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

java.lang.Object
  extended by org.apache.commons.math3.analysis.differentiation.FiniteDifferencesDifferentiator
All Implemented Interfaces:
Serializable, UnivariateFunctionDifferentiator, UnivariateMatrixFunctionDifferentiator, UnivariateVectorFunctionDifferentiator

public class FiniteDifferencesDifferentiator
extends Object
implements UnivariateFunctionDifferentiator, UnivariateVectorFunctionDifferentiator, UnivariateMatrixFunctionDifferentiator, 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:

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

Since:
3.1
Version:
$Id: FiniteDifferencesDifferentiator.java 1420666 2012-12-12 13:33:20Z erans $
See Also:
Serialized Form

Field Summary
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
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
 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


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