{-# LANGUAGE BangPatterns, ScopedTypeVariables, ForeignFunctionInterface #-}
-- |
-- Module    : Numeric.SpecFunctions.Internal
-- Copyright : (c) 2009, 2011, 2012 Bryan O'Sullivan
--
-- Maintainer  : bos@serpentine.com
-- Stability   : experimental
-- Portability : portable
--
-- Internal module with implementation of special functions.
module Numeric.SpecFunctions.Internal where

import Control.Applicative
import Data.Bits       ((.&.), (.|.), shiftR)
import Data.Int        (Int64)
import Data.Word       (Word)
import qualified Data.Vector.Unboxed as U
import           Data.Vector.Unboxed   ((!))

import Numeric.Polynomial.Chebyshev    (chebyshevBroucke)
import Numeric.Polynomial              (evaluatePolynomialL,evaluateEvenPolynomialL,evaluateOddPolynomialL)
import Numeric.RootFinding             (Root(..), newtonRaphson)
import Numeric.Series                  (sumPowerSeries,enumSequenceFrom,scanSequence,evalContFractionB)
import Numeric.MathFunctions.Constants ( m_epsilon, m_NaN, m_neg_inf, m_pos_inf
, m_sqrt_2_pi, m_ln_sqrt_2_pi, m_eulerMascheroni
, m_min_log, m_tiny
)
import Text.Printf

----------------------------------------------------------------
-- Error function
----------------------------------------------------------------

-- | 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} --
erf :: Double -> Double
{-# INLINE erf #-}
erf = c_erf

-- | 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} --
erfc :: Double -> Double
{-# INLINE erfc #-}
erfc = c_erfc

foreign import ccall "erf"  c_erf  :: Double -> Double
foreign import ccall "erfc" c_erfc :: Double -> Double

-- | Inverse of 'erf'.
invErf :: Double -- ^ /p/ ∈ [-1,1]
-> Double
invErf p = invErfc (1 - p)

-- | Inverse of 'erfc'.
invErfc :: Double -- ^ /p/ ∈ [0,2]
-> Double
invErfc p
| p == 2        = m_neg_inf
| p == 0        = m_pos_inf
| p >0 && p < 2 = if p <= 1 then r else -r
| otherwise     = modErr $"invErfc: p must be in [0,2] got " ++ show p where pp = if p <= 1 then p else 2 - p t = sqrt$ -2 * log( 0.5 * pp)
-- Initial guess
x0 = -0.70711 * ((2.30753 + t * 0.27061) / (1 + t * (0.99229 + t * 0.04481)) - t)
r  = loop 0 x0
--
loop :: Int -> Double -> Double
loop !j !x
| j >= 2    = x
| otherwise = let err = erfc x - pp
x'  = x + err / (1.12837916709551257 * exp(-x * x) - x * err) -- // Halley
in loop (j+1) x'

----------------------------------------------------------------
-- Gamma function
----------------------------------------------------------------

