Apache Commons Math 3.6.1 · org.apache.commons.math3.analysis.integration.LegendreGaussIntegrator

Implements the Legendre-Gauss quadrature formula.

Legendre-Gauss integrators are efficient integrators that can accurately integrate functions with few function evaluations. A Legendre-Gauss integrator using an n-points quadrature formula can integrate 2n-1 degree polynomials exactly.

These integrators evaluate the function on n carefully chosen abscissas in each step interval (mapped to the canonical [-1,1] interval). The evaluation abscissas are not evenly spaced and none of them are at the interval endpoints. This implies the function integrated can be undefined at integration interval endpoints.

The evaluation abscissas x_i are the roots of the degree n Legendre polynomial. The weights a_i of the quadrature formula integrals from -1 to +1 ∫ Li² where Li (x) = ∏ (x-x_k)/(x_i-x_k) for k != i.

Field Summary

Modifier and Type	Field and Description
private final double[]	abscissas Abscissas for the current method.
private static final double[]	ABSCISSAS_2 Abscissas for the 2 points method.
private static final double[]	ABSCISSAS_3 Abscissas for the 3 points method.
private static final double[]	ABSCISSAS_4 Abscissas for the 4 points method.
private static final double[]	ABSCISSAS_5 Abscissas for the 5 points method.
private final double[]	weights Weights for the current method.
private static final double[]	WEIGHTS_2 Weights for the 2 points method.
private static final double[]	WEIGHTS_3 Weights for the 3 points method.
private static final double[]	WEIGHTS_4 Weights for the 4 points method.
private static final double[]	WEIGHTS_5 Weights for the 5 points method.

Inherited from org.apache.commons.math3.analysis.integration.BaseAbstractUnivariateIntegrator:: DEFAULT_ABSOLUTE_ACCURACY DEFAULT_MAX_ITERATIONS_COUNT DEFAULT_MIN_ITERATIONS_COUNT DEFAULT_RELATIVE_ACCURACY iterations

Constructor Summary

Access	Constructor and Description
public	LegendreGaussIntegrator(final int number of points desired (must be between 2 and 5 inclusive) n, final double relative accuracy of the result relativeAccuracy, final double absolute accuracy of the result absoluteAccuracy, final int minimum number of iterations minimalIterationCount, final int maximum number of iterations maximalIterationCount) Build a Legendre-Gauss integrator with given accuracies and iterations counts.
public	LegendreGaussIntegrator(final int number of points desired (must be between 2 and 5 inclusive) n, final double relative accuracy of the result relativeAccuracy, final double absolute accuracy of the result absoluteAccuracy) Build a Legendre-Gauss integrator with given accuracies.
public	LegendreGaussIntegrator(final int number of points desired (must be between 2 and 5 inclusive) n, final int minimum number of iterations minimalIterationCount, final int maximum number of iterations maximalIterationCount) Build a Legendre-Gauss integrator with given iteration counts.

Method Summary

Modifier and Type	Method and Description
protected double	doIntegrate() Implements abstract org.apache.commons.math3.analysis.integration.BaseAbstractUnivariateIntegrator.doIntegrate. Method for implementing actual integration algorithms in derived classes.
private double	Returns: the value of n-th stage integral stage(final int number of steps n) Compute the n-th stage integral.

Inherited from org.apache.commons.math3.analysis.integration.BaseAbstractUnivariateIntegrator:: computeObjectiveValue getAbsoluteAccuracy getEvaluations getIterations getMax getMaximalIterationCount getMin getMinimalIterationCount getRelativeAccuracy incrementCount integrate setup

Field Detail

abscissas	back to summary
private final double[] abscissas Abscissas for the current method.

ABSCISSAS_2	back to summary
private static final double[] ABSCISSAS_2 Abscissas for the 2 points method.

ABSCISSAS_3	back to summary
private static final double[] ABSCISSAS_3 Abscissas for the 3 points method.

ABSCISSAS_4	back to summary
private static final double[] ABSCISSAS_4 Abscissas for the 4 points method.

ABSCISSAS_5	back to summary
private static final double[] ABSCISSAS_5 Abscissas for the 5 points method.

weights	back to summary
private final double[] weights Weights for the current method.

WEIGHTS_2	back to summary
private static final double[] WEIGHTS_2 Weights for the 2 points method.

WEIGHTS_3	back to summary
private static final double[] WEIGHTS_3 Weights for the 3 points method.

WEIGHTS_4	back to summary
private static final double[] WEIGHTS_4 Weights for the 4 points method.

WEIGHTS_5	back to summary
private static final double[] WEIGHTS_5 Weights for the 5 points method.

Constructor Detail

LegendreGaussIntegrator	back to summary
public LegendreGaussIntegrator(final int n, final double relativeAccuracy, final double absoluteAccuracy, final int minimalIterationCount, final int maximalIterationCount) throws MathIllegalArgumentException, NotStrictlyPositiveException, NumberIsTooSmallException Build a Legendre-Gauss integrator with given accuracies and iterations counts. Parameters n:int number of points desired (must be between 2 and 5 inclusive) relativeAccuracy:double relative accuracy of the result absoluteAccuracy:double absolute accuracy of the result minimalIterationCount:int minimum number of iterations maximalIterationCount:int maximum number of iterations Exceptions MathIllegalArgumentException: if number of points is out of [2; 5] NotStrictlyPositiveException: if minimal number of iterations is not strictly positive NumberIsTooSmallException: if maximal number of iterations is lesser than or equal to the minimal number of iterations

LegendreGaussIntegrator

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public LegendreGaussIntegrator(final int n, final double relativeAccuracy, final double absoluteAccuracy, final int minimalIterationCount, final int maximalIterationCount) throws MathIllegalArgumentException, NotStrictlyPositiveException, NumberIsTooSmallException

Build a Legendre-Gauss integrator with given accuracies and iterations counts.

