Numeric.Tools.Differentiation

Numerical differentiation. diffRichardson is preferred way to calculate derivative.

Differentiation

data DiffRes

Differentiation result

Constructors

DiffRes

Fields

diffRes :: Double

Derivative value

diffPrecision :: Double

Rough error estimate

Instances

Eq DiffRes

Data DiffRes

Show DiffRes

Typeable DiffRes

diffRichardson

Arguments

:: (Double -> Double)

Function

-> Double

Delta

-> Double

Point at which evaluate differential

-> DiffRes

Calculate derivative using Richaradson's deferred approach to limit. This is a preferred method for numeric differentiation since it's most precise. Function could be evaluated up to 20 times.

Initial step size should be chosen fairly big. Too small one will result reduced precision, too big one in nonsensical answer.

Fast but imprecise

diffSimple

Arguments

:: (Double -> Double)

Function to differentiate

-> Double

Delta

-> (Double, Double)

Coordinate and function value at this point

-> Double

Simplest form of differentiation. Should be used only when function evaluation is prohibitively expensive and already computed value at point x should be reused.

 f'(x) = f(x+h) - f(x) / h

diffSimmetric

Arguments

:: (Double -> Double)

Function to differentiate

-> Double

Delta

-> Double

Point at which evaluate differential

-> Double

Simple differentiation. It uses simmetric rule and provide reasonable accuracy. It's suitable when function evaluation is expensive and precision could be traded for speed.

 f'(x) = f(x-h) + f(x+h) / 2h

Utils

representableDelta

Arguments

:: Double

x

-> Double

small delta

-> Double

For number x and small h return such h' that x+h' and x differ by representable number

References

• Ridders, C.J.F. 1982, Accurate computation of F`(x) and F`(x)F``(x), Advances in Engineering Software, vol. 4, no. 2, pp. 75-76.