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!
| Modifier and Type | Field and Description |
|---|---|
| private final double | halfSampleSpan
Half sample span. |
| private final int | nbPoints
Number of points to use. |
| private static final long | serialVersionUID
Serializable UID. |
| private final double | stepSize
Step size. |
| private final double | tMax
Upper bound for independent variable. |
| private final double | tMin
Lower bound for independent variable. |
| Access | Constructor and Description |
|---|---|
| public | FiniteDifferencesDifferentiator(final int
number of points to use nbPoints, final double step size (gap between each point) stepSize)Build a differentiator with number of points and step size when independent variable is unbounded. |
| public | FiniteDifferencesDifferentiator(final int
number of points to use nbPoints, final double step size (gap between each point) stepSize, final double lower bound for independent variable (may be tLower, final double Double.NEGATIVE_INFINITY
if there are no lower bounds)upper bound for independent variable (may be tUpper)Double.POSITIVE_INFINITY
if there are no upper bounds)Build a differentiator with number of points and step size when independent variable is bounded. |
| Modifier and Type | Method and Description |
|---|---|
| public UnivariateDifferentiableFunction | differentiate(final UnivariateFunction
function to differentiate function)Implements org. Create an implementation of a |
| public UnivariateDifferentiableVectorFunction | differentiate(final UnivariateVectorFunction
function to differentiate function)Implements org. Create an implementation of a |
| public UnivariateDifferentiableMatrixFunction | differentiate(final UnivariateMatrixFunction
function to differentiate function)Implements org. Create an implementation of a |
| private DerivativeStructure | Returns: value and derivatives attevaluation abscissa value and derivatives t, final double first sample point abscissa t0, final double[] function values sample y)y[i] = f(t[i]) = f(t0 + i * stepSize)Evaluate derivatives from a sample. |
| public int | |
| public double |
| halfSampleSpan | back to summary |
|---|---|
| private final double halfSampleSpan Half sample span. | |
| nbPoints | back to summary |
|---|---|
| private final int nbPoints Number of points to use. | |
| serialVersionUID | back to summary |
|---|---|
| private static final long serialVersionUID Serializable UID. | |
| stepSize | back to summary |
|---|---|
| private final double stepSize Step size. | |
| tMax | back to summary |
|---|---|
| private final double tMax Upper bound for independent variable. | |
| tMin | back to summary |
|---|---|
| private final double tMin Lower bound for independent variable. | |
| FiniteDifferencesDifferentiator | back to summary |
|---|---|
| public FiniteDifferencesDifferentiator(final int nbPoints, final 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.
| |
| FiniteDifferencesDifferentiator | back to summary |
|---|---|
| public FiniteDifferencesDifferentiator(final int nbPoints, final double stepSize, final double tLower, final 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.
| |
| differentiate | back to summary |
|---|---|
| public UnivariateDifferentiableFunction differentiate(final UnivariateFunction function) Implements org. Doc from org. Create an implementation of a The returned object cannot compute derivatives to arbitrary orders. The
value function will throw a
| |
| differentiate | back to summary |
|---|---|
| public UnivariateDifferentiableVectorFunction differentiate(final UnivariateVectorFunction function) Implements org. Create an implementation of a The returned object cannot compute derivatives to arbitrary orders. The
value function will throw a
| |
| differentiate | back to summary |
|---|---|
| public UnivariateDifferentiableMatrixFunction differentiate(final UnivariateMatrixFunction function) Implements org. Create an implementation of a The returned object cannot compute derivatives to arbitrary orders. The
value function will throw a
| |
| evaluate | back to summary |
|---|---|
| private DerivativeStructure evaluate(final DerivativeStructure t, final double t0, final double[] y) throws NumberIsTooLargeException Evaluate derivatives from a sample. Evaluation is done using divided differences.
| |
| getNbPoints | back to summary |
|---|---|
| public int getNbPoints() Get the number of points to use.
| |
| getStepSize | back to summary |
|---|---|
| public double getStepSize() Get the step size.
| |