Package org.hipparchus.analysis.differentiation

Class 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 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:

    • 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!

    See Also:
    Serialized Form

    • Constructor Detail

      • FiniteDifferencesDifferentiator

        public FiniteDifferencesDifferentiator​(int nbPoints,
                                       double stepSize)
                                throws MathIllegalArgumentException
        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:
        MathIllegalArgumentException - if stepsize <= 0 (note that MathIllegalArgumentException extends MathIllegalArgumentException)
        MathIllegalArgumentException - nbPoint <= 1

      • FiniteDifferencesDifferentiator

        public FiniteDifferencesDifferentiator​(int nbPoints,
                                       double stepSize,
                                       double tLower,
                                       double tUpper)
                                throws MathIllegalArgumentException
        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:
        MathIllegalArgumentException - if stepsize <= 0 (note that MathIllegalArgumentException extends MathIllegalArgumentException)
        MathIllegalArgumentException - nbPoint <= 1
        MathIllegalArgumentException - stepSize * (nbPoints - 1) >= tUpper - tLower