Moduleintegration-0.2.1Haskell98

Numeric.Integration.TanhSinh

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.

1 type
11 values

Packageintegration-0.2.1
Exports12
LanguageHaskell98
LicenceBSD-3-Clause
SourceTanhSinh.hs

Quadrature methods

7 declarations

valuetrap :: (Double -> Double) -> Double -> Double -> [Result]

Integration using a truncated trapezoid rule under tanh-sinh quadrature

valuesimpson :: (Double -> Double) -> Double -> Double -> [Result]

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

valuetrap'

:: Strategy [Double]
-> Double -> Double
-> Double
-> Double
-> [Result]

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

valuesimpson'

:: Strategy [Double]
-> Double -> Double
-> Double
-> Double
-> [Result]

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

valueparTrap :: (Double -> Double) -> Double -> Double -> [Result]

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

valueparSimpson :: (Double -> Double) -> Double -> Double -> [Result]

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

datadata Result

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

Constructors

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

Instances4Eq, Ord, Read, Show

Eq ResultDefined in integration-0.2.1 · Numeric.Integration.TanhSinh
Ord ResultDefined in integration-0.2.1 · Numeric.Integration.TanhSinh
Read ResultDefined in integration-0.2.1 · Numeric.Integration.TanhSinh
Show ResultDefined in integration-0.2.1 · Numeric.Integration.TanhSinh

Estimated error bounds

2 declarations

valueabsolute :: Double -> [Result] -> Result

Filter a list of results using a specified absolute error bound

valuerelative :: Double -> [Result] -> Result

Filter a list of results using a specified relative error bound

Confidence intervals

1 declaration

valueconfidence :: Result -> (Double, Double)

Convert a Result to a confidence interval

Changes of variables

2 declarations

valuenonNegative

:: (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.

valueeverywhere

:: (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.