | 51.2 Functions and Variables for continuous distributions
| `std_normal` | 51.2 Functions and Variables for continuous distributions
| `std_pareto` | 51.2 Functions and Variables for continuous distributions
| `std_poisson` | 51.3 Functions and Variables for discrete distributions
| `std_rayleigh` | 51.2 Functions and Variables for continuous distributions
| `std_student_t` | 51.2 Functions and Variables for continuous distributions
| `std_weibull` | 51.2 Functions and Variables for continuous distributions
| `stemplot` | 49.4 Functions and Variables for statistical graphs
| `step` | 37.4 Functions and Variables for Program Flow
| `stirling` | 83.1 Functions and Variables for stirling
| `stirling1` | 35.2 Functions and Variables for Sets
| `stirling2` | 35.2 Functions and Variables for Sets
| `strim` | 84.4 String Processing
| `striml` | 84.4 String Processing
| `strimr` | 84.4 String Processing
| `string` | 5.2.2 Functions and Variables for Strings
| `string_to_octets` | 84.5 Octets and Utilities for Cryptography
| stringdisp | 5.2.2 Functions and Variables for Strings
| `stringout` | 13.3 Functions and Variables for File Input and Output
| `stringp` | 84.4 String Processing
| `strong_components` | 61.2.2 Graph properties
| structures | 5.6.2 Functions and Variables for Structures
| `struve_h` | 15.7 Struve Functions
| `struve_l` | 15.7 Struve Functions
| style | 12.4 Plotting Options
| `sublis` | 6.5 Functions and Variables for Expressions
| sublis_apply_lambda | 6.5 Functions and Variables for Expressions
| `sublist` | 5.4.2 Functions and Variables for Lists
| `sublist_indices` | 5.4.2 Functions and Variables for Lists
| `submatrix` | 23.2 Functions and Variables for Matrices and Linear Algebra
| subnumsimp | 6.5 Functions and Variables for Expressions
| `subsample` | 49.2 Functions and Variables for data manipulation
| `subset` | 35.2 Functions and Variables for Sets
| `subsetp` | 35.2 Functions and Variables for Sets
| `subst` | 6.5 Functions and Variables for Expressions
| `subst_parallel` | 85.1 Functions and Variables for to_poly_solve
| `substinpart` | 6.5 Functions and Variables for Expressions
| `substpart` | 6.5 Functions and Variables for Expressions
| `substring` | 84.4 String Processing
| `subvar` | 5.5.1 Functions and Variables for Arrays
| `subvarp` | 5.5.1 Functions and Variables for Arrays
| `sum` | 28.1 Functions and Variables for Sums and Products
| `sumcontract` | 28.1 Functions and Variables for Sums and Products
| sumexpand | 28.1 Functions and Variables for Sums and Products
| `summand_to_rec` | 81.2 Functions and Variables for solve_rec
| sumsplitfact | 10.3 Combinatorial Functions
| `supcase` | 84.4 String Processing
| `supcontext` | 11.3 Functions and Variables for Facts
| surface | 54.3.3 Scene object's options
| surface_hide | 52.2.3 Graphics options
| svg_file | 12.4 Plotting Options
| `symbolp` | 6.5 Functions and Variables for Expressions
| `symmdifference` | 35.2 Functions and Variables for Sets
| symmetric | 9.2 Functions and Variables for Simplification
| `symmetricp` | 26.2.8 Utility functions
| `system` | 32.3 Functions and Variables for Runtime Environment
T |  |
| t | 12.4 Plotting Options
| tab | 84.3 Characters
| `take_channel` | 52.3 Functions and Variables for pictures
| `take_inference` | 82.2 Functions and Variables for inference_result
| `tan` | 10.5.2 Functions and Variables for Trigonometric
| `tanh` | 10.5.2 Functions and Variables for Trigonometric
| `taylor` | 28.3 Functions and Variables for Series
| taylor_logexpand | 28.3 Functions and Variables for Series
| taylor_order_coefficients | 28.3 Functions and Variables for Series
| `taylor_simplifier` | 28.3 Functions and Variables for Series
| taylor_truncate_polynomials | 28.3 Functions and Variables for Series
| taylordepth | 28.3 Functions and Variables for Series
| `taylorinfo` | 28.3 Functions and Variables for Series
| `taylorp` | 28.3 Functions and Variables for Series
| `taytorat` | 28.3 Functions and Variables for Series
| `tcl_output` | 33.3 Functions and Variables for Miscellaneous Options
| `tcontract` | 30.2.2 Changing representations
| `tellrat` | 14.2 Functions and Variables for Polynomials
| `tellsimp` | 34.2 Functions and Variables for Rules and Patterns
| `tellsimpafter` | 34.2 Functions and Variables for Rules and Patterns
| tensorkill | 26.2.9 Variables used by `ctensor`
| `tentex` | 25.2.8 Exporting TeX expressions
| `tenth` | 5.4.2 Functions and Variables for Lists
| terminal | 52.2.3 Graphics options
| terminal | 61.2.5 Visualization
| `test_mean` | 82.3 Functions and Variables for stats
| `test_means_difference` | 82.3 Functions and Variables for stats
| `test_normality` | 82.3 Functions and Variables for stats
| `test_proportion` | 82.3 Functions and Variables for stats
| `test_proportions_difference` | 82.3 Functions and Variables for stats
| `test_rank_sum` | 82.3 Functions and Variables for stats
| `test_sign` | 82.3 Functions and Variables for stats
| `test_signed_rank` | 82.3 Functions and Variables for stats
| `test_variance` | 82.3 Functions and Variables for stats
| `test_variance_ratio` | 82.3 Functions and Variables for stats
| testsuite_files | 2.1 Functions and Variables for Bug Detection and Reporting
| `tex` | 13.4 Functions and Variables for TeX Output
| `tex1` | 13.4 Functions and Variables for TeX Output
| `tex_display` | 39.2 Functions and Variables for alt-display
| `texput` | 13.4 Functions and Variables for TeX Output
| thetaresolution | 54.3.3 Scene object's options
| `third` | 5.4.2 Functions and Variables for Lists
| `throw` | 37.4 Functions and Variables for Program Flow
| `thru` | 37.4 Functions and Variables for Program Flow
| `time` | 32.3 Functions and Variables for Runtime Environment
| `timedate` | 32.3 Functions and Variables for Runtime Environment
| `timer` | 38.3 Functions and Variables for Debugging
| timer_devalue | 38.3 Functions and Variables for Debugging
| `timer_info` | 38.3 Functions and Variables for Debugging
| title | 12.4 Plotting Options
| title | 52.2.3 Graphics options
| `tldefint` | 19.2 Functions and Variables for Integration
| `tlimit` | 17.1 Functions and Variables for Limits
| tlimswitch | 17.1 Functions and Variables for Limits
| `to_lisp` | 4.2 Functions and Variables for Command Line
| `to_poly` | 85.1 Functions and Variables for to_poly_solve
| `to_poly_solve` | 85.1 Functions and Variables for to_poly_solve
| `todd_coxeter` | 31.1 Functions and Variables for Groups
| `toeplitz` | 68.2 Functions and Variables for linearalgebra
| `tokens` | 84.4 String Processing
| `topological_sort` | 61.2.2 Graph properties
| `totaldisrep` | 14.2 Functions and Variables for Polynomials
| `totalfourier` | 28.5 Functions and Variables for Fourier series
| `totient` | 29.1 Functions and Variables for Number Theory
| `tpartpol` | 30.2.2 Changing representations
| tr | 26.2.9 Variables used by `ctensor`
| tr_array_as_ref | 36.4 Functions and Variables for Function Definition
| tr_bound_function_applyp | 36.4 Functions and Variables for Function Definition
| tr_file_tty_messagesp | 36.4 Functions and Variables for Function Definition
| tr_float_can_branch_complex | 36.4 Functions and Variables for Function Definition
| tr_function_call_default | 36.4 Functions and Variables for Function Definition
| tr_numer | 36.4 Functions and Variables for Function Definition
| tr_optimize_max_loop | 36.4 Functions and Variables for Function Definition
| tr_semicompile | 36.4 Functions and Variables for Function Definition
| tr_state_vars | 36.4 Functions and Variables for Function Definition
| tr_warn_bad_function_calls | 36.4 Functions and Variables for Function Definition
| tr_warn_fexpr | 36.4 Functions and Variables for Function Definition
| tr_warn_meval | 36.4 Functions and Variables for Function Definition
| tr_warn_mode | 36.4 Functions and Variables for Function Definition
| tr_warn_undeclared | 36.4 Functions and Variables for Function Definition
| tr_warn_undefined_variable | 36.4 Functions and Variables for Function Definition
| `tr_warnings_get` | 36.4 Functions and Variables for Function Definition
| `trace` | 38.3 Functions and Variables for Debugging
| `trace_options` | 38.3 Functions and Variables for Debugging
| `tracematrix` | 80.4 Package functs
| track | 54.3.3 Scene object's options
| transcompile | 36.4 Functions and Variables for Function Definition
| transform | 52.2.3 Graphics options
| `transform_sample` | 49.2 Functions and Variables for data manipulation
| transform_xy | 12.4 Plotting Options
| `translate` | 36.4 Functions and Variables for Function Definition
| translate_fast_arrays | 5.5.1 Functions and Variables for Arrays
| `translate_file` | 36.4 Functions and Variables for Function Definition
| transparent | 52.2.3 Graphics options
| `transpose` | 23.2 Functions and Variables for Matrices and Linear Algebra
| transrun | 36.4 Functions and Variables for Function Definition
| `tree_reduce` | 5.4.2 Functions and Variables for Lists
| `treefale` | 59.2 Definitions for IFS fractals
| `treillis` | 30.2.4 Partitions
| `treinat` | 30.2.4 Partitions
| `triangle` | 52.2.4 Graphics objects
| `triangularize` | 23.2 Functions and Variables for Matrices and Linear Algebra
| `trigexpand` | 10.5.2 Functions and Variables for Trigonometric
| trigexpandplus | 10.5.2 Functions and Variables for Trigonometric
| trigexpandtimes | 10.5.2 Functions and Variables for Trigonometric
| triginverses | 10.5.2 Functions and Variables for Trigonometric
| `trigrat` | 10.5.2 Functions and Variables for Trigonometric
| `trigreduce` | 10.5.2 Functions and Variables for Trigonometric
| trigsign | 10.5.2 Functions and Variables for Trigonometric
| `trigsimp` | 10.5.2 Functions and Variables for Trigonometric
| trivial_solutions | 87.3 General global variables
| true | 5.3.1 Functions and Variables for Constants
| `trunc` | 28.3 Functions and Variables for Series
| `truncate` | 10.1 Functions for Numbers
| `truncated_cube_graph` | 61.2.1 Building graphs
| `truncated_dodecahedron_graph` | 61.2.1 Building graphs
| `truncated_icosahedron_graph` | 61.2.1 Building graphs
| `truncated_tetrahedron_graph` | 61.2.1 Building graphs
| tstep | 54.3.1 Scene options
| ttyoff | 4.3 Functions and Variables for Display
| `tube` | 52.2.4 Graphics objects
| `tutte_graph` | 61.2.1 Building graphs
U |  |
| `ueivects` | 23.2 Functions and Variables for Matrices and Linear Algebra