data L = L {-# UNPACK #-} !Double {-# UNPACK #-} !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 &#8734; if the input is outside of the range (0 < /x/
-- &#8804; 1e305).
logGamma :: Double -> Double
logGamma x
| x <= 0    = m_pos_inf
| x <  1    = lanczos (1+x) - log x
| otherwise = lanczos x
where
-- Evaluate Lanczos approximation for γ=6
lanczos z = fini
$U.foldl' go (L 0 (z+7)) a where fini (L l _) = log (l+a0) + log m_sqrt_2_pi - z65 + (z-0.5) * log z65 go (L l t) k = L (l + k / t) (t-1) z65 = z + 6.5 -- Coefficients for Lanczos approximation a0 = 0.9999999999995183 a = U.fromList [ 0.1659470187408462e-06 , 0.9934937113930748e-05 , -0.1385710331296526 , 12.50734324009056 , -176.6150291498386 , 771.3234287757674 , -1259.139216722289 , 676.5203681218835 ] -- | Synonym for 'logGamma'. Retained for compatibility logGammaL :: Double -> Double logGammaL = logGamma -- | -- Compute the log gamma correction factor for Stirling -- approximation for @x@ &#8805; 10. This correction factor is -- suitable for an alternate (but less numerically accurate) -- definition of 'logGamma': -- -- $-- \log\Gamma(x) = \frac{1}{2}\log(2\pi) + (x-\frac{1}{2})\log x - x + \operatorname{logGammaCorrection}(x) --$ logGammaCorrection :: Double -> Double logGammaCorrection x | x < 10 = m_NaN | x < big = chebyshevBroucke (t * t * 2 - 1) coeffs / x | otherwise = 1 / (x * 12) where big = 94906265.62425156 t = 10 / x coeffs = U.fromList [ 0.1666389480451863247205729650822e+0, -0.1384948176067563840732986059135e-4, 0.9810825646924729426157171547487e-8, -0.1809129475572494194263306266719e-10, 0.6221098041892605227126015543416e-13, -0.3399615005417721944303330599666e-15, 0.2683181998482698748957538846666e-17 ] -- | 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. incompleteGamma :: Double -- ^ /z/ ∈ (0,∞) -> Double -- ^ /x/ ∈ (0,∞) -> Double -- Notation used: -- + P(a,x) - regularized lower incomplete gamma -- + Q(a,x) - regularized upper incomplete gamma incompleteGamma a x | a <= 0 || x < 0 = error$ "incompleteGamma: Domain error z=" ++ show a ++ " x=" ++ show x
| x == 0          = 0
| x == m_pos_inf  = 1
-- For very small x we use following expansion for P:
--
-- See http://functions.wolfram.com/GammaBetaErf/GammaRegularized/06/01/05/01/01/
| x < sqrt m_epsilon && a > 1
= x**a / a / exp (logGammaL a) * (1 - a*x / (a + 1))
| x < 0.5 = case () of
_| (-0.4)/log x < a  -> taylorSeriesP
| otherwise         -> taylorSeriesComplQ
| x < 1.1 = case () of
_| 0.75*x < a        -> taylorSeriesP
| otherwise         -> taylorSeriesComplQ
| a > 20 && useTemme    = uniformExpansion
| x - (1 / (3 * x)) < a = taylorSeriesP
| otherwise             = contFraction
where
mu = (x - a) / a
useTemme = (a > 200 && 20/a > mu*mu)
|| (abs mu < 0.4)
-- Gautschi's algorithm.
--
-- Evaluate series for P(a,x). See [Temme1994] Eq. 5.5
--
-- FIXME: Term exp (log x * z - x - logGamma (z+1)) doesn't give full precision
taylorSeriesP
= sumPowerSeries x (scanSequence (/) 1 $enumSequenceFrom (a+1)) * exp (log x * a - x - logGamma (a+1)) -- Series for 1-Q(a,x). See [Temme1994] Eq. 5.5 taylorSeriesComplQ = sumPowerSeries (-x) (scanSequence (/) 1 (enumSequenceFrom 1) / enumSequenceFrom a) * x**a / exp(logGammaL a) -- Legendre continued fractions contFraction = 1 - ( exp ( log x * a - x - logGamma a ) / evalContFractionB frac ) where frac = (\k -> (k*(a-k), x - a + 2*k + 1)) <$> enumSequenceFrom 0
-- Evaluation based on uniform expansions. See [Temme1994] 5.2
uniformExpansion =
let -- Coefficients f_m in paper
fm :: U.Vector Double
fm = U.fromList [ 1.00000000000000000000e+00
,-3.33333333333333370341e-01
, 8.33333333333333287074e-02
,-1.48148148148148153802e-02
, 1.15740740740740734316e-03
, 3.52733686067019369930e-04
,-1.78755144032921825352e-04
, 3.91926317852243766954e-05
,-2.18544851067999240532e-06
,-1.85406221071515996597e-06
, 8.29671134095308545622e-07
,-1.76659527368260808474e-07
, 6.70785354340149841119e-09
, 1.02618097842403069078e-08
,-4.38203601845335376897e-09
, 9.14769958223679020897e-10
,-2.55141939949462514346e-11
,-5.83077213255042560744e-11
, 2.43619480206674150369e-11
,-5.02766928011417632057e-12
, 1.10043920319561347525e-13
, 3.37176326240098513631e-13
]
y   = - log1pmx mu
eta = sqrt (2 * y) * signum mu
-- Evaluate S_α (Eq. 5.9)
loop !_  !_  u 0 = u
loop bm1 bm0 u i = let t  = (fm ! i) + (fromIntegral i + 1)*bm1 / a
u' = eta * u + t
in  loop bm0 t u' (i-1)
s_a = let n = U.length fm
in  loop (fm ! (n-1)) (fm ! (n-2)) 0 (n-3)
/ exp (logGammaCorrection a)
in 1/2 * erfc(-eta*sqrt(a/2)) - exp(-(a*y)) / sqrt (2*pi*a) * s_a

