Linear Algebra Opcodes
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## Linear Algebra Opcodes

Linear Algebra Opcodes — Scalar, vector, and matrix arithmetic on real and complex values.

## Description

These opcodes implement many linear algebra operations, from scalar, vector, and matrix arithmetic up to and including QR based eigenvalue decompositions. The opcodes are designed for digital signal processing, and of course other mathematical operations, in the Csound orchestra language.

The numerical implementation uses the gmm++ library from home.gna.org/getfem/gmm_intro.

| Warning
For applications with f-sig variables, array arithmetic must be performed only when the f-sig is "current," because f-rate is some fraction of k-rate; currency can be determined with the la_k_current_f opcode. For applications using assignments between real vectors and a-rate variables, array arithmetic must be performed only when the vectors are "current", because the size of the vector may be some integral multiple of ksmps; currency can be determined by means of the la_k_current_vr opcode.

Table 4. Linear Algebra Data Types

Mathematical Type | Code | Corresponding Csound Type or Types
--- | --- | ---
real scalar | r | i-rate or k-rate variable
complex scalar | c | pair of i-rate or k-rate variables, e.g. "kr, ki"
real vector | vr | i-rate variable holding address of array
real vector | a | a-rate variable
real vector | t | function table number
complex vector | vc | i-rate variable holding address of array
complex vector | f | fsig variable
real matrix | mr | i-rate variable holding address of array
complex matrix | mc | i-rate variable holding address of array

All arrays are 0-based; the first index iterates rows to give columns, the second index iterates columns to give elements.

All arrays are general and dense; banded, Hermitian, symmetric and sparse routines are not implemented.

An array can be of type code vr, vc, mr, or mc and is stored in an i-rate object. In orchestra code, an array is passed as a MYFLT i-rate variable that contains the address of the array object, which is actually stored in the allocator opcode instance. Although array variables are i-rate, of course their values and even shapes may change at i-rate or k-rate.

All operands must be pre-allocated; except for the creation opcodes, no opcode ever allocates any arrays. This is true even if the array appears on the left-hand side of an opcode! However, some operations may reshape arrays to hold results.

Arrays are automatically deallocated when their instrument is deallocated.

Not only for more efficient performance, but also to make it easier to remember opcode names, the performance rate, output value types, operation names, and input value types are deterministically encoded into the opcode name:

1. "la" for "linear algebra opcode family".
2. "i" or "k" for performance rate.
3. Type code(s) (see above table) for output value(s), but only if the type is not implicit from the input values.
4. Operation name: common mathematical name (preferred) or abbreviation.
5. Type code(s) for input values, if not implicit.

For additional details, see the gmm++ documentation at http://download.gna.org/getfem/doc/gmmuser.pdf.

## Syntax

#### Array Creation

    ivr                         la_i_vr_create        irows

Create a real vector with irows rows.

    ivc                         la_i_vc_create        irows

Create a complex vector with irows rows.

    imr                         la_i_mr_create        irows, icolumns  [, odiagonal]

Create a real matrix with irows rows and icolumns columns, with an optional value on the diagonal.

    imc                         la_i_mc_create        irows, icolumns  [, odiagonal_r, odiagonal_i]

Create a complex matrix with irows rows and icolumns columns, with an optional complex value on the diagonal.

#### Array Introspection

    irows                       la_i_size_vr          ivr

Return the number of rows in real vector ivr.

    irows                       la_i_size_vc          ivc

Return the number of rows in complex vector ivc.

    irows, icolumns             la_i_size_mr          imr

Return the number of rows and columns in real matrix imr.

    irows, icolumns             la_i_size_mc          imc

Return the number of rows and columns in complex matrix imc.

    kfiscurrent                 la_k_current_f        fsig

Return 1 if fsig is current, that is, if the value of fsig will change on the next kperiod.

    kvriscurrent                la_k_current_vr       ivr

Return 1 if the real vector ivr is current, that is, if Csound's current audio sample frame stands at index 0 of the vector.

                                la_i_print_vr         ivr

Print the value of real vector ivr.

                                la_i_print_vc         ivc

Print the value of complex vector ivc.

                                la_i_print_mr         imr

Print the value of real matrix imr.

                                la_i_print_mc         imc

Print the value of complex matrix imc.