| ufg | 26.2.9 Variables used by `ctensor`
| `uforget` | 86.2 Functions and Variables for Units
| ug | 26.2.9 Variables used by `ctensor`
| `ultraspherical` | 76.2 Functions and Variables for orthogonal polynomials
| und | 5.3.1 Functions and Variables for Constants
| `underlying_graph` | 61.2.1 Building graphs
| `undiff` | 25.2.3 Indicial tensor calculus
| `unicode` | 84.3 Characters
| `unicode_to_utf8` | 84.3 Characters
| `union` | 35.2 Functions and Variables for Sets
| `unique` | 5.4.2 Functions and Variables for Lists
| `unit_step` | 76.2 Functions and Variables for orthogonal polynomials
| unit_vectors | 52.2.3 Graphics options
| `uniteigenvectors` | 23.2 Functions and Variables for Matrices and Linear Algebra
| `unitp` | 56.3 Functions and Variables for ezunits
| `units` | 56.3 Functions and Variables for ezunits
| `unitvector` | 23.2 Functions and Variables for Matrices and Linear Algebra
| `unknown` | 11.4 Functions and Variables for Predicates
| `unless` | 37.4 Functions and Variables for Program Flow
| `unorder` | 6.5 Functions and Variables for Expressions
| `unsum` | 28.3 Functions and Variables for Series
| `untellrat` | 14.2 Functions and Variables for Polynomials
| `untimer` | 38.3 Functions and Variables for Debugging
| `untrace` | 38.3 Functions and Variables for Debugging
| `uppercasep` | 84.3 Characters
| uric | 26.2.9 Variables used by `ctensor`
| `uricci` | 26.2.2 The tensors of curved space
| uriem | 26.2.9 Variables used by `ctensor`
| `uriemann` | 26.2.2 The tensors of curved space
| us_ascii_only | 84.3 Characters
| use_fast_arrays | 5.5.1 Functions and Variables for Arrays
| user_preamble | 52.2.3 Graphics options
| usersetunits | 86.2 Functions and Variables for Units
| `utf8_to_unicode` | 84.3 Characters
| `uvect` | 23.2 Functions and Variables for Matrices and Linear Algebra
V |  |
| values | 4.2 Functions and Variables for Command Line
| `vandermonde_matrix` | 68.2 Functions and Variables for linearalgebra
| `var` | 49.3 Functions and Variables for descriptive statistics
| `var1` | 49.3 Functions and Variables for descriptive statistics
| `var_bernoulli` | 51.3 Functions and Variables for discrete distributions
| `var_beta` | 51.2 Functions and Variables for continuous distributions
| `var_binomial` | 51.3 Functions and Variables for discrete distributions
| `var_chi2` | 51.2 Functions and Variables for continuous distributions
| `var_continuous_uniform` | 51.2 Functions and Variables for continuous distributions
| `var_discrete_uniform` | 51.3 Functions and Variables for discrete distributions
| `var_exp` | 51.2 Functions and Variables for continuous distributions
| `var_f` | 51.2 Functions and Variables for continuous distributions
| `var_gamma` | 51.2 Functions and Variables for continuous distributions
| `var_general_finite_discrete` | 51.3 Functions and Variables for discrete distributions
| `var_geometric` | 51.3 Functions and Variables for discrete distributions
| `var_gumbel` | 51.2 Functions and Variables for continuous distributions
| `var_hypergeometric` | 51.3 Functions and Variables for discrete distributions
| `var_laplace` | 51.2 Functions and Variables for continuous distributions
| `var_logistic` | 51.2 Functions and Variables for continuous distributions
| `var_lognormal` | 51.2 Functions and Variables for continuous distributions
| `var_negative_binomial` | 51.3 Functions and Variables for discrete distributions
| `var_noncentral_chi2` | 51.2 Functions and Variables for continuous distributions
| `var_noncentral_student_t` | 51.2 Functions and Variables for continuous distributions
| `var_normal` | 51.2 Functions and Variables for continuous distributions
| `var_pareto` | 51.2 Functions and Variables for continuous distributions
| `var_poisson` | 51.3 Functions and Variables for discrete distributions
| `var_rayleigh` | 51.2 Functions and Variables for continuous distributions
| `var_student_t` | 51.2 Functions and Variables for continuous distributions
| `var_weibull` | 51.2 Functions and Variables for continuous distributions
| vect_cross | 23.2 Functions and Variables for Matrices and Linear Algebra
| `vector` | 52.2.4 Graphics objects
| `vectorpotential` | 23.2 Functions and Variables for Matrices and Linear Algebra
| `vectorsimp` | 23.2 Functions and Variables for Matrices and Linear Algebra
| `verbify` | 6.5 Functions and Variables for Expressions
| verbose | 28.3 Functions and Variables for Series
| `vers` | 80.4 Package functs
| vertex_color | 61.2.5 Visualization
| `vertex_coloring` | 61.2.2 Graph properties
| `vertex_connectivity` | 61.2.2 Graph properties
| `vertex_degree` | 61.2.2 Graph properties
| `vertex_distance` | 61.2.2 Graph properties
| `vertex_eccentricity` | 61.2.2 Graph properties
| `vertex_in_degree` | 61.2.2 Graph properties
| `vertex_out_degree` | 61.2.2 Graph properties
| vertex_partition | 61.2.5 Visualization
| vertex_size | 61.2.5 Visualization
| vertex_type | 61.2.5 Visualization
| `vertices` | 61.2.2 Graph properties
| `vertices_to_cycle` | 61.2.5 Visualization
| `vertices_to_path` | 61.2.5 Visualization
| view | 52.2.3 Graphics options
W |  |
| warnings | 87.3 General global variables
| `weyl` | 26.2.2 The tensors of curved space
| `wheel_graph` | 61.2.1 Building graphs
| `while` | 37.4 Functions and Variables for Program Flow
| width | 54.3.1 Scene options
| `wiener_index` | 61.2.2 Graph properties
| `wigner_3j` | 46.1 Functions and Variables for clebsch_gordan
| `wigner_6j` | 46.1 Functions and Variables for clebsch_gordan
| `wigner_9j` | 46.1 Functions and Variables for clebsch_gordan
| windowname | 54.3.1 Scene options
| windowtitle | 54.3.1 Scene options
| wired_surface | 52.2.3 Graphics options
| wireframe | 54.3.3 Scene object's options
| `with_stdout` | 13.3 Functions and Variables for File Input and Output
| `write_binary_data` | 73.3 Functions and Variables for binary input and output
| `write_data` | 73.2 Functions and Variables for plain-text input and output
| `writebyte` | 84.2 Input and Output
| `writefile` | 13.3 Functions and Variables for File Input and Output
| `wronskian` | 80.4 Package functs
X |  |
| x | 12.4 Plotting Options
| x_voxel | 52.2.3 Graphics options
| xaxis | 52.2.3 Graphics options
| xaxis_color | 52.2.3 Graphics options
| xaxis_secondary | 52.2.3 Graphics options
| xaxis_type | 52.2.3 Graphics options
| xaxis_width | 52.2.3 Graphics options
| xlabel | 12.4 Plotting Options
| xlabel | 52.2.3 Graphics options
| xlabel_secondary | 52.2.3 Graphics options
| xlength | 54.3.3 Scene object's options
| xrange | 52.2.3 Graphics options
| xrange_secondary | 52.2.3 Graphics options
| `xreduce` | 5.4.2 Functions and Variables for Lists
| `xthru` | 9.2 Functions and Variables for Simplification
| xtics | 12.4 Plotting Options
| xtics | 52.2.3 Graphics options
| xtics_axis | 52.2.3 Graphics options
| xtics_rotate | 52.2.3 Graphics options
| xtics_rotate_secondary | 52.2.3 Graphics options
| xtics_secondary | 52.2.3 Graphics options
| xtics_secondary_axis | 52.2.3 Graphics options
| xu_grid | 52.2.3 Graphics options
| xy_file | 52.2.3 Graphics options
| xy_scale | 12.4 Plotting Options
| xyplane | 52.2.3 Graphics options
Y |  |
| y | 12.4 Plotting Options
| y_voxel | 52.2.3 Graphics options
| yaxis | 52.2.3 Graphics options
| yaxis_color | 52.2.3 Graphics options
| yaxis_secondary | 52.2.3 Graphics options
| yaxis_type | 52.2.3 Graphics options
| yaxis_width | 52.2.3 Graphics options
| ylabel | 12.4 Plotting Options
| ylabel | 52.2.3 Graphics options
| ylabel_secondary | 52.2.3 Graphics options
| ylength | 54.3.3 Scene object's options
| yrange | 52.2.3 Graphics options
| yrange_secondary | 52.2.3 Graphics options
| ytics | 12.4 Plotting Options
| ytics | 52.2.3 Graphics options
| ytics_axis | 52.2.3 Graphics options
| ytics_rotate | 52.2.3 Graphics options
| ytics_rotate_secondary | 52.2.3 Graphics options
| ytics_secondary | 52.2.3 Graphics options
| ytics_secondary_axis | 52.2.3 Graphics options
| yv_grid | 52.2.3 Graphics options
| yx_ratio | 12.4 Plotting Options
Z |  |
| z | 12.4 Plotting Options
| z_voxel | 52.2.3 Graphics options
| zaxis | 52.2.3 Graphics options
| zaxis_color | 52.2.3 Graphics options
| zaxis_type | 52.2.3 Graphics options
| zaxis_width | 52.2.3 Graphics options
| `Zeilberger` | 87.2 Functions and Variables for zeilberger
| zeroa | 5.3.1 Functions and Variables for Constants
| zerob | 5.3.1 Functions and Variables for Constants
| zerobern | 29.1 Functions and Variables for Number Theory
| `zeroequiv` | 11.4 Functions and Variables for Predicates
| `zerofor` | 68.2 Functions and Variables for linearalgebra
| `zeromatrix` | 23.2 Functions and Variables for Matrices and Linear Algebra
| `zeromatrixp` | 68.2 Functions and Variables for linearalgebra
| `zeta` | 29.1 Functions and Variables for Number Theory
| zeta%pi | 29.1 Functions and Variables for Number Theory
| `zgeev` | 65.2 Functions and Variables for lapack
| `zheev` | 65.2 Functions and Variables for lapack
| zlabel | 12.4 Plotting Options
| zlabel | 52.2.3 Graphics options
| zlabel_rotate | 52.2.3 Graphics options
| `zlange` | 65.2 Functions and Variables for lapack
| zlength | 54.3.3 Scene object's options
| zmin | 12.4 Plotting Options
| `zn_add_table` | 29.1 Functions and Variables for Number Theory
| `zn_carmichael_lambda` | 29.1 Functions and Variables for Number Theory
| `zn_characteristic_factors` | 29.1 Functions and Variables for Number Theory
| `zn_determinant` | 29.1 Functions and Variables for Number Theory
| `zn_factor_generators` | 29.1 Functions and Variables for Number Theory
| `zn_invert_by_lu` | 29.1 Functions and Variables for Number Theory
| `zn_log` | 29.1 Functions and Variables for Number Theory
| `zn_mult_table` | 29.1 Functions and Variables for Number Theory
| `zn_nth_root` | 29.1 Functions and Variables for Number Theory
| `zn_order` | 29.1 Functions and Variables for Number Theory
| `zn_power_table` | 29.1 Functions and Variables for Number Theory
| `zn_primroot` | 29.1 Functions and Variables for Number Theory
| zn_primroot_limit | 29.1 Functions and Variables for Number Theory
| `zn_primroot_p` | 29.1 Functions and Variables for Number Theory
| zn_primroot_pretest | 29.1 Functions and Variables for Number Theory
| zn_primroot_verbose | 29.1 Functions and Variables for Number Theory
| zrange | 52.2.3 Graphics options
| ztics | 52.2.3 Graphics options
| ztics_axis | 52.2.3 Graphics options
| ztics_rotate | 52.2.3 Graphics options

