umontreal.iro.lecuyer.functions
Class MathFunctionUtil

java.lang.Object
  umontreal.iro.lecuyer.functions.MathFunctionUtil

public class MathFunctionUtil
extends Object

Provides utility methods for computing derivatives and integrals of functions.

Field Summary

static double	H Step length in x to compute derivatives.
static int	NUMINTERVALS Default number of intervals for Simpson's integral.

Method Summary

static double	derivative(MathFunction func, double x) Returns the first derivative of the function func evaluated at x.
static double	derivative(MathFunction func, double x, int n) Returns the nth derivative of function func evaluated at x.
static double	finiteCenteredDifferenceDerivative(MathFunction func, double x, double h) Returns (f (x + h) - f (x - h))/(2h), an estimate of the first derivative of f (x) using centered differences.
static double	finiteCenteredDifferenceDerivative(MathFunction func, double x, int n, double h) Computes and returns an estimate of the nth derivative of the function f (x) using finite centered differences.
static double	finiteDifferenceDerivative(MathFunction func, double x, int n, double h) Computes and returns an estimate of the nth derivative of the function f (x).
static double	gaussLobatto(MathFunction func, double a, double b, double tol) Computes and returns a numerical approximation of the integral of f (x) over [a, b], using Gauss-Lobatto adaptive quadrature with 5 nodes, with tolerance tol.
static double	gaussLobatto(MathFunction func, double a, double b, double tol, double[][] T) Similar to method gaussLobatto(MathFunction, double, double, double), but also returns in T[0] the subintervals of integration, and in T[1], the partial values of the integral over the corresponding subintervals.
static double	integral(MathFunction func, double a, double b) Returns the integral of the function func over [a, b].
static double[][]	removeNaNs(double[] x, double[] y) Removes any point (NaN, y) or (x, NaN) from x and y, and returns a 2D array containing the filtered points.
static double	simpsonIntegral(MathFunction func, double a, double b, int numIntervals) Computes and returns an approximation of the integral of func over [a, b], using the Simpson's 1/3 method with numIntervals intervals.

Methods inherited from class java.lang.Object

equals, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Field Detail

H

public static double H

Step length in x to compute derivatives. Default: 10^-6.

NUMINTERVALS

public static int NUMINTERVALS

Default number of intervals for Simpson's integral.

Method Detail

derivative

public static double derivative(MathFunction func,
                                double x)

Returns the first derivative of the function func evaluated at x. If the given function implements MathFunctionWithFirstDerivative, this method calls MathFunctionWithFirstDerivative.derivative (double). Otherwise, if the function implements MathFunctionWithDerivative, this method calls MathFunctionWithDerivative.derivative (double, int). If the function does not implement any of these two interfaces, the method uses finiteCenteredDifferenceDerivative (MathFunction, double, double) to obtain an estimate of the derivative.

Parameters:

func - the function to derivate.

x - the evaluation point.

Returns:

the first derivative.

derivative

public static double derivative(MathFunction func,
                                double x,
                                int n)

Returns the nth derivative of function func evaluated at x. If n = 0, this returns f (x). If n = 1, this calls derivative (MathFunction, double) and returns the resulting first derivative. Otherwise, if the function implements MathFunctionWithDerivative, this method calls MathFunctionWithDerivative.derivative (double, int). If the function does not implement this interface, the method uses finiteCenteredDifferenceDerivative (MathFunction, double, int, double) if n is even, or finiteDifferenceDerivative (MathFunction, double, int, double) if n is odd, to obtain a numerical approximation of the derivative.

Parameters:

func - the function to derivate.

x - the evaluation point.

n - the order of the derivative.

Returns:

the nth derivative.

finiteDifferenceDerivative

public static double finiteDifferenceDerivative(MathFunction func,
                                                double x,
                                                int n,
                                                double h)

Computes and returns an estimate of the nth derivative of the function f (x). This method estimates

,

the nth derivative of f (x) evaluated at x. This method first computes f_i = f (x + iε), for i = 0,…, n, with ε = h^1/n. The estimate is then given by Δⁿf₀/h, where Δⁿf_i = Δ^n-1f_i+1 - Δ^n-1f_i, and Δf_i = f_i+1 - f_i.

Parameters:

func - the function to derivate.

x - the evaluation point.

n - the order of the derivative.

h - the error.

