Integration using a truncated trapezoid rule under tanh-sinh quadrature
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.
Synopsis
- value trap
- value simpson
- value trap'
- value simpson'
- value parTrap
- value parSimpson
- data Result
- value absolute
- value relative
- value confidence
- value nonNegative
- value everywhere
- Packageintegration-0.2.1
- Exports12
- LanguageHaskell98
- LicenceBSD-3-Clause
- SourceTanhSinh.hs
Quadrature methods
Integration using a truncated Simpson's rule under tanh-sinh quadrature
Integration using a truncated trapezoid rule and tanh-sinh quadrature with a specified evaluation strategy
Integration using a truncated Simpson's rule under tanh-sinh quadrature with a specified evaluation strategy
Integration using a truncated trapezoid rule under tanh-sinh quadrature with buffered parallel evaluation
Integration using a truncated Simpson's rule under tanh-sinh quadrature with buffered parallel evaluation
Integral with an result and an estimate of the error such that
(result - errorEstimate, result + errorEstimate) probably bounds
the actual answer.
Constructors
Resultresult :: !DoubleerrorEstimate :: !Doubleevaluations :: !Int
Instances4Eq, Ord, Read, Show
Estimated error bounds
Filter a list of results using a specified absolute error bound
Filter a list of results using a specified relative error bound
Confidence intervals
Convert a Result to a confidence interval
Changes of variables
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.
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.