packages feed

integration (empty) → 0.1

raw patch · 5 files changed

+639/−0 lines, 5 filesdep +basedep +parallelsetup-changed

Dependencies added: base, parallel

Files

+ LICENSE view
@@ -0,0 +1,30 @@+Copyright 2012 Edward Kmett++All rights reserved.++Redistribution and use in source and binary forms, with or without+modification, are permitted provided that the following conditions+are met:++1. Redistributions of source code must retain the above copyright+   notice, this list of conditions and the following disclaimer.++2. Redistributions in binary form must reproduce the above copyright+   notice, this list of conditions and the following disclaimer in the+   documentation and/or other materials provided with the distribution.++3. Neither the name of the author nor the names of his contributors+   may be used to endorse or promote products derived from this software+   without specific prior written permission.++THIS SOFTWARE IS PROVIDED BY THE AUTHORS ``AS IS'' AND ANY EXPRESS OR+IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED+WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE+DISCLAIMED.  IN NO EVENT SHALL THE AUTHORS OR CONTRIBUTORS BE LIABLE FOR+ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL+DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS+OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)+HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT,+STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN+ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE+POSSIBILITY OF SUCH DAMAGE.
+ Numeric/Integration/TanhSinh.hs view
@@ -0,0 +1,547 @@+{-# LANGUAGE BangPatterns, PatternGuards #-}+-----------------------------------------------------------------------------+-- |+-- Module      :  Numeric.Integration.TanhSinh+-- Copyright   :  (C) 2012 Edward Kmett+-- License     :  BSD-style (see the file LICENSE)+--+-- Maintainer  :  Edward Kmett <ekmett@gmail.com>+-- Stability   :  provisional+-- Portability :  portable+--+-- An implementation of Takahashi and Mori's Tanh-Sinh+-- quadrature.+--+-- <http://en.wikipedia.org/wiki/Tanh-sinh_quadrature>+--+-- Tanh-Sinh provides good results across a wide-range+-- of functions and is pretty much as close to a+-- universal quadrature scheme as is possible. It is also+-- robust against error in the presence of singularities at+-- the endpoints of the integral.+--+-- The change of basis is precomputed, and information is+-- gained quadratically in the number of digits.+--+-- > ghci> absolute 1e-6 $ parTrap sin (pi/2) pi+-- > Result {result = 0.9999999999999312, errorEstimate = 2.721789573237518e-10, evalutions = 25}+--+-- > ghci> confidence $ absolute 1e-6 $ trap sin (pi/2) pi+-- > (0.9999999997277522,1.0000000002721101)+--+-- Unlike most quadrature schemes, this method is also fairly robust against+-- singularities at the end points.+--+-- > ghci> absolute 1e-6 $ trap (recip . sqrt . sin) 0 1+-- > Result {result = 2.03480500404275, errorEstimate = 6.349514558579017e-8, evalutions = 49}+--+-- See <http://www.johndcook.com/blog/2012/02/21/care-and-treatment-of-singularities/>+-- for a sense of how more naive quadrature schemes fare.+----------------------------------------------------------------------------+module Numeric.Integration.TanhSinh+  (+  -- * Quadrature methods+    trap -- Trapezoid rule for Tanh-Sinh quadrature+  , simpson -- Simpson's rule for Tanh-Sinh quadrature+  , trap'+  , simpson'+  , parTrap+  , parSimpson+  , Result(..)+  -- * Estimated error bounds+  , absolute -- absolute error+  , relative -- relative error+  -- * Confidence intervals+  , confidence+  ) where++import Control.Parallel.Strategies+import Data.List (foldl')++-- integral with an result and an estimate of the error such that (result - errorEstimate, result + errorEstimate) probably bounds the actual answer.+data Result = Result+  { result        :: {-# UNPACK #-} !Double+  , errorEstimate :: {-# UNPACK #-} !Double+  , evalutions    :: {-# UNPACK #-} !Int+  } deriving (Read,Show,Eq,Ord)++confidence :: Result -> (Double, Double)+confidence (Result a b _) = (a - b, a + b)++-- TanhSinh quadrature++absolute :: Double -> [Result] -> Result+absolute targetError = go where+  go [] = error "no result"+  go [r] = r+  go (r@(Result _ e _):rs)+    | e < targetError*0.1 = r+    | otherwise = absolute targetError rs++relative :: Double -> [Result] -> Result+relative _ [] = error "no result"+relative _ [r] = r+relative targetError (r'@(Result a _ _):rs') = go a r' rs' where+  go olds _ (r@(Result s e _):rs)+    | abs (s - olds) < targetError * e || s == 0 && olds == 0 = r+    | otherwise                                               = go s r rs+  go _ oldr [] = oldr++m_huge :: Double+m_huge = 1/0 -- 1.7976931348623157e308+++-- | Integration using a truncated trapezoid rule and tanh-sinh quadrature with a specified evaluation strategy+trap' :: Strategy [Double] -> (Double -> Double) -> Double -> Double -> [Result]+trap' nf f a b = go (0 :: Int) (i0+i1) (abs (i1-i0)) m_huge dd where+  go !k !t !old_delta !err (ds:dds) = res t' err' k : go (k+1) t' delta err' dds+    where+      !ht' = tr ds+      !ht = 0.5*t+      !t' = ht'+ht+      !delta = abs (ht'-ht)+      !err' | delta == 0 || old_delta == 0                         = err+            | r <- log delta / log old_delta, 1.99 < r && r < 2.01 = delta*delta+            | otherwise                                            = delta+  go !k !t !