com.imsl.math

Class HyperRectangleQuadrature

java.lang.Object

com.imsl.math.HyperRectangleQuadrature

All Implemented Interfaces:

Serializable, Cloneable

public class HyperRectangleQuadrature
extends Object
implements Serializable, Cloneable

HyperRectangleQuadrature integrates a function over a hypercube. This class is used to evaluate integrals of the form: $$ \int_{a_{n-1}}^{b_{n-1}} \cdots \int_{a_0}^{b_0} f(x_0,\ldots,x_{n-1}) \, dx_0 \ldots dx_{n-1} $$ Integration of functions over hypercubes by Monte Carlo, in which the integral is evaluated as the value of the function averaged over a sequence of randomly chosen points. Under mild assumptions on the function, this method will converge like \(1/\sqrt{n}\), where n is the number of points at which the function is evaluated.

It is possible to improve on the performance of Monte Carlo by carefully choosing the points at which the function is to be evaluated. Randomly distributed points tend to be non-uniformly distributed. The alternative to a sequence of random points is a low-discrepancy sequence. A low-discrepancy sequence is one that is highly uniform.

This function is based on the low-discrepancy Faure sequence as computed by FaureSequence.

See Also:

Example, Serialized Form

Nested Class Summary

Nested Classes

Modifier and Type	Class and Description
static interface	HyperRectangleQuadrature.Function Public interface function for the HyperRectangleQuadrature class.

Constructor Summary

Constructors

Constructor and Description
HyperRectangleQuadrature(int dim) Constructs a HyperRectangleQuadrature object.
HyperRectangleQuadrature(RandomSequence sequence) Constructs a HyperRectangleQuadrature object.

Method Summary

Modifier and Type	Method and Description
double	eval(HyperRectangleQuadrature.Function objectF) Returns the value of the integral over the unit cube.
double	eval(HyperRectangleQuadrature.Function objectF, double[] a, double[] b) Returns the value of the integral over a cube.
double	getErrorEstimate() Returns an estimate of the relative error in the computed result.
void	setAbsoluteError(double errorAbsolute) Sets the absolute error tolerance.
void	setRelativeError(double errorRelative) Sets the relative error tolerance.

Methods inherited from class java.lang.Object

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

Constructor Detail

HyperRectangleQuadrature

public HyperRectangleQuadrature(int dim)

Constructs a HyperRectangleQuadrature object.

HyperRectangleQuadrature

public HyperRectangleQuadrature(RandomSequence sequence)

Constructs a HyperRectangleQuadrature object.

Method Detail

setAbsoluteError

public void setAbsoluteError(double errorAbsolute)

Sets the absolute error tolerance.

Parameters:

errorAbsolute - a double scalar value specifying the absolute error

setRelativeError

public void setRelativeError(double errorRelative)

Sets the relative error tolerance.

Parameters:

errorRelative - a double scalar value specifying the relative error

getErrorEstimate

public double getErrorEstimate()

Returns an estimate of the relative error in the computed result.

Returns:

a double specifying an estimate of the relative error in the computed result

eval

public double eval(HyperRectangleQuadrature.Function objectF)

Returns the value of the integral over the unit cube.

Parameters:

objectF - Function containing the function to be integrated

eval

public double eval(HyperRectangleQuadrature.Function objectF,
                   double[] a,
                   double[] b)

Returns the value of the integral over a cube.

Parameters:

objectF - Function containing the function to be integrated

a - is a double specifying the lower limit of integration. If null all of the lower limits default to 0.

b - is a double specifying the upper limit of integration. If null all of the upper limits default to 1.