-- Adapted from Numerical Recipes §6.2.1

-- | Inverse incomplete gamma function. It's approximately inverse of
--   'incompleteGamma' for the same /z/. So following equality
--   approximately holds:
--
-- > invIncompleteGamma z . incompleteGamma z ≈ id
invIncompleteGamma :: Double    -- ^ /z/ ∈ (0,∞)
-> Double    -- ^ /p/ ∈ [0,1]
-> Double
invIncompleteGamma a p
| a <= 0         =
modErr $printf "invIncompleteGamma: a must be positive. a=%g p=%g" a p | p < 0 || p > 1 = modErr$ printf "invIncompleteGamma: p must be in [0,1] range. a=%g p=%g" a p
| p == 0         = 0
| p == 1         = 1 / 0
| otherwise      = loop 0 guess
where
-- Solve equation γ(a,x) = p using Halley method
loop :: Int -> Double -> Double
loop i x
| i >= 12           = x'
-- For small s derivative becomes approximately 1/x*exp(-x) and
-- skyrockets for small x. If it happens correct answer is 0.
| isInfinite f'     = 0
| abs dx < eps * x' = x'
| otherwise         = loop (i + 1) x'
where
-- Value of γ(a,x) - p
f    = incompleteGamma a x - p
-- dγ(a,x)/dx
f'   | a > 1     = afac * exp( -(x - a1) + a1 * (log x - lna1))
| otherwise = exp( -x + a1 * log x - gln)
u    = f / f'
-- Halley correction to Newton-Rapson step
corr = u * (a1 / x - 1)
dx   = u / (1 - 0.5 * min 1.0 corr)
-- New approximation to x
x'   | x < dx    = 0.5 * x -- Do not go below 0
| otherwise = x - dx
-- Calculate inital guess for root
guess
--
| a > 1   =
let t  = sqrt $-2 * log(if p < 0.5 then p else 1 - p) x1 = (2.30753 + t * 0.27061) / (1 + t * (0.99229 + t * 0.04481)) - t x2 = if p < 0.5 then -x1 else x1 in max 1e-3 (a * (1 - 1/(9*a) - x2 / (3 * sqrt a)) ** 3) -- For a <= 1 use following approximations: -- γ(a,1) ≈ 0.253a + 0.12a² -- -- γ(a,x) ≈ γ(a,1)·x^a x < 1 -- γ(a,x) ≈ γ(a,1) + (1 - γ(a,1))(1 - exp(1 - x)) x >= 1 | otherwise = let t = 1 - a * (0.253 + a*0.12) in if p < t then (p / t) ** (1 / a) else 1 - log( 1 - (p-t) / (1-t)) -- Constants a1 = a - 1 lna1 = log a1 afac = exp( a1 * (lna1 - 1) - gln ) gln = logGamma a eps = 1e-8 ---------------------------------------------------------------- -- Beta function ---------------------------------------------------------------- -- | Compute the natural logarithm of the beta function. -- -- $-- B(a,b) = \int_0^1 t^{a-1}(1-t)^{1-b}\,dt = \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)} --$ logBeta :: Double -- ^ /a/ > 0 -> Double -- ^ /b/ > 0 -> Double logBeta a b | p < 0 = m_NaN | p == 0 = m_pos_inf | p >= 10 = log q * (-0.5) + m_ln_sqrt_2_pi + logGammaCorrection p + c + (p - 0.5) * log ppq + q * log1p(-ppq) | q >= 10 = logGamma p + c + p - p * log pq + (q - 0.5) * log1p(-ppq) | otherwise = logGamma p + logGamma q - logGamma pq where p = min a b q = max a b ppq = p / pq pq = p + q c = logGammaCorrection q - logGammaCorrection pq -- | 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 :: Double -- ^ /a/ > 0 -> Double -- ^ /b/ > 0 -> Double -- ^ /x/, must lie in [0,1] range -> Double incompleteBeta p q = incompleteBeta_ (logBeta p q) p q -- | Regularized incomplete beta function. Same as 'incompleteBeta' -- but also takes logarithm of beta function as parameter. incompleteBeta_ :: 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 incompleteBeta_ beta p q x | p <= 0 || q <= 0 = modErr$ printf "incompleteBeta_: p <= 0 || q <= 0. p=%g q=%g x=%g" p q x
| x <  0 || x >  1 || isNaN x =
modErr $printf "incompleteBeta_: x out of [0,1] range. p=%g q=%g x=%g" p q x | x == 0 || x == 1 = x | p >= (p+q) * x = incompleteBetaWorker beta p q x | otherwise = 1 - incompleteBetaWorker beta q p (1 - x) -- Approximation of incomplete beta by quandrature. -- -- Note that x =< p/(p+q) incompleteBetaApprox :: Double -> Double -> Double -> Double -> Double incompleteBetaApprox beta p q x | ans > 0 = 1 - ans | otherwise = -ans where -- Constants p1 = p - 1 q1 = q - 1 mu = p / (p + q) lnmu = log mu lnmuc = log1p (-mu) -- Upper limit for integration xu = max 0$ min (mu - 10*t) (x - 5*t)
where
t = sqrt $p*q / ( (p+q) * (p+q) * (p + q + 1) ) -- Calculate incomplete beta by quadrature go y w = let t = x + (xu - x) * y in w * exp( p1 * (log t - lnmu) + q1 * (log(1-t) - lnmuc) ) s = U.sum$ U.zipWith go coefY coefW
ans = s * (xu - x) * exp( p1 * lnmu + q1 * lnmuc - beta )

