dev.nm.analysis.integration.univariate.riemann.gaussian

Class GaussianQuadrature

java.lang.Object

dev.nm.analysis.integration.univariate.riemann.gaussian.GaussianQuadrature

All Implemented Interfaces:

Integrator

Direct Known Subclasses:

GaussChebyshevQuadrature, GaussHermiteQuadrature, GaussLaguerreQuadrature, GaussLegendreQuadrature

public class GaussianQuadrature
extends Object
implements Integrator

A quadrature rule is a method of numerical integration in which we approximate the integral of a function by a weighted sum of sample points. That is, \[ \int_{a}^b f(x)\,dx \approx \sum_{i=1}^n c_i f(x_i). \]

By a suitable choice of the points \(x_i\) and the weights \(c_i\), an n-point weighted sum will yield an exact result for polynomials of degree (2n-1) or less, which is known as a Gaussian quadrature rule.

Gaussian quadrature will only produce accurate results if \(f(x)\) is well approximated by a polynomial function within the range [-1,1]. However, if the integrated function can be written as \(f(x) = W(x) g(x)\) where \(g(x)\) is approximately polynomial and \(W(x)\) is known, then there are alternative weights \(w_i'\) such that \[ \int_{a}^b f(x)\,dx =\int_{a}^b W(x) g(x)\,dx \approx \sum_{i=1}^n w_i' g(x_i). \]

Note that the precision for Gaussian quadrature rules cannot be specified, and hence calling Integrator.getPrecision() will throw an UnsupportedOperationException.

See Also:

Wikipedia: Gaussian quadrature

Constructor Summary

Constructors

Constructor and Description
GaussianQuadrature(GaussianQuadratureRule rule) Create a Gaussian quadrature integrator with the given quadrature rule.

Method Summary

Modifier and Type	Method and Description
double	getPrecision() Get the convergence threshold.
double	integrate(UnivariateRealFunction f, double a, double b) Integrate function f from a to b, \[ \int_a^b\!

Methods inherited from class java.lang.Object

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

Constructor Detail

GaussianQuadrature

public GaussianQuadrature(GaussianQuadratureRule rule)

Create a Gaussian quadrature integrator with the given quadrature rule.

Parameters:

rule - the Gaussian quadrature rule

Method Detail

integrate

public double integrate(UnivariateRealFunction f,
                        double a,
                        double b)

Description copied from interface: Integrator

Integrate function f from a to b, \[ \int_a^b\! f(x)\, dx \]

Specified by:

integrate in interface Integrator

Parameters:

f - a univariate function

a - the lower limit

b - the upper limit

Returns:

\(\int_a^b\! f(x)\, dx\)

getPrecision

public double getPrecision()

Description copied from interface: Integrator

Get the convergence threshold. The usage depends on the specific integrator. For example, for an IterativeIntegrator, the integral is considered converged if the relative error of two successive sums is less than the threshold.

Specified by:

getPrecision in interface Integrator

Returns:

the precision