# Gamma and Beta function

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gammaln :: Double -> Double | gammaln :: Double -> Double | ||

gammaln xx = let tmp' = (xx+5.5) - (xx+0.5)*log(xx+5.5) | gammaln xx = let tmp' = (xx+5.5) - (xx+0.5)*log(xx+5.5) | ||

− | ser' = | + | ser' = ser + sum $ zipWith (/) cof [xx+1..] |

− | in -tmp' + log(2.5066282746310005 * ser' / xx) | + | in -tmp' + log(2.5066282746310005 * ser' / xx) |

</haskell> | </haskell> | ||

− | the beta function relates to the gamma function by <math> | + | the beta function relates to the gamma function by <math>\Beta(z,w)= \frac{\Gamma(z)\cdot\Gamma(w)}{\Gamma(z+w)}</math>, so we can compute the Beta function using gammaln like this: |

<haskell> | <haskell> | ||

− | beta z w = exp | + | beta z w = exp (gammaln z + gammaln w - gammaln (z+w)) |

</haskell> | </haskell> | ||

[[Category:Code]] | [[Category:Code]] | ||

[[Category:Mathematics]] | [[Category:Mathematics]] |

## Latest revision as of 10:18, 13 December 2009

The Gamma and Beta function as described in 'Numerical Recipes in C++', the approximation is taken from [Lanczos, C. 1964 SIAM Journal on Numerical Analysis, ser. B, vol. 1, pp. 86-96]

cof :: [Double] cof = [76.18009172947146,-86.50532032941677,24.01409824083091,-1.231739572450155,0.001208650973866179,-0.000005395239384953] ser :: Double ser = 1.000000000190015 gammaln :: Double -> Double gammaln xx = let tmp' = (xx+5.5) - (xx+0.5)*log(xx+5.5) ser' = ser + sum $ zipWith (/) cof [xx+1..] in -tmp' + log(2.5066282746310005 * ser' / xx)

the beta function relates to the gamma function by , so we can compute the Beta function using gammaln like this:

beta z w = exp (gammaln z + gammaln w - gammaln (z+w))