diff --git a/Numeric/MathFunctions/Comparison.hs b/Numeric/MathFunctions/Comparison.hs --- a/Numeric/MathFunctions/Comparison.hs +++ b/Numeric/MathFunctions/Comparison.hs @@ -19,6 +19,7 @@ -- * Ulps-based comparison , addUlps , ulpDistance + , ulpDelta , within ) where @@ -36,7 +37,7 @@ -- | -- Calculate relative error of two numbers: -- --- > |a - b| / max |a| |b| +-- \[ \frac{|a - b|}{\max(|a|,|b|)} \] -- -- It lies in [0,1) interval for numbers with same sign and (1,2] for -- numbers with different sign. If both arguments are zero or negative @@ -49,9 +50,9 @@ -- | Check that relative error between two numbers @a@ and @b@. If -- 'relativeError' returns NaN it returns @False@. -eqRelErr :: Double -- ^ @eps@ relative error should be in [0,1) range - -> Double -- ^ @a@ - -> Double -- ^ @b@ +eqRelErr :: Double -- ^ /eps/ relative error should be in [0,1) range + -> Double -- ^ /a/ + -> Double -- ^ /b/ -> Bool eqRelErr eps a b = relativeError a b < eps @@ -80,7 +81,8 @@ -- | -- Measure distance between two @Double@s in ULPs (units of least --- precision). +-- precision). Note that it's different from @abs (ulpDelta a b)@ +-- since it returns correct result even when 'ulpDelta' overflows. ulpDistance :: Double -> Double -> Word64 @@ -100,6 +102,31 @@ d | ai > bi = ai - bi | otherwise = bi - ai return $! d + +-- | +-- Measure signed distance between two @Double@s in ULPs (units of least +-- precision). Note that unlike 'ulpDistance' it can overflow. +-- +-- > >>> ulpDelta 1 (1 + m_epsilon) +-- > 1 +ulpDelta :: Double + -> Double + -> Int64 +ulpDelta a b = runST $ do + buf <- newByteArray 8 + ai0 <- writeByteArray buf 0 a >> readByteArray buf 0 + bi0 <- writeByteArray buf 0 b >> readByteArray buf 0 + -- IEEE754 floats use most significant bit as sign bit (not + -- 2-complement) and we need to rearrange representations of float + -- number so that they could be compared lexicographically as + -- Word64. + let big = 0x8000000000000000 :: Word64 + order i | i < big = i + big + | otherwise = maxBound - i + ai = order ai0 + bi = order bi0 + return $! fromIntegral $ bi - ai + -- | Compare two 'Double' values for approximate equality, using -- Dawson's method. diff --git a/Numeric/MathFunctions/Constants.hs b/Numeric/MathFunctions/Constants.hs --- a/Numeric/MathFunctions/Constants.hs +++ b/Numeric/MathFunctions/Constants.hs @@ -13,12 +13,15 @@ ( -- * IEE754 constants m_epsilon + , m_sqrt_eps , m_huge , m_tiny , m_max_exp , m_pos_inf , m_neg_inf , m_NaN + , m_max_log + , m_min_log -- * Mathematical constants , m_1_sqrt_2 , m_2_sqrt_pi @@ -32,11 +35,12 @@ -- IEE754 constants ---------------------------------------------------------------- --- | A very large number. +-- | Largest representable finite value. m_huge :: Double m_huge = 1.7976931348623157e308 {-# INLINE m_huge #-} +-- | The smallest representable positive normalized value. m_tiny :: Double m_tiny = 2.2250738585072014e-308 {-# INLINE m_tiny #-} @@ -61,8 +65,17 @@ m_NaN = 0/0 {-# INLINE m_NaN #-} +-- | Maximum possible finite value of @log x@ +m_max_log :: Double +m_max_log = 709.782712893384 +{-# INLINE m_max_log #-} +-- | Logarithm of smallest normalized double ('m_tiny') +m_min_log :: Double +m_min_log = -708.3964185322641 +{-# INLINE m_min_log #-} + ---------------------------------------------------------------- -- Mathematical constants ---------------------------------------------------------------- @@ -91,6 +104,12 @@ m_epsilon :: Double m_epsilon = encodeFloat (signif+1) expo - 1.0 where (signif,expo) = decodeFloat (1.0::Double) + +-- | @sqrt m_epsilon@ +-- +-- @since 0.3.3.0 +m_sqrt_eps :: Double +m_sqrt_eps = 1.4901161193847656e-8 -- | @log(sqrt((2*pi))@ m_ln_sqrt_2_pi :: Double diff --git a/Numeric/Polynomial.hs b/Numeric/Polynomial.hs --- a/Numeric/Polynomial.hs +++ b/Numeric/Polynomial.hs @@ -69,7 +69,7 @@ -- $lists -- -- When all coefficients are known statically it's more convenient to --- pass coefficient in a list instad of vector. Functions below +-- pass coefficient in a list instead of vector. Functions below -- provide just that functionality. If list is known statically it -- will be inlined anyway. diff --git a/Numeric/Polynomial/Chebyshev.hs b/Numeric/Polynomial/Chebyshev.hs --- a/Numeric/Polynomial/Chebyshev.hs +++ b/Numeric/Polynomial/Chebyshev.hs @@ -27,9 +27,12 @@ -- A Chebyshev polynomial of the first kind is defined by the -- following recurrence: -- --- > t 0 _ = 1 --- > t 1 x = x --- > t n x = 2 * x * t (n-1) x - t (n-2) x +-- \[\begin{aligned} +-- T_0(x) &= 1 \\ +-- T_1(x) &= x \\ +-- T_{n+1}(x) &= 2xT_n(x) - T_{n-1}(x) \\ +-- \end{aligned} +-- \] data C = C {-# UNPACK #-} !Double {-# UNPACK #-} !Double @@ -68,7 +71,7 @@ -- -- * Broucke, R. (1973) Algorithm 446: Ten subroutines for the -- manipulation of Chebyshev series. /Communications of the ACM/ --- 16(4):254–256. +-- 16(4):254–256. -- -- * Clenshaw, C.W. (1962) Chebyshev series for mathematical -- functions. /National Physical Laboratory Mathematical Tables 5/, diff --git a/Numeric/RootFinding.hs b/Numeric/RootFinding.hs new file mode 100644 --- /dev/null +++ b/Numeric/RootFinding.hs @@ -0,0 +1,387 @@ +{-# LANGUAGE BangPatterns #-} +{-# LANGUAGE DeriveDataTypeable #-} +{-# LANGUAGE DeriveFoldable #-} +{-# LANGUAGE DeriveGeneric #-} +{-# LANGUAGE DeriveTraversable #-} +{-# LANGUAGE TypeFamilies #-} +-- | +-- Module : Numeric.RootFinding +-- Copyright : (c) 2011 Bryan O'Sullivan, 2018 Alexey Khudyakov +-- License : BSD3 +-- +-- Maintainer : bos@serpentine.com +-- Stability : experimental +-- Portability : portable +-- +-- Haskell functions for finding the roots of real functions of real +-- arguments. These algorithms are iterative so we provide both +-- function returning root (or failure to find root) and list of +-- iterations. +module Numeric.RootFinding + ( -- * Data types + Root(..) + , fromRoot + , Tolerance(..) + , withinTolerance + , IterationStep(..) + , findRoot + -- * Ridders algorithm + , RiddersParam(..) + , ridders + , riddersIterations + , RiddersStep(..) + -- * Newton-Raphson algorithm + , NewtonParam(..) + , newtonRaphson + , newtonRaphsonIterations + , NewtonStep(..) + -- * References + -- $references + ) where + +import Control.Applicative (Alternative(..)) +import Control.Monad (MonadPlus(..), ap) +import Control.DeepSeq (NFData(..)) +import Data.Data (Data, Typeable) +import Data.Default.Class +import GHC.Generics (Generic) +import Numeric.MathFunctions.Comparison (within,eqRelErr) +import Numeric.MathFunctions.Constants (m_epsilon) + + + +---------------------------------------------------------------- +-- Data types +---------------------------------------------------------------- + +-- | The result of searching for a root of a mathematical function. +data Root a + = NotBracketed + -- ^ The function does not have opposite signs when + -- evaluated at the lower and upper bounds of the search. + | SearchFailed + -- ^ The search failed to converge to within the given + -- error tolerance after the given number of iterations. + | Root !a + -- ^ A root was successfully found. + deriving (Eq, Read, Show, Typeable, Data, Foldable, Traversable, Functor, Generic) + +instance (NFData a) => NFData (Root a) where + rnf NotBracketed = () + rnf SearchFailed = () + rnf (Root a) = rnf a + +instance Applicative Root where + pure = Root + (<*>) = ap + +instance Monad Root where + NotBracketed >>= _ = NotBracketed + SearchFailed >>= _ = SearchFailed + Root a >>= f = f a + return = pure + +instance MonadPlus Root where + mzero = empty + mplus = (<|>) + +instance Alternative Root where + empty = NotBracketed + r@Root{} <|> _ = r + _ <|> r@Root{} = r + NotBracketed <|> r = r + r <|> NotBracketed = r + _ <|> r = r + +-- | Returns either the result of a search for a root, or the default +-- value if the search failed. +fromRoot :: a -- ^ Default value. + -> Root a -- ^ Result of search for a root. + -> a +fromRoot _ (Root a) = a +fromRoot a _ = a + + +-- | Error tolerance for finding root. It describes when root finding +-- algorithm should stop trying to improve approximation. +data Tolerance + = RelTol !Double + -- ^ Relative error tolerance. Given @RelTol ε@ two values are + -- considered approximately equal if + -- \[ \frac{|a - b|}{|\operatorname{max}(a,b)} < \varepsilon \] + | AbsTol !Double + -- ^ Absolute error tolerance. Given @AbsTol δ@ two values are + -- considered approximately equal if \[ |a - b| < \delta \]. + -- Note that @AbsTol 0@ could be used to require to find + -- approximation within machine precision. + deriving (Eq, Read, Show, Typeable, Data, Generic) + +-- | Check that two values are approximately equal. In addition to +-- specification values are considered equal if they're within 1ulp +-- of precision. No further improvement could be done anyway. +withinTolerance :: Tolerance -> Double -> Double -> Bool +withinTolerance _ a b + | within 1 a b = True +withinTolerance (RelTol eps) a b = eqRelErr eps a b +withinTolerance (AbsTol tol) a b = abs (a - b) < tol + +-- | Type class for checking whether iteration converged already. +class IterationStep a where + -- | Return @Just root@ is current iteration converged within + -- required error tolerance. Returns @Nothing@ otherwise. + matchRoot :: Tolerance -> a -> Maybe (Root Double) + +-- | Find root in lazy list of iterations. +findRoot :: IterationStep a + => Int -- ^ Maximum + -> Tolerance -- ^ Error tolerance + -> [a] + -> Root Double +findRoot maxN tol = go 0 + where + go !i _ | i >= maxN = SearchFailed + go !_ [] = SearchFailed + go i (x:xs) = case matchRoot tol x of + Just r -> r + Nothing -> go (i+1) xs +{-# INLINABLE findRoot #-} +{-# SPECIALIZE findRoot :: Int -> Tolerance -> [RiddersStep] -> Root Double #-} +{-# SPECIALIZE findRoot :: Int -> Tolerance -> [NewtonStep] -> Root Double #-} + + +---------------------------------------------------------------- +-- Attaching information to roots +---------------------------------------------------------------- + +-- | Parameters for 'ridders' root finding +data RiddersParam = RiddersParam + { riddersMaxIter :: !Int + -- ^ Maximum number of iterations. Default = 100 + , riddersTol :: !Tolerance + -- ^ Error tolerance for root approximation. Default is relative + -- error 4·ε, where ε is machine precision. + } + deriving (Eq, Read, Show, Typeable, Data, Generic) + +instance Default RiddersParam where + def = RiddersParam + { riddersMaxIter = 100 + , riddersTol = RelTol (4 * m_epsilon) + } + +-- | Single Ridders step. It's a bracket of root +data RiddersStep + = RiddersStep !Double !Double + -- ^ Ridders step. Parameters are bracket for the root + | RiddersBisect !Double !Double + -- ^ Bisection step. It's fallback which is taken when Ridders + -- update takes us out of bracket + | RiddersRoot !Double + -- ^ Root found + | RiddersNoBracket + -- ^ Root is not bracketed + deriving (Eq, Read, Show, Typeable, Data, Generic) + +instance NFData RiddersStep where + rnf x = x `seq` () + +instance IterationStep RiddersStep where + matchRoot tol r = case r of + RiddersRoot x -> Just $ Root x + RiddersNoBracket -> Just NotBracketed + RiddersStep a b + | withinTolerance tol a b -> Just $ Root ((a + b) / 2) + | otherwise -> Nothing + RiddersBisect a b + | withinTolerance tol a b -> Just $ Root ((a + b) / 2) + | otherwise -> Nothing + + +-- | Use the method of Ridders[Ridders1979] to compute a root of a +-- function. It doesn't require derivative and provide quadratic +-- convergence (number of significant digits grows quadratically +-- with number of iterations). +-- +-- The function must have opposite signs when evaluated at the lower +-- and upper bounds of the search (i.e. the root must be +-- bracketed). If there's more that one root in the bracket +-- iteration will converge to some root in the bracket. +ridders + :: RiddersParam -- ^ Parameters for algorithms. @def@ + -- provides reasonable defaults + -> (Double,Double) -- ^ Bracket for root + -> (Double -> Double) -- ^ Function to find roots + -> Root Double +ridders p bracket fun + = findRoot (riddersMaxIter p) (riddersTol p) + $ riddersIterations bracket fun + +-- | List of iterations for Ridders methods. See 'ridders' for +-- documentation of parameters +riddersIterations :: (Double,Double) -> (Double -> Double) -> [RiddersStep] +riddersIterations (lo,hi) f + | flo == 0 = [RiddersRoot lo] + | fhi == 0 = [RiddersRoot hi] + -- root is not bracketed + | flo*fhi > 0 = [RiddersNoBracket] + -- Ensure that a= b = case () of + _| fm*fa < 0 -> recBisect a fa m fm + | otherwise -> recBisect m fm b fb + | fn*fm < 0 = recRidders n fn m fm + | fn*fa < 0 = recRidders a fa n fn + | otherwise = recRidders n fn b fb + where + recBisect x fx y fy = RiddersBisect x y : go x fx y fy + recRidders x fx y fy = RiddersStep x y : go x fx y fy + -- + dm = (b - a) * 0.5 + -- Mean point + m = (a + b) / 2 + fm = f m + -- Ridders update + n = m - signum (fb - fa) * dm * fm / sqrt(fm*fm - fa*fb) + fn = f n + + + +---------------------------------------------------------------- +-- Newton-Raphson algorithm +---------------------------------------------------------------- + +-- | Parameters for 'ridders' root finding +data NewtonParam = NewtonParam + { newtonMaxIter :: !Int + -- ^ Maximum number of iterations. Default = 50 + , newtonTol :: !Tolerance + -- ^ Error tolerance for root approximation. Default is relative + -- error 4·ε, where ε is machine precision + } + deriving (Eq, Read, Show, Typeable, Data, Generic) + +instance Default NewtonParam where + def = NewtonParam + { newtonMaxIter = 50 + , newtonTol = RelTol (4 * m_epsilon) + } + +-- | Steps for Newton iterations +data NewtonStep + = NewtonStep !Double !Double + -- ^ Normal Newton-Raphson update. Parameters are: old guess, new guess + | NewtonBisection !Double !Double + -- ^ Bisection fallback when Newton-Raphson iteration doesn't + -- work. Parameters are bracket on root + | NewtonRoot !Double + -- ^ Root is found + | NewtonNoBracket + -- ^ Root is not bracketed + deriving (Eq, Read, Show, Typeable, Data, Generic) + +instance NFData NewtonStep where + rnf x = x `seq` () + +instance IterationStep NewtonStep where + matchRoot tol r = case r of + NewtonRoot x -> Just (Root x) + NewtonNoBracket -> Just NotBracketed + NewtonStep x x' + | withinTolerance tol x x' -> Just (Root x') + | otherwise -> Nothing + NewtonBisection a b + | withinTolerance tol a b -> Just (Root ((a + b) / 2)) + | otherwise -> Nothing + {-# INLINE matchRoot #-} + + +-- | Solve equation using Newton-Raphson iterations. +-- +-- This method require both initial guess and bounds for root. If +-- Newton step takes us out of bounds on root function reverts to +-- bisection. +newtonRaphson + :: NewtonParam -- ^ Parameters for algorithm. @def@ + -- provide reasonable defaults. + -> (Double,Double,Double) -- ^ Triple of @(low bound, initial + -- guess, upper bound)@. If initial + -- guess if out of bracket middle + -- of bracket is taken as + -- approximation + -> (Double -> (Double,Double)) -- ^ Function to find root of. It + -- returns pair of function value and + -- its first derivative + -> Root Double +newtonRaphson p guess fun + = findRoot (newtonMaxIter p) (newtonTol p) + $ newtonRaphsonIterations guess fun + +-- | List of iteration for Newton-Raphson algorithm. See documentation +-- for 'newtonRaphson' for meaning of parameters. +newtonRaphsonIterations :: (Double,Double,Double) -> (Double -> (Double,Double)) -> [NewtonStep] +newtonRaphsonIterations (lo,guess,hi) function + | flo == 0 = [NewtonRoot lo] + | fhi == 0 = [NewtonRoot hi] + | flo*fhi > 0 = [NewtonNoBracket] + -- Ensure that function value on low bound is negative + | flo > 0 = go hi guess' lo + | otherwise = go lo guess hi + where + (flo,_) = function lo + (fhi,_) = function hi + -- Ensure that initial guess is within bracket + guess' + | guess >= lo && guess <= hi = guess + | guess >= hi && guess <= lo = guess + | otherwise = (lo + hi) / 2 + -- Newton iterations. Invariant: + -- > f xA < 0 + -- > f xB > 0 + go xA x xB + | f == 0 = [NewtonRoot x] + | f' == 0 = bisectionStep + -- Accept Newton step since it stays within bracket. + | (x' - xA) * (x' - xB) < 0 = newtonStep + -- Otherwise bracket root and pick new approximation as + -- midpoint. + | otherwise = bisectionStep + where + -- Calculate Newton step + (f,f') = function x + x' = x - f / f' + -- Newton step + newtonStep + | f > 0 = NewtonStep x x' : go xA x' x + | otherwise = NewtonStep x x' : go x x' xB + -- Fallback bisection step + bisectionStep + | f > 0 = NewtonBisection xA x : go xA ((xA + x) / 2) x + | otherwise = NewtonBisection x xB : go x ((x + xB) / 2) xB + + + +---------------------------------------------------------------- +-- Internal functions +---------------------------------------------------------------- + +-- $references +-- +-- * Ridders, C.F.J. (1979) A new algorithm for computing a single +-- root of a real continuous function. +-- /IEEE Transactions on Circuits and Systems/ 26:979–980. +-- +-- * Press W.H.; Teukolsky S.A.; Vetterling W.T.; Flannery B.P. +-- (2007). \"Section 9.2.1. Ridders' Method\". /Numerical Recipes: The +-- Art of Scientific Computing (3rd ed.)