org.apache.commons.math3.analysis.differentiation

Class FiniteDifferencesDifferentiator

java.lang.Object

org.apache.commons.math3.analysis.differentiation.FiniteDifferencesDifferentiator

All Implemented Interfaces:

java.io.Serializable, UnivariateFunctionDifferentiator, UnivariateMatrixFunctionDifferentiator, UnivariateVectorFunctionDifferentiator

public class FiniteDifferencesDifferentiator
extends java.lang.Object
implements UnivariateFunctionDifferentiator, UnivariateVectorFunctionDifferentiator, UnivariateMatrixFunctionDifferentiator, java.io.Serializable

Univariate functions differentiator using finite differences.

This class creates some wrapper objects around regular univariate functions (or univariate vector functions or univariate matrix functions). These wrapper objects compute derivatives in addition to function value.

The wrapper objects work by calling the underlying function on a sampling grid around the current point and performing polynomial interpolation. A finite differences scheme with n points is theoretically able to compute derivatives up to order n-1, but it is generally better to have a slight margin. The step size must also be small enough in order for the polynomial approximation to be good in the current point neighborhood, but it should not be too small because numerical instability appears quickly (there are several differences of close points). Choosing the number of points and the step size is highly problem dependent.

As an example of good and bad settings, lets consider the quintic polynomial function f(x) = (x-1)*(x-0.5)*x*(x+0.5)*(x+1). Since it is a polynomial, finite differences with at least 6 points should theoretically recover the exact same polynomial and hence compute accurate derivatives for any order. However, due to numerical errors, we get the following results for a 7 points finite differences for abscissae in the [-10, 10] range:

• step size = 0.25, second order derivative error about 9.97e-10
• step size = 0.25, fourth order derivative error about 5.43e-8
• step size = 1.0e-6, second order derivative error about 148
• step size = 1.0e-6, fourth order derivative error about 6.35e+14

This example shows that the small step size is really bad, even simply for second order derivative!

Since:

3.1

See Also:

Serialized Form

Field Summary

Fields

Modifier and Type	Field and Description
private double	halfSampleSpan Half sample span.
private int	nbPoints Number of points to use.
private static long	serialVersionUID Serializable UID.
private double	stepSize Step size.
private double	tMax Upper bound for independent variable.
private double	tMin Lower bound for independent variable.

Constructor Summary

Constructors

Constructor and Description
FiniteDifferencesDifferentiator(int nbPoints, double stepSize) Build a differentiator with number of points and step size when independent variable is unbounded.
FiniteDifferencesDifferentiator(int nbPoints, double stepSize, double tLower, double tUpper) Build a differentiator with number of points and step size when independent variable is bounded.

Method Summary

Methods

Modifier and Type	Method and Description
UnivariateDifferentiableFunction	differentiate(UnivariateFunction function) Create an implementation of a differential from a regular function.
UnivariateDifferentiableMatrixFunction	differentiate(UnivariateMatrixFunction function) Create an implementation of a differential from a regular matrix function.
UnivariateDifferentiableVectorFunction	differentiate(UnivariateVectorFunction function) Create an implementation of a differential from a regular vector function.
private DerivativeStructure	evaluate(DerivativeStructure t, double t0, double[] y) Evaluate derivatives from a sample.
int	getNbPoints() Get the number of points to use.
double	getStepSize() Get the step size.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Field Detail

serialVersionUID

private static final long serialVersionUID

Serializable UID.

See Also:

Constant Field Values

nbPoints

private final int nbPoints

Number of points to use.

stepSize

private final double stepSize

Step size.

halfSampleSpan

private final double halfSampleSpan

Half sample span.

tMin

private final double tMin

Lower bound for independent variable.

tMax

private final double tMax

Upper bound for independent variable.

Constructor Detail

FiniteDifferencesDifferentiator

public FiniteDifferencesDifferentiator(int nbPoints,
                               double stepSize)
                                throws NotPositiveException,
                                       NumberIsTooSmallException

Build a differentiator with number of points and step size when independent variable is unbounded.