Returns:

the estimate of the derivative.

finiteCenteredDifferenceDerivative

public static double finiteCenteredDifferenceDerivative(MathFunction func,
                                                        double x,
                                                        double h)

Returns (f (x + h) - f (x - h))/(2h), an estimate of the first derivative of f (x) using centered differences.

Parameters:

func - the function to derivate.

x - the evaluation point.

h - the error.

Returns:

the estimate of the first derivative.

finiteCenteredDifferenceDerivative

public static double finiteCenteredDifferenceDerivative(MathFunction func,
                                                        double x,
                                                        int n,
                                                        double h)

Computes and returns an estimate of the nth derivative of the function f (x) using finite centered differences. If n is even, this method returns finiteDifferenceDerivative (func, x - ε*n/2, n, h), with h = εⁿ.

Parameters:

func - the function to derivate.

x - the evaluation point.

n - the order of the derivative.

h - the error.

Returns:

the estimate of the derivative.

removeNaNs

public static double[][] removeNaNs(double[] x,
                                    double[] y)

Removes any point (NaN, y) or (x, NaN) from x and y, and returns a 2D array containing the filtered points. This method filters each pair (x[i], y[i]) containing at least one NaN element. It constructs a 2D array containing the two filtered arrays, whose size is smaller than or equal to x.length.

Parameters:

x - the X coordinates.

y - the Y coordinates.

Returns:

the filtered X and Y arrays.

integral

public static double integral(MathFunction func,
                              double a,
                              double b)

Returns the integral of the function func over [a, b]. If the given function implements MathFunctionWithIntegral, this returns MathFunctionWithIntegral.integral (double, double). Otherwise, this calls simpsonIntegral (MathFunction, double, double, int) with NUMINTERVALS intervals.

Parameters:

func - the function to integrate.

a - the lower bound.

b - the upper bound.

Returns:

the value of the integral.

simpsonIntegral

public static double simpsonIntegral(MathFunction func,
                                     double a,
                                     double b,
                                     int numIntervals)

Computes and returns an approximation of the integral of func over [a, b], using the Simpson's 1/3 method with numIntervals intervals. This method estimates

∫_a^bf (x)dx,

where f (x) is the function defined by func evaluated at x, by dividing [a, b] in n = numIntervals intervals of length h = (b - a)/n. The integral is estimated by

(f (a) + 4f (a + h) + 2f (a + 2h) + 4f (a + 3h) + ^... + f (b))

This method assumes that a <= b < ∞, and n is even.

Parameters:

func - the function being integrated.

a - the left bound

b - the right bound.

numIntervals - the number of intervals.

Returns:

the approximate value of the integral.

gaussLobatto

public static double gaussLobatto(MathFunction func,
                                  double a,
                                  double b,
                                  double tol)

Computes and returns a numerical approximation of the integral of f (x) over [a, b], using Gauss-Lobatto adaptive quadrature with 5 nodes, with tolerance tol. This method estimates

∫_a^bf (x)dx,

where f (x) is the function defined by func. Whenever the estimated error is larger than tol, the interval [a, b] will be halved in two smaller intervals, and the method will recursively call itself on the two smaller intervals until the estimated error is smaller than tol.

Parameters:

func - the function being integrated.

a - the left bound

b - the right bound.

tol - error.

Returns:

the approximate value of the integral.

gaussLobatto

public static double gaussLobatto(MathFunction func,
                                  double a,
                                  double b,
                                  double tol,
                                  double[][] T)

Similar to method gaussLobatto(MathFunction, double, double, double), but also returns in T[0] the subintervals of integration, and in T[1], the partial values of the integral over the corresponding subintervals. Thus T[0][0] = x₀ = a and T[0][n] = x_n = b; T[1][i] contains the value of the integral over the subinterval [x_i-1, x_i]; we also have T[1][0] = 0. The sum over all T[1][i], for i = 1,…, n gives the value of the integral over [a, b], which is the value returned by this method. WARNING: The user must reserve the 2 elements of the first dimension (T[0] and T[1]) before calling this method.

Parameters:

func - function being integrated

a - left bound of interval

b - right bound of interval

tol - error

T - (x, y) = (values of partial intervals,partial values of integral)

Returns:

value of the integral