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# B. Documentation Categories

Category: Airy functions

Introduction to Special Functions airy_ai airy_bi airy_dai airy_dbi

Category: Algebraic equations

%rnum %rnum_list algepsilon algexact algsys backsubst breakup dispflag eliminate bf_find_root find_root find_root_abs find_root_error find_root_rel funcsolve multiplicities newton programmode realonly solve solvedecomposes solveexplicit solvefactors solvenullwarn solveradcan solvetrigwarn

Category: Arrays

array arrayapply arrayinfo arraymake arrays arraysetapply fillarray listarray make_array rearray remarray subvar translate_fast_arrays use_fast_arrays

Category: Bessel functions

%f %s Introduction to Special Functions bessel_i bessel_j bessel_k bessel_simplify bessel_y besselexpand hankel_1 hankel_2 scaled_bessel_i scaled_bessel_i0 scaled_bessel_i1 spherical_bessel_j spherical_bessel_y spherical_hankel1 spherical_hankel2

Category: Binary operations

bit_and bit_length bit_lsh bit_not bit_onep bit_or bit_rsh bit_xor

Category: clebsch_gordan

clebsch_gordan racah_v racah_w wigner_3j wigner_6j wigner_9j

Category: Complex variables

Introduction to Numbers cabs carg cbffac conjugate demoivre exponentialize imagpart plog polarform polartorect realpart rectform recttopolar residue

Category: Console interaction

% %th ? ?? Documentation Interrupts _ appendfile closefile demo describe entermatrix example ibase kill labels labels linenum myoptions obase optionset playback printfile prompt quit ratprint read readonly refcheck setcheck setcheckbreak setval to_lisp writefile

Category: Constants

%e %gamma %i %phi %pi constant constantp false ind inf infinity minf true und zeroa zerob

Category: Continued fractions

cf cfdisrep cfexpand cflength

Category: Debugging

Keyword Commands Source Level Debugging backtrace break bug_report build_info debugmode error_size error_syms room run_testsuite share_testsuite_files showtime testsuite_files time timer timer_devalue timer_info trace trace_options untimer untrace

Category: Declarations and inferences

activate activecontexts additive alias aliases alphabetic antisymmetric askexp askinteger asksign assume assume_pos assume_pos_pred assumescalar atvalue commutative compare constant context contexts deactivate declare decreasing increasing dependencies depends even odd facts featurep features forget get gradef gradefs infix infolists integer noninteger integervalued is killcontext lassociative linear mainvar matchdeclare maybe multiplicative newcontext nonscalar numerval outative posfun printprops properties props propvars put qput rassociative irrational rational complex imaginary real rem remove scalar sign supcontext symmetric

Category: Descriptive statistics

Introduction to descriptive

Category: Differential calculus

atomgrad del depends derivabbrev derivdegree derivlist derivsubst diff dscalar express gradef gradefs hessian implicit_derivative jacobian vect_cross wronskian

Category: Differential equations

Introduction to contrib_ode Introduction to drawdf Introduction to numerical solution of differential equations Lindstedt at atvalue bc2 desolve ic1 ic2 laplace ode2 plotdf ploteq rk

Category: Differential geometry

cartan

Category: Display flags and variables

%edispflag absboxchar dispflag display2d display_format_internal error_size error_syms exptdispflag fpprintprec grind inchar labels leftjust linechar linel linenum lispdisp lmxchar negsumdispflag noundisp obase outchar pfeformat powerdisp psexpand rmxchar showtime sqrtdispflag stardisp stringdisp ttyoff

Category: Display functions

disp dispcon dispfun display disprule dispterms engineering_format_floats grind labels ldisp ldisplay playback print printpois printprops reveal show showratvars