-- Worker for incomplete beta function. It is separate function to
-- avoid confusion with parameter during parameter swapping
incompleteBetaWorker :: Double -> Double -> Double -> Double -> Double
incompleteBetaWorker beta p q x
-- For very large p and q this method becomes very slow so another
-- method is used.
| p > 3000 && q > 3000 = incompleteBetaApprox beta p q x
| otherwise            = loop (p+q) (truncate $q + cx * (p+q)) 1 1 1 where -- Constants eps = 1e-15 cx = 1 - x -- Common multiplies for expansion. Accurate calculation is a bit -- tricky. Performing calculation in log-domain leads to slight -- loss of precision for small x, while using ** prone to -- underflows. -- -- If either beta function of x**p·(1-x)**(q-1) underflows we -- switch to log domain. It could waste work but there's no easy -- switch criterion. factor | beta < m_min_log || prod < m_tiny = exp( p * log x + (q - 1) * log cx - beta) | otherwise = prod / exp beta where prod = x**p * cx**(q - 1) -- Soper's expansion of incomplete beta function loop !psq (ns :: Int) ai term betain | done = betain' * factor / p | otherwise = loop psq' (ns - 1) (ai + 1) term' betain' where -- New values term' = term * fact / (p + ai) betain' = betain + term' fact | ns > 0 = (q - ai) * x/cx | ns == 0 = (q - ai) * x | otherwise = psq * x -- Iterations are complete done = db <= eps && db <= eps*betain' where db = abs term' psq' = if ns < 0 then psq + 1 else psq -- | Compute inverse of regularized incomplete beta function. Uses -- initial approximation from AS109, AS64 and Halley method to solve -- equation. invIncompleteBeta :: Double -- ^ /a/ > 0 -> Double -- ^ /b/ > 0 -> Double -- ^ /x/ ∈ [0,1] -> Double invIncompleteBeta p q a | p <= 0 || q <= 0 = modErr$ printf "invIncompleteBeta p <= 0 || q <= 0.  p=%g q=%g a=%g" p q a
| a <  0 || a >  1 =
modErr $printf "invIncompleteBeta x must be in [0,1]. p=%g q=%g a=%g" p q a | a == 0 || a == 1 = a | otherwise = invIncompleteBetaWorker (logBeta p q) p q a invIncompleteBetaWorker :: Double -> Double -> Double -> Double -> Double invIncompleteBetaWorker beta a b p = loop (0::Int) (invIncBetaGuess beta a b p) where a1 = a - 1 b1 = b - 1 -- Solve equation using Halley method loop !i !x -- We cannot continue at this point so we simply return x' | x == 0 || x == 1 = x -- When derivative becomes infinite we cannot continue -- iterations. It can only happen in vicinity of 0 or 1. It's -- hardly possible to get good answer in such circumstances but -- x' is already reasonable. | isInfinite f' = x -- Iterations limit reached. Most of the time solution will -- converge to answer because of discreteness of Double. But -- solution have good precision already. | i >= 10 = x -- Solution converges | abs dx <= 16 * m_epsilon * x = x' | otherwise = loop (i+1) x' where -- Calculate Halley step. f = incompleteBeta_ beta a b x - p f' = exp$ a1 * log x + b1 * log1p (-x) - beta
u   = f / f'
-- We bound Halley correction to Newton-Raphson to (-1,1) range
corr | d > 1     = 1
| d < -1    = -1
| isNaN d   = 0
| otherwise = d
where
d = u * (a1 / x - b1 / (1 - x))
dx  = u / (1 - 0.5 * corr)
-- Next approximation. If Halley step leads us out of [0,1]
-- range we revert to bisection.
x'  | z < 0     = x / 2
| z > 1     = (x + 1) / 2
| otherwise = z
where z = x - dx

