math-functions-0.2.1.0: Special functions and Chebyshev polynomials

Copyright | (c) 2009, 2011, 2012 Bryan O'Sullivan
License | BSD3
Maintainer | bos@serpentine.com
Stability | experimental
Portability | portable
Safe Haskell | None
Language | Haskell2010

Numeric.SpecFunctions

Contents

- Error function
- Gamma function
- Beta function
- Sinc
- Logarithm
- Exponent
- Factorial
- Combinatorics
- References

Description

Special functions and factorials.

Synopsis

- erf :: Double -> Double
- erfc :: Double -> Double
- invErf :: Double -> Double
- invErfc :: Double -> Double
- logGamma :: Double -> Double
- logGammaL :: Double -> Double
- incompleteGamma :: Double -> Double -> Double
- invIncompleteGamma :: Double -> Double -> Double
- digamma :: Double -> Double
- logBeta :: Double -> Double -> Double
- incompleteBeta :: Double -> Double -> Double -> Double
- incompleteBeta_ :: Double -> Double -> Double -> Double -> Double
- invIncompleteBeta :: Double -> Double -> Double -> Double
- sinc :: Double -> Double
- log1p :: Floating a => a -> a
- log1pmx :: Double -> Double
- log2 :: Int -> Int
- expm1 :: Floating a => a -> a
- factorial :: Int -> Double
- logFactorial :: Integral a => a -> Double
- stirlingError :: Double -> Double
- choose :: Int -> Int -> Double
- logChoose :: Int -> Int -> Double

# Error function

erf :: Double -> Double #

Error function.

\[ \operatorname{erf}(x) = \frac{2}{\sqrt{\pi}} \int_{0}^{x} \exp(-t^2) dt \]

Function limits are:

[ begin{aligned} &operatorname{erf}(-infty) &=& -1 -- &operatorname{erf}(0) &=& phantom{-},0 -- &operatorname{erf}(+infty) &=& phantom{-},1 -- end{aligned}

erfc :: Double -> Double #

Complementary error function.

\[ \operatorname{erfc}(x) = 1 - \operatorname{erf}(x) \]

Function limits are:

[ begin{aligned} &operatorname{erf}(-infty) &=&, 2 -- &operatorname{erf}(0) &=&, 1 -- &operatorname{erf}(+infty) &=&, 0 -- end{aligned}

invErf #

Arguments

:: Double | p ∈ [-1,1]
-> Double |

Inverse of `erf`.

invErfc #

Arguments

:: Double | p ∈ [0,2]
-> Double |

Inverse of `erfc`.

# Gamma function

logGamma :: Double -> Double #

Compute the logarithm of the gamma function, Γ(x).

\[ \Gamma(x) = \int_0^{\infty}t^{x-1}e^{-t}\,dt = (x - 1)! \]

This implementation uses Lanczos approximation. It gives 14 or more significant decimal digits, except around x = 1 and x = 2, where the function goes to zero.

Returns ∞ if the input is outside of the range (0 < x ≤ 1e305).

logGammaL :: Double -> Double #

Synonym for `logGamma`. Retained for compatibility

incompleteGamma #

Arguments

:: Double | z ∈ (0,∞)
-> Double | x ∈ (0,∞)
-> Double |

Compute the normalized lower incomplete gamma function γ(z,x). Normalization means that γ(z,∞)=1

\[ \gamma(z,x) = \frac{1}{\Gamma(z)}\int_0^{x}t^{z-1}e^{-t}\,dt \]

Uses Algorithm AS 239 by Shea.

invIncompleteGamma #

Arguments

:: Double | z ∈ (0,∞)
-> Double | p ∈ [0,1]
-> Double |

Inverse incomplete gamma function. It's approximately inverse of `incompleteGamma` for the same z. So following equality approximately holds:

    invIncompleteGamma z . incompleteGamma z ≈ id

digamma :: Double -> Double #

Compute ψ(x), the first logarithmic derivative of the gamma function.

\[ \psi(x) = \frac{d}{dx} \ln \left(\Gamma(x)\right) = \frac{\Gamma'(x)}{\Gamma(x)} \]

Uses Algorithm AS 103 by Bernardo, based on Minka's C implementation.