./ New York: Cambridge +-- University Press. ISBN 978-0-521-88068-8. diff --git a/Numeric/Series.hs b/Numeric/Series.hs new file mode 100644 --- /dev/null +++ b/Numeric/Series.hs @@ -0,0 +1,180 @@ +{-# LANGUAGE BangPatterns #-} +{-# LANGUAGE ExistentialQuantification #-} +-- | +-- Module : Numeric.Series +-- Copyright : (c) 2016 Alexey Khudyakov +-- License : BSD3 +-- +-- Maintainer : alexey.skladnoy@gmail.com, bos@serpentine.com +-- Stability : experimental +-- Portability : portable +-- +-- Functions for working with infinite sequences. In particular +-- summation of series and evaluation of continued fractions. +module Numeric.Series ( + -- * Data type for infinite sequences. + Sequence(..) + -- * Constructors + , enumSequenceFrom + , enumSequenceFromStep + , scanSequence + -- * Summation of series + , sumSeries + , sumPowerSeries + , sequenceToList + -- * Evaluation of continued fractions + , evalContFractionB + ) where + +import Control.Applicative +import Data.List (unfoldr) + +import Numeric.MathFunctions.Constants (m_epsilon) + + +---------------------------------------------------------------- + +-- | Infinite series. It's represented as opaque state and step +-- function. +data Sequence a = forall s. Sequence s (s -> (a,s)) + +instance Functor Sequence where + fmap f (Sequence s0 step) = Sequence s0 (\s -> let (a,s') = step s in (f a, s')) + {-# INLINE fmap #-} + +instance Applicative Sequence where + pure a = Sequence () (\() -> (a,())) + Sequence sA fA <*> Sequence sB fB = Sequence (sA,sB) $ \(!sa,!sb) -> + let (a,sa') = fA sa + (b,sb') = fB sb + in (a b, (sa',sb')) + {-# INLINE pure #-} + {-# INLINE (<*>) #-} + +-- | Elementwise operations with sequences +instance Num a => Num (Sequence a) where + (+) = liftA2 (+) + (*) = liftA2 (*) + (-) = liftA2 (-) + {-# INLINE (+) #-} + {-# INLINE (*) #-} + {-# INLINE (-) #-} + abs = fmap abs + signum = fmap signum + fromInteger = pure . fromInteger + {-# INLINE abs #-} + {-# INLINE signum #-} + {-# INLINE fromInteger #-} + +-- | Elementwise operations with sequences +instance Fractional a => Fractional (Sequence a) where + (/) = liftA2 (/) + recip = fmap recip + fromRational = pure . fromRational + {-# INLINE (/) #-} + {-# INLINE recip #-} + {-# INLINE fromRational #-} + + + +---------------------------------------------------------------- +-- Constructors +---------------------------------------------------------------- + +-- | @enumSequenceFrom x@ generate sequence: +-- +-- \[ a_n = x + n \] +enumSequenceFrom :: Num a => a -> Sequence a +enumSequenceFrom i = Sequence i (\n -> (n,n+1)) +{-# INLINE enumSequenceFrom #-} + +-- | @enumSequenceFromStep x d@ generate sequence: +-- +-- \[ a_n = x + nd \] +enumSequenceFromStep :: Num a => a -> a -> Sequence a +enumSequenceFromStep n d = Sequence n (\i -> (i,i+d)) +{-# INLINE enumSequenceFromStep #-} + +-- | Analog of 'scanl' for sequence. +scanSequence :: (b -> a -> b) -> b -> Sequence a -> Sequence b +{-# INLINE scanSequence #-} +scanSequence f b0 (Sequence s0 step) = Sequence (b0,s0) $ \(b,s) -> + let (a,s') = step s + b' = f b a + in (b,(b',s')) + + +---------------------------------------------------------------- +-- Evaluation of series +---------------------------------------------------------------- + +-- | Calculate sum of series +-- +-- \[ \sum_{i=0}^\infty a_i \] +-- +-- Summation is stopped when +-- +-- \[ a_{n+1} < \varepsilon\sum_{i=0}^n a_i \] +-- +-- where ε is machine precision ('m_epsilon') +sumSeries :: Sequence Double -> Double +{-# INLINE sumSeries #-} +sumSeries (Sequence sInit step) + = go x0 s0 + where + (x0,s0) = step sInit + go x s | abs (d/x) >= m_epsilon = go x' s' + | otherwise = x' + where (d,s') = step s + x' = x + d + +-- | Calculate sum of series +-- +-- \[ \sum_{i=0}^\infty x^ia_i \] +-- +-- Calculation is stopped when next value in series is less than +-- ε·sum. +sumPowerSeries :: Double -> Sequence Double -> Double +sumPowerSeries x ser = sumSeries $ liftA2 (*) (scanSequence (*) 1 (pure x)) ser +{-# INLINE sumPowerSeries #-} + +-- | Convert series to infinite list +sequenceToList :: Sequence a -> [a] +sequenceToList (Sequence s f) = unfoldr (Just . f) s + + + +---------------------------------------------------------------- +-- Evaluation of continued fractions +---------------------------------------------------------------- + +-- | +-- Evaluate continued fraction using modified Lentz algorithm. +-- Sequence contain pairs (a[i],b[i]) which form following expression: +-- +-- \[ +-- b_0 + \frac{a_1}{b_1+\frac{a_2}{b_2+\frac{a_3}{b_3 + \cdots}}} +-- \] +-- +-- Modified Lentz algorithm is described in Numerical recipes 5.2 +-- "Evaluation of Continued Fractions" +evalContFractionB :: Sequence (Double,Double) -> Double +{-# INLINE evalContFractionB #-} +evalContFractionB (Sequence sInit step) + = let ((_,b0),s0) = step sInit + f0 = maskZero b0 + in go f0 f0 0 s0 + where + tiny = 1e-60 + maskZero 0 = tiny + maskZero x = x + + go f c d s + | abs (delta - 1) >= m_epsilon = go f' c' d' s' + | otherwise = f' + where + ((a,b),s') = step s + d' = recip $ maskZero $ b + a*d + c' = maskZero $ b + a/c + delta = c'*d' + f' = f*delta diff --git a/Numeric/SpecFunctions.hs b/Numeric/SpecFunctions.hs --- a/Numeric/SpecFunctions.hs +++ b/Numeric/SpecFunctions.hs @@ -1,4 +1,3 @@ -{-# LANGUAGE BangPatterns, ScopedTypeVariables #-} -- | -- Module : Numeric.SpecFunctions -- Copyright : (c) 2009, 2011, 2012 Bryan O'Sullivan @@ -29,8 +28,12 @@ -- * Sinc , sinc -- * Logarithm + -- $log1p , log1p + , log1pmx , log2 + -- * Exponent + , expm1 -- * Factorial , factorial , logFactorial @@ -44,6 +47,13 @@ import Numeric.SpecFunctions.Internal +-- $log1p +-- +-- 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@. + -- $references -- -- * Bernardo, J. (1976) Algorithm AS 103: Psi (digamma) @@ -81,6 +91,11 @@ -- Vol. 22, No. 3 (1973), pp. 411-414 -- -- --- * Shea, B. (1988) Algorithm AS 239: Chi-squared and incomplete --- gamma integral. /Applied Statistics/ --- 37(3):466–473. +-- * 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. + diff --git a/Numeric/SpecFunctions/Compat.hs b/Numeric/SpecFunctions/Compat.hs new file mode 100644 --- /dev/null +++ b/Numeric/SpecFunctions/Compat.hs @@ -0,0 +1,167 @@ +{-# LANGUAGE CPP #-} +{-# LANGUAGE ForeignFunctionInterface #-} +-- | +-- Functions which have different implementations on different platforms +module Numeric.SpecFunctions.Compat ( + erf + , erfc + , log1p + , expm1 + ) where + +#if !defined(USE_SYSTEM_ERF) || !defined(USE_SYSTEM_EXPM1) +import qualified Data.Vector.Unboxed as U +#endif + +#if !defined(USE_SYSTEM_ERF) +import Numeric.Polynomial.Chebyshev (chebyshev) +import Numeric.Polynomial (evaluateOddPolynomial) +#endif + +#if !defined(USE_SYSTEM_EXPM1) +import Control.Applicative (liftA2) +import Numeric.Polynomial.Chebyshev (chebyshevBroucke) +import Numeric.Series (scanSequence,sumSeries,enumSequenceFrom) +import Numeric.MathFunctions.Constants +#endif + +#if defined(USE_SYSTEM_EXPM1) +import GHC.Float (log1p,expm1) +#endif + + +---------------------------------------------------------------- +-- erf & erfc +-- +-- We provide pure haskell implementation for GHCJS and accessible on +-- GHC via flag +---------------------------------------------------------------- + +#if defined(USE_SYSTEM_ERF) + +erf :: Double -> Double +erf = c_erf +{-# INLINE erf #-} + +erfc :: Double -> Double +erfc = c_erfc +{-# INLINE erfc #-} + +foreign import ccall unsafe "erf" c_erf :: Double -> Double +foreign import ccall unsafe "erfc" c_erfc :: Double -> Double + +#else + +erf :: Double -> Double +erf x + -- Computing erf as 1-erfc loses precision near 0 so we switch to + -- Taylor expansion here + | abs x < 0.1 = 0.56418958354775629 + * evaluateOddPolynomial x erfTaylorSeries + | x < 0 = (-1) + erfcCheb (-x) + | otherwise = 1 - erfcCheb x + +erfTaylorSeries :: U.Vector Double +{-# NOINLINE erfTaylorSeries #-} +erfTaylorSeries = U.fromList + [ 2 + , -2/3 + , 1/5 + , -1/21 + , 1/108 + , -1/660 + , 1/4680 + ] + +erfc :: Double -> Double +erfc x | x < 0 = 2 - erfcCheb (-x) + | otherwise = erfcCheb x + +-- Adapted from Numerical Recipes §6.2.2 +erfcCheb :: Double -> Double +erfcCheb z + = t * exp( -z * z + chebyshev ty erfcCoef ) + where + -- We're using approximation: + -- + -- erfc(z) ≈ t·exp(-z² + P(t)) + -- t = 2 / (2 + z) + t = 2 / (2 + z) + ty = 2 * t - 1 + + +erfcCoef :: U.Vector Double +{-# NOINLINE erfcCoef #-} +erfcCoef = U.fromList + [ -0.6513268598908546 , 6.4196979235649026e-1 , 1.9476473204185836e-2 + , -9.561514786808631e-3 , -9.46595344482036e-4 , 3.66839497852761e-4 + , 4.2523324806907e-5 , -2.0278578112534e-5 , -1.624290004647e-6 + , 1.303655835580e-6 , 1.5626441722e-8 , -8.5238095915e-8 + , 6.529054439e-9 , 5.059343495e-9 , -9.91364156e-10 + , -2.27365122e-10 , 9.6467911e-11 , 2.394038e-12 + , -6.886027e-12 , 8.94487e-13 , 3.13092e-13 + , -1.12708e-13 , 3.81e-16 , 7.106e-15 + , -1.523e-15 , -9.4e-17 , 1.21e-16 + , -2.8e-17 + ] + +#endif + + +---------------------------------------------------------------- +-- expm1 & log1p +-- +-- We use one provided by base of for GHCJS use hand-coded one +---------------------------------------------------------------- + +#if !defined(USE_SYSTEM_EXPM1) + +-- | Compute @exp x - 1@ without loss of accuracy for x near zero. +expm1 :: Double -> Double +-- NOTE: this is simplest implementation and not terribly efficient. +expm1 x + | x < (-37.42994775023705) = -1 + | x > m_max_log = m_pos_inf + | abs x > 0.5 = exp x - 1 + | otherwise = sumSeries $ liftA2 (*) (scanSequence (*) x (pure x)) + (1 / scanSequence (*) 1 (enumSequenceFrom 2)) +-- | 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 + ] +#endif diff --git a/Numeric/SpecFunctions/Extra.hs b/Numeric/SpecFunctions/Extra.hs --- a/Numeric/SpecFunctions/Extra.hs +++ b/Numeric/SpecFunctions/Extra.hs @@ -12,10 +12,12 @@ bd0 , chooseExact , logChooseFast + , logGammaAS245 + , logGammaCorrection ) where -import Numeric.MathFunctions.Constants (m_NaN) -import Numeric.SpecFunctions.Internal (chooseExact,logChooseFast) +import Numeric.MathFunctions.Constants (m_NaN,m_pos_inf) +import Numeric.SpecFunctions.Internal (chooseExact,logChooseFast,logGammaCorrection) -- | Evaluate the deviance term @x log(x/np) + np - x@. bd0 :: Double -- ^ @x@ @@ -34,3 +36,61 @@ loop j ej s = case s + ej/(2*j+1) of s' | s' == s -> s' -- FIXME: Comparing Doubles for equality! | otherwise -> loop (j+1) (ej*vv) s' + + + +-- | 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). +logGammaAS245 :: Double -> Double +-- Adapted from http://people.sc.fsu.edu/~burkardt/f_src/asa245/asa245.html +logGammaAS245 x + | x <= 0 = m_pos_inf + -- Handle positive infinity. logGamma overflows before 1e308 so + -- it's safe + | x > 1e308 = m_pos_inf + -- Normal cases + | x < 1.5 = a + c * + ((((r1_4 * b + r1_3) * b + r1_2) * b + r1_1) * b + r1_0) / + ((((b + r1_8) * b + r1_7) * b + r1_6) * b + r1_5) + | x < 4 = (x - 2) * + ((((r2_4 * x + r2_3) * x + r2_2) * x + r2_1) * x + r2_0) / + ((((x + r2_8) * x + r2_7) * x + r2_6) * x + r2_5) + | x < 12 = ((((r3_4 * x + r3_3) * x + r3_2) * x + r3_1) * x + r3_0) / + ((((x + r3_8) * x + r3_7) * x + r3_6) * x + r3_5) + | x > 3e6 = k + | otherwise = k + x1 * + ((r4_2 * x2 + r4_1) * x2 + r4_0) / + ((x2 + r4_4) * x2 + r4_3) + where + (a , b , c) + | x < 0.5 = (-y , x + 1 , x) + | otherwise = (0 , x , x - 1) + + y = log x + k = x * (y-1) - 0.5 * y + alr2pi + alr2pi = 0.918938533204673 + + x1 = 1 / x + x2 = x1 * x1 + + r1_0 = -2.66685511495; r1_1 = -24.4387534237; r1_2 = -21.9698958928 + r1_3 = 11.1667541262; r1_4 = 3.13060547623; r1_5 = 0.607771387771 + r1_6 = 11.9400905721; r1_7 = 31.4690115749; r1_8 = 15.2346874070 + + r2_0 = -78.3359299449; r2_1 = -142.046296688; r2_2 = 137.519416416 + r2_3 = 78.6994924154; r2_4 = 4.16438922228; r2_5 = 47.0668766060 + r2_6 = 313.399215894; r2_7 = 263.505074721; r2_8 = 43.3400022514 + + r3_0 = -2.12159572323e5; r3_1 = 2.30661510616e5; r3_2 = 2.74647644705e4 + r3_3 = -4.02621119975e4; r3_4 = -2.29660729780e3; r3_5 = -1.16328495004e5 + r3_6 = -1.46025937511e5; r3_7 = -2.42357409629e4; r3_8 = -5.70691009324e2 + + r4_0 = 0.279195317918525; r4_1 = 0.4917317610505968; + r4_2 = 0.0692910599291889; r4_3 = 3.350343815022304 + r4_4 = 6.012459259764103 diff --git a/Numeric/SpecFunctions/Internal.hs b/Numeric/SpecFunctions/Internal.hs --- a/Numeric/SpecFunctions/Internal.hs +++ b/Numeric/SpecFunctions/Internal.hs @@ -1,813 +1,1381 @@ {-# LANGUAGE BangPatterns, ScopedTypeVariables #-} --- | --- Module : Numeric.SpecFunctions.Internal --- Copyright : (c) 2009, 2011, 2012 Bryan O'Sullivan --- License : BSD3 --- --- Maintainer : bos@serpentine.com --- Stability : experimental --- Portability : portable --- --- Internal module with implementation of special functions. -module Numeric.SpecFunctions.Internal where - -import Data.Bits ((.&.), (.|.), shiftR) -import Data.Int (Int64) -import qualified Data.Number.Erf as Erf (erfc,erf) -import qualified Data.Vector.Unboxed as U - -import Numeric.Polynomial.Chebyshev (chebyshevBroucke) -import Numeric.Polynomial (evaluateEvenPolynomialL,evaluateOddPolynomialL) -import Numeric.MathFunctions.Constants ( m_epsilon, m_NaN, m_neg_inf, m_pos_inf - , m_sqrt_2_pi, m_ln_sqrt_2_pi, m_sqrt_2 - , m_eulerMascheroni - ) -import Text.Printf - - ----------------------------------------------------------------- --- Error function ----------------------------------------------------------------- - --- | Error function. --- --- > erf -∞ = -1 --- > erf 0 = 0 --- > erf +∞ = 1 -erf :: Double -> Double -{-# INLINE erf #-} -erf = Erf.erf - --- | Complementary error function. --- --- > erfc -∞ = 2 --- > erfc 0 = 1 --- > errc +∞ = 0 -erfc :: Double -> Double -{-# INLINE erfc #-} -erfc = Erf.erfc - - --- | 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 ----------------------------------------------------------------- - --- Adapted from http://people.sc.fsu.edu/~burkardt/f_src/asa245/asa245.html - --- | 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). -logGamma :: Double -> Double -logGamma x - | x <= 0 = m_pos_inf - -- Handle positive infinity. logGamma overflows before 1e308 so - -- it's safe - | x > 1e308 = m_pos_inf - -- Normal cases - | x < 1.5 = a + c * - ((((r1_4 * b + r1_3) * b + r1_2) * b + r1_1) * b + r1_0) / - ((((b + r1_8) * b + r1_7) * b + r1_6) * b + r1_5) - | x < 4 = (x - 2) * - ((((r2_4 * x + r2_3) * x + r2_2) * x + r2_1) * x + r2_0) / - ((((x + r2_8) * x + r2_7) * x + r2_6) * x + r2_5) - | x < 12 = ((((r3_4 * x + r3_3) * x + r3_2) * x + r3_1) * x + r3_0) / - ((((x + r3_8) * x + r3_7) * x + r3_6) * x + r3_5) - | x > 3e6 = k - | otherwise = k + x1 * - ((r4_2 * x2 + r4_1) * x2 + r4_0) / - ((x2 + r4_4) * x2 + r4_3) - where - (a , b , c) - | x < 0.5 = (-y , x + 1 , x) - | otherwise = (0 , x , x - 1) - - y = log x - k = x * (y-1) - 0.5 * y + alr2pi - alr2pi = 0.918938533204673 - - x1 = 1 / x - x2 = x1 * x1 - - r1_0 = -2.66685511495; r1_1 = -24.4387534237; r1_2 = -21.9698958928 - r1_3 = 11.1667541262; r1_4 = 3.13060547623; r1_5 = 0.607771387771 - r1_6 = 11.9400905721; r1_7 = 31.4690115749; r1_8 = 15.2346874070 - - r2_0 = -78.3359299449; r2_1 = -142.046296688; r2_2 = 137.519416416 - r2_3 = 78.6994924154; r2_4 = 4.16438922228; r2_5 = 47.0668766060 - r2_6 = 313.399215894; r2_7 = 263.505074721; r2_8 = 43.3400022514 - - r3_0 = -2.12159572323e5; r3_1 = 2.30661510616e5; r3_2 = 2.74647644705e4 - r3_3 = -4.02621119975e4; r3_4 = -2.29660729780e3; r3_5 = -1.16328495004e5 - r3_6 = -1.46025937511e5; r3_7 = -2.42357409629e4; r3_8 = -5.70691009324e2 - - r4_0 = 0.279195317918525; r4_1 = 0.4917317610505968; - r4_2 = 0.0692910599291889; r4_3 = 3.350343815022304 - r4_4 = 6.012459259764103 - - - -data L = L {-# UNPACK #-} !Double {-# UNPACK #-} !Double - --- | Compute the logarithm of the gamma function, Γ(/x/). Uses a --- Lanczos approximation. --- --- This function is slower than 'logGamma', but gives 14 or more --- significant decimal digits of accuracy, 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 -logGammaL x - | x <= 0 = m_pos_inf - -- Lanroz approximation loses precision for small arguments - | x <= 1e-3 = logGamma x - | otherwise = fini . U.foldl' go (L 0 (x+7)) $ a - where fini (L l _) = log (l+a0) + log m_sqrt_2_pi - x65 + (x-0.5) * log x65 - go (L l t) k = L (l + k / t) (t-1) - x65 = x + 6.5 - a0 = 0.9999999999995183 - a = U.fromList [ 0.1659470187408462e-06 - , 0.9934937113930748e-05 - , -0.1385710331296526 - , 12.50734324009056 - , -176.6150291498386 - , 