-- Calculate initial guess for inverse incomplete beta function.
invIncBetaGuess :: Double -> Double -> Double -> Double -> Double
-- Calculate initial guess. for solving equation for inverse incomplete beta.
-- It's really hodgepodge of different approximations accumulated over years.
--
-- Equations are referred to by name of paper and number e.g. [AS64 2]
-- In AS64 papers equations are not numbered so they are refered to by
-- number of appearance starting from definition of incomplete beta.
invIncBetaGuess beta a b p
-- If both a and b are less than 1 incomplete beta have inflection
-- point.
--
-- > x = (1 - a) / (2 - a - b)
--
-- We approximate incomplete beta by neglecting one of factors under
-- integral and then rescaling result of integration into [0,1]
-- range.
| a < 1 && b < 1 =
let x_infl = (1 - a) / (2 - a - b)
p_infl = incompleteBeta a b x_infl
x | p < p_infl = let xg = (a * p     * exp beta) ** (1/a) in xg / (1+xg)
| otherwise  = let xg = (b * (1-p) * exp beta) ** (1/b) in 1 - xg/(1+xg)
in x
-- If both a and b larger or equal that 1 but not too big we use
-- same approximation as above but calculate it a bit differently
| a+b <= 6 && a>1 && b>1 =
let x_infl = (a - 1) / (a + b - 2)
p_infl = incompleteBeta a b x_infl
x | p < p_infl = exp ((log(p * a) + beta) / a)
| otherwise  = 1 - exp((log((1-p) * b) + beta) / b)
in x
-- For small a and not too big b we use approximation from boost.
| b < 5 && a <= 1 =
let x | p**(1/a) < 0.5 = (p * a * exp beta) ** (1/a)
| otherwise      = 1 - (1 - p ** (b * exp beta))**(1/b)
in x
-- When a>>b and both are large approximation from [Temme1992],
-- section 4 "the incomplete gamma function case" used. In this
-- region it greatly improves over other approximation (AS109, AS64,
-- "Numerical Recipes")
--
-- FIXME: It could be used when b>>a too but it require inverse of
--        upper incomplete gamma to be precise enough. In current
--        implementation it loses precision in horrible way (40
--        order of magnitude off for sufficiently small p)
| a+b > 5 &&  a/b > 4 =
let -- Calculate initial approximation to eta using eq 4.1
eta0 = invIncompleteGamma b (1-p) / a
mu   = b / a            -- Eq. 4.3
-- A lot of helpers for calculation of
w    = sqrt(1 + mu)     -- Eq. 4.9
w_2  = w * w
w_3  = w_2 * w
w_4  = w_2 * w_2
w_5  = w_3 * w_2
w_6  = w_3 * w_3
w_7  = w_4 * w_3
w_8  = w_4 * w_4
w_9  = w_5 * w_4
w_10 = w_5 * w_5
d    = eta0 - mu
d_2  = d * d
d_3  = d_2 * d
d_4  = d_2 * d_2
w1   = w + 1
w1_2 = w1 * w1
w1_3 = w1 * w1_2
w1_4 = w1_2 * w1_2
-- Evaluation of eq 4.10
e1 = (w + 2) * (w - 1) / (3 * w)
+ (w_3 + 9 * w_2 + 21 * w + 5) * d
/ (36 * w_2 * w1)
- (w_4 - 13 * w_3 + 69 * w_2 + 167 * w + 46) * d_2
/ (1620 * w1_2 * w_3)
- (7 * w_5 + 21 * w_4 + 70 * w_3 + 26 * w_2 - 93 * w - 31) * d_3
/ (6480 * w1_3 * w_4)
- (75 * w_6 + 202 * w_5 + 188 * w_4 - 888 * w_3 - 1345 * w_2 + 118 * w + 138) * d_4
/ (272160 * w1_4 * w_5)
e2 = (28 * w_4 + 131 * w_3 + 402 * w_2 + 581 * w + 208) * (w - 1)
/ (1620 * w1 * w_3)
- (35 * w_6 - 154 * w_5 - 623 * w_4 - 1636 * w_3 - 3983 * w_2 - 3514 * w - 925) * d
/ (12960 * w1_2 * w_4)
- ( 2132 * w_7 + 7915 * w_6 + 16821 * w_5 + 35066 * w_4 + 87490 * w_3
+ 141183 * w_2 + 95993 * w + 21640
) * d_2
/ (816480 * w_5 * w1_3)
- ( 11053 * w_8 + 53308 * w_7 + 117010 * w_6 + 163924 * w_5 + 116188 * w_4
- 258428 * w_3 - 677042 * w_2 - 481940 * w - 105497
) * d_3
/ (14696640 * w1_4 * w_6)
e3 = -( (3592 * w_7 + 8375 * w_6 - 1323 * w_5 - 29198 * w_4 - 89578 * w_3
- 154413 * w_2 - 116063 * w - 29632
) * (w - 1)
)
/ (816480 * w_5 * w1_2)
- ( 442043 * w_9 + 2054169 * w_8 + 3803094 * w_7 + 3470754 * w_6 + 2141568 * w_5
- 2393568 * w_4 - 19904934 * w_3 - 34714674 * w_2 - 23128299 * w - 5253353
) * d
/ (146966400 * w_6 * w1_3)
- ( 116932 * w_10 + 819281 * w_9 + 2378172 * w_8 + 4341330 * w_7 + 6806004 * w_6
+ 10622748 * w_5 + 18739500 * w_4 + 30651894 * w_3 + 30869976 * w_2
+ 15431867 * w + 2919016
) * d_2
/ (146966400 * w1_4 * w_7)