Category: Dynamical systems

The dynamics package

Category: Elliptic functions

Introduction to Elliptic Functions and Integrals inverse_jacobi_cd inverse_jacobi_cn inverse_jacobi_cs inverse_jacobi_dc inverse_jacobi_dn inverse_jacobi_ds inverse_jacobi_nc inverse_jacobi_nd inverse_jacobi_ns inverse_jacobi_sc inverse_jacobi_sd inverse_jacobi_sn jacobi_cd jacobi_cn jacobi_cs jacobi_dc jacobi_dn jacobi_ds jacobi_nc jacobi_nd jacobi_ns jacobi_sc jacobi_sd jacobi_sn

Category: Elliptic integrals

elliptic_e elliptic_ec elliptic_eu elliptic_f elliptic_kc elliptic_pi

Category: Error messages

0 to a negative exponent Comma is not a prefix operator Illegal use of delimiter No such list element Only symbols can be bound Operators of arguments must all be the same VTK is not installed argument must be a non-atomic expression cannot assign to function loadfile failed to load makelist second argument must evaluate to a number out of memory part fell off the end undefined variable during plotting undefined variable during translation

Category: Evaluation flags

%enumer detout eval evflag evfun float infeval noeval nouns numer pred simp

Category: Evaluation

' " : :: Nouns and Verbs at derivlist ev kill refcheck remvalue values

Category: Exponential and logarithm functions

%e_to_numlog %edispflag %emode %enumer exp exptsubst li log logabs logarc logconcoeffp logcontract logexpand lognegint logsimp plog polarform taylor_logexpand

Category: Exponential Integrals

expintegral_chi expintegral_ci expintegral_e expintegral_e1 expintegral_e_simplify expintegral_ei expintegral_li expintegral_shi expintegral_si expintexpand expintrep

Category: Expressions

Package facexp Package rducon Package scifac append args arrayapply arraymake arraysetapply assoc atom block box boxchar collapse collectterms combine concat cons copy delete denom derivdegree derivsubst disolate dispform distrib dontfactor dpart eighth endcons expand expandwrt expandwrt_denom expandwrt_factored expon expop exptdispflag exptisolate exptsubst facsum facsum_combine factorfacsum factorout factorsum fifth first firstn fourth freeof fullmap fullmapl funmake gcfac gfactorsum hipow inflag inpart intosum isolate isolate_wrt_times last lastn length lfreeof lhs linear listconstvars listdummyvars listofvars lopow lpart m1pbranch mainvar maxnegex maxposex member multiplicative multthru nextlayerfactor ninth nonarray nonzeroandfreeof nterms num numfactor op opsubst opsubst optimize optimprefix ordergreat orderless ordergreatp orderlessp part partition partswitch pickapart piece pop psubst push rational reduce_consts rembox rempart rest reveal reverse rhs rncombine rootsconmode sconcat second seventh sixth sqrtdenest sublis sublis_apply_lambda subnumsimp subst substinpart substpart subvar tenth third unorder xthru

Category: File input

Introduction to numericalio batch batchload close demo directory file_search file_search_demo file_search_lisp file_search_maxima file_search_tests file_search_usage file_type filename_merge flength fposition load load_pathname loadfile loadprint openr openr_binary pathname_directory pathname_name pathname_type printfile read_array read_binary_array read_binary_list read_binary_matrix read_hashed_array read_list read_matrix read_nested_list readbyte readchar readline setup_autoload

Category: File output

Introduction to numericalio appendfile close closefile draw draw2d draw3d draw_file file_output_append filename_merge flength flush_output fposition freshline multiplot_mode newline opena opena_binary openw openw_binary printf save stringout tex with_stdout write_binary_data write_data writebyte writefile

Category: Fourier transform

Introduction to Fourier series Introduction to fast Fourier transform

Category: Function application

apply fullmap fullmapl funmake macroexpand macroexpand1 macroexpansion map maperror maplist mapprint maxapplydepth maxapplyheight outermap scanmap

Category: Function definition

::= := Function buildq define dispfun functions fundef lambda local macros remfunction setup_autoload splice

Category: Gamma and factorial functions

!! beta beta_args_sum_to_integer beta_expand beta_incomplete beta_incomplete_generalized beta_incomplete_regularized bffac bfpsi bfpsi0 cbffac factcomb factlim ! factorial factorial_expand gamma gamma_expand gamma_incomplete gamma_incomplete_generalized gamma_incomplete_lower gamma_incomplete_regularized gammalim genfact log_gamma makefact makegamma maxpsifracdenom maxpsifracnum maxpsinegint maxpsiposint pochhammer pochhammer_max_index psi stirling sumsplitfact

Category: Generating functions

ggf

Category: Global flags

debugmode derivabbrev engineering_format_floats file_output_append listarith loadprint logabs m1pbranch macroexpansion nolabels optionset ratvarswitch refcheck setcheck setcheckbreak timer_devalue use_fast_arrays

Category: Global variables

% %% _ __ adim aform aliases arrays asymbol dependencies file_search_demo file_search_lisp file_search_maxima file_search_tests file_search_usage functions infolists macros manual_demo maxima_tempdir maxima_userdir myoptions newline opproperties prompt props setval share_testsuite_files space structures tab testsuite_files us_ascii_only values

Category: Groebner bases

Introduction to Affine Introduction to grobner

Category: Group theory

Introduction to Symmetries todd_coxeter

Category: Help

? ?? apropos demo describe example manual_demo

Category: Hyperbolic functions

%iargs acosh acoth acsch asech asinh atanh cosh coth csch demoivre exponentialize sech sinh tanh

Category: Hypergeometric functions

hypergeometric_simp

Category: Integers

divisors gcfactor integer_partitions intfaclim modulus moebius multinomial_coeff num_distinct_partitions num_partitions stirling1 stirling2

Category: Integral calculus

Introduction to QUADPACK absint antid antidiff changevar dblint defint erfflag intanalysis integrate integrate_use_rootsof integration_constant integration_constant_counter ldefint logabs residue risch tldefint

Category: Integral equations

ieqn ieqnprint

Category: Laplace transform

delta desolve ilt laplace specint

Category: Limits

lhospitallim limit limsubst tlimit tlimswitch

Category: Linear algebra

Introduction to linearalgebra

Category: Linear equations

augcoefmatrix coefmatrix dgesv echelon globalsolve linsolve linsolve_params linsolvewarn triangularize

Category: Linear recurrences

Introduction to solve_rec

Category: Lists

[ ] append assoc cons copylist create_list delete eighth endcons fifth first firstn flatten fourth fullsetify join last lastn length listarith listp lmax lmin lreduce makelist member ninth permut permutations pop push random_permutation rest reverse rreduce second setify seventh sixth some sort sublist sublist_indices tenth third tree_reduce xreduce

Category: Mathematical functions

abs agd ceiling charfun combination covers delta entier exsec fix floor gaussprob gd hav isqrt lmax lmin max min mod permutation round signum sqrt sqrtdispflag truncate unit_step vers

Category: Matrices

addcol addrow adjoint augcoefmatrix cauchy_matrix charpoly coefmatrix col columnvector covect copymatrix determinant detout diag diagmatrix doallmxops domxexpt domxmxops domxnctimes doscmxops doscmxplus echelon eigen ematrix entermatrix genmatrix ident invert list_matrix_entries lmxchar matrix matrix_element_add matrix_element_mult matrix_element_transpose matrixexp matrixmap matrixp mattrace minor ncharpoly newdet nonscalar nonscalarp permanent rank ratmx row scalarmatrixp scalarp setelmx sparse submatrix tracematrix transpose triangularize zeromatrix

Category: Matrix decompositions

cholesky eigens_by_jacobi lu_factor

Category: Nouns and verbs

Nouns and Verbs noun noundisp nounify nouns verbify

Category: Number theory

bern bernpoly bfhzeta bfzeta binomial bit_and bit_length bit_lsh bit_not bit_onep bit_or bit_rsh bit_xor burn chinese divsum euler factors_only fib fibtophi ifactors igcdex inrt inv_mod jacobi lcm lucas minfactorial next_prime power_mod prev_prime primep primep_number_of_tests primes qunit totient zerobern zeta zeta%pi zn_add_table zn_carmichael_lambda zn_characteristic_factors zn_determinant zn_factor_generators zn_invert_by_lu zn_log zn_mult_table zn_nth_root zn_order zn_power_table zn_primroot zn_primroot_limit zn_primroot_p zn_primroot_pretest zn_primroot_verbose

Category: Numerical evaluation

bffac bfhzeta bfloat bfloatp bfpsi bfpsi0 bftorat bftrunc bfzeta cbffac float float2bf floatnump fpprec fpprintprec keepfloat numer numer_pbranch numerval ratepsilon rationalize ratprint stats_numer