#### Array Assignment and Conversion

    ivr                         la_i_assign_vr        ivr

Assign the value of the real vector on the right-hand side to the real vector on the left-hand side, at i-rate.

    ivr                         la_k_assign_vr        ivr

Assign the value of the real vector on the right-hand side to the real vector on the left-hand side, at k-rate.

    ivc                         la_i_assign_vc        ivc

    ivc                         la_k_assign_vc        ivr

    imr                         la_i_assign_mr        imr

    imr                         la_k_assign_mr        imr

    imc                         la_i_assign_mc        imc

    imc                         la_k_assign_mc        imr

| Warning
Assignments to vectors from tables or fsigs may resize the vectors. Assignments to vectors from a-rate variables, or to a-rate variables from vectors, will be performed incrementally, one chunk of ksmps elements per kperiod. Therefore, array arithmetic on such vectors should only be performed when the vectors are current, as determined by the la_k_currrent_vr opcode.

    ivr                         la_k_assign_a         asig

    ivr                         la_i_assign_t         itablenumber

    ivr                         la_k_assign_t         itablenumber

    ivc                         la_k_assign_f         fsig

    asig                        la_k_a_assign         ivr

    itablenum                   la_i_t_assign         ivr

    itablenum                   la_k_t_assign         ivr

    fsig                        la_k_f_assign         ivc

#### Fill Arrays with Random Elements

    ivr                         la_i_random_vr        [ifill_fraction]

    ivr                         la_k_random_vr        [kfill_fraction]

    ivc                         la_i_random_vc        [ifill_fraction]

    ivc                         la_k_random_vc        [kfill_fraction]

    imr                         la_i_random_mr        [ifill_fraction]

    imr                         la_k_random_mr        [kfill_fraction]

    imc                         la_i_random_mc        [ifill_fraction]

    imc                         la_k_random_mc        [kfill_fraction]

#### Array Element Access

    ivr                         la_i_vr_set           irow, ivalue

    kvr                         la_k_vr_set           krow, kvalue

    ivc                         la_i_vc_set           irow, ivalue_r, ivalue_i

    kvc                         la_k_vc_set           krow, kvalue_r, kvalue_i

    imr                         la_i mr_set           irow, icolumn, ivalue

    kmr                         la_k mr_set           krow, kcolumn, ivalue

    imc                         la_i_mc_set           irow, icolumn, ivalue_r, ivalue_i

    kmc                         la_k_mc_set           krow, kcolumn, kvalue_r, kvalue_i