eta = evaluatePolynomialL (1/a) [eta0, e1, e2, e3]
-- Now we solve eq 4.2 to recover x using Newton iterations
u       = eta - mu * log eta + (1 + mu) * log(1 + mu) - mu
cross   = 1 / (1 + mu);
lower   = if eta < mu then cross else 0
upper   = if eta < mu then 1     else cross
x_guess = (lower + upper) / 2
func x  = ( u + log x + mu*log(1 - x)
, 1/x - mu/(1-x)
)
Root x0 = newtonRaphson 1e-8 (lower, x_guess, upper) func
in x0
-- For large a and b approximation from AS109 (Carter
-- approximation). It's reasonably good in this region
| a > 1 && b > 1 =
let r = (y*y - 3) / 6
s = 1 / (2*a - 1)
t = 1 / (2*b - 1)
h = 2 / (s + t)
w = y * sqrt(h + r) / h - (t - s) * (r + 5/6 - 2 / (3 * h))
in a / (a + b * exp(2 * w))
-- Otherwise we revert to approximation from AS64 derived from
-- [AS64 2] when it's applicable.
--
-- It slightly reduces average number of iterations when a' and
-- b' have different magnitudes.
| chi2 > 0 && ratio > 1 = 1 - 2 / (ratio + 1)
-- If all else fails we use approximation from "Numerical
-- Recipes". It's very similar to approximations [AS64 4,5] but
-- it never goes out of [0,1] interval.
| otherwise = case () of
_| p < t / w  -> (a * p * w) ** (1/a)
| otherwise  -> 1 - (b * (1 - p) * w) ** (1/b)
where
lna = log $a / (a+b) lnb = log$ b / (a+b)
t   = exp( a * lna ) / a
u   = exp( b * lnb ) / b
w   = t + u
where
-- Formula [AS64 2]
ratio = (4*a + 2*b - 2) / chi2
-- Quantile of chi-squared distribution. Formula [AS64 3].
chi2 = 2 * b * (1 - t + y * sqrt t) ** 3
where
t   = 1 / (9 * b)
-- y' is Hasting's approximation of p'th quantile of standard
-- normal distribution.
y   = r - ( 2.30753 + 0.27061 * r )
/ ( 1.0 + ( 0.99229 + 0.04481 * r ) * r )
where
r = sqrt $- 2 * log p ---------------------------------------------------------------- -- Sinc function ---------------------------------------------------------------- -- | Compute sinc function @sin(x)\/x@ sinc :: Double -> Double sinc x | ax < eps_0 = 1 | ax < eps_2 = 1 - x2/6 | ax < eps_4 = 1 - x2/6 + x2*x2/120 | otherwise = sin x / x where ax = abs x x2 = x*x -- For explanation of choice see doc/sinc.hs' eps_0 = 1.8250120749944284e-8 -- sqrt (6ε/4) eps_2 = 1.4284346431400855e-4 -- (30ε)**(1/4) / 2 eps_4 = 4.043633626430947e-3 -- (1206ε)**(1/6) / 2 ---------------------------------------------------------------- -- Logarithm ---------------------------------------------------------------- -- | Compute the natural logarithm of 1 + @x@. This is accurate even -- for values of @x@ near zero, where use of @log(1+x)@ would lose -- precision. log1p :: Double -> Double log1p x | x == 0 = 0 | x == -1 = m_neg_inf | x < -1 = m_NaN | x' < m_epsilon * 0.5 = x | (x >= 0 && x < 1e-8) || (x >= -1e-9 && x < 0) = x * (1 - x * 0.5) | x' < 0.375 = x * (1 - x * chebyshevBroucke (x / 0.375) coeffs) | otherwise = log (1 + x) where x' = abs x coeffs = U.fromList [ 0.10378693562743769800686267719098e+1, -0.13364301504908918098766041553133e+0, 0.19408249135520563357926199374750e-1, -0.30107551127535777690376537776592e-2, 0.48694614797154850090456366509137e-3, -0.81054881893175356066809943008622e-4, 0.13778847799559524782938251496059e-4, -0.23802210894358970251369992914935e-5, 0.41640416213865183476391859901989e-6, -0.73595828378075994984266837031998e-7, 0.13117611876241674949152294345011e-7, -0.23546709317742425136696092330175e-8, 0.42522773276034997775638052962567e-9, -0.77190894134840796826108107493300e-10, 0.14075746481359069909215356472191e-10, -0.25769072058024680627537078627584e-11, 0.47342406666294421849154395005938e-12, -0.87249012674742641745301263292675e-13, 0.16124614902740551465739833119115e-13, -0.29875652015665773006710792416815e-14, 0.55480701209082887983041321697279e-15, -0.10324619158271569595141333961932e-15 ] -- | Compute log(1+x)-x: log1pmx :: Double -> Double log1pmx x | x < -1 = error "Domain error" | x == -1 = m_neg_inf | ax > 0.95 = log(1 + x) - x | ax < m_epsilon = -(x * x) /2 | otherwise = - x * x * sumPowerSeries (-x) (recip <$> enumSequenceFrom 2)
where
ax = abs x