Category: Numerical methods

Introduction to QUADPACK Introduction to cobyla Introduction to fast Fourier transform Introduction to interpol Introduction to lapack Introduction to lbfgs Introduction to minpack Introduction to mnewton Introduction to numerical solution of differential equations Introduction to simplex allroots augmented_lagrangian_method bfallroots bf_find_root find_root find_root_abs find_root_error find_root_rel horner linear_program lsquares_estimates lsquares_estimates_approximate maximize_lp minimize_lp newton nroots plotdf ploteq plsquares polyfactor quad_control quad_qag quad_qagi quad_qagp quad_qags quad_qawc quad_qawf quad_qawo quad_qaws random realroots rk romberg rootsepsilon

Category: Operators

!! # ' " ** * + - / ^ . : :: ::= := < <= > >= = @ Introduction to operators [ ] ^^ additive and antisymmetric commutative equal express ! factorial infix lassociative linear matchfix nary nofix not notequal op operatorp opproperties or outative posfun postfix prefix rassociative symmetric | ~

Category: Optimization

Introduction to cobyla Introduction to lbfgs Introduction to minpack Introduction to simplex augmented_lagrangian_method

Category: Orthogonal polynomials

Introduction to orthogonal polynomials

Category: Package absimp

Package absimp

Category: Package affine

Introduction to Affine all_dotsimp_denoms check_overlaps declare_weights dotsimp extract_linear_equations fast_central_elements fast_linsolve grobner_basis list_nc_monomials mono monomial_dimensions nc_degree set_up_dot_simplifications

Category: Package alt-display

define_alt_display info_display mathml_display multi_display_for_texinfo reset_displays set_alt_display set_prompt tex_display

Category: Package antid

linear

Category: Package asympa

asympa

Category: Package atensor

Introduction to atensor abasep adim af aform alg_type asymbol atensimp av init_atensor sf

Category: Package atrig1

atrig1

Category: Package augmented_lagrangian

augmented_lagrangian_method

Category: Package bitwise

bit_and bit_length bit_lsh bit_not bit_onep bit_or bit_rsh bit_xor

Category: Package bode

bode_gain bode_phase

Category: Package cobyla

Introduction to cobyla

Category: Package contrib_ode

%c %k1 %k2 Introduction to contrib_ode bessel_simplify contrib_ode dgauss_a dgauss_b dkummer_m dkummer_u expintegral_e_simplify gauss_a gauss_b kummer_m kummer_u method ode_check odelin

Category: Package ctensor

Introduction to ctensor bdvac bimetric cdisplay cframe_flag cgeodesic checkdiv christof cmetric cnonmet_flag cograd contortion contragrad csetup ct_coords ct_coordsys ctaylor ctaypov ctaypt ctayswitch ctayvar ctorsion_flag ctransform ctrgsimp deleten diagmatrixp diagmetric dim dscalar dscalar einstein fb findde frame_bracket gdet ic_convert init_ctensor invariant1 invariant2 kinvariant kt leinstein lfg lg lriem lriemann nm nmc nonmetricity np npi nptetrad ntermst petrov psi ratchristof rateinstein ratriemann ratweyl ric ricci riem riemann rinvariant scurvature symmetricp tensorkill tr ufg ug uric uricci uriem uriemann weyl weyl

Category: Package descriptive

Introduction to descriptive barsplot barsplot_description boxplot boxplot_description build_sample cdf_empirical central_moment continuous_freq cor cov cov1 cv discrete_freq geometric_mean global_variances harmonic_mean histogram histogram_description km kurtosis list_correlations mean mean_deviation median median_deviation noncentral_moment pearson_skewness piechart piechart_description principal_components qrange quantile quartile_skewness range scatterplot scatterplot_description skewness smax smin standardize starplot starplot_description std std1 stemplot subsample transform_sample var var1

Category: Package diag

JF ModeMatrix diag dispJordan jordan mat_function minimalPoly

Category: Package distrib

Introduction to distrib cdf_bernoulli cdf_beta cdf_binomial cdf_cauchy cdf_chi2 cdf_continuous_uniform cdf_discrete_uniform cdf_exp cdf_f cdf_gamma cdf_general_finite_discrete cdf_geometric cdf_gumbel cdf_hypergeometric cdf_laplace cdf_logistic cdf_lognormal cdf_negative_binomial cdf_noncentral_chi2 cdf_noncentral_student_t cdf_normal cdf_pareto cdf_poisson cdf_rayleigh cdf_student_t cdf_weibull kurtosis_bernoulli kurtosis_beta kurtosis_binomial kurtosis_chi2 kurtosis_continuous_uniform kurtosis_discrete_uniform kurtosis_exp kurtosis_f kurtosis_gamma kurtosis_general_finite_discrete kurtosis_geometric kurtosis_gumbel kurtosis_gumbel kurtosis_hypergeometric kurtosis_laplace kurtosis_logistic kurtosis_lognormal kurtosis_negative_binomial kurtosis_noncentral_chi2 kurtosis_noncentral_student_t kurtosis_normal kurtosis_pareto kurtosis_poisson kurtosis_rayleigh kurtosis_student_t kurtosis_weibull mean_bernoulli mean_beta mean_binomial mean_chi2 mean_continuous_uniform mean_discrete_uniform mean_exp mean_f mean_gamma mean_general_finite_discrete mean_geometric mean_gumbel mean_hypergeometric mean_laplace mean_logistic mean_lognormal mean_negative_binomial mean_noncentral_chi2 mean_noncentral_student_t mean_normal mean_pareto mean_poisson mean_rayleigh mean_student_t mean_weibull pdf_bernoulli pdf_beta pdf_binomial pdf_cauchy pdf_chi2 pdf_continuous_uniform pdf_discrete_uniform pdf_exp pdf_f pdf_gamma pdf_general_finite_discrete pdf_geometric pdf_gumbel pdf_hypergeometric pdf_laplace pdf_logistic pdf_lognormal pdf_negative_binomial pdf_noncentral_chi2 pdf_noncentral_student_t pdf_normal pdf_pareto pdf_poisson pdf_rayleigh pdf_student_t pdf_weibull quantile_bernoulli quantile_beta quantile_binomial quantile_cauchy quantile_chi2 quantile_continuous_uniform quantile_discrete_uniform quantile_exp quantile_f quantile_gamma quantile_general_finite_discrete quantile_geometric quantile_gumbel quantile_hypergeometric quantile_laplace quantile_logistic quantile_lognormal quantile_negative_binomial quantile_noncentral_chi2 quantile_noncentral_student_t quantile_normal quantile_pareto quantile_poisson quantile_rayleigh quantile_student_t quantile_weibull random_bernoulli random_beta random_binomial random_cauchy random_chi2 random_continuous_uniform random_discrete_uniform random_exp random_f random_gamma random_general_finite_discrete random_geometric random_gumbel random_hypergeometric random_laplace random_logistic random_lognormal random_negative_binomial random_noncentral_chi2 random_noncentral_student_t random_normal random_pareto random_poisson random_rayleigh random_student_t random_weibull skewness_bernoulli skewness_beta skewness_binomial skewness_chi2 skewness_continuous_uniform skewness_discrete_uniform skewness_exp skewness_f skewness_gamma skewness_general_finite_discrete skewness_geometric skewness_gumbel skewness_hypergeometric skewness_laplace skewness_logistic skewness_lognormal skewness_negative_binomial skewness_noncentral_chi2 skewness_noncentral_student_t skewness_normal skewness_pareto skewness_poisson skewness_rayleigh skewness_student_t skewness_weibull std_bernoulli std_beta std_binomial std_chi2 std_continuous_uniform std_discrete_uniform std_exp std_f std_gamma std_general_finite_discrete std_geometric std_gumbel std_hypergeometric std_laplace std_logistic std_lognormal std_negative_binomial std_noncentral_chi2 std_noncentral_student_t std_normal std_pareto std_poisson std_rayleigh std_student_t std_weibull var_bernoulli var_beta var_binomial var_chi2 var_continuous_uniform var_discrete_uniform var_exp var_f var_gamma var_general_finite_discrete var_geometric var_gumbel var_hypergeometric var_laplace var_logistic var_lognormal var_negative_binomial var_noncentral_chi2 var_noncentral_student_t var_normal var_pareto var_poisson var_rayleigh var_student_t var_weibull

Category: Package drawdf

Introduction to drawdf drawdf

Category: Package draw

Introduction to draw Introduction to drawdf adapt_depth allocation axis_3d axis_bottom axis_left axis_right axis_top background_color bars border boundaries_array capping cbrange cbtics color colorbox columns contour contour_levels cylindrical data_file_name delay dimensions draw draw2d draw3d draw_file draw_realpart elevation_grid ellipse enhanced3d errors explicit file_name fill_color filled_func font font_size geomap get_pixel gnuplot_file_name gr2d gr3d grid head_angle head_both head_length head_type image implicit interpolate_color ip_grid ip_grid_in key key_pos label label_alignment label_orientation line_type line_width logcb logx logx_secondary logy logy_secondary logz make_level_picture make_poly_continent make_poly_country make_polygon make_rgb_picture mesh multiplot_mode negative_picture nticks numbered_boundaries palette parametric parametric_surface picture_equalp picturep point_size point_type points points_joined polar polygon proportional_axes quadrilateral read_xpm rectangle region_boundaries region_boundaries_plus rgb2level set_draw_defaults spherical surface_hide take_channel terminal title transform transparent triangle tube unit_vectors user_preamble vector view wired_surface x_voxel xaxis xaxis_color xaxis_secondary xaxis_type xaxis_width xlabel xlabel_secondary xrange xrange_secondary xtics xtics_axis xtics_rotate xtics_rotate_secondary xtics_secondary xtics_secondary_axis xu_grid xy_file xyplane y_voxel yaxis yaxis_color yaxis_secondary yaxis_type yaxis_width ylabel ylabel_secondary yrange yrange_secondary ytics ytics_axis ytics_rotate ytics_rotate_secondary ytics_secondary ytics_secondary_axis yv_grid z_voxel zaxis zaxis_color zaxis_type zaxis_width zlabel zlabel_rotate zrange ztics ztics_axis ztics_rotate