# Beta function

logBeta #

Arguments

:: Double | a > 0
-> Double | b > 0
-> Double |

Compute the natural logarithm of the beta function.

\[ B(a,b) = \int_0^1 t^{a-1}(1-t)^{1-b}\,dt = \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)} \]

incompleteBeta #

Arguments

:: Double | a > 0
-> Double | b > 0
-> Double | x, must lie in [0,1] range
-> Double |

Regularized incomplete beta function.

\[ I(x;a,b) = \frac{1}{B(a,b)} \int_0^x t^{a-1}(1-t)^{1-b}\,dt \]

Uses algorithm AS63 by Majumder and Bhattachrjee and quadrature approximation for large p and q.

incompleteBeta_ #

Arguments

:: Double | logarithm of beta function for given p and q
-> Double | a > 0
-> Double | b > 0
-> Double | x, must lie in [0,1] range
-> Double |

Regularized incomplete beta function. Same as `incompleteBeta` but also takes logarithm of beta function as parameter.

invIncompleteBeta #

Arguments

:: Double | a > 0
-> Double | b > 0
-> Double | x ∈ [0,1]
-> Double |

Compute inverse of regularized incomplete beta function. Uses initial approximation from AS109, AS64 and Halley method to solve equation.

# Sinc

sinc :: Double -> Double #

Compute sinc function `sin(x)/x`

# Logarithm

Base starting from `4.9.0` (GHC 8.0) provides `log1p` and `expm1` as method of class `Floating`. In this case we simply reexport these function. Otherwise we provide our own with more restrictive signature `Double → Double`.

log1p :: Floating a => a -> a #

``log1p` x` computes ``log` (1 + x)`, but provides more precise results for small (absolute) values of `x` if possible.

Since: 4.9.0.0

log1pmx :: Double -> Double #

Compute log(1+x)-x:

log2 :: Int -> Int #

O(log n) Compute the logarithm in base 2 of the given value.

# Exponent

expm1 :: Floating a => a -> a #

``expm1` x` computes ``exp` x - 1`, but provides more precise results for small (absolute) values of `x` if possible.

Since: 4.9.0.0

# Factorial

factorial :: Int -> Double #

Compute the factorial function n!. Returns +∞ if the input is above 170 (above which the result cannot be represented by a 64-bit `Double`).

logFactorial :: Integral a => a -> Double #

Compute the natural logarithm of the factorial function. Gives 16 decimal digits of precision.

stirlingError :: Double -> Double #

Calculate the error term of the Stirling approximation. This is only defined for non-negative values.

\[ \operatorname{stirlingError}(n) = \log(n!) - \log(\sqrt{2\pi n}\frac{n}{e}^n) \]

# Combinatorics

choose :: Int -> Int -> Double #

Compute the binomial coefficient n ```choose``` k. For values of k > 50, this uses an approximation for performance reasons. The approximation is accurate to 12 decimal places in the worst case

Example:

    7 `choose` 3 == 35

logChoose :: Int -> Int -> Double #

Compute logarithm of the binomial coefficient.

# References

- Bernardo, J. (1976) Algorithm AS 103: Psi (digamma) function. /Journal of the Royal Statistical Society. Series C (Applied Statistics)/ 25(3):315-317. http://www.jstor.org/stable/2347257
- Cran, G.W., Martin, K.J., Thomas, G.E. (1977) Remark AS R19 and Algorithm AS 109: A Remark on Algorithms: AS 63: The Incomplete Beta Integral AS 64: Inverse of the Incomplete Beta Function Ratio. /Journal of the Royal Statistical Society. Series C (Applied Statistics)/ Vol. 26, No. 1 (1977), pp. 111-114 http://www.jstor.org/pss/2346887
- Lanczos, C. (1964) A precision approximation of the gamma function. SIAM Journal on Numerical Analysis B 1:86–96. http://www.jstor.org/stable/2949767
- Loader, C. (2000) Fast and Accurate Computation of Binomial Probabilities. http://projects.scipy.org/scipy/raw-attachment/ticket/620/loader2000Fast.pdf
- Macleod, A.J. (1989) Algorithm AS 245: A robust and reliable algorithm for the logarithm of the gamma function. Journal of the Royal Statistical Society, Series C (Applied Statistics) 38(2):397–402. http://www.jstor.org/stable/2348078
- Majumder, K.L., Bhattacharjee, G.P. (1973) Algorithm AS 63: The Incomplete Beta Integral. /Journal of the Royal Statistical Society. Series C (Applied Statistics)/ Vol. 22, No. 3 (1973), pp. 409-411. http://www.jstor.org/pss/2346797
- Majumder, K.L., Bhattacharjee, G.P. (1973) Algorithm AS 64: Inverse of the Incomplete Beta Function Ratio. /Journal of the Royal Statistical Society. Series C (Applied Statistics)/ Vol. 22, No. 3 (1973), pp. 411-414 http://www.jstor.org/pss/2346798
- Temme, N.M. (1992) Asymptotic inversion of the incomplete beta function. /Journal of Computational and Applied Mathematics 41(1992) 145-157.
- Temme, N.M. (1994) A set of algorithms for the incomplete gamma functions. /Probability in the Engineering and Informational Sciences/, 8, 1994, 291-307. Printed in the U.S.A.