-- | Compute @exp x - 1@ without loss of accuracy for x near zero.
expm1 :: Double -> Double
expm1 = c_expm1

foreign import ccall "expm1" c_expm1 :: Double -> Double

-- | /O(log n)/ Compute the logarithm in base 2 of the given value.
log2 :: Int -> Int
log2 v0
| v0 <= 0   = modErr $"log2: nonpositive input, got " ++ show v0 | otherwise = go 5 0 v0 where go !i !r !v | i == -1 = r | v .&. b i /= 0 = let si = U.unsafeIndex sv i in go (i-1) (r .|. si) (v shiftR si) | otherwise = go (i-1) r v b = U.unsafeIndex bv !bv = U.fromList [ 0x02, 0x0c, 0xf0, 0xff00 , fromIntegral (0xffff0000 :: Word) , fromIntegral (0xffffffff00000000 :: Word)] !sv = U.fromList [1,2,4,8,16,32] ---------------------------------------------------------------- -- Factorial ---------------------------------------------------------------- -- | Compute the factorial function /n/!. Returns +∞ if the -- input is above 170 (above which the result cannot be represented by -- a 64-bit 'Double'). factorial :: Int -> Double factorial n | n < 0 = error "Numeric.SpecFunctions.factorial: negative input" | n <= 1 = 1 | n <= 170 = U.product$ U.map fromIntegral $U.enumFromTo 2 n | otherwise = m_pos_inf -- | Compute the natural logarithm of the factorial function. Gives -- 16 decimal digits of precision. logFactorial :: Integral a => a -> Double logFactorial n | n < 0 = error "Numeric.SpecFunctions.logFactorial: negative input" | n <= 14 = log$ factorial $fromIntegral n -- N.B. Γ(n+1) = n! -- -- We use here asymptotic series for gamma function. See -- http://mathworld.wolfram.com/StirlingsSeries.html | otherwise = (x - 0.5) * log x - x + m_ln_sqrt_2_pi + evaluateOddPolynomialL (1/x) [1/12, -1/360, 1/1260, -1/1680] where x = fromIntegral n + 1 {-# SPECIALIZE logFactorial :: Int -> Double #-} -- | Calculate the error term of the Stirling approximation. This is -- only defined for non-negative values. -- -- $-- \operatorname{stirlingError}(n) = \log(n!) - \log(\sqrt{2\pi n}\frac{n}{e}^n) --$ stirlingError :: Double -> Double stirlingError n | n <= 15.0 = case properFraction (n+n) of (i,0) -> sfe U.unsafeIndex i _ -> logGamma (n+1.0) - (n+0.5) * log n + n - m_ln_sqrt_2_pi | n > 500 = evaluateOddPolynomialL (1/n) [s0,-s1] | n > 80 = evaluateOddPolynomialL (1/n) [s0,-s1,s2] | n > 35 = evaluateOddPolynomialL (1/n) [s0,-s1,s2,-s3] | otherwise = evaluateOddPolynomialL (1/n) [s0,-s1,s2,-s3,s4] where s0 = 0.083333333333333333333 -- 1/12 s1 = 0.00277777777777777777778 -- 1/360 s2 = 0.00079365079365079365079365 -- 1/1260 s3 = 0.000595238095238095238095238 -- 1/1680 s4 = 0.0008417508417508417508417508 -- 1/1188 sfe = U.fromList [ 0.0, 0.1534264097200273452913848, 0.0810614667953272582196702, 0.0548141210519176538961390, 0.0413406959554092940938221, 0.03316287351993628748511048, 0.02767792568499833914878929, 0.02374616365629749597132920, 0.02079067210376509311152277, 0.01848845053267318523077934, 0.01664469118982119216319487, 0.01513497322191737887351255, 0.01387612882307074799874573, 0.01281046524292022692424986, 0.01189670994589177009505572, 0.01110455975820691732662991, 0.010411265261972096497478567, 0.009799416126158803298389475, 0.009255462182712732917728637, 0.008768700134139385462952823, 0.008330563433362871256469318, 0.007934114564314020547248100, 0.007573675487951840794972024, 0.007244554301320383179543912, 0.006942840107209529865664152, 0.006665247032707682442354394, 0.006408994188004207068439631, 0.006171712263039457647532867, 0.005951370112758847735624416, 0.005746216513010115682023589, 0.005554733551962801371038690 ] ---------------------------------------------------------------- -- Combinatorics ---------------------------------------------------------------- -- | -- 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
logChooseFast :: Double -> Double -> Double
logChooseFast n k = -log (n + 1) - logBeta (n - k + 1) (k + 1)

