math-functions-0.2.0.1: Special functions and Chebyshev polynomials

Copyright (c) 2009, 2011 Bryan O'Sullivan BSD3 bos@serpentine.com experimental portable None Haskell2010

Numeric.SpecFunctions.Extra

Description

Less common mathematical functions.

Synopsis

# Documentation

Arguments

 :: Double x -> Double np -> Double

Evaluate the deviance term x log(x/np) + np - x.

Calculate binomial coefficient using exact formula

Quickly compute the natural logarithm of n choose k, with no checking.

Less numerically stable:

exp \$ lg (n+1) - lg (k+1) - lg (n-k+1)
where lg = logGamma . fromIntegral

Compute the logarithm of the gamma function Γ(x). Uses Algorithm AS 245 by Macleod.

Gives an accuracy of 10-12 significant decimal digits, except for small regions around x = 1 and x = 2, where the function goes to zero. For greater accuracy, use logGammaL.

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

Compute the log gamma correction factor for Stirling approximation for x ≥ 10. This correction factor is suitable for an alternate (but less numerically accurate) definition of logGamma:

$\log\Gamma(x) = \frac{1}{2}\log(2\pi) + (x-\frac{1}{2})\log x - x + \operatorname{logGammaCorrection}(x)$