{-# LANGUAGE ParallelListComp #-} -- |Stirling's approximation to the gamma function and utility functions for -- selecting coefficients. module Math.Gamma.Stirling ( lnGammaStirling , cs , s , abs_s , terms) where import qualified Data.Vector as V -- |Convergent when Re(z) > 0. The first argument is the c_n series to use -- ('cs' is an ineffecient but generic definition of the full infinite series. -- Some precomputed finite prefix of 'cs' should be fed to this function, the -- length of which will determine the accuracy achieved.) {-# INLINE lnGammaStirling #-} lnGammaStirling :: Floating a => [a] -> a -> a lnGammaStirling ks z = (z - 0.5) * log z - z + 0.5 * log (2*pi) + sum (zipWith (/) ks (risingPowers (z+1))) {-# INLINE risingPowers #-} risingPowers :: Num a => a -> [a] risingPowers = scanl1 (*) . iterate (1+) -- |The c_n series in the convergent version of Stirling's approximation given -- on wikipedia at -- http:\/\/en.wikipedia.org\/wiki\/Stirling%27s_approximation#A_convergent_version_of_Stirling.27s_formula -- as fetched on 11 June 2010. cs :: (Fractional a, Ord a) => [a] cs = map c [1..] c :: (Fractional a, Ord a) => Int -> a c n = 0.5 * recip n' * sum [k' * fromInteger (abs_s n k) / ((k' + 1) * (k' + 2)) | k <- [1..n], let k' = fromIntegral k] where n' = fromIntegral n -- |The (signed) Stirling numbers of the first kind. s :: Int -> Int -> Integer s n k | n < 0 = error "s n k: n < 0" | k < 0 = error "s n k: k < 0" | k > n = error "s n k: k > n" | otherwise = s' n k where table = [V.generate (i+1) $ s' i | i <- [0..]] s' 0 0 = 1 s' _ 0 = 0 s' n' k' | n' == k' = 1 | otherwise = s'' (n'-1) (k'-1) - (toInteger n'-1) * s'' (n'-1) k' where s'' n'' k'' = table !! n'' V.! k'' -- |The (unsigned) Stirling numbers of the first kind. abs_s :: Int -> Int -> Integer abs_s n k | n < 0 = error "abs_s n k: n < 0" | k < 0 = error "abs_s n k: k < 0" | k > n = error "abs_s n k: k > n" | otherwise = abs_s' n k where table = [V.generate (n''+1) $ \k'' -> abs_s' n'' k'' | n'' <- [0..]] abs_s' 0 0 = 1 abs_s' _ 0 = 0 abs_s' n' k' | n' == k' = 1 | otherwise = abs_s'' (n'-1) (k'-1) + (toInteger n'-1) * abs_s'' (n'-1) k' where abs_s'' n'' k'' = table !! n'' V.! k'' -- |Compute the number of terms required to achieve a given precision for a -- given value of z. The mamimum will typically (always?) be around 1, and -- seems to be more or less independent of the precision desired (though not -- of the machine epsilon - essentially, near zero I think this method is -- extremely numerically unstable). terms :: (Num t, Floating a, Ord a) => a -> a -> t terms prec z = stepsNeeded (eps z) (f z) where cs' = cs f x = scanl1 (+) (zipWith (/) cs' (risingPowers (x+1))) -- (eps is 0 at z=0.86639115674955 and z=2.087930091329227) eps x = prec * abs ((x - 0.5) * log x - x + 0.5 * log (2*pi)) stepsNeeded e xs = go 1 xs where go n (x:y:zs) | abs(x-y)<=e = n | otherwise = go (n+1) (y:zs) go n _ = n -- this case should be impossible