-- | Calculate binomial coefficient using exact formula
chooseExact :: Int -> Int -> Double
n chooseExact k
= U.foldl' go 1 $U.enumFromTo 1 k where go a i = a * (nk + j) / j where j = fromIntegral i :: Double nk = fromIntegral (n - k) -- | Compute logarithm of the binomial coefficient. logChoose :: Int -> Int -> Double n logChoose k | k > n = (-1) / 0 -- For very large N exact algorithm overflows double so we -- switch to beta-function based one | k' < 50 && (n < 20000000) = log$ chooseExact n k'
| otherwise                 = logChooseFast (fromIntegral n) (fromIntegral k)
where
k' = min k (n-k)

-- | 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
choose :: Int -> Int -> Double
n choose k
| k  > n         = 0
| k' < 50        = chooseExact n k'
| approx < max64 = fromIntegral . round64 $approx | otherwise = approx where k' = min k (n-k) approx = exp$ logChooseFast (fromIntegral n) (fromIntegral k')
max64          = fromIntegral (maxBound :: Int64)
round64 x      = round x :: Int64

-- | 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.
digamma :: Double -> Double
digamma x
| isNaN x || isInfinite x                  = m_NaN
-- FIXME:
--   This is ugly. We are testing here that number is in fact
--   integer. It's somewhat tricky question to answer. When ε for
--   given number becomes 1 or greater every number is represents
--   an integer. We also must make sure that excess precision
--   won't bite us.
| x <= 0 && fromIntegral (truncate x :: Int64) == x = m_neg_inf
-- Jeffery's reflection formula
| x < 0     = digamma (1 - x) + pi / tan (negate pi * x)
| x <= 1e-6 = - γ - 1/x + trigamma1 * x
| x' < c    = r
-- De Moivre's expansion
| otherwise = let s = 1/x'
in  evaluateEvenPolynomialL s
[   r + log x' - 0.5 * s
, - 1/12
,   1/120
, - 1/252
,   1/240
, - 1/132
,  391/32760
]
where
γ  = m_eulerMascheroni
c  = 12
-- Reduce to digamma (x + n) where (x + n) >= c
(r, x') = reduce 0 x
where
reduce !s y
| y < c     = reduce (s - 1 / y) (y + 1)
| otherwise = (s, y)

----------------------------------------------------------------
-- Constants
----------------------------------------------------------------

-- Coefficients for 18-point Gauss-Legendre integration. They are
-- used in implementation of incomplete gamma and beta functions.
coefW,coefY :: U.Vector Double
coefW = U.fromList [ 0.0055657196642445571, 0.012915947284065419, 0.020181515297735382
, 0.027298621498568734,  0.034213810770299537, 0.040875750923643261
, 0.047235083490265582,  0.053244713977759692, 0.058860144245324798
, 0.064039797355015485,  0.068745323835736408, 0.072941885005653087
, 0.076598410645870640,  0.079687828912071670, 0.082187266704339706
, 0.084078218979661945,  0.085346685739338721, 0.085983275670394821
]
coefY = U.fromList [ 0.0021695375159141994, 0.011413521097787704, 0.027972308950302116
, 0.051727015600492421,  0.082502225484340941, 0.12007019910960293
, 0.16415283300752470,   0.21442376986779355,  0.27051082840644336
, 0.33199876341447887,   0.39843234186401943,  0.46931971407375483
, 0.54413605556657973,   0.62232745288031077,  0.70331500465597174
, 0.78649910768313447,   0.87126389619061517,  0.95698180152629142
]
{-# NOINLINE coefW #-}
{-# NOINLINE coefY #-}

trigamma1 :: Double
trigamma1 = 1.6449340668482264365 -- pi**2 / 6

modErr :: String -> a
modErr msg = error \$ "Numeric.SpecFunctions." ++ msg