771.3234287757674 - , -1259.139216722289 - , 676.5203681218835 - ] - - - --- | Compute the log gamma correction factor for @x@ ≥ 10. This --- correction factor is suitable for an alternate (but less --- numerically accurate) definition of 'logGamma': --- --- >lgg x = 0.5 * log(2*pi) + (x-0.5) * log x - x + 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(/s/,/x/). Normalization means that --- γ(/s/,∞)=1. Uses Algorithm AS 239 by Shea. -incompleteGamma :: Double -- ^ /s/ ∈ (0,∞) - -> Double -- ^ /x/ ∈ (0,∞) - -> Double -incompleteGamma p x - | isNaN p || isNaN x = m_NaN - | x < 0 || p <= 0 = m_pos_inf - | x == 0 = 0 - -- For very large `p' normal approximation gives <1e-10 error - | p >= 2e5 = norm (3 * sqrt p * ((x/p) ** (1/3) + 1/(9*p) - 1)) - | p >= 500 = approx - -- Dubious approximation - | x >= 1e8 = 1 - | x <= 1 || x < p = let a = p * log x - x - logGamma (p + 1) - g = a + log (pearson p 1 1) - in if g > limit then exp g else 0 - | otherwise = let g = p * log x - x - logGamma p + log cf - in if g > limit then 1 - exp g else 1 - where - -- CDF for standard normal distributions - norm a = 0.5 * erfc (- a / m_sqrt_2) - -- For large values of `p' we use 18-point Gauss-Legendre - -- integration. - approx - | ans > 0 = 1 - ans - | otherwise = -ans - where - -- Set upper limit for integration - xu | x > p1 = (p1 + 11.5*sqrtP1) `max` (x + 6*sqrtP1) - | otherwise = max 0 $ (p1 - 7.5*sqrtP1) `min` (x - 5*sqrtP1) - s = U.sum $ U.zipWith go coefY coefW - go y w = let t = x + (xu - x)*y - in w * exp( -(t-p1) + p1*(log t - lnP1) ) - ans = s * (xu - x) * exp( p1 * (lnP1 - 1) - logGamma p) - -- - p1 = p - 1 - lnP1 = log p1 - sqrtP1 = sqrt p1 - -- - pearson !a !c !g - | c' <= tolerance = g' - | otherwise = pearson a' c' g' - where a' = a + 1 - c' = c * x / a' - g' = g + c' - cf = let a = 1 - p - b = a + x + 1 - p3 = x + 1 - p4 = x * b - in contFrac a b 0 1 x p3 p4 (p3/p4) - contFrac !a !b !c !p1 !p2 !p3 !p4 !g - | abs (g - rn) <= min tolerance (tolerance * rn) = g - | otherwise = contFrac a' b' c' (f p3) (f p4) (f p5) (f p6) rn - where a' = a + 1 - b' = b + 2 - c' = c + 1 - an = a' * c' - p5 = b' * p3 - an * p1 - p6 = b' * p4 - an * p2 - rn = p5 / p6 - f n | abs p5 > overflow = n / overflow - | otherwise = n - limit = -88 - tolerance = 1e-14 - overflow = 1e37 - - - --- Adapted from Numerical Recipes §6.2.1 - --- | Inverse incomplete gamma function. It's approximately inverse of --- 'incompleteGamma' for the same /s/. So following equality --- approximately holds: --- --- > invIncompleteGamma s . incompleteGamma s = id -invIncompleteGamma :: Double -- ^ /s/ ∈ (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. -logBeta :: Double -> Double -> 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. Uses algorithm AS63 by --- Majumder and Bhattachrjee and quadrature approximation for large --- /p/ and /q/. -incompleteBeta :: Double -- ^ /p/ > 0 - -> Double -- ^ /q/ > 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 -- ^ /p/ > 0 - -> Double -- ^ /q/ > 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 "incompletBeta_: 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 = log (1 - 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 - -- Loop - loop !psq (ns :: Int) ai term betain - | done = betain' * exp( p * log x + (q - 1) * log cx - beta) / 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 -- ^ /p/ > 0 - -> Double -- ^ /q/ > 0 - -> Double -- ^ /a/ ∈ [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 - | a > 0.5 = 1 - invIncompleteBetaWorker (logBeta p q) q p (1 - a) - | otherwise = invIncompleteBetaWorker (logBeta p q) p q a - - -invIncompleteBetaWorker :: Double -> Double -> Double -> Double -> Double --- NOTE: p <= 0.5. -invIncompleteBetaWorker beta a b p = loop (0::Int) guess - 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 * log (1 - x) - beta - u = f / f' - dx = u / (1 - 0.5 * min 1 (u * (a1 / x - b1 / (1 - x)))) - -- 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. Approximations from AS64, AS109 and - -- Numerical recipes are used. - -- - -- 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. - guess - -- In this region we use approximation from AS109 (Carter - -- approximation). It's reasonably good (2 iterations on - -- average) - | 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 [2] - ratio = (4*a + 2*b - 2) / chi2 - -- Quantile of chi-squared distribution. Formula [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 - ] - - --- | /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 [0x2, 0xc, 0xf0, 0xff00, 0xffff0000, 0xffffffff00000000] - !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. --- --- > stirlingError @n@ = @log(n!) - log(sqrt(2*pi*n)*(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 ψ0(/x/), the first logarithmic derivative of the gamma --- function. 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 +{-# OPTIONS_HADDOCK hide #-} +-- | +-- Module : Numeric.SpecFunctions.Internal +-- Copyright : (c) 2009, 2011, 2012 Bryan O'Sullivan +-- License : BSD3 +-- +-- Maintainer : bos@serpentine.com +-- Stability : experimental +-- Portability : portable +-- +-- Internal module with implementation of special functions. +module Numeric.SpecFunctions.Internal + ( module Numeric.SpecFunctions.Internal + , Compat.log1p + , Compat.expm1 + ) where + +import Data.Bits ((.&.), (.|.), shiftR) +import Data.Int (Int64) +import Data.Default.Class +import qualified Data.Vector.Unboxed as U +import Data.Vector.Unboxed ((!)) +import Text.Printf + +import Numeric.Polynomial.Chebyshev (chebyshevBroucke) +import Numeric.Polynomial (evaluatePolynomial, evaluatePolynomialL, evaluateEvenPolynomialL + ,evaluateOddPolynomialL) +import Numeric.RootFinding (Root(..), newtonRaphson, NewtonParam(..), Tolerance(..)) +import Numeric.Series +import Numeric.MathFunctions.Constants +import Numeric.SpecFunctions.Compat (log1p) +import qualified Numeric.SpecFunctions.Compat as Compat + +---------------------------------------------------------------- +-- 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 +erf = Compat.erf +{-# INLINE 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 +erfc = Compat.erfc +{-# INLINE erfc #-} + +-- | Inverse of 'erf'. +invErf :: Double -- ^ /p/ ∈ [-1,1] + -> Double +invErf p + | p == 1 = m_pos_inf + | p == -1 = m_neg_inf + | p < 1 && p > -1 = if p > 0 then r else -r + | otherwise = error "invErf: p must in [-1,1] range" + where + -- We solve equation same as in invErfc. We're able to ruse same + -- Halley step by solving equation: + -- > pp - erf x = 0 + -- instead of + -- > erf x - pp = 0 + pp = abs p + r = step $ step $ guessInvErfc $ 1 - pp + step x = invErfcHalleyStep (pp - erf x) x + +-- | 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 | p <= 1 = p + | otherwise = 2 - p + -- We perform 2 Halley steps in order to get to solution + r = step $ step $ guessInvErfc pp + step x = invErfcHalleyStep (erfc x - pp) x + +-- Initial guess for invErfc & invErf +guessInvErfc :: Double -> Double +guessInvErfc p + = -0.70711 * ((2.30753 + t * 0.27061) / (1 + t * (0.99229 + t * 0.04481)) - t) + where + t = sqrt $ -2 * log( 0.5 * p) + +-- Halley step for solving invErfc +invErfcHalleyStep :: Double -> Double -> Double +invErfcHalleyStep err x + = x + err / (1.12837916709551257 * exp(-x * x) - x * err) + +---------------------------------------------------------------- +-- 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 ∞ if the input is outside of the range (0 < /x/ +-- ≤ 1e305). +logGamma :: Double -> Double +logGamma z + | z <= 0 = m_pos_inf + -- For very small values z we can just use Laurent expansion + | z < m_sqrt_eps = log (1/z - m_eulerMascheroni) + -- For z<1 we use recurrence. Γ(z+1) = z·Γ(z) Note that in order to + -- avoid precision loss we have to compute parameter to + -- approximations here: + -- + -- > (z + 1) - 1 = z + -- > (z + 1) - 2 = z - 1 + -- + -- Simple passing (z + 1) to piecewise approximations and computing + -- difference leads to bad loss of precision near 1. + -- This is reason lgamma1_15 & lgamma15_2 have three parameters + | z < 0.5 = lgamma1_15 z (z - 1) - log z + | z < 1 = lgamma15_2 z (z - 1) - log z + -- Piecewise polynomial approximations + | z <= 1.5 = lgamma1_15 (z - 1) (z - 2) + | z < 2 = lgamma15_2 (z - 1) (z - 2) + | z < 15 = lgammaSmall z + -- Otherwise we switch to Lanczos approximation + | otherwise = lanczosApprox z + + +-- | Synonym for 'logGamma'. Retained for compatibility +logGammaL :: Double -> Double +logGammaL = logGamma +{-# DEPRECATED logGammaL "Use logGamma instead" #-} + + + +-- Polynomial expansion used in interval (1,1.5] +-- +-- > logΓ(z) = (z-1)(z-2)(Y + R(z-1)) +lgamma1_15 :: Double -> Double -> Double +lgamma1_15 zm1 zm2 + = r * y + r * ( evaluatePolynomial zm1 tableLogGamma_1_15P + / evaluatePolynomial zm1 tableLogGamma_1_15Q + ) + where + r = zm1 * zm2 + y = 0.52815341949462890625 + +tableLogGamma_1_15P,tableLogGamma_1_15Q :: U.Vector Double +tableLogGamma_1_15P = U.fromList + [ 0.490622454069039543534e-1 + , -0.969117530159521214579e-1 + , -0.414983358359495381969e0 + , -0.406567124211938417342e0 + , -0.158413586390692192217e0 + , -0.240149820648571559892e-1 + , -0.100346687696279557415e-2 + ] +{-# NOINLINE tableLogGamma_1_15P #-} +tableLogGamma_1_15Q = U.fromList + [ 1 + , 0.302349829846463038743e1 + , 0.348739585360723852576e1 + , 0.191415588274426679201e1 + , 0.507137738614363510846e0 + , 0.577039722690451849648e-1 + , 0.195768102601107189171e-2 + ] +{-# NOINLINE tableLogGamma_1_15Q #-} + + + +-- Polynomial expansion used in interval (1.5,2) +-- +-- > logΓ(z) = (2-z)(1-z)(Y + R(2-z)) +lgamma15_2 :: Double -> Double -> Double +lgamma15_2 zm1 zm2 + = r * y + r * ( evaluatePolynomial (-zm2) tableLogGamma_15_2P + / evaluatePolynomial (-zm2) tableLogGamma_15_2Q + ) + where + r = zm1 * zm2 + y = 0.452017307281494140625 + +tableLogGamma_15_2P,tableLogGamma_15_2Q :: U.Vector Double +tableLogGamma_15_2P = U.fromList + [ -0.292329721830270012337e-1 + , 0.144216267757192309184e0 + , -0.142440390738631274135e0 + , 0.542809694055053558157e-1 + , -0.850535976868336437746e-2 + , 0.431171342679297331241e-3 + ] +{-# NOINLINE tableLogGamma_15_2P #-} +tableLogGamma_15_2Q = U.fromList + [ 1 + , -0.150169356054485044494e1 + , 0.846973248876495016101e0 + , -0.220095151814995745555e0 + , 0.25582797155975869989e-1 + , -0.100666795539143372762e-2 + , -0.827193521891290553639e-6 + ] +{-# NOINLINE tableLogGamma_15_2Q #-} + + + +-- Polynomial expansion used in interval (2,3) +-- +-- > logΓ(z) = (z - 2)(z + 1)(Y + R(z-2)) +lgamma2_3 :: Double -> Double +lgamma2_3 z + = r * y + r * ( evaluatePolynomial zm2 tableLogGamma_2_3P + / evaluatePolynomial zm2 tableLogGamma_2_3Q + ) + where + r = zm2 * (z + 1) + zm2 = z - 2 + y = 0.158963680267333984375e0 + + +tableLogGamma_2_3P,tableLogGamma_2_3Q :: U.Vector Double +tableLogGamma_2_3P = U.fromList + [ -0.180355685678449379109e-1 + , 0.25126649619989678683e-1 + , 0.494103151567532234274e-1 + , 0.172491608709613993966e-1 + , -0.259453563205438108893e-3 + , -0.541009869215204396339e-3 + , -0.324588649825948492091e-4 + ] +{-# NOINLINE tableLogGamma_2_3P #-} +tableLogGamma_2_3Q = U.fromList + [ 1 + , 0.196202987197795200688e1 + , 0.148019669424231326694e1 + , 0.541391432071720958364e0 + , 0.988504251128010129477e-1 + , 0.82130967464889339326e-2 + , 0.224936291922115757597e-3 + , -0.223352763208617092964e-6 + ] +{-# NOINLINE tableLogGamma_2_3Q #-} + + + +-- For small z we can just use Gamma function recurrence and reduce +-- problem to interval [2,3] and use polynomial approximation +-- there. Surprisingly it gives very good precision +lgammaSmall :: Double -> Double +lgammaSmall = go 0 + where + go acc z | z < 3 = acc + lgamma2_3 z + | otherwise = go (acc + log zm1) zm1 + where + zm1 = z - 1 + + +-- Lanczos approximation for gamma function. +-- +-- > Γ(z) = sqrt(2π)(z + g - 0.5)^(z - 0.5)·exp{-(z + g - 0.5)}·A_g(z) +-- +-- Coefficients are taken from boost. Constants are absorbed into +-- polynomial's coefficients. +lanczosApprox :: Double -> Double +lanczosApprox z + = (log (z + g - 0.5) - 1) * (z - 0.5) + + log (evalRatio tableLanczos z) + where + g = 6.024680040776729583740234375 + +tableLanczos :: U.Vector (Double,Double) +{-# NOINLINE tableLanczos #-} +tableLanczos = U.fromList + [ (56906521.91347156388090791033559122686859 , 0) + , (103794043.1163445451906271053616070238554 , 39916800) + , (86363131.28813859145546927288977868422342 , 120543840) + , (43338889.32467613834773723740590533316085 , 150917976) + , (14605578.08768506808414169982791359218571 , 105258076) + , (3481712.15498064590882071018964774556468 , 45995730) + , (601859.6171681098786670226533699352302507 , 13339535) + , (75999.29304014542649875303443598909137092 , 2637558) + , (6955.999602515376140356310115515198987526 , 357423) + , (449.9445569063168119446858607650988409623 , 32670) + , (19.51992788247617482847860966235652136208 , 1925) + , (0.5098416655656676188125178644804694509993 , 66) + , (0.006061842346248906525783753964555936883222 , 1) + ] + +-- Evaluate rational function. Polynomials in both numerator and +-- denominator must have same order. Function seems to be too specific +-- so it's not exposed +-- +-- Special care taken in order to avoid overflow for large values of x +evalRatio :: U.Vector (Double,Double) -> Double -> Double +evalRatio coef x + | x > 1 = fini $ U.foldl' stepL (L 0 0) coef + | otherwise = fini $ U.foldr' stepR (L 0 0) coef + where + fini (L num den) = num / den + stepR (a,b) (L num den) = L (num * x + a) (den * x + b) + stepL (L num den) (a,b) = L (num * rx + a) (den * rx + b) + rx = recip x + + + +-- | +-- 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': +-- +-- \[ +-- \log\Gamma(x) = \frac{1}{2}\log(2\pi) + (x-\frac{1}{2})\log x - x + \operatorname{logGammaCorrection}(x) +-- \] +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 (logGamma 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 and [NOTE: + -- incompleteGamma.taylorP] + factorP + | a < 10 = x ** a + / (exp x * exp (logGamma (a + 1))) + | a < 1182.5 = (x * exp 1 / a) ** a + / exp x + / sqrt (2*pi*a) + / exp (logGammaCorrection a) + | otherwise = (x * exp 1 / a * exp (-x/a)) ** a + / sqrt (2*pi*a) + / exp (logGammaCorrection a) + taylorSeriesP + = sumPowerSeries x (scanSequence (/) 1 $ enumSequenceFrom (a+1)) + * factorP + -- Series for 1-Q(a,x). See [Temme1994] Eq. 5.5 + taylorSeriesComplQ + = sumPowerSeries (-x) (scanSequence (/) 1 (enumSequenceFrom 1) / enumSequenceFrom a) + * x**a / exp(logGamma 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 initial 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. +-- +-- \[ +-- B(a,b) = \int_0^1 t^{a-1}(1-t)^{b-1}\,dt = \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)} +-- \] +logBeta + :: Double -- ^ /a/ > 0 + -> Double -- ^ /b/ > 0 + -> Double +logBeta a b + | p < 0 = m_NaN + | p == 0 = m_pos_inf + | p >= 10 = allStirling + | q >= 10 = twoStirling + -- This order of summands marginally improves precision + | otherwise = logGamma p + (logGamma q - logGamma pq) + where + p = min a b + q = max a b + ppq = p / pq + pq = p + q + -- When both parameters are large than 10 we can use Stirling + -- approximation with correction. It's more precise than sum of + -- logarithms of gamma functions + allStirling + = log q * (-0.5) + + m_ln_sqrt_2_pi + + logGammaCorrection p + + (logGammaCorrection q - logGammaCorrection pq) + + (p - 0.5) * log ppq + + q * log1p(-ppq) + -- Otherwise only two of three gamma functions use Stirling + -- approximation + twoStirling + = logGamma p + + (logGammaCorrection q - logGammaCorrection pq) + + p + - p * log pq + + (q - 0.5) * log1p(-ppq) + + +-- | Regularized incomplete beta function. +-- +-- \[ +-- I(x;a,b) = \frac{1}{B(a,b)} \int_0^x t^{a-1}(1-t)^{b-1}\,dt +-- \] +-- +-- 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 quadrature. +-- +-- 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 referred 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 def{newtonTol=RelTol 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 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 + +-- | /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 > 170 = m_pos_inf + | otherwise = U.unsafeIndex factorialTable n + +-- | 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" + -- For smaller inputs we just look up table + | n <= 170 = log $ U.unsafeIndex factorialTable (fromIntegral n) + -- Otherwise we use asymptotic Stirling's series. Number of terms + -- necessary depends on the argument. + | n < 1500 = stirling + rx * ((1/12) - (1/360)*rx*rx) + | otherwise = stirling + (1/12)*rx + where + stirling = (x - 0.5) * log x - x + m_ln_sqrt_2_pi + x = fromIntegral n + 1 + rx = 1 / x +{-# 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 + +factorialTable :: U.Vector Double +{-# NOINLINE factorialTable #-} +factorialTable = U.fromListN 171 + [ 1.0 + , 1.0 + , 2.0 + , 6.0 + , 24.0 + , 120.0 + , 720.0 + , 5040.0 + , 40320.0 + , 362880.0 + , 3628800.0 + , 3.99168e7 + , 4.790016e8 + , 6.2270208e9 + , 8.71782912e10 + , 1.307674368e12 + , 2.0922789888e13 + , 3.55687428096e14 + , 6.402373705728e15 + , 1.21645100408832e17 + , 2.43290200817664e18 + , 5.109094217170944e19 + , 1.1240007277776077e21 + , 2.5852016738884974e22 + , 6.204484017332394e23 + , 1.5511210043330984e25 + , 4.032914611266056e26 + , 1.0888869450418352e28 + , 3.0488834461171384e29 + , 8.841761993739702e30 + , 2.6525285981219103e32 + , 8.222838654177922e33 + , 2.631308369336935e35 + , 8.683317618811886e36 + , 2.9523279903960412e38 + , 1.0333147966386144e40 + , 3.719933267899012e41 + , 1.3763753091226343e43 + , 5.23022617466601e44 + , 2.0397882081197442e46 + , 8.159152832478977e47 + , 3.3452526613163803e49 + , 1.4050061177528798e51 + , 6.041526306337383e52 + , 2.6582715747884485e54 + , 1.1962222086548019e56 + , 5.5026221598120885e57 + , 2.5862324151116818e59 + , 1.2413915592536073e61 + , 6.082818640342675e62 + , 3.0414093201713376e64 + , 1.5511187532873822e66 + , 8.065817517094388e67 + , 4.2748832840600255e69 + , 2.308436973392414e71 + , 1.2696403353658275e73 + , 7.109985878048634e74 + , 4.0526919504877214e76 + , 2.3505613312828785e78 + , 1.386831185456898e80 + , 8.32098711274139e81 + , 5.075802138772247e83 + , 3.146997326038793e85 + , 1.9826083154044399e87 + , 1.2688693218588415e89 + , 8.24765059208247e90 + , 5.44344939077443e92 + , 3.647111091818868e94 + , 2.4800355424368305e96 + , 1.711224524281413e98 + , 1.197857166996989e100 + , 8.504785885678623e101 + , 6.1234458376886085e103 + , 4.470115461512684e105 + , 3.307885441519386e107 + , 2.4809140811395396e109 + , 1.88549470166605e111 + , 1.4518309202828586e113 + , 1.1324281178206297e115 + , 8.946182130782974e116 + , 7.15694570462638e118 + , 5.797126020747368e120 + , 4.753643337012841e122 + , 3.9455239697206583e124 + , 3.314240134565353e126 + , 2.81710411438055e128 + , 2.422709538367273e130 + , 2.1077572983795275e132 + , 1.8548264225739844e134 + , 1.650795516090846e136 + , 1.4857159644817613e138 + , 1.352001527678403e140 + , 1.2438414054641305e142 + , 1.1567725070816416e144 + , 1.087366156656743e146 + , 1.0329978488239058e148 + , 9.916779348709496e149 + , 9.619275968248211e151 + , 9.426890448883246e153 + , 9.332621544394413e155 + , 9.332621544394415e157 + , 9.425947759838358e159 + , 9.614466715035125e161 + , 9.902900716486179e163 + , 1.0299016745145626e166 + , 1.0813967582402908e168 + , 1.1462805637347082e170 + , 1.2265202031961378e172 + , 1.3246418194518288e174 + , 1.4438595832024934e176 + , 1.5882455415227428e178 + , 1.7629525510902446e180 + , 1.974506857221074e182 + , 2.2311927486598134e184 + , 2.543559733472187e186 + , 2.9250936934930154e188 + , 3.393108684451898e190 + , 3.9699371608087206e192 + , 4.68452584975429e194 + , 5.574585761207606e196 + , 6.689502913449126e198 + , 8.094298525273443e200 + , 9.875044200833601e202 + , 1.214630436702533e205 + , 1.5061417415111406e207 + , 1.8826771768889257e209 + , 2.372173242880047e211 + , 3.0126600184576594e213 + , 3.856204823625804e215 + , 4.974504222477286e217 + , 6.466855489220473e219 + , 8.471580690878819e221 + , 1.1182486511960041e224 + , 1.4872707060906857e226 + , 1.9929427461615188e228 + , 2.6904727073180504e230 + , 3.6590428819525483e232 + , 5.012888748274991e234 + , 6.917786472619488e236 + , 9.615723196941088e238 + , 1.3462012475717523e241 + , 1.898143759076171e243 + , 2.6953641378881624e245 + , 3.8543707171800725e247 + , 5.5502938327393044e249 + , 8.047926057471992e251 + , 1.1749972043909107e254 + , 1.7272458904546386e256 + , 2.5563239178728654e258 + , 3.808922637630569e260 + , 5.713383956445854e262 + , 8.62720977423324e264 + , 1.3113358856834524e267 + , 2.0063439050956823e269 + , 3.0897696138473508e271 + , 4.789142901463393e273 + , 7.471062926282894e275 + , 1.1729568794264143e278 + , 1.8532718694937346e280 + , 2.946702272495038e282 + , 4.714723635992061e284 + , 7.590705053947218e286 + , 1.2296942187394494e289 + , 2.0044015765453023e291 + , 3.287218585534296e293 + , 5.423910666131589e295 + , 9.003691705778436e297 + , 1.5036165148649988e300 + , 2.526075744973198e302 + , 4.269068009004705e304 + , 7.257415615307998e306 + ] + + +-- [NOTE: incompleteGamma.taylorP] +-- +-- Incompltete gamma uses several algorithms for different parts of +-- parameter space. Most troublesome is P(a,x) Taylor series +-- [Temme1994,Eq.5.5] which requires to evaluate rather nasty +-- expression: +-- +-- x^a x^a +-- ------------- = ------------- +-- exp(x)·Γ(a+1) exp(x)·a·Γ(a) +-- +-- Conditions: +-- | 0.5 155 - x^a could overflow +-- · a > 1182.5 - exp(x) could overflow +-- +-- Usual way to avoid overflow problem is to perform calculations in +-- the log domain. It however doesn't work very well in this case +-- since we encounter catastrophic cancellations and could easily lose +-- up to 6(!) digits for large `a`. +-- +-- So we take another approach and use Stirling approximation with +-- correction (logGammaCorrection). +-- +-- x^a / x·e \^a 1 +-- ≈ ------------------------- = | --- | · ---------------- +-- exp(x)·sqrt(2πa)·(a/e)^a) \ a / exp(x)·sqrt(2πa) +-- +-- We're using this approach as soon as logGammaCorrection starts +-- working (a>10) because we don't have implementation for gamma +-- function and exp(logGamma z) results in errors for large a. +-- +-- Once we get into region when exp(x) could overflow we rewrite +-- expression above once more: +-- +-- / x·e \^a 1 +-- | --- · e^(-x/a) | · --------- +-- \ a / sqrt(2πa) +-- +-- This approach doesn't work very well but it's still big improvement +-- over calculations in the log domain. diff --git a/Numeric/Sum.hs b/Numeric/Sum.hs --- a/Numeric/Sum.hs +++ b/Numeric/Sum.hs @@ -1,5 +1,5 @@ -{-# LANGUAGE BangPatterns, CPP, DeriveDataTypeable, FlexibleContexts, - MultiParamTypeClasses, TemplateHaskell, TypeFamilies #-} +{-# LANGUAGE BangPatterns, DeriveDataTypeable, FlexibleContexts, + MultiParamTypeClasses, TypeFamilies #-} {-# OPTIONS_GHC -fno-warn-name-shadowing #-} -- | -- Module : Numeric.Sum @@ -53,14 +53,14 @@ import Control.DeepSeq (NFData(..)) import Data.Bits (shiftR) import Data.Data (Typeable, Data) -import Data.Vector.Generic (Vector(..), foldl') -import Data.Vector.Unboxed.Deriving (derivingUnbox) --- Needed for GHC 7.2 & 7.4 to derive Unbox instances +import Data.Semigroup (Semigroup(..)) -- Needed for GHC <8.4 +import Data.Vector.Generic (Vector(..)) import Data.Vector.Generic.Mutable (MVector(..)) import qualified Data.Foldable as F import qualified Data.Vector as V import qualified Data.Vector.Generic as G +import qualified Data.Vector.Generic.Mutable as GM import qualified Data.Vector.Unboxed as U -- | A class for summation of floating point numbers. @@ -74,7 +74,7 @@ -- | Sum a collection of values. -- -- Example: - -- @foo = 'sum' 'kbn' [1,2,3]@ + -- @foo = 'Numeric.Sum.sum' 'kbn' [1,2,3]@ sum :: (F.Foldable f) => (s -> Double) -> f Double -> Double sum f = f . F.foldl' add zero {-# INLINE sum #-} @@ -94,11 +94,54 @@ data KahanSum = KahanSum {-# UNPACK #-} !Double {-# UNPACK #-} !Double deriving (Eq, Show, Typeable, Data) -derivingUnbox "KahanSum" - [t| KahanSum -> (Double, Double) |] - [| \ (KahanSum a b) -> (a, b) |] - [| \ (a, b) -> KahanSum a b |] +instance U.Unbox KahanSum +newtype instance U.MVector s KahanSum = MV_KahanSum (U.MVector s (Double, Double)) +instance MVector U.MVector KahanSum where + {-# INLINE GM.basicLength #-} + {-# INLINE GM.basicUnsafeSlice #-} + {-# INLINE basicOverlaps #-} + {-# INLINE basicUnsafeNew #-} + {-# INLINE basicInitialize #-} + {-# INLINE basicUnsafeReplicate #-} + {-# INLINE basicUnsafeRead #-} + {-# INLINE basicUnsafeWrite #-} + {-# INLINE basicClear #-} + {-# INLINE basicSet #-} + {-# INLINE GM.basicUnsafeCopy #-} + {-# INLINE basicUnsafeMove #-} + {-# INLINE basicUnsafeGrow #-} + basicLength (MV_KahanSum mvec) = GM.basicLength mvec + basicUnsafeSlice idx len (MV_KahanSum mvec) = MV_KahanSum (GM.basicUnsafeSlice idx len mvec) + basicOverlaps (MV_KahanSum mvec) (MV_KahanSum mvec') = basicOverlaps mvec mvec' + basicUnsafeNew len = MV_KahanSum <$> basicUnsafeNew len + basicInitialize (MV_KahanSum mvec) = basicInitialize mvec + basicUnsafeReplicate len val = MV_KahanSum <$> basicUnsafeReplicate len ((\ (KahanSum a b) -> (a, b)) val) + basicUnsafeRead (MV_KahanSum mvec) idx = (\ (a, b) -> KahanSum a b) <$> basicUnsafeRead mvec idx + basicUnsafeWrite (MV_KahanSum mvec) idx val = basicUnsafeWrite mvec idx ((\ (KahanSum a b) -> (a, b)) val) + basicClear (MV_KahanSum mvec) = basicClear mvec + basicSet (MV_KahanSum mvec) val = basicSet mvec ((\ (KahanSum a b) -> (a, b)) val) + basicUnsafeCopy (MV_KahanSum mvec) (MV_KahanSum mvec') = GM.basicUnsafeCopy mvec mvec' + basicUnsafeMove (MV_KahanSum mvec) (MV_KahanSum mvec') = basicUnsafeMove mvec mvec' + basicUnsafeGrow (MV_KahanSum mvec) len = MV_KahanSum <$> basicUnsafeGrow mvec len +newtype instance U.Vector KahanSum = V_KahanSum (U.Vector (Double, Double)) +instance Vector U.Vector KahanSum where + {-# INLINE basicUnsafeFreeze #-} + {-# INLINE basicUnsafeThaw #-} + {-# INLINE G.basicLength #-} + {-# INLINE G.basicUnsafeSlice #-} + {-# INLINE basicUnsafeIndexM #-} + {-# INLINE G.basicUnsafeCopy #-} + {-# INLINE elemseq #-} + basicUnsafeFreeze (MV_KahanSum mvec) = V_KahanSum <$> basicUnsafeFreeze mvec + basicUnsafeThaw (V_KahanSum vec) = MV_KahanSum <$> basicUnsafeThaw vec + basicLength (V_KahanSum vec) = G.basicLength vec + basicUnsafeSlice idx len (V_KahanSum vec) = V_KahanSum (G.basicUnsafeSlice idx len vec) + basicUnsafeIndexM (V_KahanSum vec) idx = (\ (a, b) -> KahanSum a b) <$> basicUnsafeIndexM vec idx + basicUnsafeCopy (MV_KahanSum mvec) (V_KahanSum vec) = G.basicUnsafeCopy mvec vec + elemseq (V_KahanSum vec) val = elemseq vec ((\ (KahanSum a b) -> (a, b)) val) + + instance Summation KahanSum where zero = KahanSum 0 0 add = kahanAdd @@ -106,6 +149,15 @@ instance NFData KahanSum where rnf !_ = () +-- | @since 0.3.0.0 +instance Monoid KahanSum where + mempty = zero + mappend = (<>) + +-- | @since 0.3.0.0 +instance Semigroup KahanSum where + s <> KahanSum s' _ = add s s' + kahanAdd :: KahanSum -> Double -> KahanSum kahanAdd (KahanSum sum c) x = KahanSum sum' c' where sum' = sum + y @@ -122,11 +174,54 @@ data KBNSum = KBNSum {-# UNPACK #-} !Double {-# UNPACK #-} !Double deriving (Eq, Show, Typeable, Data) -derivingUnbox "KBNSum" - [t| KBNSum -> (Double, Double) |] - [| \ (KBNSum a b) -> (a, b) |] - [| \ (a, b) -> KBNSum a b |] +instance U.Unbox KBNSum +newtype instance U.MVector s KBNSum = MV_KBNSum (U.MVector s (Double, Double)) +instance MVector U.MVector KBNSum where + {-# INLINE GM.basicLength #-} + {-# INLINE GM.basicUnsafeSlice #-} + {-# INLINE basicOverlaps #-} + {-# INLINE basicUnsafeNew #-} + {-# INLINE basicInitialize #-} + {-# INLINE basicUnsafeReplicate #-} + {-# INLINE basicUnsafeRead #-} + {-# INLINE basicUnsafeWrite #-} + {-# INLINE basicClear #-} + {-# INLINE basicSet #-} + {-# INLINE GM.basicUnsafeCopy #-} + {-# INLINE basicUnsafeMove #-} + {-# INLINE basicUnsafeGrow #-} + basicLength (MV_KBNSum mvec) = GM.basicLength mvec + basicUnsafeSlice idx len (MV_KBNSum mvec) = MV_KBNSum (GM.basicUnsafeSlice idx len mvec) + basicOverlaps (MV_KBNSum mvec) (MV_KBNSum mvec') = basicOverlaps mvec mvec' + basicUnsafeNew len = MV_KBNSum <$> basicUnsafeNew len + basicInitialize (MV_KBNSum mvec) = basicInitialize mvec + basicUnsafeReplicate len val = MV_KBNSum <$> basicUnsafeReplicate len ((\ (KBNSum a b) -> (a, b)) val) + basicUnsafeRead (MV_KBNSum mvec) idx = (\ (a, b) -> KBNSum a b) <$> basicUnsafeRead mvec idx + basicUnsafeWrite (MV_KBNSum mvec) idx val = basicUnsafeWrite mvec idx ((\ (KBNSum a b) -> (a, b)) val) + basicClear (MV_KBNSum mvec) = basicClear mvec + basicSet (MV_KBNSum mvec) val = basicSet mvec ((\ (KBNSum a b) -> (a, b)) val) + basicUnsafeCopy (MV_KBNSum mvec) (MV_KBNSum mvec') = GM.basicUnsafeCopy mvec mvec' + basicUnsafeMove (MV_KBNSum mvec) (MV_KBNSum mvec') = basicUnsafeMove mvec mvec' + basicUnsafeGrow (MV_KBNSum mvec) len = MV_KBNSum <$> basicUnsafeGrow mvec len +newtype instance U.Vector KBNSum = V_KBNSum (U.Vector (Double, Double)) +instance Vector U.Vector KBNSum where + {-# INLINE basicUnsafeFreeze #-} + {-# INLINE basicUnsafeThaw #-} + {-# INLINE G.basicLength #-} + {-# INLINE G.basicUnsafeSlice #-} + {-# INLINE basicUnsafeIndexM #-} + {-# INLINE G.basicUnsafeCopy #-} + {-# INLINE elemseq #-} + basicUnsafeFreeze (MV_KBNSum mvec) = V_KBNSum <$> basicUnsafeFreeze mvec + basicUnsafeThaw (V_KBNSum vec) = MV_KBNSum <$> basicUnsafeThaw vec + basicLength (V_KBNSum vec) = G.basicLength vec + basicUnsafeSlice idx len (V_KBNSum vec) = V_KBNSum (G.basicUnsafeSlice idx len vec) + basicUnsafeIndexM (V_KBNSum vec) idx = (\ (a, b) -> KBNSum a b) <$> basicUnsafeIndexM vec idx + basicUnsafeCopy (MV_KBNSum mvec) (V_KBNSum vec) = G.basicUnsafeCopy mvec vec + elemseq (V_KBNSum vec) val = elemseq vec ((\ (KBNSum a b) -> (a, b)) val) + + instance Summation KBNSum where zero = KBNSum 0 0 add = kbnAdd @@ -134,6 +229,16 @@ instance NFData KBNSum where rnf !_ = () +-- | @since 0.3.0.0 +instance Monoid KBNSum where + mempty = zero + mappend = (<>) + +-- | @since 0.3.0.0 +instance Semigroup KBNSum where + s <> KBNSum s' c' = add (add s s') c' + + kbnAdd :: KBNSum -> Double -> KBNSum kbnAdd (KBNSum sum c) x = KBNSum sum' c' where c' | abs sum >= abs x = c + ((sum - sum') + x) @@ -157,11 +262,53 @@ {-# UNPACK #-} !Double deriving (Eq, Show, Typeable, Data) -derivingUnbox "KB2Sum" - [t| KB2Sum -> (Double, Double, Double) |] - [| \ (KB2Sum a b c) -> (a, b, c) |] - [| \ (a, b, c) -> KB2Sum a b c |] +instance U.Unbox KB2Sum +newtype instance U.MVector s KB2Sum = MV_KB2Sum (U.MVector s (Double, Double, Double)) +instance MVector U.MVector KB2Sum where + {-# INLINE GM.basicLength #-} + {-# INLINE GM.basicUnsafeSlice #-} + {-# INLINE basicOverlaps #-} + {-# INLINE basicUnsafeNew #-} + {-# INLINE basicInitialize #-} + {-# INLINE basicUnsafeReplicate #-} + {-# INLINE basicUnsafeRead #-} + {-# INLINE basicUnsafeWrite #-} + {-# INLINE basicClear #-} + {-# INLINE basicSet #-} + {-# INLINE GM.basicUnsafeCopy #-} + {-# INLINE basicUnsafeMove #-} + {-# INLINE basicUnsafeGrow #-} + basicLength (MV_KB2Sum mvec) = GM.basicLength mvec + basicUnsafeSlice idx len (MV_KB2Sum mvec) = MV_KB2Sum (GM.basicUnsafeSlice idx len mvec) + basicOverlaps (MV_KB2Sum mvec) (MV_KB2Sum mvec') = basicOverlaps mvec mvec' + basicUnsafeNew len = MV_KB2Sum <$> basicUnsafeNew len + basicInitialize (MV_KB2Sum mvec) = basicInitialize mvec + basicUnsafeReplicate len val = MV_KB2Sum <$> basicUnsafeReplicate len ((\ (KB2Sum a b c) -> (a, b, c)) val) + basicUnsafeRead (MV_KB2Sum mvec) idx = (\ (a, b, c) -> KB2Sum a b c) <$> basicUnsafeRead mvec idx + basicUnsafeWrite (MV_KB2Sum mvec) idx val = basicUnsafeWrite mvec idx ((\ (KB2Sum a b c) -> (a, b, c)) val) + basicClear (MV_KB2Sum mvec) = basicClear mvec + basicSet (MV_KB2Sum mvec) val = basicSet mvec ((\ (KB2Sum a b c) -> (a, b, c)) val) + basicUnsafeCopy (MV_KB2Sum mvec) (MV_KB2Sum mvec') = GM.basicUnsafeCopy mvec mvec' + basicUnsafeMove (MV_KB2Sum mvec) (MV_KB2Sum mvec') = basicUnsafeMove mvec mvec' + basicUnsafeGrow (MV_KB2Sum mvec) len = MV_KB2Sum <$> basicUnsafeGrow mvec len +newtype instance U.Vector KB2Sum = V_KB2Sum (U.Vector (Double, Double, Double)) +instance Vector U.Vector KB2Sum where + {-# INLINE basicUnsafeFreeze #-} + {-# INLINE basicUnsafeThaw #-} + {-# INLINE G.basicLength #-} + {-# INLINE G.basicUnsafeSlice #-} + {-# INLINE basicUnsafeIndexM #-} + {-# INLINE G.basicUnsafeCopy #-} + {-# INLINE elemseq #-} + basicUnsafeFreeze (MV_KB2Sum mvec) = V_KB2Sum <$> basicUnsafeFreeze mvec + basicUnsafeThaw (V_KB2Sum vec) = MV_KB2Sum <$> basicUnsafeThaw vec + basicLength (V_KB2Sum vec) = G.basicLength vec + basicUnsafeSlice idx len (V_KB2Sum vec) = V_KB2Sum (G.basicUnsafeSlice idx len vec) + basicUnsafeIndexM (V_KB2Sum vec) idx = (\ (a, b, c) -> KB2Sum a b c) <$> basicUnsafeIndexM vec idx + basicUnsafeCopy (MV_KB2Sum mvec) (V_KB2Sum vec) = G.basicUnsafeCopy mvec vec + elemseq (V_KB2Sum vec) val = elemseq vec ((\ (KB2Sum a b c) -> (a, b, c)) val) + instance Summation KB2Sum where zero = KB2Sum 0 0 0 add = kb2Add @@ -169,6 +316,16 @@ instance NFData KB2Sum where rnf !