Category: Package dynamics

The dynamics package animation azimuth background capping center chaosgame color cone cube cylinder elevation endphi endtheta evolution evolution2d height height ifs julia linewidth mandelbrot opacity orbits orientation origin phiresolution points pointsize position radius resolution restart scale scene sphere staircase startphi starttheta surface thetaresolution track tstep width windowname windowtitle wireframe xlength ylength zlength

Category: Package eigen

eigen eigenvalues eivals eigenvectors eivects gramschmidt innerproduct inprod similaritytransform simtran ueivects uniteigenvectors unitvector uvect

Category: Package ezunits

Introduction to ezunits ` `` constvalue declare_constvalue declare_dimensions declare_fundamental_dimensions fundamental_dimensions remove_fundamental_dimensions declare_fundamental_units remove_fundamental_units declare_qty declare_unit_conversion declare_units dimensionless dimensions dimensions_as_list fundamental_units natural_unit qty remove_constvalue remove_dimensions unitp units

Category: Package f90

f90

Category: Package facexp

Package facexp collectterms facsum facsum_combine factorfacsum nextlayerfactor

Category: Package fft

Introduction to fast Fourier transform bf_fft bf_inverse_fft bf_inverse_real_fft bf_real_fft fft inverse_fft inverse_real_fft polartorect real_fft recttopolar

Category: Package finance

amortization annuity_fv annuity_pv arit_amortization benefit_cost days360 fv geo_amortization geo_annuity_fv geo_annuity_pv graph_flow irr npv pv saving

Category: Package fourie

Introduction to Fourier series absint cosnpiflag equalp fourcos fourexpand fourier fourint fourintcos fourintsin foursimp foursin funp remfun sinnpiflag totalfourier

Category: Package fractals

fernfale hilbertmap julia_parameter julia_set julia_sin mandelbrot_set sierpinskiale sierpinskimap snowmap treefale

Category: Package functs

Package functs agd arithmetic arithsum combination covers exsec gaussprob gcdivide gd geometric geosum harmonic hav nonzeroandfreeof permutation rational rempart tracematrix vers wronskian

Category: Package ggf

GGFCFMAX GGFINFINITY ggf

Category: Package graphs - constructions

circulant_graph clebsch_graph complement_graph complete_bipartite_graph complete_graph copy_graph cube_graph cuboctahedron_graph cycle_digraph cycle_graph dodecahedron_graph empty_graph flower_snark from_adjacency_matrix frucht_graph graph_product graph_union great_rhombicosidodecahedron_graph great_rhombicuboctahedron_graph grid_graph grotzch_graph heawood_graph icosahedron_graph icosidodecahedron_graph induced_subgraph line_graph make_graph mycielski_graph new_graph path_digraph path_graph petersen_graph random_bipartite_graph random_digraph random_graph random_graph1 random_network random_regular_graph random_tournament random_tree small_rhombicosidodecahedron_graph small_rhombicuboctahedron_graph snub_cube_graph snub_dodecahedron_graph truncated_cube_graph truncated_dodecahedron_graph truncated_icosahedron_graph truncated_tetrahedron_graph tutte_graph underlying_graph wheel_graph

Category: Package graphs - draw_graphs options

draw_graph_program edge_color edge_coloring edge_partition edge_type edge_width file_name fixed_vertices head_angle head_length label_alignment program redraw show_edge_color show_edge_type show_edge_width show_edges show_id show_label show_vertex_color show_vertex_size show_vertex_type show_vertices show_weight spring_embedding_depth terminal vertex_color vertex_coloring vertex_partition vertex_size vertex_type

Category: Package graphs - io

dimacs_export dimacs_import graph6_decode graph6_encode graph6_export graph6_import sparse6_decode sparse6_encode sparse6_export sparse6_import

Category: Package graphs - modifications

add_edge add_edges add_vertex add_vertices connect_vertices contract_edge remove_edge

Category: Package graphs - properties

adjacency_matrix average_degree biconnected_components bipartition chromatic_index chromatic_number clear_edge_weight clear_vertex_label connected_components degree_sequence diameter edge_coloring edge_connectivity edges get_edge_weight get_vertex_label girth graph_center graph_charpoly graph_eigenvalues graph_order graph_periphery graph_size hamilton_cycle hamilton_path in_neighbors is_biconnected is_bipartite is_connected is_digraph is_edge_in_graph is_graph is_graph_or_digraph is_isomorphic is_planar is_sconnected is_tree is_vertex_in_graph isomorphism laplacian_matrix max_clique max_degree max_flow max_independent_set max_matching min_degree min_edge_cut min_vertex_cover min_vertex_cut minimum_spanning_tree neighbors odd_girth out_neighbors planar_embedding radius set_edge_weight set_vertex_label shortest_path shortest_weighted_path strong_components topological_sort vertex_coloring vertex_connectivity vertex_degree vertex_distance vertex_eccentricity vertex_in_degree vertex_out_degree vertices wiener_index

Category: Package graphs

Introduction to graphs add_edge add_edges add_vertex add_vertices adjacency_matrix average_degree biconnected_components bipartition chromatic_index chromatic_number circulant_graph clear_edge_weight clear_vertex_label clebsch_graph complement_graph complete_bipartite_graph complete_graph connect_vertices connected_components contract_edge copy_graph create_graph cube_graph cuboctahedron_graph cycle_digraph cycle_graph degree_sequence diameter dimacs_export dimacs_import dodecahedron_graph draw_graph draw_graph_program edge_color edge_coloring edge_coloring edge_connectivity edge_partition edge_type edge_width edges empty_graph file_name fixed_vertices flower_snark from_adjacency_matrix frucht_graph get_edge_weight get_vertex_label girth graph6_decode graph6_encode graph6_export graph6_import graph_center graph_charpoly graph_eigenvalues graph_order graph_periphery graph_product graph_size graph_union great_rhombicosidodecahedron_graph great_rhombicuboctahedron_graph grid_graph grotzch_graph hamilton_cycle hamilton_path head_angle head_length heawood_graph icosahedron_graph icosidodecahedron_graph in_neighbors induced_subgraph is_biconnected is_bipartite is_connected is_digraph is_edge_in_graph is_graph is_graph_or_digraph is_isomorphic is_planar is_sconnected is_tree is_vertex_in_graph isomorphism label_alignment laplacian_matrix line_graph make_graph max_clique max_degree max_flow max_independent_set max_matching min_degree min_edge_cut min_vertex_cover min_vertex_cut minimum_spanning_tree mycielski_graph neighbors new_graph odd_girth out_neighbors path_digraph path_graph petersen_graph planar_embedding print_graph program radius random_bipartite_graph random_digraph random_graph random_graph1 random_network random_regular_graph random_tournament random_tree redraw remove_edge remove_vertex set_edge_weight set_vertex_label shortest_path shortest_weighted_path show_edge_color show_edge_type show_edge_width show_edges show_id show_label show_vertex_color show_vertex_size show_vertex_type show_vertices show_weight small_rhombicosidodecahedron_graph small_rhombicuboctahedron_graph snub_cube_graph snub_dodecahedron_graph sparse6_decode sparse6_encode sparse6_export sparse6_import spring_embedding_depth strong_components terminal topological_sort truncated_cube_graph truncated_dodecahedron_graph truncated_icosahedron_graph truncated_tetrahedron_graph tutte_graph underlying_graph vertex_color vertex_coloring vertex_coloring vertex_connectivity vertex_degree vertex_distance vertex_eccentricity vertex_in_degree vertex_out_degree vertex_partition vertex_size vertex_type vertices vertices_to_cycle vertices_to_path wheel_graph wiener_index

Category: Package grobner

Introduction to grobner poly_add poly_buchberger poly_buchberger_criterion poly_coefficient_ring poly_colon_ideal poly_content poly_depends_p poly_elimination_ideal poly_elimination_order poly_exact_divide poly_expand poly_expt poly_gcd poly_grobner poly_grobner_algorithm poly_grobner_debug poly_grobner_equal poly_grobner_member poly_grobner_subsetp poly_ideal_intersection poly_ideal_polysaturation poly_ideal_polysaturation1 poly_ideal_saturation poly_ideal_saturation1 poly_lcm poly_minimization poly_monomial_order poly_multiply poly_normal_form poly_normalize poly_normalize_list poly_polysaturation_extension poly_primary_elimination_order poly_primitive_part poly_pseudo_divide poly_reduced_grobner poly_reduction poly_return_term_list poly_s_polynomial poly_saturation_extension poly_secondary_elimination_order poly_subtract poly_top_reduction_only

