User Manual 3.3 Numerical differentiation and integration

User Manual 3.3 Numerical differentiation and integration

Introduction

Scope

This section détails numerical differentialtion and integration (not to be misunderstood with numerical integration of ODE). A focus is realised on:

• differentiation methods of real univariate functions: Ridders and finite difference.
• integration methods of real univariate functions : Trapezoidal and Simpson.

Javadoc

The numerical differentiator objects are available in the package org.apache.commons.math3.analysis.differentiation in the Commons Math Library. The numerical integrator objects are available in the package org.apache.commons.math3.analysis.integration in the Commons Math Library.

|=(% colspan="3" %)Library|=(% colspan="6" %)Javadoc |(% colspan="3" %)Commons Math|(% colspan="6" %)Package org.apache.commons.math3.analysis.differentiation |(% colspan="3" %)Commons Math|(% colspan="6" %)Package org.apache.commons.math3.analysis.integration

Links

Links to the implemented integration methods :

http://mathworld.wolfram.com/TrapezoidalRule.html

http://mathworld.wolfram.com/SimpsonsRule.html

Package Overview

The numerical differentiation conception is described hereafter :

Fichier:Differentiators.bmp

Accuracy of integration method (all examples are based on sinus function):

|=(% colspan="3" %)Default setting|=(% colspan="3" %)Value |(% colspan="3" %)Maximum absolute error|(% colspan="6" %)1.0e-15 |(% colspan="3" %)Maximum relative error|(% colspan="6" %)1.0e-6 |(% colspan="3" %)Maximum number of iterations|(% colspan="6" %)64 |(% colspan="3" %)Minimum number of iterations|(% colspan="6" %)3

Note : the accuracy of the results and the computing time are directly dependent on the value of maximum relative error.

Features Description

Trapezoidal method

The trapezoidal rule works by approximating the region under the graph of the function as a trapezoid and calculating its area.

UnivariateFunction f = new SinFunction();
UnivariateIntegrator integrator = new TrapezoidIntegrator();
double r = integrator.integrate(10000, f, 0, FastMath.PI);

Result : r = 1.9999996078171345 (exact result = 2)

And :

integrator = new TrapezoidIntegrator(1.e-12, 1.0e-15, 3, 64);
r = integrator.integrate(10000000, f, 0, FastMath.PI);

Result : r = 1.9999999999904077

But :

integrator = new TrapezoidIntegrator(1.e-13, 1.0e-15, 3, 64);
r = integrator.integrate(10000000, f, 0, FastMath.PI);

Result : r = no result (infinite time !)

Simpson method

Simpson's rule is a Newton-Cotes formula for approximating the integral of a function using quadratic polynomials (i.e., parabolic arcs instead of the straight line segments used in the trapezoidal rule). In general, this method has faster convergence than the trapezoidal rule for functions which are twice continuously differentiable, though not in all specific cases.

UnivariateFunction f = new SinFunction();
UnivariateIntegrator integrator = new SimpsonIntegrator();
double r = integrator.integrate(10000, f, 0, FastMath.PI);

Result : r = 2.000000064530001 (exact result = 2)

Contents

Interfaces

|=(% colspan="3" %)Interface|=(% colspan="6" %)Summary|=(% colspan="1" %)Javadoc |(% colspan="3" %)UnivariateFunctionDifferentiator|(% colspan="6" %)This interface represents a generic numerical differentiator.|... |(% colspan="3" %)UnivariateIntegrator|(% colspan="6" %)Interface for univariate integration algorithms.|...

Classes

|=(% colspan="3" %)Class|=(% colspan="6" %)Summary|=(% colspan="1" %)Javadoc |(% colspan="3" %)FiniteDifferencesDifferentiator|(% colspan="6" %)Apply the finite difference method to differentiate a function.|... |(% colspan="3" %)RiddersDifferentiator|(% colspan="6" %)Apply the Ridders algorithm to differentiate a function.|... |(% colspan="3" %)TrapezoidIntegrator|(% colspan="6" %)The class implements the Trapezoidal rule|(% colspan="1" %)... |(% colspan="3" %)SimpsonIntegrator|(% colspan="6" %)The class implements the Simpson rule|(% colspan="1" %)...