    ivalue                      la_i_get_vr           ivr, irow

    kvalue                      la_k_get_vr           ivr, krow

    ivalue_r, ivalue_i          la_i_get_vc           ivc, irow

    kvalue_r, kvalue_i          la_k_get_vc           ivc, krow

    ivalue                      la_i_get_mr           imr, irow, icolumn

    kvalue                      la_k_get_mr           imr, krow, kcolumn

    ivalue_r, ivalue_i          la_i_get_mc           imc, irow, icolumn

    kvalue_r, kvalue_i          la_k_get_mc           imc, krow, kcolumn

#### Single Array Operations

    imr                         la_i_transpose_mr     imr

    imr                         la_k_transpose_mr     imr

    imc                         la_i_transpose_mc     imc

    imc                         la_k_transpose_mc     imc

    ivr                         la_i_conjugate_vr     ivr

    ivr                         la_k_conjugate_vr     ivr

    ivc                         la_i_conjugate_vc     ivc

    ivc                         la_k_conjugate_vc     ivc

    imr                         la_i_conjugate_mr     imr

    imr                         la_k_conjugate_mr     imr

    imc                         la_i_conjugate_mc     imc

    imc                         la_k_conjugate_mc     imc

#### Scalar Operations

    ir                          la_i_norm1_vr         ivr

    kr                          la_k_norm1_vr         ivc

    ir                          la_i_norm1_vc         ivc

    kr                          la_k_norm1_vc         ivc

    ir                          la_i_norm1_mr         imr

    kr                          la_k_norm1_mr         imr

    ir                          la_i_norm1_mc         imc

    kr                          la_k_norm1_mc         imc

    ir                          la_i_norm_euclid_vr   ivr

    kr                          la_k_norm_euclid_vr   ivr

    ir                          la_i_norm_euclid_vc   ivc

    kr                          la_k_norm_euclid_vc   ivc

    ir                          la_i_norm_euclid_mr   mvr

    kr                          la_k_norm_euclid_mr   mvr

    ir                          la_i_norm_euclid_mc   mvc

    kr                          la_k_norm_euclid_mc   mvc

    ir                          la_i_distance_vr      ivr

    kr                          la_k_distance_vr      ivr

    ir                          la_i_distance_vc      ivc

    kr                          la_k_distance_vc      ivc

    ir                          la_i_norm_max         imr

    kr                          la_k_norm_max         imc

    ir                          la_i_norm_max         imr

    kr                          la_k_norm_max         imc

    ir                          la_i_norm_inf_vr      ivr

    kr                          la_k_norm_inf_vr      ivr

    ir                          la_i_norm_inf_vc      ivc

    kr                          la_k_norm_inf_vc      ivc

    ir                          la_i_norm_inf_mr      imr

    kr                          la_k_norm_inf_mr      imr

    ir                          la_i_norm_inf_mc      imc

    kr                          la_k_norm_inf_mc      imc

    ir                          la_i_trace_mr         imr

    kr                          la_k_trace_mr         imr

    ir, ii                      la_i_trace_mc         imc

    kr, ki                      la_k_trace_mc         imc

    ir                          la_i_lu_det           imr

    kr                          la_k_lu_det           imr

    ir                          la_i_lu_det           imc

    kr                          la_k_lu_det           imc

#### Elementwise Array-Array Operations

    ivr                         la_i_add_vr           ivr_a, ivr_b

    ivc                         la_k_add_vc           ivc_a, ivc_b

    imr                         la_i_add_mr           imr_a, imr_b

    imc                         la_k_add_mc           imc_a, imc_b

    ivr                         la_i_subtract_vr      ivr_a, ivr_b

    ivc                         la_k_subtract_vc      ivc_a, ivc_b

    imr                         la_i_subtract_mr      imr_a, imr_b

    imc                         la_k_subtract_mc      imc_a, imc_b

    ivr                         la_i_multiply_vr      ivr_a, ivr_b

    ivc                         la_k_multiply_vc      ivc_a, ivc_b

    imr                         la_i_multiply_mr      imr_a, imr_b

    imc                         la_k_multiply_mc      imc_a, imc_b

    ivr                         la_i_divide_vr        ivr_a, ivr_b

    ivc                         la_k_divide_vc        ivc_a, ivc_b

    imr                         la_i_divide_mr        imr_a, imr_b

    imc                         la_k_divide_mc        imc_a, imc_b

#### Inner Products

    ir                          la_i_dot_vr           ivr_a, ivr_b

    kr                          la_k_dot_vr           ivr_a, ivr_b

    ir, ii                      la_i_dot_vc           ivc_a, ivc_b

    kr, ki                      la_k_dot_vc           ivc_a, ivc_b

    imr                         la_i_dot_mr           imr_a, imr_b

    imr                         la_k_dot_mr           imr_a, imr_b

    imc                         la_i_dot_mc           imc_a, imc_b

    imc                         la_k_dot_mc           imc_a, imc_b

    ivr                         la_i_dot_mr_vr        imr_a, ivr_b

    ivr                         la_k_dot_mr_vr        imr_a, ivr_b

    ivc                         la_i_dot_mc_vc        imc_a, ivc_b

    ivc                         la_k_dot_mc_vc        imc_a, ivc_b

#### Matrix Inversion

    imr, icondition             la_i_invert_mr        imr

    imr, kcondition             la_k_invert_mr        imr

    imc, icondition             la_i_invert_mc        imc

    imc, kcondition             la_k_invert_mc        imc

#### Matrix Decompositions and Solvers

    ivr                         la_i_upper_solve_mr   imr [, j_1_diagonal]

    ivr                         la_k_upper_solve_mr   imr [, j_1_diagonal]

    ivc                         la_i_upper_solve_mc   imc [, j_1_diagonal]

    ivc                         la_k_upper_solve_mc   imc [, j_1_diagonal]

    ivr                         la_i_lower_solve_mr   imr [, j_1_diagonal]

    ivr                         la_k_lower_solve_mr   imr [, j_1_diagonal]

    ivc                         la_i_lower_solve_mc   imc [, j_1_diagonal]

    ivc                         la_k_lower_solve_mc   imc [, j_1_diagonal]

    imr, ivr_pivot, isize       la_i_lu_factor_mr     imr

    imr, ivr_pivot, ksize       la_k_lu_factor_mr     imr

    imc, ivr_pivot, isize       la_i_lu_factor_mc     imc

    imc, ivr_pivot, ksize       la_k_lu_factor_mc     imc

    ivr_x                       la_i_lu_solve_mr      imr, ivr_b

    ivr_x                       la_k_lu_solve_mr      imr, ivr_b

    ivc_x                       la_i_lu_solve_mc      imc, ivc_b

    ivc_x                       la_k_lu_solve_mc      imc, ivc_b

    imr_q, imr_r                la_i_qr_factor_mr     imr

    imr_q, imr_r                la_k_qr_factor_mr     imr

    imc_q, imc_r                la_i_qr_factor_mc     imc

    imc_q, imc_r                la_k_qr_factor_mc     imc

    ivr_eig_vals                la_i_qr_eigen_mr      imr, i_tolerance

    ivr_eig_vals                la_k_qr_eigen_mr      imr, k_tolerance

    ivr_eig_vals                la_i_qr_eigen_mc      imc, i_tolerance

    ivr_eig_vals                la_k_qr_eigen_mc      imc, k_tolerance

| Warning
Matrix must be Hermitian in order to compute eigenvectors.

    ivr_eig_vals, imr_eig_vecs  la_i_qr_sym_eigen_mr  imr, i_tolerance

    ivr_eig_vals, imr_eig_vecs  la_k_qr_sym_eigen_mr  imr, k_tolerance

    ivc_eig_vals, imc_eig_vecs  la_i_qr_sym_eigen_mc  imc, i_tolerance

    ivc_eig_vals, imc_eig_vecs  la_k_qr_sym_eigen_mc  imc, k_tolerance

## Credits

Michael Gogins

New in Csound version 5.09

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