Overview

We use dispatch to determine the mode of finite difference we use to perform numerical differentiation. The modes are specified using the following types:

  • ForwardMode
  • BackwardMode
  • CentralMode
  • ComplexMode
  • HessianMode

Given a mode and a value of x at which to differentiate f(x), the following functions are used to determine the step-size to use for finite difference approximations of derivatives:

  • step_size(::ForwardMode, x::AbstractFloat)
  • step_size(::BackwardMode, x::AbstractFloat)
  • step_size(::CentralMode, x::AbstractFloat)
  • step_size(::ComplexMode, x::AbstractFloat)
  • step_size(::HessianMode, x::AbstractFloat)

Unless there are bugs in the definitions of the core functions, the quality of an approximate derivative is almost entirely determined by these step-sizes. These step sizes are chosen to reach a reasonable compromise between errors introduced by floating-point truncation (which could be minimized by using larger step-sizes) and errors introduced by the use of an truncated Taylor series (which could be minimized by using smaller step-sizes). The exact compromise depends on the mode of finite differences we use.

Detailed Method-Level Documentation

# FiniteDiff.step_size — Method.

step_size(::ForwardMode, x::AbstractFloat)

Description

Determine the step-size to use for forward-mode finite differences.

Arguments

  • ::ForwardMode: An instance of the ForwardMode type.
  • x::AbstractFloat: A point at which the derivative of a function will be approximated. Its type must implement eps.

Returns

  • ϵ::Real: The step-size to use for finite differences at x.

References

See Section 5.7 of Numerical Recipes in C for the mathematical justification for working with sqrt(eps(typeof(x))).

Examples

import FiniteDiff: step_size, ForwardMode
ϵ = step_size(ForwardMode(), 0.0)

source

# FiniteDiff.step_size — Method.

step_size(::BackwardMode, x::AbstractFloat)

Description

Determine the step-size to use for backward-mode finite differences.

Arguments

  • ::BackwardMode: An instance of the BackwardMode type.
  • x::AbstractFloat: A point at which the derivative of a function will be approximated. Its type must implement eps.

Returns

  • ϵ::Real: The step-size to use for finite differences at x.

References

See Section 5.7 of Numerical Recipes in C for the mathematical justification for working with sqrt(eps(typeof(x))).

Examples

import FiniteDiff: step_size, BackwardMode
ϵ = step_size(BackwardMode(), 0.0)

source

# FiniteDiff.step_size — Method.

step_size(::CentralMode, x::AbstractFloat)

Description

Determine the step-size to use for central-mode finite differences.

Arguments

  • ::CentralMode: An instance of the CentralMode type.
  • x::AbstractFloat: A point at which the derivative of a function will be approximated. Its type must implement eps.

Returns

  • ϵ::Real: The step-size to use for finite differences at x.

References

See Section 5.7 of Numerical Recipes in C for the mathematical justification for working with cbrt(eps(typeof(x))).

Examples

import FiniteDiff: step_size, CentralMode
ϵ = step_size(CentralMode(), 0.0)

source

# FiniteDiff.step_size — Method.

step_size(::ComplexMode, x::AbstractFloat)

Description

Determine the step-size to use for complex-mode finite differences.

Arguments

  • ::ComplexMode: An instance of the ComplexMode type.
  • x::AbstractFloat: A point at which the derivative of a function will be approximated. Its type must implement eps.

Returns

  • ϵ::Real: The step-size to use for finite differences at x.

References

See "The Complex-Step Derivative Approximation" by Martins, Sturdza and Alonso (2003) for the mathematical justification for working with eps(x).

Examples

import FiniteDiff: step_size, ComplexMode
ϵ = step_size(ComplexMode(), 0.0)

source

# FiniteDiff.step_size — Method.

step_size(::HessianMode, x::AbstractFloat)

Description

Determine the step-size to use for finite differences of hessians.

Arguments

  • ::HessianMode: An instance of the HessianMode type.
  • x::AbstractFloat: A point at which the derivative of a function will be approximated. Its type must implement eps.

Returns

  • ϵ::Real: The step-size to use for finite differences at x.

References

See Section 5.7 of Numerical Recipes in C for the mathematical justification for working with eps(x)^(1 // 4).

Examples

import FiniteDiff: step_size, HessianMode
ϵ = step_size(HessianMode(), 0.0)

source