_ = () +-- | @since 0.3.0.0 +instance Monoid KB2Sum where + mempty = zero + mappend = (<>) + +-- | @since 0.3.0.0 +instance Semigroup KB2Sum where + s <> KB2Sum s' c' cc' = add (add (add s s') c') cc' + + kb2Add :: KB2Sum -> Double -> KB2Sum kb2Add (KB2Sum sum c cc) x = KB2Sum sum' c' cc' where sum' = sum + x @@ -185,7 +342,7 @@ -- | /O(n)/ Sum a vector of values. sumVector :: (Vector v Double, Summation s) => (s -> Double) -> v Double -> Double -sumVector f = f . foldl' add zero +sumVector f = f . G.foldl' add zero {-# INLINE sumVector #-} -- | /O(n)/ Sum a vector of values using pairwise summation. @@ -223,7 +380,7 @@ -- where s' = 'add' s x -- @ -- --- In most instances, you can simply use the much more general 'sum' +-- In most instances, you can simply use the much more general 'Numeric.Sum.sum' -- function instead of writing a summation function by hand. -- -- @ @@ -231,7 +388,7 @@ -- import Prelude hiding (sum) -- -- -- betterSumList :: [Double] -> Double --- betterSumList xs = 'sum' 'kbn' xs +-- betterSumList xs = 'Numeric.Sum.sum' 'kbn' xs -- @ -- Note well the use of 'seq' in the example above to force the diff --git a/README.markdown b/README.markdown --- a/README.markdown +++ b/README.markdown @@ -1,7 +1,21 @@ -# math-functions: efficient, special purpose mathematical functions +# math-functions: collection of tools for numeric computations -This package provides a number of special-purpose mathematical -functions used in statistical and numerical computing. +[![Build Status](https://travis-ci.org/Shimuuar/math-functions.png?branch=master)](https://travis-ci.org/Shimuuar/math-functions) +[![Build status](https://ci.appveyor.com/api/projects/status/6xexxj9g6rnbg2q4/branch/master?svg=true)](https://ci.appveyor.com/project/Shimuuar/math-functions/branch/master) + +This package provides collection of various tools for numeric +computations. Namely: + + - Number pure haskell implementations of special function which are used in + statistical and numerical computing. + + - Compensated summation (Kahan summation) which allows to + + - Root finding for functions of single real variable + + - Series summation + + - Functions for comparing IEEE754 numbers Where possible, we give citations and computational complexity estimates for the algorithms used. diff --git a/bench/bench.hs b/bench/bench.hs new file mode 100644 --- /dev/null +++ b/bench/bench.hs @@ -0,0 +1,126 @@ +{-# LANGUAGE NumDecimals #-} +import Data.Default.Class +import qualified Data.Vector.Unboxed as U +import Text.Printf +import Test.Tasty.Bench +import System.Random (randomIO) + +import qualified Numeric.Sum as Sum +import Numeric.SpecFunctions +import Numeric.Polynomial +import Numeric.RootFinding + + + +-- Uniformly sample logGamma performance between 10^-6 to 10^6 +benchmarkLogGamma :: (Double -> Double) -> [Benchmark] +benchmarkLogGamma logG = + [ bench (printf "%.3g" x) $ nf logG x + | x <- [ m * 10**n | n <- [ -8 .. 8 ] + , m <- [ 10**(i / tics) | i <- [0 .. tics-1] ] + ] + ] + where tics = 3 +{-# INLINE benchmarkLogGamma #-} + + +-- Power of polynomial to be evaluated (In other words length of coefficients vector) +coef_size :: [Int] +coef_size = [ 1,2,3,4,5,6,7,8,9 + , 10, 30 + , 100, 300 + , 1000, 3000 + , 10000, 30000 + ] +{-# INLINE coef_size #-} + +-- Precalculated coefficients +coef_list :: [U.Vector Double] +coef_list = [ U.replicate n 1.2 | n <- coef_size] +{-# NOINLINE coef_list #-} + + + +main :: IO () +main = do + v <- U.replicateM 1e6 randomIO :: IO (U.Vector Double) + defaultMain + [ bgroup "logGamma" $ + benchmarkLogGamma logGamma + , bgroup "incompleteGamma" $ + [ bench (show p) $ nf (incompleteGamma p) p + | p <- [ 0.1 + , 1, 3 + , 10, 30 + , 100, 300 + , 999, 1000 + ] + ] + , bgroup "factorial" + [ bench (show n) $ nf factorial n + | n <- [ 0, 1, 3, 6, 9, 11, 15 + , 20, 30, 40, 50, 60, 70, 80, 90, 100 + ] + ] + , bgroup "incompleteBeta" + [ bench (show (p,q,x)) $ nf (incompleteBeta p q) x + | (p,q,x) <- [ (10, 10, 0.5) + , (101, 101, 0.5) + , (1010, 1010, 0.5) + , (10100, 10100, 0.5) + , (100100, 100100, 0.5) + , (1001000, 1001000, 0.5) + , (10010000,10010000,0.5) + ] + ] + , bgroup "log1p" + [ bench (show x) $ nf log1p x + | x <- [ -0.9 + , -0.5 + , -0.1 + , 0.1 + , 0.5 + , 1 + , 10 + , 100 + ] :: [Double] + ] + , bgroup "sinc" $ + bench "sin" (nf sin (0.55 :: Double)) + : [ bench (show x) $ nf sinc x + | x <- [0, 1e-6, 1e-3, 0.5] + ] + , bgroup "erf & Co" + [ bgroup "erf" + [ bench (show x) $ nf erf x + | x <- [0, 1.1, 100, 1000] + ] + , bgroup "erfc" + [ bench (show x) $ nf erfc x + | x <- [0, 1.1, 100, 1000] + ] + , bgroup "invErfc" + [ bench (show x) $ nf erfc x + | x <- [1e-9, 1e-6, 1e-3, 0.1, 1] + ] + ] + , bgroup "expm1" + [ bench (show x) $ nf expm1 (x :: Double) + | x <- [-0.1, 0, 1, 19] + ] + , bgroup "poly" + $ [ bench ("vector_"++show (U.length coefs)) $ nf (\x -> evaluatePolynomial x coefs) (1 :: Double) + | coefs <- coef_list ] + ++ [ bench ("unpacked_"++show n) $ nf (\x -> evaluatePolynomialL x (map fromIntegral [1..n])) (1 :: Double) + | n <- coef_size ] + , bgroup "RootFinding" + [ bench "ridders sin" $ nf (ridders def (0,pi/2)) (\x -> sin x - 0.525) + , bench "newton sin" $ nf (newtonRaphson def (0,1.2,pi/2)) (\x -> (sin x - 0.525,cos x)) + ] + , bgroup "Sum" + [ bench "naive" $ whnf U.sum v + , bench "kahan" $ whnf (Sum.sumVector Sum.kahan) v + , bench "kbn" $ whnf (Sum.sumVector Sum.kbn) v + , bench "kb2" $ whnf (Sum.sumVector Sum.kb2) v + ] + ] diff --git a/benchmark/Summation.hs b/benchmark/Summation.hs deleted file mode 100644 --- a/benchmark/Summation.hs +++ /dev/null @@ -1,15 +0,0 @@ -import Criterion.Main -import Numeric.Sum as Sum -import System.Random.MWC -import qualified Data.Vector.Unboxed as U - -main = do - gen <- createSystemRandom - v <- uniformVector gen 10000000 :: IO (U.Vector Double) - defaultMain [ - bench "naive" $ whnf U.sum v - , bench "pairwise" $ whnf pairwiseSum v - , bench "kahan" $ whnf (sumVector kahan) v - , bench "kbn" $ whnf (sumVector kbn) v - , bench "kb2" $ whnf (sumVector kb2) v - ] diff --git a/benchmark/bench.hs b/benchmark/bench.hs deleted file mode 100644 --- a/benchmark/bench.hs +++ /dev/null @@ -1,89 +0,0 @@ -import Criterion.Main -import qualified Data.Vector.Unboxed as U -import Numeric.SpecFunctions -import Numeric.Polynomial -import Text.Printf - --- Uniformly sample logGamma performance between 10^-6 to 10^6 -benchmarkLogGamma logG = - [ bench (printf "%.3g" x) $ nf logG x - | x <- [ m * 10**n | n <- [ -8 .. 8 ] - , m <- [ 10**(i / tics) | i <- [0 .. tics-1] ] - ] - ] - where tics = 3 -{-# INLINE benchmarkLogGamma #-} - - --- Power of polynomial to be evaluated (In other words length of coefficients vector) -coef_size :: [Int] -coef_size = [ 1,2,3,4,5,6,7,8,9 - , 10, 30 - , 100, 300 - , 1000, 3000 - , 10000, 30000 - ] -{-# INLINE coef_size #-} - --- Precalculated coefficients -coef_list :: [U.Vector Double] -coef_list = [ U.replicate n 1.2 | n <- coef_size] -{-# NOINLINE coef_list #-} - - - -main :: IO () -main = defaultMain - [ bgroup "logGamma" $ - benchmarkLogGamma logGamma - , bgroup "logGammaL" $ - benchmarkLogGamma logGammaL - , bgroup "incompleteGamma" $ - [ bench (show p) $ nf (incompleteGamma p) p - | p <- [ 0.1 - , 1, 3 - , 10, 30 - , 100, 300 - , 999, 1000 - ] - ] - , bgroup "factorial" - [ bench (show n) $ nf factorial n - | n <- [ 0, 1, 3, 6, 9, 11, 15 - , 20, 30, 40, 50, 60, 70, 80, 90, 100 - ] - ] - , bgroup "incompleteBeta" - [ bench (show (p,q,x)) $ nf (incompleteBeta p q) x - | (p,q,x) <- [ (10, 10, 0.5) - , (101, 101, 0.5) - , (1010, 1010, 0.5) - , (10100, 10100, 0.5) - , (100100, 100100, 0.5) - , (1001000, 1001000, 0.5) - , (10010000,10010000,0.5) - ] - ] - , bgroup "log1p" - [ bench (show x) $ nf log1p x - | x <- [ -0.9 - , -0.5 - , -0.1 - , 0.1 - , 0.5 - , 1 - , 10 - , 100 - ] - ] - , bgroup "sinc" $ - bench "sin" (nf sin (0.55 :: Double)) - : [ bench (show x) $ nf sinc x - | x <- [0, 1e-6, 1e-3, 0.5] - ] - , bgroup "poly" - $ [ bench ("vector_"++show (U.length coefs)) $ nf (\x -> evaluatePolynomial x coefs) (1 :: Double) - | coefs <- coef_list ] - ++ [ bench ("unpacked_"++show n) $ nf (\x -> evaluatePolynomialL x (map fromIntegral [1..n])) (1 :: Double) - | n <- coef_size ] - ] diff --git a/changelog.md b/changelog.md --- a/changelog.md +++ b/changelog.md @@ -1,5 +1,136 @@ -Changes in 0.1.7.0 +## Changes in 0.3.4.4 + * Support for future versions of GHC. Monoid and Monad instances now have + canonical form + + +## Changes in 0.3.4.3 + + * Support for `QuickCheck >= 2.14`. Test no longer fail + + * Support for GHC<8.0 dropped + + +## Changes in 0.3.4.2 + + * Fixed crash in `invIncompleteBeta` (#68) for some inputs initial approximation + was computed incorrectly. + + +## Changes in 0.3.4.1 + + * Precision of `incompleteGamma` improved. + + +## Changes in 0.3.4.0 + + * Dependency on `vector-th-unbox` is dropped. All instances are written by + hand now. + + +## Changes in 0.3.3.0 + + * New implementation for `logGamma`. Now it's precise within 2 ulps at full + range including zeros at 1 and 2. + + * Bug with precision loss of `invErf` for parameters near zero is fixed. + + * Fixed loss of precision in bundled `erf` near zero. (Affect primarily GHCJS) + + * `factorial` for now uses lookup table + + * `logFactorial` is optimized a bit (less number of terms is used) + + * `m_sqrt_eps` constant added. + + * Module `Numeric.SpecFunctions.Internal` is exposed. + + * Many improvements for test suite. Tables of expected function values are + generated with mpmath, error estimates improved. Test suite itself is + migrated to `tasty` from `test-framework`. + + +## Changes in 0.3.2.1 + + * Fixes build on windows for GHC<8.0 + + +## Changes in 0.3.2.0 + + * GHCJS is now supported + + * Flag `system-expm1` is set to true by default. Only affects GHC<8.0 + +## Changes in 0.3.1.0 + + * Exported data types for iteration steps in root finding + + * Defaults for root finding algorithm are documented + + +## Changes in 0.3.0.2 + + * Fix license field in cabal file + + +## Changes in 0.3.0.0 + + * `Semigroup` and `Monoid` instances added for data types from `Numeric.Sum` + + * API for finding roots of real functions reworked. 1) All algorithm + parameters are now tweakable. 2) Functions for getting list of iterations + added. + + * `Foldable` and `Traversable` instances for `Root` were added. + +## Changes in 0.2.1.0 + + * `log1p` and `expm1` are simply reexported from `GHC.Float`. They're methods + of `Floating` type class. + + * On windows `expm1` is implemented in pure haskell for older GHCs. + + +## Changes in 0.2.0.0 + + * Bug fixes and documentation tweaks + + +## Changes in 0.2.0.0 + + * `logGamma` now uses Lancsoz approximation and same as `logGammaL`. Old + implementation of `logGamma` moved to `Numeric.SpecFunctions.Extra.logGammaAS245`. + + * Precision of `logGamma` for z<1 improved. + + * New much more precise implementation for `incompleteGamma` + + * Dependency on `erf` package dropped. `erf` and `erfc` just do direct calls + to C. + + * `Numeric.SpecFunctions.expm1` added + + * `Numeric.SpecFunctions.log1pmx` added. + + * `logGammaCorrection` exported in `Numeric.SpecFunctions.Extra`. + + * Module `Numeric.Series` added for working with infinite sequences, series + summation and evaluation of continued fractions. + + * Module `statistics: Statistics.Math.RootFinding` copied to + `Numeric.RootFinding`. Instances for `binary` and `aeson` dropped. + + * Root-finding using Newton-Raphson added + + * `Numeric.MathFunctions.Comparison.ulpDelta` added. It calculates signed + distance between two doubles. + + * Other bug fixes. + + + +## Changes in 0.1.7.0 + * Module `statistics: Statistics.Function.Comparison` moved to `Numeric.MathFunctions.Comparison`. Old implementation if `within` compared negative numbers incorrectly. @@ -11,7 +142,7 @@ * Precision of `logFactorial` is slightly improved. -Changes in 0.1.6.0 +## Changes in 0.1.6.0 * `logChoose` added for calculation of logarithm of binomial coefficient @@ -20,17 +151,17 @@ * `sinc` added -Changes in 0.1.5.3 +## Changes in 0.1.5.3 * Fix for test suite on 32bit platform -Changes in 0.1.5 +## Changes in 0.1.5 * Numeric.Sum: new module adds accurate floating point summation. -Changes in 0.1.4 +## Changes in 0.1.4 * logFactorial type is genberalized. It accepts any `Integral` type @@ -38,7 +169,7 @@ are store in lists added -Changes in 0.1.3 +## Changes in 0.1.3 * Error function and its inverse added. diff --git a/math-functions.cabal b/math-functions.cabal --- a/math-functions.cabal +++ b/math-functions.cabal @@ -1,50 +1,100 @@ name: math-functions -version: 0.1.7.0 -cabal-version: >= 1.8 -license: BSD3 +version: 0.3.4.4 +cabal-version: >= 1.10 +license: BSD2 license-file: LICENSE author: Bryan O'Sullivan , - Aleksey Khudyakov -maintainer: Bryan O'Sullivan -homepage: https://github.com/bos/math-functions -bug-reports: https://github.com/bos/math-functions/issues + Alexey Khudyakov +maintainer: Alexey Khudyakov +homepage: https://github.com/haskell/math-functions +bug-reports: https://github.com/haskell/math-functions/issues category: Math, Numeric build-type: Simple -synopsis: Special functions and Chebyshev polynomials +synopsis: Collection of tools for numeric computations description: - This library provides implementations of special mathematical - functions and Chebyshev polynomials. These functions are often - useful in statistical and numerical computing. + This library contain collection of various utilities for numerical + computing. So far there're special mathematical functions, + compensated summation algorithm, summation of series, root finding + for real functions, polynomial summation and Chebyshev + polynomials. + +tested-with: + GHC ==8.0.2 + || ==8.2.2 + || ==8.4.4 + || ==8.6.5 + || ==8.8.4 + || ==8.10.7 + || ==9.0.2 + || ==9.2.7 + || ==9.4.5 + || ==9.6.2 + + extra-source-files: - changelog.md - README.markdown - benchmark/*.hs - tests/*.hs - tests/Tests/*.hs - tests/Tests/SpecFunctions/gen.py - doc/sinc.hs + changelog.md + README.markdown + tests/Tests/SpecFunctions/gen.py + tests/tables/generate.py + tests/tables/shell.nix + tests/tables/*.dat + tests/tables/inputs/*.dat + doc/sinc.hs +flag system-expm1 + description: Use expm1 provided by GHC. On GHCJS we don't have one so we + have to use hand-coded one. + default: True + manual: True + +flag system-erf + description: Use erf and erfc provided by system. On GHCJS + version provided by library is used regardless of + flag for that lack of libc. + default: True + manual: True + library - ghc-options: -Wall - build-depends: base >=3 && <5, - deepseq, - erf >= 2, - vector >= 0.7, - primitive, - vector-th-unbox + default-language: Haskell2010 + other-extensions: + BangPatterns + CPP + DeriveDataTypeable + FlexibleContexts + MultiParamTypeClasses + ScopedTypeVariables + TypeFamilies + DeriveGeneric + + ghc-options: -Wall -O2 + build-depends: base >= 4.9 && < 5 + , deepseq + , data-default-class >= 0.1.2.0 + , vector >= 0.11 + , primitive + if flag(system-expm1) && !impl(ghcjs) + cpp-options: -DUSE_SYSTEM_EXPM1 + if flag(system-erf) && !impl(ghcjs) + cpp-options: -DUSE_SYSTEM_ERF exposed-modules: Numeric.MathFunctions.Constants Numeric.MathFunctions.Comparison Numeric.Polynomial Numeric.Polynomial.Chebyshev + Numeric.RootFinding Numeric.SpecFunctions Numeric.SpecFunctions.Extra + Numeric.SpecFunctions.Internal + Numeric.Series Numeric.Sum other-modules: - Numeric.SpecFunctions.Internal + Numeric.SpecFunctions.Compat -test-suite tests +test-suite math-function-tests + default-language: Haskell2010 + other-extensions: ViewPatterns + type: exitcode-stdio-1.0 ghc-options: -Wall -threaded if arch(i386) @@ -56,27 +106,50 @@ Tests.Helpers Tests.Chebyshev Tests.Comparison + Tests.RootFinding Tests.SpecFunctions Tests.SpecFunctions.Tables Tests.Sum - build-depends: - math-functions, - base >=3 && <5, - deepseq, - primitive, - vector >= 0.7, - vector-th-unbox, - erf, - HUnit >= 1.2, - QuickCheck >= 2.7, - test-framework, - test-framework-hunit, - test-framework-quickcheck2 + if flag(system-erf) && !impl(ghcjs) + cpp-options: -DUSE_SYSTEM_ERF + build-depends: base >=4.5 && <5 + , math-functions + , data-default-class >= 0.1.2.0 + , deepseq + , primitive + , vector >= 0.7 + , vector-th-unbox + , erf + , QuickCheck >= 2.7 + , tasty >= 1.2 + , tasty-hunit >= 0.10 + , tasty-quickcheck >= 0.10 +benchmark math-functions-bench + type: exitcode-stdio-1.0 + if impl(ghcjs) + buildable: False + default-language: Haskell2010 + other-extensions: + BangPatterns + CPP + DeriveDataTypeable + FlexibleContexts + MultiParamTypeClasses + ScopedTypeVariables + TemplateHaskell + TypeFamilies + DeriveGeneric + ghc-options: -Wall -O2 + hs-source-dirs: bench + Main-is: bench.hs + build-depends: base >= 4.5 && < 5 + , math-functions + , data-default-class + , vector + , random + , tasty-bench >=0.3.4 + source-repository head type: git location: https://github.com/bos/math-functions - -source-repository head - type: mercurial - location: https://bitbucket.org/bos/math-functions diff --git a/tests/Tests/Chebyshev.hs b/tests/Tests/Chebyshev.hs --- a/tests/Tests/Chebyshev.hs +++ b/tests/Tests/Chebyshev.hs @@ -1,12 +1,10 @@ {-# OPTIONS_GHC -fno-warn-type-defaults #-} -module Tests.Chebyshev ( - tests - ) where +module Tests.Chebyshev ( tests ) where import Data.Vector.Unboxed (fromList) -import Test.Framework -import Test.Framework.Providers.QuickCheck2 +import Test.Tasty +import Test.Tasty.QuickCheck (testProperty) import Test.QuickCheck (Arbitrary(..),counterexample,Property) import Tests.Helpers @@ -14,7 +12,7 @@ import Numeric.MathFunctions.Comparison -tests :: Test +tests :: TestTree tests = testGroup "Chebyshev polynomials" [ testProperty "Chebyshev 0" $ \a0 (Ch x) -> testCheb [a0] x diff --git a/tests/Tests/Comparison.hs b/tests/Tests/Comparison.hs --- a/tests/Tests/Comparison.hs +++ b/tests/Tests/Comparison.hs @@ -2,15 +2,15 @@ -- Tests for approximate comparison module Tests.Comparison (tests) where -import Test.Framework -import Test.Framework.Providers.QuickCheck2 +import Test.Tasty +import Test.Tasty.QuickCheck import Tests.Helpers import Numeric.MathFunctions.Comparison import Numeric.MathFunctions.Constants (m_epsilon) -tests :: Test +tests :: TestTree tests = testGroup "Comparison" [ testProperty "addUlps 0" $ \x -> x == addUlps 0 x , testProperty "addUlps sym" $ \i x -> x == (addUlps (-i) . addUlps i) x diff --git a/tests/Tests/Helpers.hs b/tests/Tests/Helpers.hs --- a/tests/Tests/Helpers.hs +++ b/tests/Tests/Helpers.hs @@ -15,9 +15,8 @@ import Data.Complex import Data.Typeable -import qualified Test.HUnit as HU -import Test.Framework -import Test.Framework.Providers.HUnit +import Test.Tasty +import Test.Tasty.HUnit import Numeric.MathFunctions.Comparison @@ -71,8 +70,8 @@ -- HUnit helpers ---------------------------------------------------------------- -testAssertion :: String -> Bool -> Test -testAssertion str cont = testCase str $ HU.assertBool str cont +testAssertion :: String -> Bool -> TestTree +testAssertion str cont = testCase str $ assertBool str cont -testEquality :: (Show a, Eq a) => String -> a -> a -> Test -testEquality msg a b = testCase msg $ HU.assertEqual msg a b +testEquality :: (Show a, Eq a) => String -> a -> a -> TestTree +testEquality msg a b = testCase msg $ assertEqual msg a b diff --git a/tests/Tests/RootFinding.hs b/tests/Tests/RootFinding.hs new file mode 100644 --- /dev/null +++ b/tests/Tests/RootFinding.hs @@ -0,0 +1,44 @@ +-- | +module Tests.RootFinding ( tests ) where + +import Data.Default.Class +import Test.Tasty +import Test.Tasty.HUnit + +import Numeric.RootFinding +import Tests.Helpers + + +tests :: TestTree +tests = testGroup "Root finding" + [ testGroup "Ridders" + [ testAssertion "sin x - 0.525 [exact]" $ testRiddersSin0_525 (AbsTol 0) + , testAssertion "sin x - 0.525 [abs 1e-12]" $ testRiddersSin0_525 (AbsTol 1e-12) + , testAssertion "sin x - 0.525 [abs 1e-6]" $ testRiddersSin0_525 (AbsTol 1e-6) + , testAssertion "sin x - 0.525 [rel 1e-12]" $ testRiddersSin0_525 (RelTol 1e-12) + , testAssertion "sin x - 0.525 [rel 1e-6]" $ testRiddersSin0_525 (RelTol 1e-6) + ] + , testGroup "Newton-Raphson" + [ testAssertion "sin x - 0.525 [rel 1e-12]" $ testNewtonSin0_525 (RelTol 1e-12) + , testAssertion "sin x - 0.525 [rel 1e-6]" $ testNewtonSin0_525 (RelTol 1e-6) + , testAssertion "sin x - 0.525 [abs 1e-12]" $ testNewtonSin0_525 (AbsTol 1e-12) + , testAssertion "sin x - 0.525 [abs 1e-6]" $ testNewtonSin0_525 (AbsTol 1e-6) + , testAssertion "1/x - 0.5 [0]" $ + let Root r = newtonRaphson def{newtonTol=RelTol 0} (1,1000,1000) + (\x -> (1/x - 0.5, -1/(x*x))) + in r == 2 + ] + ] + where + -- Exact root for equation: sin x - 0.525 = 0 + exactRoot = 0.5527151130967832 + -- + testRiddersSin0_525 tol + = withinTolerance tol r exactRoot + where + Root r = ridders def{riddersTol = tol} (0, pi/2) (\x -> sin x - 0.525) + -- + testNewtonSin0_525 tol + = withinTolerance tol r exactRoot + where + Root r = newtonRaphson def{newtonTol=tol} (0, pi/4, pi/2) (\x -> (sin x - 0.525, cos x)) diff --git a/tests/Tests/SpecFunctions.hs b/tests/Tests/SpecFunctions.hs --- a/tests/Tests/SpecFunctions.hs +++ b/tests/Tests/SpecFunctions.hs @@ -1,106 +1,373 @@ +{-# LANGUAGE BangPatterns #-} +{-# LANGUAGE CPP #-} {-# LANGUAGE ViewPatterns #-} -- | Tests for Statistics.Math module Tests.SpecFunctions ( tests ) where +import Control.Monad +import Data.List +import Data.Maybe import qualified Data.Vector as V import Data.Vector ((!)) +import qualified Data.Vector.Unboxed as U import Test.QuickCheck hiding (choose,within) -import Test.Framework -import Test.Framework.Providers.QuickCheck2 +import Test.Tasty +import Test.Tasty.QuickCheck (testProperty) +import Test.Tasty.HUnit import Tests.Helpers import Tests.SpecFunctions.Tables import Numeric.SpecFunctions -import Numeric.MathFunctions.Comparison (within) +import Numeric.SpecFunctions.Internal (factorialTable) +import Numeric.MathFunctions.Comparison (within,ulpDistance) +import Numeric.MathFunctions.Constants (m_epsilon,m_tiny) +erfTol,erfcTol,erfcLargeTol :: Int +-- Pure haskell implementation is not very good +#if !defined(USE_SYSTEM_ERF) || defined(__GHCJS__) +erfTol = 4 +erfcTol = 4 +erfcLargeTol = 64 +-- Macos's erf is slightly worse that GNU one +#elif defined(darwin_HOST_OS) +erfTol = 2 +erfcTol = 2 +erfcLargeTol = 2 +-- Windows' one is not very good too +#elif defined(mingw32_HOST_OS) +erfTol = 2 +erfcTol = 2 +erfcLargeTol = 4 +#else +erfTol = 1 +erfcTol = 2 +erfcLargeTol = 2 +#endif -tests :: Test +isGHCJS :: Bool +#if defined(__GHCJS__) +isGHCJS = True +#else +isGHCJS = False +#endif + +isWindows :: Bool +#if defined(mingw32_HOST_OS) +isWindows = True +#else +isWindows = False +#endif + + +tests :: TestTree tests = testGroup "Special functions" - [ testProperty "Gamma(x+1) = x*Gamma(x) [logGamma]" $ gammaReccurence logGamma 3e-8 - , testProperty "Gamma(x+1) = x*Gamma(x) [logGammaL]" $ gammaReccurence logGammaL 2e-13 - , testProperty "gamma(1,x) = 1 - exp(-x)" $ incompleteGammaAt1Check - , testProperty "0 <= gamma <= 1" $ incompleteGammaInRange - , testProperty "0 <= I[B] <= 1" $ incompleteBetaInRange - -- XXX FIXME DISABLED due to failures - -- , testProperty "invIncompleteGamma = gamma^-1" $ invIGammaIsInverse - -- , testProperty "invIncompleteBeta = B^-1" $ invIBetaIsInverse - -- , testProperty "gamma - increases" $ - -- \s x y -> s > 0 && x > 0 && y > 0 ==> monotonicallyIncreases (incompleteGamma s) x y - , testProperty "invErfc = erfc^-1" $ invErfcIsInverse - , testProperty "invErf = erf^-1" $ invErfIsInverse - -- Unit tests - , testAssertion "Factorial is expected to be precise at 1e-15 level" - $ and [ eq 1e-15 (factorial (fromIntegral n :: Int)) - (fromIntegral (factorial' n)) - |n <- [0..170]] - , testAssertion "Log factorial is expected to be precise at 1e-15 level" - $ and [ eq 1e-15 (logFactorial (fromIntegral n :: Int)) - (log $ fromIntegral $ factorial' n) - | n <- [2..170]] - , testAssertion "logGamma is expected to be precise at 1e-9 level [integer points]" - $ and [ eq 1e-9 (logGamma (fromIntegral n)) - (logFactorial (n-1)) - | n <- [3..10000::Int]] - , testAssertion "logGamma is expected to be precise at 1e-9 level [fractional points]" - $ and [ eq 1e-9 (logGamma x) lg | (x,lg) <- tableLogGamma ] - , testAssertion "logGammaL is expected to be precise at 1e-15 level" - $ and [ eq 1e-15 (logGammaL (fromIntegral n)) - (logFactorial (n-1)) - | n <- [3..10000::Int]] - -- FIXME: Too low! - , testAssertion "logGammaL is expected to be precise at 1e-10 level [fractional points]" - $ and [ eq 1e-10 (logGammaL x) lg | (x,lg) <- tableLogGamma ] - -- FIXME: loss of precision when logBeta p q ≈ 0. - -- Relative error doesn't work properly in this case. - , testAssertion "logBeta is expected to be precise at 1e-6 level" - $ and [ eq 1e-6 (logBeta p q) - (logGammaL p + logGammaL q - logGammaL (p+q)) - | p <- [0.1,0.2 .. 0.9] ++ [2 .. 20] - , q <- [0.1,0.2 .. 0.9] ++ [2 .. 20] - ] - , testAssertion "digamma is expected to be precise at 1e-14 [integers]" - $ digammaTestIntegers 1e-14 - -- Relative precision is lost when digamma(x) ≈ 0 - , testAssertion "digamma is expected to be precise at 1e-12" - $ and [ eq 1e-12 r (digamma x) | (x,r) <- tableDigamma ] - -- FIXME: Why 1e-8? Is it due to poor precision of logBeta? - , testAssertion "incompleteBeta is expected to be precise at 1e-8 level" - $ and [ eq 1e-8 (incompleteBeta p q x) ib | (p,q,x,ib) <- tableIncompleteBeta ] - , testAssertion "incompleteBeta with p > 3000 and q > 3000" - $ and [ eq 1e-11 (incompleteBeta p q x) ib | (x,p,q,ib) <- - [ (0.495, 3001, 3001, 0.2192546757957825068677527085659175689142653854877723) - , (0.501, 3001, 3001, 0.5615652382981522803424365187631195161665429270531389) - , (0.531, 3500, 3200, 0.9209758089734407825580172472327758548870610822321278) - , (0.501, 13500, 13200, 0.0656209987264794057358373443387716674955276089622780) - ] - ] - , testAssertion "choose is expected to precise at 1e-12 level" - $ and [ eq 1e-12 (choose (fromIntegral n) (fromIntegral k)) (fromIntegral $ choose' n k) - | n <- [0..1000], k <- [0..n]] - , testAssertion "logChoose == log . choose" - $ and [ let n' = fromIntegral n - k' = fromIntegral k - in within 2 (logChoose n' k') (log $ choose n' k') - | n <- [0..1000], k <- [0..n]] + [ testGroup "erf" + [ -- implementation from numerical recipes loses precision for + -- large arguments + testCase "erfc table" $ + forTable "tests/tables/erfc.dat" $ \[x, exact] -> + checkTabularPure erfcTol (show x) exact (erfc x) + , testCase "erfc table [large]" $ + forTable "tests/tables/erfc-large.dat" $ \[x, exact] -> + checkTabularPure erfcLargeTol (show x) exact (erfc x) + -- + , testCase "erf table" $ + forTable "tests/tables/erf.dat" $ \[x, exact] -> do + checkTabularPure erfTol (show x) exact (erf x) + , testProperty "id = erfc . invErfc" invErfcIsInverse + , testProperty "id = invErfc . erfc" invErfcIsInverse2 + , testProperty "invErf = erf^-1" invErfIsInverse + ] + -- + , testGroup "log1p & Co" + [ testCase "expm1 table" $ + forTable "tests/tables/expm1.dat" $ \[x, exact] -> + checkTabularPure 2 (show x) exact (expm1 x) + , testCase "log1p table" $ + forTable "tests/tables/log1p.dat" $ \[x, exact] -> + checkTabularPure 1 (show x) exact (log1p x) + ] + ---------------- + , testGroup "gamma function" + [ testCase "logGamma table [fractional points" $ + forTable "tests/tables/loggamma.dat" $ \[x, exact] -> do + checkTabularPure 2 (show x) exact (logGamma x) + , testProperty "Gamma(x+1) = x*Gamma(x)" $ gammaReccurence + , testCase "logGamma is expected to be precise at 1e-15 level" $ + forM_ [3..10000::Int] $ \n -> do + let exact = logFactorial (n-1) + val = logGamma (fromIntegral n) + checkTabular 8 (show n) exact val + ] + ---------------- + , testGroup "incomplete gamma" + [ testCase "incompleteGamma table" $ + forTable "tests/tables/igamma.dat" $ \[a,x,exact] -> do + let err | a < 10 = 16 + | a <= 101 = case () of + _| isGHCJS -> 64 + | isWindows -> 64 + | otherwise -> 32 + | a == 201 = 200 + | otherwise = 32 + checkTabularPure err (show (a,x)) exact (incompleteGamma a x) + , testProperty "incomplete gamma - increases" $ + \(abs -> s) (abs -> x) (abs -> y) -> s > 0 ==> monotonicallyIncreases (incompleteGamma s) x y + , testProperty "0 <= gamma <= 1" incompleteGammaInRange + , testProperty "gamma(1,x) = 1 - exp(-x)" incompleteGammaAt1Check + , testProperty "invIncompleteGamma = gamma^-1" invIGammaIsInverse + ] + ---------------- + , testGroup "beta function" + [ testCase "logBeta table" $ + forTable "tests/tables/logbeta.dat" $ \[p,q,exact] -> + let errEst + -- For Stirling approx. errors are very good + | b > 10 = 2 + -- Partial Stirling approx + | a > 10 = case () of + _| b >= 1 -> 4 + | otherwise -> 2 * est + -- sum of logGamma + | otherwise = case () of + _| a <= 1 && b <= 1 -> 8 + | a >= 1 && b >= 1 -> 8 + | otherwise -> 2 * est + where + a = max p q + b = min p q + -- + est = ceiling + $ abs (logGamma a) + abs (logGamma b) + abs (logGamma (a + b)) + / abs (logBeta a b) + in checkTabularPure errEst (show (p,q)) exact (logBeta p q) + , testCase "logBeta factorial" betaFactorial + , testProperty "beta(1,p) = 1/p" beta1p + -- , testProperty "beta recurrence" betaRecurrence + ] + ---------------- + , testGroup "incomplete beta" + [ testCase "incompleteBeta table" $ + forM_ tableIncompleteBeta $ \(p,q,x,exact) -> + checkTabular 64 (show (x,p,q)) (incompleteBeta p q x) exact + , testCase "incompleteBeta table with p > 3000 and q > 3000" $ + forM_ tableIncompleteBetaP3000 $ \(x,p,q,exact) -> + checkTabular 7000 (show (x,p,q)) (incompleteBeta p q x) exact + -- + , testProperty "0 <= I[B] <= 1" incompleteBetaInRange + , testProperty "ibeta symmetry" incompleteBetaSymmetry + , testCase "Regression #68" $ do + let a = 1 + b = 0.3 + p = 0.3 + x = invIncompleteBeta a b p + assertBool "Inversion OK" $ incompleteBeta a b x `ulpDistance` p < 4 + -- XXX FIXME DISABLED due to failures + -- , testProperty "invIncompleteBeta = B^-1" $ invIBetaIsInverse + ] + ---------------- + , testGroup "digamma" + [ testAssertion "digamma is expected to be precise at 1e-14 [integers]" + $ digammaTestIntegers 1e-14 + -- Relative precision is lost when digamma(x) ≈ 0 + , testCase "digamma is expected to be precise at 1e-12" $ + forTable "tests/tables/digamma.dat" $ \[x, exact] -> + checkTabularPure 2048 + (show x) (digamma x) exact + ] + ---------------- + , testGroup "factorial" + [ testCase "Factorial table" $ + forM_ [0 .. 170] $ \n -> do + checkTabular 1 + (show n) + (fromIntegral (factorial' n)) + (factorial (fromIntegral n :: Int)) + -- + , testCase "Log factorial from integer" $ + forM_ [2 .. 170] $ \n -> do + checkTabular 1 + (show n) + (log $ fromIntegral $ factorial' n) + (logFactorial (fromIntegral n :: Int)) + , testAssertion "Factorial table is OK" + $ U.length factorialTable == 171 + , testCase "Log factorial table" $ + forTable "tests/tables/factorial.dat" $ \[i,exact] -> + checkTabularPure 3 + (show i) (logFactorial (round i :: Int)) exact + ] + ---------------- + , testGroup "combinatorics" + [ testCase "choose table" $ + forM_ [0 .. 1000] $ \n -> + forM_ [0 .. n] $ \k -> do + checkTabular (if isWindows then 3072 else 2048) + (show (n,k)) + (fromIntegral $ choose' n k) + (choose (fromInteger n) (fromInteger k)) + -- + , testCase "logChoose == log . choose" $ + forM_ [0 .. 1000] $ \n -> + forM_ [0 .. n] $ \k -> do + checkTabular 2 + (show (n,k)) + (log $ choose n k) + (logChoose n k) + ] ---------------------------------------------------------------- -- Self tests - , testProperty "Self-test: 0 <= range01 <= 1" $ \x -> let f = range01 x in f <= 1 && f >= 0 + , testGroup "self-test" + [ testProperty "Self-test: 0 <= range01 <= 1" $ \x -> let f = range01 x in f <= 1 && f >= 0 + ] ] ---------------------------------------------------------------- +-- efr tests +---------------------------------------------------------------- + +roundtrip_erfc_invErfc, + roundtrip_invErfc_erfc, + roundtrip_erf_invErf + :: (Double,Double) +#if !defined(USE_SYSTEM_ERF) || defined(__GHCJS__) +roundtrip_erfc_invErfc = (2,8) +roundtrip_invErfc_erfc = (8,4) +roundtrip_erf_invErf = (128,128) +#elif defined(darwin_HOST_OS) +roundtrip_erfc_invErfc = (4,4) +roundtrip_invErfc_erfc = (4,4) +roundtrip_erf_invErf = (2,2) +#elif defined(mingw32_HOST_OS) +roundtrip_erfc_invErfc = (4,4) +roundtrip_invErfc_erfc = (4,4) +roundtrip_erf_invErf = (4,4) +#else +roundtrip_erfc_invErfc = (2,2) +roundtrip_invErfc_erfc = (2,2) +roundtrip_erf_invErf = (1,1) +#endif + +-- id ≈ erfc . invErfc +invErfcIsInverse :: Double -> Property +invErfcIsInverse ((*2) . range01 -> x) + = (not $ isInfinite x) ==> + ( counterexample ("x = " ++ show x ) + $ counterexample ("y = " ++ show y ) + $ counterexample ("x' = " ++ show x') + $ counterexample ("calc.err = " ++ show (delta, delta-e')) + $ counterexample ("ulps = " ++ show (ulpDistance x x')) + $ ulpDistance x x' <= round delta + ) + where + (e,e') = roundtrip_erfc_invErfc + delta = e' + e * abs ( y / x * 2 / sqrt pi * exp( -y*y )) + y = invErfc x + x' = erfc y + +-- id ≈ invErfc . erfc +invErfcIsInverse2 :: Double -> Property +invErfcIsInverse2 x + = (not $ isInfinite x') ==> + (y > m_tiny) ==> + (x /= 0) ==> + counterexample ("x = " ++ show x ) + $ counterexample ("y = " ++ show y ) + $ counterexample ("x' = " ++ show x') + $ counterexample ("calc.err = " ++ show delta) + $ counterexample ("ulps = " ++ show (ulpDistance x x')) + $ ulpDistance x x' <= delta + where + (e,e') = roundtrip_invErfc_erfc + delta = round + $ e' + e * abs (y / x / (2 / sqrt pi * exp( -x*x ))) + y = erfc x + x' = invErfc y + +-- id ≈ erf . invErf +invErfIsInverse :: Double -> Property +invErfIsInverse a + = (x /= 0) ==> + counterexample ("x = " ++ show x ) + $ counterexample ("y = " ++ show y ) + $ counterexample ("x' = " ++ show x') + $ counterexample ("calc.err = " ++ show delta) + $ counterexample ("ulps = " ++ show (ulpDistance x x')) + $ ulpDistance x x' <= delta + where + (e,e') = roundtrip_erf_invErf + delta = round + $ e + e' * abs (y / x * 2 / sqrt pi * exp ( -y * y )) + x | a < 0 = - range01 a + | otherwise = range01 a + y = invErf x + x' = erf y + +---------------------------------------------------------------- -- QC tests ---------------------------------------------------------------- +-- B(p,q) = (x - 1)!