_ !err [] = [res t err k]+  res i e k = Result (i*c) (e*c) (1 + 12*(2^k))+  c  = 0.5 * (b - a)+  d  = 0.5 * (a + b)+  i0 = w0 * f d + tr dd0+  i1 = tr dd1+  tr xs = foldl' (+) 0 (map (\(DD i w) -> let !ci = c * i in w*(f(d+ci)+f(d-ci))) xs `using` nf)++-- | Integration using a truncated trapezoid rule under tanh-sinh quadrature+trap :: (Double -> Double) -> Double -> Double -> [Result]+trap = trap' r0++-- | Integration using a truncated trapezoid rule under tanh-sinh quadrature with buffered parallel evaluation+parTrap :: (Double -> Double) -> Double -> Double -> [Result]+parTrap = trap' (parBuffer 32 rseq)++-- | Integration using a truncated Simpson's rule under tanh-sinh quadrature with a specified evaluation strategy+simpson' :: Strategy [Double] -> (Double -> Double) -> Double -> Double -> [Result]+simpson' nf f a b = go (0 :: Int) i01 (i01*4/3) (abs (i1-i0)) m_huge dd where+  go !k !t !s !old_delta !err (ds:dds) = res s' err' k : go (k+1) t' s' delta err' dds+    where+      !ht' = tr ds+      !ht = 0.5*t+      !t' = ht'+ht+      !s' = (4*t'-t)/3+      !delta = abs (s'-s)+      !err' | delta == 0 || old_delta == 0                         = err+            | r <- log delta / log old_delta, 1.99 < r && r < 2.01 = delta*delta+            | otherwise                                            = delta+  go !k _ !s !_ !err [] = [res s err k]+  res i e k = Result (i*c) (e*c) (1 + 12*(2^k))+  c  = 0.5 * (b - a)+  d  = 0.5 * (a + b)+  i0 = w0 * f d + tr dd0+  i1 = tr dd1+  i01 = i0 + i1+  tr xs = foldl' (+) 0 (map (\(DD i w) -> let !ci = c * i in w*(f(d+ci)+f(d-ci))) xs `using` nf)++-- | Integration using a truncated Simpson's rule under tanh-sinh quadrature+simpson :: (Double -> Double) -> Double -> Double -> [Result]+simpson = simpson' r0++-- | Integration using a truncated Simpson's rule under tanh-sinh quadrature with buffered parallel evaluation+parSimpson  :: (Double -> Double) -> Double -> Double -> [Result]+parSimpson = simpson' (parBuffer 32 rseq)++data DD = DD {-# UNPACK #-} !Double {-# UNPACK #-} !Double+  deriving Show++w0 :: Double+w0 = 0.7853981633974483++dd0, dd1 :: [DD]+dd0 = [DD 0.9513679640727469 0.11501119725739434,DD 0.9999774771924616 1.3310025687635846e-4,DD 0.999999999999957 3.395446068634773e-13]+dd1 = [DD 0.6742714922484359 0.4829882897061506,DD 0.9975148564572244 9.171583494963921e-3,DD 0.9999999888756649 1.071560227847152e-7]++dd :: [[DD]]+dd = [+ [DD 0.3772097381640342 0.3474036898118141,+  DD 0.8595690586898966 0.1327695688570135,+  DD 0.9870405605073769 1.9096435892708076e-2,+  DD 0.9996882640283532 7.256294369753284e-4,+  DD 0.9999992047371147 2.99592534079268e-6,+  DD 0.9999999999528565 2.9077914535639456e-10],++ [DD 0.19435700332493544 0.19041046482933816,+  DD 0.5391467053879677 0.14918287823114462,+  DD 0.7806074389832003 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+ Setup.hs view
@@ -0,0 +1,2 @@+import Distribution.Simple+main = defaultMain
+ codegen/TableGen.hs view
@@ -0,0 +1,30 @@+{-# LANGUAGE ParallelListComp #-}++import Data.Number.CReal+import Numeric.AD+import Text.Groom++import Data.List (foldl')++data DD = DD {-# UNPACK #-} !Double {-# UNPACK #-} !Double+  deriving Show++f :: Floating a => a -> (a, a)+f = diff' x where x t = tanh (0.5 * pi * sinh t)++weight :: Double -> Double -> DD+weight w d = DD (dbl x) (w * dbl x')+  where (x,x') = f (realToFrac d :: CReal)+        dbl = read . showCReal 80++dz = weight 0.5 0+dd0 = map (weight 0.5) [1..3]+dd1 = map (weight 0.5) [0.5,1.5,2.5]++dd :: [[DD]]+dd = +   [ map (\x -> weight offset (x*scale + offset)) [0..u] +   | u <- [5,11,23,47,95,191,383,767,1535] | scale <- iterate (/2) 0.5 | offset <- iterate (/2) 0.25+   ]++groomed_dd = groom dd
+ integration.cabal view
@@ -0,0 +1,30 @@+name:                integration+version:             0.1+stability:           experimental+synopsis:            Fast robust numeric integration via tanh-sinh quadrature+description:         Fast robust numeric integration via tanh-sinh quadrature+homepage:            https://github.com/ekmett/integration+license:             BSD3+license-file:        LICENSE+author:              Edward Kmett+maintainer:          Edward Kmett <ekmett@gmail.com>+category:            Graphics+build-type:          Simple+cabal-version:       >=1.6++extra-source-files: codegen/TableGen.hs++Library+  -- Packages needed in order to build this package.+  build-depends:+     base                   >= 4 && < 5,+     parallel               >= 3.2 && < 3.3++  exposed-modules: Numeric.Integration.TanhSinh++  ghc-options: -O2 -fexcess-precision -threaded -Wall -O2 -fspec-constr -fliberate-case -fstatic-argument-transformation -fspec-constr-count=10++source-repository head+  type: git+  location: git://github.com/ekmett/integration.git+