Category: Package impdiff

implicit_derivative

Category: Package implicit_plot

implicit_plot

Category: Package ineq

Package ineq

Category: Package interpol

Introduction to interpol charfun2 cspline lagrange linearinterpol ratinterpol

Category: Package itensor

Introduction to itensor allsym canform canten changename components concan conmetderiv contract coord covdiff decsym defcon diff entertensor evundiff extdiff flipflag flush flush1deriv flushd flushnd hodge ic_convert icc1 icc2 ichr1 ichr2 icounter icurvature idiff idim idummy idummyx ifb ifc1 ifc2 ifg ifgi ifr iframe_bracket_form iframes ifri igeodesic_coords igeowedge_flag ikt1 ikt2 imetric indexed_tensor indices inm inmc1 inmc2 ishow itr kdels kdelta lc2kdt lc_l lc_u levi_civita liediff listoftens lorentz_gauge makebox rediff remcomps remcon remcoord remsym rename show showcomps simpmetderiv tentex undiff | ~

Category: Package lapack

Introduction to lapack dgeev dgemm dgeqrf dgesv dgesvd dlange zlange zgeev zheev

Category: Package lbfgs

Introduction to lbfgs lbfgs lbfgs_ncorrections lbfgs_nfeval_max

Category: Package lindstedt

Lindstedt

Category: Package linearalgebra

Introduction to linearalgebra addmatrices blockmatrixp cholesky columnop columnspace columnswap ctranspose diag_matrix dotproduct eigens_by_jacobi get_lu_factors hankel hessian hilbert_matrix identfor invert_by_lu jacobian kronecker_product listp locate_matrix_entry lu_backsub lu_factor mat_cond mat_fullunblocker mat_norm mat_trace mat_unblocker matrix_size matrixp nonnegintegerp nullity nullspace orthogonal_complement polynomialp polytocompanion ptriangularize rank rowop rowswap toeplitz vandermonde_matrix zerofor zeromatrixp

Category: Package lsquares

Introduction to lsquares lsquares_estimates lsquares_estimates_approximate lsquares_estimates_exact lsquares_mse lsquares_residual_mse lsquares_residuals plsquares

Category: Package makeOrders

makeOrders

Category: Package minpack

Introduction to minpack

Category: Package mnewton

Introduction to mnewton mnewton newtonepsilon newtonmaxiter

Category: Package nchrpl

mattrace ncharpoly

Category: Package ntrig

ntrig

Category: Package numericalio

Introduction to numericalio assume_external_byte_order opena_binary openr_binary openw_binary read_array read_binary_array read_binary_list read_binary_matrix read_hashed_array read_list read_matrix read_nested_list write_binary_data write_data

Category: Package opsubst

opsubst

Category: Package orthopoly

Introduction to orthogonal polynomials assoc_legendre_p assoc_legendre_q chebyshev_t chebyshev_u gen_laguerre hermite intervalp jacobi_p laguerre legendre_p legendre_q orthopoly_recur orthopoly_returns_intervals orthopoly_weight pochhammer pochhammer_max_index spherical_bessel_j spherical_bessel_y spherical_hankel1 spherical_hankel2 spherical_harmonic ultraspherical unit_step

Category: Package physical_constants

Introduction to physical_constants

Category: Package quadpack

Introduction to QUADPACK quad_control quad_qag quad_qagi quad_qagp quad_qags quad_qawc quad_qawf quad_qawo quad_qaws

Category: Package ratpow

ratp_coeffs ratp_dense_coeffs ratp_hipow ratp_lopow

Category: Package rducon

Package rducon reduce_consts

Category: Package romberg

romberg rombergabs rombergit rombergmin rombergtol

Category: Package scifac

Package scifac gcfac

Category: Package simplex

Introduction to simplex epsilon_lp linear_program maximize_lp minimize_lp nonegative_lp pivot_count_sx pivot_max_sx scale_lp

Category: Package solve_rec

Introduction to solve_rec product_use_gamma reduce_order simplify_products simplify_sum solve_rec solve_rec_rat summand_to_rec

Category: Package sqdnst

sqrtdenest

Category: Package stats

Introduction to stats cdf_rank_sum cdf_signed_rank inference_result inferencep items_inference linear_regression pdf_rank_sum pdf_signed_rank stats_numer take_inference test_mean test_means_difference test_normality test_proportion test_proportions_difference test_rank_sum test_sign test_signed_rank test_variance test_variance_ratio

Category: Package stirling

stirling

Category: Package stringproc

Introduction to String Processing adjust_external_format alphacharp alphanumericp ascii base64 base64_decode cequal cequalignore cgreaterp cgreaterpignore charat charlist charp cint clessp clesspignore close constituent crc24sum digitcharp eval_string flength flush_output fposition freshline get_output_stream_string lowercasep make_string_input_stream make_string_output_stream md5sum mgf1_sha1 newline newline number_to_octets octets_to_number octets_to_oid octets_to_string oid_to_octets opena openr openw parse_string printf readbyte readchar readline scopy sdowncase sequal sequalignore sexplode sha1sum sha256sum simplode sinsert sinvertcase slength smake smismatch space split sposition sprint sremove sremovefirst sreverse ssearch ssort ssubst ssubstfirst strim striml strimr string_to_octets stringp substring supcase tab tokens unicode unicode_to_utf8 uppercasep us_ascii_only utf8_to_unicode writebyte

Category: Package sym

Introduction to Symmetries comp2pui cont2part contract direct ele2comp ele2polynome ele2pui elem explose kostka lgtreillis ltreillis mon2schur multi_elem multi_orbit multi_pui multinomial multsym orbit part2cont partpol permut polynome2ele prodrac pui pui2comp pui2ele pui2polynome pui_direct puireduc resolvante resolvante_alternee1 resolvante_bipartite resolvante_diedrale resolvante_klein resolvante_klein3 resolvante_produit_sym resolvante_unitaire resolvante_vierer schur2comp somrac tcontract tpartpol treillis treinat

Category: Package unit

%unitexpand Introduction to Units convert metricexpandall setunits uforget usersetunits

Category: Package vect

Vectors scalefactors vect_cross vectorpotential vectorsimp

Category: Package zeilberger

AntiDifference Gosper GosperSum Gosper_in_Zeilberger Introduction to zeilberger MAX_ORD Zeilberger Zeilberger ev_point linear_solver mod_big_prime mod_test mod_threshold modular_linear_solver parGosper parGosper simplified_output trivial_solutions warnings

Category: Physical units

Introduction to Units Introduction to ezunits Introduction to physical_constants

Category: Plotting

Introduction to Plotting Introduction to draw Introduction to drawdf Introduction to numerical solution of differential equations Introduction to orthogonal polynomials Plotting Formats adapt_depth animation axes azimuth azimuth background barsplot barsplot_description bode_gain bode_phase box boxplot boxplot_description capping center chaosgame color color color_bar color_bar_tics cone contour_plot cube cylinder elevation elevation endphi endtheta evolution evolution2d geomview_command gnuplot_close gnuplot_command gnuplot_curve_styles gnuplot_curve_titles gnuplot_default_term_command gnuplot_dumb_term_command gnuplot_file_args gnuplot_out_file gnuplot_pdf_term_command gnuplot_pm3d gnuplot_png_term_command gnuplot_postamble gnuplot_preamble gnuplot_ps_term_command gnuplot_replot gnuplot_reset gnuplot_restart gnuplot_start gnuplot_svg_term_command gnuplot_term gnuplot_view_args grid grid2d height height histogram histogram_description ifs implicit_plot iterations julia label legend linewidth logx logy make_transform mandelbrot mesh_lines_color nticks opacity orbits orientation origin palette pdf_file phiresolution piechart piechart_description plot2d plot3d plot_format plot_options plot_realpart plotdf ploteq png_file point_type points pointsize polar_to_xy position ps_file radius remove_plot_option resolution restart run_viewer same_xy same_xyz scale scatterplot scatterplot_description scene set_plot_option sphere spherical_to_xyz staircase starplot starplot_description startphi starttheta stemplot style surface svg_file t thetaresolution title track transform_xy tstep width windowname windowtitle wireframe x xlabel xlength xtics xy_scale y ylabel ylength ytics yx_ratio z zlabel zlength zmin

Category: Poisson series

intopois outofpois poisdiff poisexpt poisint poislim poismap poisplus poissimp poisson poissubst poistimes poistrim printpois