(y-1)! / (x + y - 1)! +betaFactorial :: IO () +betaFactorial = do + forM_ prod $ \(p,q,facP,facQ,facProd) -> do + let exact = fromIntegral (facQ * facP) + / fromIntegral facProd + checkTabular 16 (show (p,q)) + (logBeta (fromIntegral p) (fromIntegral q)) + (log exact) + where + prod = [ (p,q,facP,facQ, factorial' (p + q - 1)) + | (p,facP) <- facList + , (q,facQ) <- facList + , p + q < 170 + , not (p == 1 && q== 1) + ] + facList = [(p,factorial' (p-1)) | p <- [1 .. 170]] + +-- B(1,p) = 1/p +beta1p :: Double -> Property +beta1p (abs -> p) + = p > 2 ==> + counterexample ("p = " ++ show p) + $ counterexample ("logB = " ++ show lb) + $ counterexample ("err = " ++ show d) + $ d <= 24 + where + lb = logBeta 1 p + d = ulpDistance lb (- log p) + +{- +-- B(p+1,q) = B(p,q) · p/(p+q) +betaRecurrence :: Double -> Double -> Property +betaRecurrence (abs -> p) (abs -> q) + = p > 0 && q > 0 ==> + counterexample ("p = " ++ show p) + $ counterexample ("q = " ++ show q) + $ counterexample ("log B(p,q) = " ++ show (logBeta p q)) + $ counterexample ("log B(p+1,q) = " ++ show (logBeta (p+1) q)) + $ counterexample ("err = " ++ show d) + $ d <= 128 + where + logB = logBeta p q + log (p / (p + q)) + logB' = logBeta (p + 1) q + d = ulpDistance logB logB' +-} + -- Γ(x+1) = x·Γ(x) -gammaReccurence :: (Double -> Double) -> Double -> Double -> Property -gammaReccurence logG ε x = - (x > 0 && x < 100) ==> (abs (g2 - g1 - log x) < ε) +gammaReccurence :: Double -> Property +gammaReccurence x + = x > 0 ==> err < errEst where - g1 = logG x - g2 = logG (x+1) + g1 = logGamma x + g2 = logGamma (x+1) + err = abs (g2 - g1 - log x) + -- logGamma apparently is not as precise for small x. See #59 for details + errEst = max 1e-14 + $ 2 * m_epsilon * sum (map abs [ g1 , g2 , log x ]) -- γ(s,x) is in [0,1] range incompleteGammaInRange :: Double -> Double -> Property @@ -109,55 +376,52 @@ -- γ(1,x) = 1 - exp(-x) -- Since Γ(1) = 1 normalization doesn't make any difference -incompleteGammaAt1Check :: Double -> Property +incompleteGammaAt1Check :: Double -> Bool incompleteGammaAt1Check (abs -> x) = - x > 0 ==> (incompleteGamma 1 x + exp(-x)) ≈ 1 - where - (≈) = eq 1e-13 + ulpDistance (incompleteGamma 1 x) (-expm1(-x)) < 16 -- invIncompleteGamma is inverse of incompleteGamma invIGammaIsInverse :: Double -> Double -> Property invIGammaIsInverse (abs -> a) (range01 -> p) = - a > 0 && p > 0 && p < 1 ==> ( counterexample ("a = " ++ show a ) - $ counterexample ("p = " ++ show p ) - $ counterexample ("x = " ++ show x ) - $ counterexample ("p' = " ++ show p') - $ counterexample ("Δp = " ++ show (p - p')) - $ abs (p - p') <= 1e-12 - ) + a > m_tiny && p > m_tiny && p < 1 && x > m_tiny ==> + ( counterexample ("a = " ++ show a ) + $ counterexample ("p = " ++ show p ) + $ counterexample ("x = " ++ show x ) + $ counterexample ("p' = " ++ show p') + $ counterexample ("err = " ++ show (ulpDistance p p')) + $ counterexample ("est = " ++ show est) + $ ulpDistance p p' <= est + ) where x = invIncompleteGamma a p + f' = exp ( log x * (a-1) - x - logGamma a) p' = incompleteGamma a x - --- invErfc is inverse of erfc -invErfcIsInverse :: Double -> Property -invErfcIsInverse ((*2) . range01 -> p) - = counterexample ("p = " ++ show p ) - $ counterexample ("x = " ++ show x ) - $ counterexample ("p' = " ++ show p') - $ abs (p - p') <= 1e-14 - where - x = invErfc p - p' = erfc x - --- invErf is inverse of erf -invErfIsInverse :: Double -> Property -invErfIsInverse a - = counterexample ("p = " ++ show p ) - $ counterexample ("x = " ++ show x ) - $ counterexample ("p' = " ++ show p') - $ abs (p - p') <= 1e-14 - where - x = invErf p - p' = erf x - p | a < 0 = - range01 a - | otherwise = range01 a + -- FIXME: Test should be rechecked when #42 is fixed + (e,e') = (32,32) + est = round + $ e' + e * abs (x / p * f') --- B(s,x) is in [0,1] range +-- I(x;p,q) is in [0,1] range incompleteBetaInRange :: Double -> Double -> Double -> Property incompleteBetaInRange (abs -> p) (abs -> q) (range01 -> x) = p > 0 && q > 0 ==> let i = incompleteBeta p q x in i >= 0 && i <= 1 +-- I(0.5; p,p) = 0.5 +incompleteBetaSymmetry :: Double -> Property +incompleteBetaSymmetry (abs -> p) = + p > 0 ==> + counterexample ("p = " ++ show p) + $ counterexample ("ib = " ++ show ib) + $ counterexample ("err = " ++ show d) + $ counterexample ("est = " ++ show est) + $ d <= est + where + est | p < 1 = 80 + | p < 10 = 200 + | otherwise = round $ 6 * p + d = ulpDistance ib 0.5 + ib = incompleteBeta p p 0.5 + -- invIncompleteBeta is inverse of incompleteBeta invIBetaIsInverse :: Double -> Double -> Double -> Property invIBetaIsInverse (abs -> p) (abs -> q) (range01 -> x) = @@ -210,3 +474,33 @@ -- Truncate double to [0,1] range01 :: Double -> Double range01 = abs . (snd :: (Integer, Double) -> Double) . properFraction + + +readTable :: FilePath -> IO [[Double]] +readTable + = fmap (fmap (fmap read . words) . lines) + . readFile + +forTable :: FilePath -> ([Double] -> Maybe String) -> IO () +forTable path fun = do + rows <- readTable path + case mapMaybe fun rows of + [] -> return () + errs -> assertFailure $ intercalate "---\n" errs + +checkTabular :: Int -> String -> Double -> Double -> IO () +checkTabular prec x exact val = + case checkTabularPure prec x exact val of + Nothing -> return () + Just s -> assertFailure s + +checkTabularPure :: Int -> String -> Double -> Double -> Maybe String +checkTabularPure prec x exact val + | within prec exact val = Nothing + | otherwise = Just $ unlines + [ " x = " ++ x + , " expected = " ++ show exact + , " got = " ++ show val + , " ulps diff = " ++ show (ulpDistance exact val) + , " err.est. = " ++ show prec + ] diff --git a/tests/Tests/SpecFunctions/Tables.hs b/tests/Tests/SpecFunctions/Tables.hs --- a/tests/Tests/SpecFunctions/Tables.hs +++ b/tests/Tests/SpecFunctions/Tables.hs @@ -1,38 +1,5 @@ module Tests.SpecFunctions.Tables where -tableLogGamma :: [(Double,Double)] -tableLogGamma = - [(0.000001250000000, 13.592366285131767256) - , (0.000068200000000, 9.5930266308318756785) - , (0.000246000000000, 8.3100370767447948595) - , (0.000880000000000, 7.0350813373524845318) - , (0.003120000000000, 5.7681293583655666168) - , (0.026700000000000, 3.6082588918892972707) - , (0.077700000000000, 2.5148371858768232556) - , (0.234000000000000, 1.3579557559432757774) - , (0.860000000000000, 0.098146578027685588141) - , (1.340000000000000, -0.11404757557207759189) - , (1.890000000000000, -0.042511642297870140539) - , (2.450000000000000, 0.25014296569217620014) - , (3.650000000000000, 1.3701041997380685178) - , (4.560000000000000, 2.5375143317949575561) - , (6.660000000000000, 5.9515377269550207018) - , (8.250000000000000, 9.0331869196051215454) - , (11.300000000000001, 15.814180681373947834) - , (25.600000000000001, 56.711261598328121636) - , (50.399999999999999, 146.12815158702164808) - , (123.299999999999997, 468.85500075897556371) - , (487.399999999999977, 2526.9846647543727158) - , (853.399999999999977, 4903.9359135978220365) - , (2923.300000000000182, 20402.93198938705973) - , (8764.299999999999272, 70798.268343590112636) - , (12630.000000000000000, 106641.7726498250704) - , (34500.000000000000000, 325976.34838781820145) - , (82340.000000000000000, 849629.79603036714252) - , (234800.000000000000000, 2668846.4390507955104) - , (834300.000000000000000, 10540830.912557533011) - , (1230000.000000000000000, 16017699.322315014899) - ] tableIncompleteBeta :: [(Double,Double,Double,Double)] tableIncompleteBeta = [(2.000000000000000, 3.000000000000000, 0.030000000000000, 0.0051864299999999988189) @@ -45,106 +12,12 @@ , (13.100000000000000, 9.800000000000001, 0.420000000000000, 0.071321857831804780226) , (13.100000000000000, 9.800000000000001, 0.920000000000000, 0.99999578339197070509) ] -tableDigamma :: [(Double,Double)] -tableDigamma = - [(10.0261172557341425, 2.2544954834170942704) - , (0.9070101446062873, -0.74152778337908598072) - , (3.4679213262860156, 1.0925031389314479036) - , (28.5703089405901878, 3.3347652650101657912) - , (5.9700184459319399, 1.7006665338476731897) - , (20.5303177686997920, 2.9973508205248808878) - , (5.6622605630542511, 1.6429280447671743559) - , (4.4741465342999014, 1.3824198603491071324) - , (21.4416006516504787, 3.0418326144933285349) - , (47.6946291432301663, 3.8542988022858128971) - , (11.2357450115053670, 2.37393979612347783) - , (0.3352840110772935, -3.1124447967622668187) - , (2.5037441860153118, 0.70499097759044615508) - , (0.5241560861477529, -1.8489960634174653631) - , (0.1972018552655726, -5.3635382066874592866) - , (0.8289440927562556, -0.90024805153750442344) - , (2.0717397641759350, 0.4680412969073853291) - , (9.1173553049782452, 2.1543380160183831507) - , (1.1815938184339669, -0.31262126373727594508) - , (7.3600347508772019, 1.9265946441432049152) - , (19.7457045917841398, 2.9574003365402390386) - , (4.1956416643620571, 1.3101672771843546617) - , (7.3868205159465790, 1.9304848277860633399) - , (1.2786090750546355, -0.19373178842778399078) - , (10.6498308581562604, 2.3178608134278069208) - , (10.6750266252851169, 2.3203381265132185796) - , (10.6883248506773985, 2.3216431742802625671) - , (14.3373372205836365, 2.6275879484098640937) - , (3.3932538441985769, 1.0672611106295626371) - , (11.4168205413938768, 2.3906538776946248959) - , (3.2500957742991048, 1.0170253699094919941) - , (2.7573211981404855, 0.82209952378707851217) - , (21.8943170241258827, 3.063216323919045081) - , (16.7950471612825254, 2.7910180230044043803) - , (9.2578640399661225, 2.1704940538770385317) - , (5.3213868642873896, 1.5748408574979930741) - , (9.4381079039564071, 2.1908443398518979706) - , (13.1568457441413429, 2.538458049596743038) - , (10.6478950333943825, 2.3176702242110884811) - , (6.4894496431749733, 1.7911554320176725774) - , (20.3998669454332315, 2.9908182167188113176) - , (3.6989463639934752, 1.1668268193484248041) - , (3.4716258279958572, 1.093739186127963281) - , (24.7013029455164919, 3.1864775907749920414) - , (1.1608524325026863, -0.33982067949719851896) - , (1.9482800424522431, 0.3888762195060542215) - , (30.4956621109554185, 3.4010990755913685923) - , (16.3105956379859052, 2.7608468922073350349) - , (10.6908820268137070, 2.3218939328714371939) - , (3.4369121607821915, 1.082096765647714065) - , (2.2914619096171260, 0.5953971130541900747) - , (24.1273989930028883, 3.1624816269998849982) - , (14.9455957898231535, 2.6705890837495616097) - , (32.2002179941400826, 3.4563650137673369578) - , (1.7232417075599473, 0.22682264125689588496) - , (9.9662376350778192, 2.248195612105357899) - , (10.9702870318273966, 2.348920912357223223) - , (18.8934063317711676, 2.912115343761407793) - , (8.6720493874148570, 2.1013420151521415846) - , (20.4905634096258815, 2.9953645521238549954) - , (1.4654265058258678, 0.0036653372399428492921) - , (15.4401781010745509, 2.7042406258657996077) - , (13.6688064138713390, 2.5780909087521290957) - , (2.4073661551765566, 0.65668881914974130964) - , (0.8108729056729371, -0.94026521559981879328) - , (29.5024809785193902, 3.367430902728568487) - , (7.5321882978878660, 1.9513375601887514854) - , (3.3716588961200955, 1.0598414578703589939) - , (2.9310065630306474, 0.89516303667430119351) - , (7.2023118361897769, 1.9033764996201536501) - , (3.1362387322050900, 0.97520764792577085966) - , (6.5709053027851487, 1.8046329737306385788) - , (3.7348491113356177, 1.1779005641199544741) - , (1.2328105814385013, -0.24823346907893503732) - , (7.9098387372709587, 2.0035651569967258823) - , (2.8590898311999715, 0.86554629114604864082) - , (2.1964374279534344, 0.54225028515290207842) - , (3.8933394033155189, 1.2253803767351847398) - , (10.7410508007627694, 2.3268008547643748152) - , (2.4921048837305193, 0.69927782909414781809) - , (2.2101710538553756, 0.55010424351998354897) - , (14.0357118427322334, 2.6055587167248708269) - , (4.1320729121597584, 1.2929216807716104043) - , (0.2766365979680845, -3.8108738889017752527) - , (27.9448247140513644, 3.3122329205038494315) - , (9.3081256750537182, 2.1762105230057038341) - , (1.4222181352589696, -0.038843893649701873028) - , (1.5107587188614726, 0.046499571962236106726) - , (3.3467578222470555, 1.0512176183500512305) - , (12.2373583939228876, 2.4630788434421742039) - , (0.9385094944630431, -0.68317598609698348966) - , (5.8655552400886410, 1.6814385243672138603) - , (17.1377048621110468, 2.8118219246156086477) - , (4.0502102843199079, 1.2702685434611069581) - , (2.2041235084734976, 0.54665320805956585382) - , (0.9498749870396368, -0.66283138696545962354) - , (5.5020466797149687, 1.6115010556650317675) - , (1.8741725410778542, 0.33826100356492333487) - , (14.1730624058772161, 2.6156503142962224118) - , (1.0704026637921555, -0.46701211139417769802) + +-- (x, p, q, exact) +tableIncompleteBetaP3000 :: [(Double,Double,Double,Double)] +tableIncompleteBetaP3000 = + [ (0.495, 3001, 3001, 0.2192546757957825068677527085659175689142653854877723) + , (0.501, 3001, 3001, 0.5615652382981522803424365187631195161665429270531389) + , (0.531, 3500, 3200, 0.9209758089734407825580172472327758548870610822321278) + , (0.501, 13500, 13200, 0.0656209987264794057358373443387716674955276089622780) ] diff --git a/tests/Tests/SpecFunctions_flymake.hs b/tests/Tests/SpecFunctions_flymake.hs deleted file mode 100644 --- a/tests/Tests/SpecFunctions_flymake.hs +++ /dev/null @@ -1,206 +0,0 @@ -{-# LANGUAGE ViewPatterns #-} --- | Tests for Statistics.Math -module Tests.SpecFunctions ( - tests - ) where - -import qualified Data.Vector as V -import Data.Vector ((!)) - -import Test.QuickCheck hiding (choose) -import Test.Framework -import Test.Framework.Providers.QuickCheck2 - -import Tests.Helpers -import Tests.SpecFunctions.Tables -import Numeric.SpecFunctions - - -tests :: Test -tests = testGroup "Special functions" - [ testProperty "Gamma(x+1) = x*Gamma(x) [logGamma]" $ gammaReccurence logGamma 3e-8 - , testProperty "Gamma(x+1) = x*Gamma(x) [logGammaL]" $ gammaReccurence logGammaL 2e-13 - , testProperty "gamma(1,x) = 1 - exp(-x)" $ incompleteGammaAt1Check - , testProperty "0 <= gamma <= 1" $ incompleteGammaInRange - , testProperty "0 <= I[B] <= 1" $ incompleteBetaInRange - -- XXX FIXME DISABLED due to failures - -- , testProperty "invIncompleteGamma = gamma^-1" $ invIGammaIsInverse - -- , testProperty "invIncompleteBeta = B^-1" $ invIBetaIsInverse - -- , testProperty "gamma - increases" $ - -- \s x y -> s > 0 && x > 0 && y > 0 ==> monotonicallyIncreases (incompleteGamma s) x y - , testProperty "invErfc = erfc^-1" $ invErfcIsInverse - , testProperty "invErf = erf^-1" $ invErfIsInverse - -- Unit tests - , testAssertion "Factorial is expected to be precise at 1e-15 level" - $ and [ eq 1e-15 (factorial (fromIntegral n :: Int)) - (fromIntegral (factorial' n)) - |n <- [0..170]] - , testAssertion "Log factorial is expected to be precise at 1e-15 level" - $ and [ eq 1e-15 (logFactorial (fromIntegral n :: Int)) - (log $ fromIntegral $ factorial' n) - | n <- [2..170]] - , testAssertion "logGamma is expected to be precise at 1e-9 level [integer points]" - $ and [ eq 1e-9 (logGamma (fromIntegral n)) - (logFactorial (n-1)) - | n <- [3..10000::Int]] - , testAssertion "logGamma is expected to be precise at 1e-9 level [fractional points]" - $ and [ eq 1e-9 (logGamma x) lg | (x,lg) <- tableLogGamma ] - , testAssertion "logGammaL is expected to be precise at 1e-15 level" - $ and [ eq 1e-15 (logGammaL (fromIntegral n)) - (logFactorial (n-1)) - | n <- [3..10000::Int]] - -- FIXME: Too low! - , testAssertion "logGammaL is expected to be precise at 1e-10 level [fractional points]" - $ and [ eq 1e-10 (logGammaL x) lg | (x,lg) <- tableLogGamma ] - -- FIXME: loss of precision when logBeta p q ≈ 0. - -- Relative error doesn't work properly in this case. - , testAssertion "logBeta is expected to be precise at 1e-6 level" - $ and [ eq 1e-6 (logBeta p q) - (logGammaL p + logGammaL q - logGammaL (p+q)) - | p <- [0.1,0.2 .. 0.9] ++ [2 .. 20] - , q <- [0.1,0.2 .. 0.9] ++ [2 .. 