Copyright	(C) 2012-2015 Edward Kmett
License	BSD-style (see the file LICENSE)
Maintainer	Edward Kmett <ekmett@gmail.com>
Stability	provisional
Portability	portable
Safe Haskell	None
Language	Haskell98

Numeric.Integration.TanhSinh

Contents

Quadrature methods
Estimated error bounds
Confidence intervals
Changes of variables

Description

An implementation of Takahashi and Mori's 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, evaluations = 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, evaluations = 49}

See John D. Cook's "Care and Treatment of Singularities" for a sense of how more naïve quadrature schemes fare.

Synopsis

Quadrature methods

trap :: (Double -> Double) -> Double -> Double -> [Result] #

Integration using a truncated trapezoid rule under tanh-sinh quadrature

simpson :: (Double -> Double) -> Double -> Double -> [Result] #

Integration using a truncated Simpson's rule under tanh-sinh quadrature

trap' :: Strategy [Double] -> (Double -> Double) -> Double -> Double -> [Result] #

Integration using a truncated trapezoid rule and tanh-sinh quadrature with a specified evaluation strategy

simpson' :: Strategy [Double] -> (Double -> Double) -> Double -> Double -> [Result] #

Integration using a truncated Simpson's rule under tanh-sinh quadrature with a specified evaluation strategy

parTrap :: (Double -> Double) -> Double -> Double -> [Result] #

Integration using a truncated trapezoid rule under tanh-sinh quadrature with buffered parallel evaluation

parSimpson :: (Double -> Double) -> Double -> Double -> [Result] #

Integration using a truncated Simpson's rule under tanh-sinh quadrature with buffered parallel evaluation

data Result #

Integral with an result and an estimate of the error such that (result - errorEstimate, result + errorEstimate) probably bounds the actual answer.

Constructors

Result
Fields result :: !Double errorEstimate :: !Double evaluations :: !Int

Instances

Eq Result #
Methods (==) :: Result -> Result -> Bool # (/=) :: Result -> Result -> Bool #
Ord Result #
Methods compare :: Result -> Result -> Ordering # (<) :: Result -> Result -> Bool # (<=) :: Result -> Result -> Bool # (>) :: Result -> Result -> Bool # (>=) :: Result -> Result -> Bool # max :: Result -> Result -> Result # min :: Result -> Result -> Result #
Read Result #
Methods readsPrec :: Int -> ReadS Result # readList :: ReadS [Result] # readPrec :: ReadPrec Result # readListPrec :: ReadPrec [Result] #
Show Result #
Methods showsPrec :: Int -> Result -> ShowS # show :: Result -> String # showList :: [Result] -> ShowS #

Estimated error bounds

absolute :: Double -> [Result] -> Result #

Filter a list of results using a specified absolute error bound

relative :: Double -> [Result] -> Result #

Filter a list of results using a specified relative error bound

Confidence intervals

confidence :: Result -> (Double, Double) #

Convert a Result to a confidence interval

Changes of variables

nonNegative :: ((Double -> Double) -> Double -> Double -> r) -> (Double -> Double) -> r #

Integrate a function from 0 to infinity by using the change of variables x = t/(1-t)

This works much better than just clipping the interval at some arbitrary large number.

everywhere :: ((Double -> Double) -> Double -> Double -> r) -> (Double -> Double) -> r #

Integrate from -inf to inf using tanh-sinh quadrature after using the change of variables x = tan t

everywhere trap (\x -> exp(-x*x))

This works much better than just clipping the interval at arbitrary large and small numbers.