Category: Polynomials

Introduction to Affine Introduction to Polynomials Introduction to Symmetries allroots berlefact bezout bfallroots bothcoef coeff content divide eliminate ezgcd facexpand factor factorflag fasttimes gcd gcdex gcdivide gfactor lratsubst makeOrders multiplicities nroots nthroot polydecomp polyfactor polymod programmode quotient ratcoef realroots remainder resultant resultant rootsepsilon savefactors sqfr tellrat untellrat

Category: Power series

deftaylor maxtayorder pade powerseries revert revert2 taylor taylor_logexpand taylor_order_coefficients taylor_simplifier taylor_truncate_polynomials taylordepth taylorinfo taylorp taytorat trunc verbose

Category: Predicate functions

abasep alphacharp alphanumericp atom bfloatp blockmatrixp cequal cequalignore cgreaterp cgreaterpignore charp clessp clesspignore constantp constituent diagmatrixp digitcharp disjointp elementp emptyp evenp featurep floatnump if integerp intervalp is listp listp lowercasep mapatom matrixp matrixp maybe member nonnegintegerp nonscalarp numberp oddp operatorp ordergreatp orderlessp picture_equalp picturep poly_depends_p poly_grobner_subsetp polynomialp prederror primep ratnump ratp scalarp sequal sequalignore setequalp setp stringp subsetp subvarp symbolp symmetricp taylorp unknown uppercasep zeroequiv zeromatrixp zn_primroot_p

Category: Programming

Function Lisp and Maxima block catch do for from in next step thru unless while errcatch error errormsg errormsg go if local prederror return sstatus status throw

Category: Random numbers

make_random_state random random_bernoulli random_beta random_binomial random_cauchy random_chi2 random_continuous_uniform random_discrete_uniform random_exp random_f random_gamma random_general_finite_discrete random_geometric random_gumbel random_hypergeometric random_laplace random_logistic random_lognormal random_negative_binomial random_noncentral_chi2 random_noncentral_student_t random_normal random_pareto random_poisson random_rayleigh random_student_t random_weibull set_random_state

Category: Rational expressions

Introduction to Polynomials fullratsimp fullratsubst gcd gcdex lratsubst rat ratcoef ratdenom ratdenomdivide ratdiff ratdisrep ratepsilon ratexpand ratfac ratmx ratnumer ratnump ratp ratp_coeffs ratp_dense_coeffs ratp_hipow ratp_lopow ratprint ratsimp ratsimpexpons ratsubst ratvars ratvarswitch ratweight ratweights ratwtlvl showratvars taytorat tellrat totaldisrep untellrat

Category: Rules and patterns

Introduction to Rules and Patterns Package absimp Package ineq apply1 apply2 applyb1 clear_rules current_let_rule_package default_let_rule_package defmatch defrule disprule let let_rule_packages letrat letrules letsimp matchdeclare remlet remrule tellsimp tellsimpafter

Category: Session management

Introduction for Runtime Environment batch batchload kill load loadfile myoptions nolabels optionset reset save stringout

Category: Sets

Introduction to Sets adjoin belln cardinality cartesian_product disjoin disjointp elementp emptyp equiv_classes every extremal_subset flatten full_listify intersect intersection listify lmax lmin makeset partition_set permutations powerset random_permutation set_partitions setdifference setequalp setp some subset subsetp symmdifference tree_reduce union xreduce

Category: Share packages

Introduction to Affine Introduction to Fourier series Introduction to QUADPACK Introduction to String Processing Introduction to Symmetries Introduction to Units Introduction to atensor Introduction to cobyla Introduction to contrib_ode Introduction to ctensor Introduction to descriptive Introduction to distrib Introduction to draw Introduction to drawdf Introduction to ezunits Introduction to fast Fourier transform Introduction to graphs Introduction to grobner Introduction to interpol Introduction to itensor Introduction to lapack Introduction to lbfgs Introduction to linearalgebra Introduction to lsquares Introduction to minpack Introduction to mnewton Introduction to numericalio Introduction to orthogonal polynomials Introduction to physical_constants Introduction to simplex Introduction to solve_rec Introduction to stats Introduction to zeilberger Lindstedt Package absimp Package facexp Package functs Package ineq Package rducon Package scifac The dynamics package Vectors augmented_lagrangian_method diag dimension eigen engineering_format_floats f90 ggf implicit_derivative implicit_plot makeOrders opsubst sqrtdenest stirling

Category: Simplification flags and variables

%e_to_numlog %emode %iargs %piargs algebraic besselexpand beta_args_sum_to_integer beta_expand ctrgsimp distribute_over domain dot0nscsimp dot0simp dot1simp dotassoc dotconstrules dotdistrib dotexptsimp dotident dotscrules evflag gamma_expand gammalim halfangles logarc logconcoeffp logexpand lognegint logsimp negdistrib radexpand radsubstflag ratalgdenom ratdenomdivide ratsimpexpons rootsconmode scalarmatrixp simpproduct simpsum sumexpand sumsplitfact trigexpandplus trigexpandtimes triginverses trigsign

Category: Simplification functions

Package absimp Package ineq atensimp foursimp fullratsimp hypergeometric_simp logarc radcan ratsimp rootscontract scsimp simplify_sum trigexpand trigrat trigreduce trigsimp unknown vectorsimp

Category: Simplification

Introduction to Rules and Patterns lassociative linear multiplicative

Category: Special functions

%f %m %s %w Introduction to Special Functions airy_ai airy_bi airy_dai airy_dbi bessel_i bessel_j bessel_k bessel_simplify bessel_y besselexpand erf erf_generalized erfc erfi expintegral_chi expintegral_ci expintegral_e expintegral_e1 expintegral_e_simplify expintegral_ei expintegral_li expintegral_shi expintegral_si fresnel_c fresnel_s gamma gamma_incomplete gamma_incomplete_generalized gamma_incomplete_lower gamma_incomplete_regularized generalized_lambert_w hankel_1 hankel_2 hypergeometric_simp lambert_w log_gamma nzeta nzetai nzetar parabolic_cylinder_d scaled_bessel_i0 scaled_bessel_i1 struve_h struve_l

Category: Statistical estimation

Introduction to lsquares linear_regression

Category: Statistical functions

Introduction to distrib

Category: Statistical inference

Introduction to stats

Category: Strings

Introduction to String Processing concat sconcat string

Category: Structures

@ defstruct new structures

Category: Sums and products

Introduction to zeilberger arithmetic arithsum bashindices cauchysum genindex gensumnum geometric geosum harmonic lsum niceindices niceindicespref nusum product simplify_sum simpproduct simpsum sum sumcontract sumexpand unsum

Category: Syntax

Comments Identifiers Introduction to Strings Introduction to operators infix matchfix nary nofix postfix prefix

Category: Tensors

Introduction to atensor Introduction to ctensor Introduction to itensor

Category: TeX output

get_tex_environment set_tex_environment get_tex_environment_default set_tex_environment_default tentex tex texput

Category: Time and date functions

absolute_real_time decode_time elapsed_real_time elapsed_run_time encode_time parse_timedate timedate

Category: Translation and compilation

compfile compile compile_file declare_translated define_variable f90 fortindent fortran fortspaces mode_declare mode_identity tr_warnings_get translate translate_file

Category: Translation flags and variables

mode_check_errorp mode_check_warnp mode_checkp packagefile savedef tr_array_as_ref tr_bound_function_applyp tr_file_tty_messagesp tr_float_can_branch_complex tr_function_call_default tr_numer tr_optimize_max_loop tr_semicompile tr_state_vars tr_warn_bad_function_calls tr_warn_fexpr tr_warn_meval tr_warn_mode tr_warn_undeclared tr_warn_undefined_variable transcompile translate_fast_arrays transrun

Category: Trigonometric functions

%iargs %piargs Introduction to Trigonometric acos acot acsc asec asin atan atan2 atrig1 cos cot csc demoivre exponentialize foursimp halfangles ntrig sec sin tan trigexpand trigexpandplus trigexpandtimes triginverses trigrat trigreduce trigsign trigsimp

Category: Vectors

Vectors eigen express nonscalar nonscalarp scalarp

Category: Warning messages

replaced x by y

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# Footnotes

### (1)

Applied Mathematics and Programming Division, K.U. Leuven

### (2)

Applied Mathematics and Programming Division, K.U. Leuven

### (3)

Institut für Mathematik, T.U. Wien

### (4)

National Bureau of Standards, Washington, D.C., U.S.A

### (5)

http://www.netlib.org/slatec

### (6)

http://www.netlib.org/quadpack

### (7)

R. Piessens, E. de Doncker-Kapenga, C.W. Uberhuber, and D.K. Kahaner. QUADPACK: A Subroutine Package for Automatic Integration. Berlin: Springer-Verlag, 1983, ISBN 0387125531.

### (8)

Readers using the `info` reader in `Emacs` will see the actual prompt strings; other readers will see the colorized output

[Top] | [Contents] | [Index] | [ ? ]

# Table of Contents

[Top] | [Contents] | [Index] | [ ? ]

# Short Table of Contents

[Top] | [Contents] | [Index] | [ ? ]

# About This Document

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