public class FiniteDifferencesDifferentiator extends Object implements UnivariateFunctionDifferentiator, UnivariateVectorFunctionDifferentiator, UnivariateMatrixFunctionDifferentiator, Serializable
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
values.
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!
| 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.
|
| 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. |
int |
getNbPoints()
Get the number of points to use.
|
double |
getStepSize()
Get the step size.
|
public FiniteDifferencesDifferentiator(int nbPoints,
double stepSize)
throws NotPositiveException,
NumberIsTooSmallException
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.
nbPoints - number of points to usestepSize - step size (gap between each point)NotPositiveException - if stepsize <= 0 (note that
NotPositiveException extends NumberIsTooSmallException)NumberIsTooSmallException - nbPoint <= 1public FiniteDifferencesDifferentiator(int nbPoints,
double stepSize,
double tLower,
double tUpper)
throws NotPositiveException,
NumberIsTooSmallException,
NumberIsTooLargeException
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.
nbPoints - number of points to usestepSize - 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)NotPositiveException - if stepsize <= 0 (note that
NotPositiveException extends NumberIsTooSmallException)NumberIsTooSmallException - nbPoint <= 1NumberIsTooLargeException - stepSize * (nbPoints - 1) >= tUpper - tLowerpublic int getNbPoints()
public double getStepSize()
public UnivariateDifferentiableFunction differentiate(UnivariateFunction function)
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.
differentiate in interface UnivariateFunctionDifferentiatorfunction - function to differentiatepublic UnivariateDifferentiableVectorFunction differentiate(UnivariateVectorFunction function)
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.
differentiate in interface UnivariateVectorFunctionDifferentiatorfunction - function to differentiatepublic UnivariateDifferentiableMatrixFunction differentiate(UnivariateMatrixFunction function)
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.
differentiate in interface UnivariateMatrixFunctionDifferentiatorfunction - function to differentiateCopyright © 2003–2016 The Apache Software Foundation. All rights reserved.