20] - ] - , testAssertion "digamma is expected to be precise at 1e-14 [integers]" - $ digammaTestIntegers 1e-14 - -- Relative precision is lost when digamma(x) ≈ 0 - , testAssertion "digamma is expected to be precise at 1e-12" - $ and [ eq 1e-12 r (digamma x) | (x,r) <- tableDigamma ] - -- FIXME: Why 1e-8? Is it due to poor precision of logBeta? - , testAssertion "incompleteBeta is expected to be precise at 1e-8 level" - $ and [ eq 1e-8 (incompleteBeta p q x) ib | (p,q,x,ib) <- tableIncompleteBeta ] - , testAssertion "incompleteBeta with p > 3000 and q > 3000" - $ and [ eq 1e-11 (incompleteBeta p q x) ib | (x,p,q,ib) <- - [ (0.495, 3001, 3001, 0.2192546757957825068677527085659175689142653854877723) - , (0.501, 3001, 3001, 0.5615652382981522803424365187631195161665429270531389) - , (0.531, 3500, 3200, 0.9209758089734407825580172472327758548870610822321278) - , (0.501, 13500, 13200, 0.0656209987264794057358373443387716674955276089622780) - ] - ] - , testAssertion "choose is expected to precise at 1e-12 level" - $ and [ eq 1e-12 (choose (fromIntegral n) (fromIntegral k)) (fromIntegral $ choose' n k) - | n <- [0..300], k <- [0..n]] - ---------------------------------------------------------------- - -- Self tests - , testProperty "Self-test: 0 <= range01 <= 1" $ \x -> let f = range01 x in f <= 1 && f >= 0 - ] - ----------------------------------------------------------------- --- QC tests ----------------------------------------------------------------- - --- Γ(x+1) = x·Γ(x) -gammaReccurence :: (Double -> Double) -> Double -> Double -> Property -gammaReccurence logG ε x = - (x > 0 && x < 100) ==> (abs (g2 - g1 - log x) < ε) - where - g1 = logG x - g2 = logG (x+1) - --- γ(s,x) is in [0,1] range -incompleteGammaInRange :: Double -> Double -> Property -incompleteGammaInRange (abs -> s) (abs -> x) = - x >= 0 && s > 0 ==> let i = incompleteGamma s x in i >= 0 && i <= 1 - --- γ(1,x) = 1 - exp(-x) --- Since Γ(1) = 1 normalization doesn't make any difference -incompleteGammaAt1Check :: Double -> Property -incompleteGammaAt1Check (abs -> x) = - x > 0 ==> (incompleteGamma 1 x + exp(-x)) ≈ 1 - where - (≈) = eq 1e-13 - --- invIncompleteGamma is inverse of incompleteGamma -invIGammaIsInverse :: Double -> Double -> Property -invIGammaIsInverse (abs -> a) (range01 -> p) = - a > 0 && p > 0 && p < 1 ==> ( printTestCase ("a = " ++ show a ) - $ printTestCase ("p = " ++ show p ) - $ printTestCase ("x = " ++ show x ) - $ printTestCase ("p' = " ++ show p') - $ printTestCase ("Δp = " ++ show (p - p')) - $ abs (p - p') <= 1e-12 - ) - where - x = invIncompleteGamma a p - p' = incompleteGamma a x - --- invErfc is inverse of erfc -invErfcIsInverse :: Double -> Property -invErfcIsInverse ((*2) . range01 -> p) - = printTestCase ("p = " ++ show p ) - $ printTestCase ("x = " ++ show x ) - $ printTestCase ("p' = " ++ show p') - $ abs (p - p') <= 1e-14 - where - x = invErfc p - p' = erfc x - --- invErf is inverse of erf -invErfIsInverse :: Double -> Property -invErfIsInverse a - = printTestCase ("p = " ++ show p ) - $ printTestCase ("x = " ++ show x ) - $ printTestCase ("p' = " ++ show p') - $ abs (p - p') <= 1e-14 - where - x = invErf p - p' = erf x - p | a < 0 = - range01 a - | otherwise = range01 a - --- B(s,x) is in [0,1] range -incompleteBetaInRange :: Double -> Double -> Double -> Property -incompleteBetaInRange (abs -> p) (abs -> q) (range01 -> x) = - p > 0 && q > 0 ==> let i = incompleteBeta p q x in i >= 0 && i <= 1 - --- invIncompleteBeta is inverse of incompleteBeta -invIBetaIsInverse :: Double -> Double -> Double -> Property -invIBetaIsInverse (abs -> p) (abs -> q) (range01 -> x) = - p > 0 && q > 0 ==> ( printTestCase ("p = " ++ show p ) - $ printTestCase ("q = " ++ show q ) - $ printTestCase ("x = " ++ show x ) - $ printTestCase ("x' = " ++ show x') - $ printTestCase ("a = " ++ show a) - $ printTestCase ("err = " ++ (show $ abs $ (x - x') / x)) - $ abs (x - x') <= 1e-12 - ) - where - x' = incompleteBeta p q a - a = invIncompleteBeta p q x - --- Table for digamma function: --- --- Uses equality ψ(n) = H_{n-1} - γ where --- H_{n} = Σ 1/k, k = [1 .. n] - harmonic number --- γ = 0.57721566490153286060 - Euler-Mascheroni number -digammaTestIntegers :: Double -> Bool -digammaTestIntegers eps - = all (uncurry $ eq eps) $ take 3000 digammaInt - where - ok approx exact = approx - -- Harmonic numbers starting from 0 - harmN = scanl (\a n -> a + 1/n) 0 [1::Rational .. ] - gam = 0.57721566490153286060 - -- Digamma values - digammaInt = zipWith (\i h -> (digamma i, realToFrac h - gam)) [1..] harmN - - ----------------------------------------------------------------- --- Unit tests ----------------------------------------------------------------- - --- Lookup table for fact factorial calculation. It has fixed size --- which is bad but it's OK for this particular case -factorial_table :: V.Vector Integer -factorial_table = V.generate 2000 (\n -> product [1..fromIntegral n]) - --- Exact implementation of factorial -factorial' :: Integer -> Integer -factorial' n = factorial_table ! fromIntegral n - --- Exact albeit slow implementation of choose -choose' :: Integer -> Integer -> Integer -choose' n k = factorial' n `div` (factorial' k * factorial' (n-k)) - --- Truncate double to [0,1] -range01 :: Double -> Double -range01 = abs . (snd :: (Integer, Double) -> Double) . properFraction diff --git a/tests/Tests/Sum.hs b/tests/Tests/Sum.hs --- a/tests/Tests/Sum.hs +++ b/tests/Tests/Sum.hs @@ -4,54 +4,68 @@ import Control.Applicative ((<$>)) import Numeric.Sum as Sum +import Numeric.MathFunctions.Comparison import Prelude hiding (sum) -import Test.Framework (Test, testGroup) -import Test.Framework.Providers.QuickCheck2 (testProperty) +import Test.Tasty (TestTree, testGroup) +import Test.Tasty.QuickCheck import Test.QuickCheck (Arbitrary(..)) import qualified Prelude -t_sum :: ([Double] -> Double) -> [Double] -> Bool -t_sum f xs = f xs == trueSum xs +-- Test that summation result is same as exact sum. That should pass +-- if we're effectively working with quad precision +t_sum :: ([Double] -> Double) -> [Double] -> Property +t_sum f xs + = counterexample ("APPROX = " ++ show approx) + $ counterexample ("EXACT = " ++ show exact) + $ counterexample ("DELTA = " ++ show (approx - exact)) + $ counterexample ("ULPS = " ++ show (ulpDistance approx exact)) + $ approx == exact + where + approx = f xs + exact = trueSum xs -t_sum_error :: ([Double] -> Double) -> [Double] -> Bool -t_sum_error f xs = abs (ts - f xs) <= abs (ts - Prelude.sum xs) - where ts = trueSum xs +-- Test that summation has smaller error than naive summation or no +-- worse than given number of ulps. If we're close enough to exact +-- answer naive may get ahead +t_sum_error :: ([Double] -> Double) -> [Double] -> Property +t_sum_error f xs + = counterexample ("APPROX = " ++ show approx) + $ counterexample ("NAIVE = " ++ show naive) + $ counterexample ("EXACT = " ++ show exact) + $ counterexample ("A-EXACT = " ++ show (approx - exact)) + $ counterexample ("N-EXACT = " ++ show (naive - exact)) + $ counterexample ("ULPS[A] = " ++ show (ulpDistance approx exact)) + $ counterexample ("ULPS[N] = " ++ show (ulpDistance naive exact)) + $ abs (exact - approx) <= abs (exact - naive) + where + naive = Prelude.sum xs + approx = f xs + exact = trueSum xs -t_sum_shifted :: ([Double] -> Double) -> [Double] -> Bool +t_sum_shifted :: ([Double] -> Double) -> [Double] -> Property t_sum_shifted f = t_sum_error f . zipWith (+) badvec trueSum :: (Fractional b, Real a) => [a] -> b trueSum xs = fromRational . Prelude.sum . map toRational $ xs badvec :: [Double] -badvec = cycle [1,1e16,-1e16] - -tests :: Test -tests = testGroup "Summation" [ - testGroup "ID" [ - -- plain summation loses precision quickly - -- testProperty "t_sum" $ t_sum (sum id) - - -- tautological tests: - -- testProperty "t_sum_error" $ t_sum_error (sum id) - -- testProperty "t_sum_shifted" $ t_sum_shifted (sum id) - ] - , testGroup "Kahan" [ - -- tests that cannot pass: - -- testProperty "t_sum" $ t_sum (sum kahan) - -- testProperty "t_sum_error" $ t_sum_error (sum kahan) +badvec = cycle [1, 1e14, -1e14] - -- kahan summation only beats normal summation with large values +tests :: TestTree +tests = testGroup "Summation" + [ testGroup "Kahan" [ + -- Kahan summation only beats naive summation when truly + -- catastrophic cancellation occurs testProperty "t_sum_shifted" $ t_sum_shifted (sum kahan) ] , testGroup "KBN" [ - testProperty "t_sum" $ t_sum (sum kbn) - , testProperty "t_sum_error" $ t_sum_error (sum kbn) + testProperty "t_sum" $ t_sum (sum kbn) + , testProperty "t_sum_error" $ t_sum_error (sum kbn) , testProperty "t_sum_shifted" $ t_sum_shifted (sum kbn) ] , testGroup "KB2" [ - testProperty "t_sum" $ t_sum (sum kb2) - , testProperty "t_sum_error" $ t_sum_error (sum kb2) + testProperty "t_sum" $ t_sum (sum kb2) + , testProperty "t_sum_error" $ t_sum_error (sum kb2) , testProperty "t_sum_shifted" $ t_sum_shifted (sum kb2) ] ] diff --git a/tests/tables/digamma.dat b/tests/tables/digamma.dat new file mode 100644 --- /dev/null +++ b/tests/tables/digamma.dat @@ -0,0 +1,100 @@ +0.197201855265572601 -5.36353820668745935029747601615 +0.276636597968084486 -3.81087388890177491030754285155 +0.335284011077293498 -3.1124447967622667468330567897 +0.524156086147752887 -1.84899606341746529768044821951 +0.810872905672937128 -0.940265215599818752300070120231 +0.828944092756255579 -0.900248051537504354273935908761 +0.907010144606287261 -0.741527783379085906959358391336 +0.938509494463043104 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a/tests/tables/erf.dat b/tests/tables/erf.dat new file mode 100644 --- /dev/null +++ b/tests/tables/erf.dat @@ -0,0 +1,68 @@ +-3 -0.99997790950300141455862722387 +-2.5 -0.99959304798255504106043578426 +-2.39999999999999991 -0.99931148610335492111445002519 +-2.29999999999999982 -0.998856823402643347524080071797 +-2.20000000000000018 -0.998137153702018110141441361147 +-2.10000000000000009 -0.997020533343667015714309417558 +-2 -0.995322265018952734162069256367 +-1.89999999999999991 -0.992790429235257467237215967132 +-1.80000000000000004 -0.989090501635730716146239560398 +-1.69999999999999996 -0.983790458590774560841312945809 +-1.60000000000000009 -0.976348383344644015521780192085 +-1.5 -0.966105146475310727066976261646 +-1.39999999999999991 -0.952285119762648796399623216609 +-1.30000000000000004 -0.934007944940652445850185361578 +-1.19999999999999996 -0.910313978229635368365931793847 +-1.10000000000000009 -0.880205069574081729657259509256 +-1 -0.842700792949714869341220635083 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acc.append(float(s)) + for _ in nm: + yield(acc) + + + +def load_inputs(path): + "Load inputs for function from tsv file" + with open(path) as f: + for s in skip_comments(f): + yield tuple(map(float, s.split())) + + +def load_inputs_cartesian(path): + "Load inputs for several variables where we want to generate all pair" + with open(path) as f: + for x in itertools.product(*tokenize_stream(skip_comments(f))): + yield x + +def fmt(f, inputs, out): + # Here we set up output. We hope that refcounting will collect fds + # promptly + if out is None: + out = sys.stdout + else: + out = open(out, 'w') + for xs in inputs: + param = ["%.18g" % x for x in xs] + sOut = mpmath.nstr( f(*xs), mpmath.mp.dps ) + print( '\t'.join( param + [sOut]), + file=out) + + +## ================================================================ + +fmt( mpmath.erf, + load_inputs('inputs/erf.dat'), + 'erf.dat') +fmt( mpmath.erf, + load_inputs('inputs/erf.dat'), + 'erf.dat') +fmt( mpmath.erfc, + load_inputs('inputs/erfc.dat'), + 'erfc.dat') +fmt( mpmath.erfc, + load_inputs('inputs/erfc-large.dat'), + 'erfc-large.dat') +fmt( mpmath.loggamma, + load_inputs('inputs/loggamma.dat'), + 'loggamma.dat') +fmt( mpmath.digamma, + load_inputs('inputs/digamma.dat'), + 'digamma.dat') +fmt( mpmath.expm1, + load_inputs('inputs/expm1.dat'), + 'expm1.dat') +fmt( mpmath.log1p, + load_inputs('inputs/log1p.dat'), + 'log1p.dat') +fmt( lambda x: mpmath.log(mpmath.factorial(x)), + map(lambda x: (x,), range(0, 20000)), + 'factorial.dat') +fmt( lambda a, x: mpmath.gammainc(z=a, a=0, b=x, regularized=True), + load_inputs_cartesian('inputs/igamma.dat'), + 'igamma.dat') +fmt( lambda p, q: mpmath.log(mpmath.beta(p,q)), + load_inputs_cartesian('inputs/logbeta.dat'), + 'logbeta.dat') diff --git a/tests/tables/igamma.dat b/tests/tables/igamma.dat new file mode 100644 --- /dev/null +++ b/tests/tables/igamma.dat @@ -0,0 +1,180 @@ +0.000100000000000000005 9.99999999999999955e-07 0.998677038073159095961395592937 +0.000100000000000000005 1.00000000000000008e-05 0.998907017536745023882308685182 +0.000100000000000000005 0.00100000000000000002 0.99936703825416550859567037094 +0.000100000000000000005 0.0100000000000000002 0.999596280434128887942384540059 +0.000100000000000000005 0.100000000000000006 0.999817716935473516734296177546 +0.000100000000000000005 1 0.999978059361861853366809113793 +0.000100000000000000005 10 0.999999999584179852012789127241 +0.000100000000000000005 100 1.0 +0.000100000000000000005 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+50.3999999999999986 146.128151587021650625708161324 +77 256.221135550009525456082846319 +123.299999999999997 468.855000758975584910395998022 +222 975.612353993036060800198673231 +487.399999999999977 2526.98466475437276484523752546 +555 2949.78690775573651020653356334 +853.399999999999977 4903.93591359782221336869490715 +888 5138.13145428259548143327765439 +2923.30000000000018 20402.9319893870607273552436914 +3333 23699.9195344962473524769527089 +8764.29999999999927 70798.268343590118940583038718 +12630 106641.772649825079076675672666 +34500 325976.34838781821769124632363 +82340 849629.79603036717685116333246 +234800 2668846.43905079586519344403509 +834300 10540830.9125575340577530715314 +1230000 16017699.3223150155655909214342 diff --git a/tests/tables/shell.nix b/tests/tables/shell.nix new file mode 100644 --- /dev/null +++ b/tests/tables/shell.nix @@ -0,0 +1,10 @@ +let + pkgs = import {}; + pyp = pkgs.python3.withPackages (py: with py; + [ ipython + mpmath + ]); +in +pkgs.mkShell { + buildInputs = [ pyp ]; +} diff --git a/tests/tests.hs b/tests/tests.hs --- a/tests/tests.hs +++ b/tests/tests.hs @@ -1,13 +1,16 @@ -import Test.Framework (defaultMain) -import qualified Tests.SpecFunctions +import Test.Tasty (defaultMain,testGroup) import qualified Tests.Chebyshev -import qualified Tests.Sum import qualified Tests.Comparison +import qualified Tests.RootFinding +import qualified Tests.SpecFunctions +import qualified Tests.Sum main :: IO () -main = defaultMain [ Tests.SpecFunctions.tests - -- FIXME: tests for chebyshev polynomials fail intermittently - -- , Tests.Chebyshev.tests - , Tests.Sum.tests - , Tests.Comparison.tests - ] +main = defaultMain $ testGroup "math-functions" + [ Tests.SpecFunctions.tests + -- FIXME: tests for chebyshev polynomials fail intermittently + -- , Tests.Chebyshev.tests + , Tests.Sum.tests + , Tests.Comparison.tests + , Tests.RootFinding.tests + ] diff --git a/tests/view.hs b/tests/view.hs deleted file mode 100644 --- a/tests/view.hs +++ /dev/null @@ -1,102 +0,0 @@ -{-# LANGUAGE OverloadedStrings #-} -import Control.Applicative -import Control.Monad -import Numeric.SpecFunctions -import Numeric.MathFunctions.Constants -import CPython.Sugar -import CPython.MPMath -import qualified CPython as Py - -import HEP.ROOT.Plot - - ----------------------------------------------------------------- - - -viewBetaDelta = runPy $ do - addToPythonPath "." - m <- loadMPMath - mpmSetDps m 100 - xs <- forM pqBeta $ \(p,q) -> do x <- fromMPNum =<< mpmLog m =<< mpmBeta m (MPDouble p) (MPDouble q) - return (p,q, relErr x (logBeta p q)) - draws $ do - -- let xs = [ (p,q, logBeta p q `relErr` (logGammaL p + logGammaL q - logGammaL (q+p))) - -- | (p,q) <- pqBeta - -- ] - add $ Graph2D xs - - -pqBeta = [ (p,q) - | p <- logRange 50 0.3 0.6 - , q <- logRange 50 5 6 - ] - where - - - - -viewIBeta x = runPy $ do - addToPythonPath "." - m <- loadMPMath - mpmSetDps m 30 - -- - let n = 40 - let pq = (,) - <$> logRange n 100 1000 - <*> logRange n 100 1000 - -- - xs <- forM pq $ \(p,q) -> do - i <- fromMPNum =<< mpmIncompleteBeta m (MPDouble p) (MPDouble q) (MPDouble x) - return (p,q, incompleteBeta p q x `relErr` i) - -- - draws $ do - add $ Graph2D xs - - -go = runPy $ do - addToPythonPath "." - m <- loadMPMath - mpmSetDps m 16 - -- - print =<< fromMPNum =<< mpmIncompleteBeta m (MPDouble 10) (MPDouble 10) (MPDouble 0.4) - print $ incompleteBeta 10 10 0.4 - - - - -viewLancrox = runPy $ do - addToPythonPath "." - m <- loadMPMath - mpmSetDps m 50 - -- - let xs = logRange 10000 (1e-8) (1e-1) - pl <- forM xs $ \x -> do y0 <- fromMPNum =<< mpmLog m =<< mpmGamma m (MPDouble x) - return (x, y0) - draws $ do - add $ Graph $ [ (x, abs $ y `relErr` logGammaL x) | (x,y) <- pl ] - set $ lineColor RED - -- - add $ Graph $ [ (x, abs $ y `relErr` logGamma x) | (x,y) <- pl ] - set $ lineColor BLUE - -- - set $ xaxis $ logScale ON - -- set $ yaxis $ logScale ON - -- - add $ HLine m_epsilon - add $ HLine $ negate m_epsilon - - ----------------------------------------------------------------- - -relErr :: Double -> Double -> Double -relErr 0 0 = 0 -relErr x y = (x - y) / max (abs x) (abs y) - - - -logRange :: Int -> Double -> Double -> [Double] -logRange n a b - = [ a * r^i | i <- [0 .. n] ] - where - r = (b / a) ** (1 / fromIntegral n) -