Beware that wrong settings for the finite differences differentiator can lead to highly unstable and inaccurate results, especially for high derivation orders. Using very small step sizes is often a bad idea.

Parameters:

nbPoints - number of points to use

stepSize - step size (gap between each point)

Throws:

NotPositiveException - if stepsize <= 0 (note that NotPositiveException extends NumberIsTooSmallException)

NumberIsTooSmallException - nbPoint <= 1

FiniteDifferencesDifferentiator

public FiniteDifferencesDifferentiator(int nbPoints,
                               double stepSize,
                               double tLower,
                               double tUpper)
                                throws NotPositiveException,
                                       NumberIsTooSmallException,
                                       NumberIsTooLargeException

Build a differentiator with number of points and step size when independent variable is bounded.

When the independent variable is bounded (tLower < t < tUpper), the sampling points used for differentiation will be adapted to ensure the constraint holds even near the boundaries. This means the sample will not be centered anymore in these cases. At an extreme case, computing derivatives exactly at the lower bound will lead the sample to be entirely on the right side of the derivation point.

Note that the boundaries are considered to be excluded for function evaluation.

Beware that wrong settings for the finite differences differentiator can lead to highly unstable and inaccurate results, especially for high derivation orders. Using very small step sizes is often a bad idea.

Parameters:

nbPoints - number of points to use

stepSize - step size (gap between each point)

tLower - lower bound for independent variable (may be Double.NEGATIVE_INFINITY if there are no lower bounds)

tUpper - upper bound for independent variable (may be Double.POSITIVE_INFINITY if there are no upper bounds)

Throws:

NotPositiveException - if stepsize <= 0 (note that NotPositiveException extends NumberIsTooSmallException)

NumberIsTooSmallException - nbPoint <= 1

NumberIsTooLargeException - stepSize * (nbPoints - 1) >= tUpper - tLower

Method Detail

getNbPoints

public int getNbPoints()

Get the number of points to use.

Returns:

number of points to use

getStepSize

public double getStepSize()

Get the step size.

Returns:

step size

evaluate

private DerivativeStructure evaluate(DerivativeStructure t,
                           double t0,
                           double[] y)
                              throws NumberIsTooLargeException

Evaluate derivatives from a sample.

Evaluation is done using divided differences.

Parameters:

t - evaluation abscissa value and derivatives

t0 - first sample point abscissa

y - function values sample y[i] = f(t[i]) = f(t0 + i * stepSize)

Returns:

value and derivatives at t

Throws:

NumberIsTooLargeException - if the requested derivation order is larger or equal to the number of points

differentiate

public UnivariateDifferentiableFunction differentiate(UnivariateFunction function)

Create an implementation of a differential from a regular function.

The returned object cannot compute derivatives to arbitrary orders. The value function will throw a NumberIsTooLargeException if the requested derivation order is larger or equal to the number of points.

Specified by:

differentiate in interface UnivariateFunctionDifferentiator

Parameters:

function - function to differentiate

Returns:

differential function

differentiate

public UnivariateDifferentiableVectorFunction differentiate(UnivariateVectorFunction function)

Create an implementation of a differential from a regular vector function.

The returned object cannot compute derivatives to arbitrary orders. The value function will throw a NumberIsTooLargeException if the requested derivation order is larger or equal to the number of points.

Specified by:

differentiate in interface UnivariateVectorFunctionDifferentiator

Parameters:

function - function to differentiate

Returns:

differential function

differentiate

public UnivariateDifferentiableMatrixFunction differentiate(UnivariateMatrixFunction function)

Create an implementation of a differential from a regular matrix function.

The returned object cannot compute derivatives to arbitrary orders. The value function will throw a NumberIsTooLargeException if the requested derivation order is larger or equal to the number of points.

Specified by:

differentiate in interface UnivariateMatrixFunctionDifferentiator

Parameters:

function - function to differentiate

Returns:

differential function