Package dev.nm.analysis.differentiation

Class Ridders

java.lang.Object
dev.nm.analysis.function.rn2r1.AbstractRealScalarFunction
dev.nm.analysis.differentiation.Ridders
All Implemented Interfaces:
Function<Vector,Double>, RealScalarFunction
public class Ridders extends AbstractRealScalarFunction
Ridders' method computes the numerical derivative of a function. In general it gives a higher precision than the simple finite differencing method, c.f., FiniteDifference. Ridders' method tries a sequence of decreasing h's to compute the derivatives, and then extrapolate to zero using Neville's algorithm. The choice of the initial h is critical. If h is too big, the value computed could be inaccurate. If h is too small, due to rounding error, we might be computing the "same" value over and over again for different h's.

  • Constructor Details

    • Ridders

      public Ridders(UnivariateRealFunction f, int order, double rate, int discretization)
      Construct the derivative function of a univariate function using Ridder's method.
      Parameters:
      f - the UnivariateRealFunction to take derivative of
      order - the order of differentiation
      rate - the rate at which the increment h decreases; rate should be a simple number such as 0.75, not like 0.66666666666...
      discretization - the number of points for extrapolation

    • Ridders

      public Ridders(UnivariateRealFunction f, int order)
      Construct the derivative function of a univariate function using Ridder's method.
      Parameters:
      f - the UnivariateRealFunction to take derivative of
      order - the order of the derivative

    • Ridders

      public Ridders(RealScalarFunction f, int[] varidx, double rate, int discretization)
      Construct the derivative function of a vector-valued function using Ridder's method.

      By convention, varidx = new int[]{1, 2} means \[ f_{x_1,x_2} = {\partial^2 f \over \partial x_1 \partial x_2} = {\partial \over \partial x_2}{\partial \over \partial x_1} \]

      The indices count from 1 up to the number of variables of f, i.e., the domain dimension of f.

      Parameters:
      f - the multivariate function to take derivative of
      varidx - specify the variable indices, numbering from 1 up to the domain dimension of f
      rate - rate should be a simple number, not like 0.66666666666...
      discretization - the number of points used for extrapolation

    • Ridders

      public Ridders(RealScalarFunction f, int[] varidx)
      Construct the derivative function of a vector-valued function using Ridder's method.

      By convention, varidx = new int[]{1, 2} means \[ f_{x_1,x_2} = {\partial^2 f \over \partial x_1 \partial x_2} = {\partial \over \partial x_2}{\partial \over \partial x_1} \]

      The indices count from 1 up to the number of variables of f, i.e., the domain dimension of f.

      Parameters:
      f - the real multivariate function to take derivative of
      varidx - specify the variable indices, numbering from 1 up to the domain dimension of f

  • Method Details

    • evaluate

      public Double evaluate(Vector x)
      Evaluate the function f at x, where x is from the domain.

      Make sure that h and x+h are representable in floating point precision so that the difference between x+h and x is exactly h, the step size.

      Parameters:
      x - the point to evaluate the derivative of f at
      Returns:
      f'(x), the numerical derivative of f at point x using Ridders' method
      See Also:

    • evaluate

      public double evaluate(double x)
      Evaluate f'(x), where f is a UnivariateRealFunction.
      Parameters:
      x - the point to evaluate the derivative of f at
      Returns:
      f'(x), the numerical derivative of f at point x using Ridders' method
      See Also:

    • evaluate

      public double evaluate(Vector x, double h)
      Evaluate numerically the derivative of f at point x, f'(x), with step size h. It could be challenging to automatically determine the step size h, esp. when |x| is near 0. It may, for example, require an analysis that involves f' and f''. The user may want to experiment with different hs by calling this function.
      Parameters:
      x - the point to evaluate f at
      h - the step size
      Returns:
      f'(x), the numerical derivative of f at point x with step size h