Parameters

n:int: number of points desired (must be between 2 and 5 inclusive)
relativeAccuracy:double: relative accuracy of the result
absoluteAccuracy:double: absolute accuracy of the result
minimalIterationCount:int: minimum number of iterations
maximalIterationCount:int: maximum number of iterations

Exceptions

MathIllegalArgumentException:: if number of points is out of [2; 5]
NotStrictlyPositiveException:: if minimal number of iterations is not strictly positive
NumberIsTooSmallException:: if maximal number of iterations is lesser than or equal to the minimal number of iterations

LegendreGaussIntegrator	back to summary
public LegendreGaussIntegrator(final int n, final double relativeAccuracy, final double absoluteAccuracy) throws MathIllegalArgumentException Build a Legendre-Gauss integrator with given accuracies. Parameters n:int number of points desired (must be between 2 and 5 inclusive) relativeAccuracy:double relative accuracy of the result absoluteAccuracy:double absolute accuracy of the result Exceptions MathIllegalArgumentException: if number of points is out of [2; 5]

LegendreGaussIntegrator

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public LegendreGaussIntegrator(final int n, final double relativeAccuracy, final double absoluteAccuracy) throws MathIllegalArgumentException

Build a Legendre-Gauss integrator with given accuracies.

Parameters

n:int: number of points desired (must be between 2 and 5 inclusive)
relativeAccuracy:double: relative accuracy of the result
absoluteAccuracy:double: absolute accuracy of the result

Exceptions

MathIllegalArgumentException:: if number of points is out of [2; 5]

LegendreGaussIntegrator	back to summary
public LegendreGaussIntegrator(final int n, final int minimalIterationCount, final int maximalIterationCount) throws MathIllegalArgumentException Build a Legendre-Gauss integrator with given iteration counts. Parameters n:int number of points desired (must be between 2 and 5 inclusive) minimalIterationCount:int minimum number of iterations maximalIterationCount:int maximum number of iterations Exceptions MathIllegalArgumentException: if number of points is out of [2; 5] NotStrictlyPositiveException: if minimal number of iterations is not strictly positive NumberIsTooSmallException: if maximal number of iterations is lesser than or equal to the minimal number of iterations

LegendreGaussIntegrator

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public LegendreGaussIntegrator(final int n, final int minimalIterationCount, final int maximalIterationCount) throws MathIllegalArgumentException

Build a Legendre-Gauss integrator with given iteration counts.

Parameters

n:int: number of points desired (must be between 2 and 5 inclusive)
minimalIterationCount:int: minimum number of iterations
maximalIterationCount:int: maximum number of iterations

Exceptions

MathIllegalArgumentException:: if number of points is out of [2; 5]
NotStrictlyPositiveException:: if minimal number of iterations is not strictly positive
NumberIsTooSmallException:: if maximal number of iterations is lesser than or equal to the minimal number of iterations

Method Detail

doIntegrate	back to summary
protected double doIntegrate() throws MathIllegalArgumentException, TooManyEvaluationsException, MaxCountExceededException Implements abstract org.apache.commons.math3.analysis.integration.BaseAbstractUnivariateIntegrator.doIntegrate. Doc from org.apache.commons.math3.analysis.integration.BaseAbstractUnivariateIntegrator.doIntegrate. Method for implementing actual integration algorithms in derived classes. Returns:double the root. Annotations @Override Exceptions TooManyEvaluationsException: if the maximal number of evaluations is exceeded. MaxCountExceededException: if the maximum iteration count is exceeded or the integrator detects convergence problems otherwise

doIntegrate

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protected double doIntegrate() throws MathIllegalArgumentException, TooManyEvaluationsException, MaxCountExceededException

Implements abstract org.apache.commons.math3.analysis.integration.BaseAbstractUnivariateIntegrator.doIntegrate.

Doc from org.apache.commons.math3.analysis.integration.BaseAbstractUnivariateIntegrator.doIntegrate.

Method for implementing actual integration algorithms in derived classes.

Returns:double

the root.

Annotations

@Override

Exceptions

TooManyEvaluationsException:: if the maximal number of evaluations is exceeded.
MaxCountExceededException:: if the maximum iteration count is exceeded or the integrator detects convergence problems otherwise

stage	back to summary
private double stage(final int n) throws TooManyEvaluationsException Compute the n-th stage integral. Parameters n:int number of steps Returns:double the value of n-th stage integral Exceptions TooManyEvaluationsException: if the maximum number of evaluations is exceeded.

stage

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private double stage(final int n) throws TooManyEvaluationsException

Compute the n-th stage integral.

Parameters

n:int: number of steps

Returns:double

the value of n-th stage integral

Exceptions

TooManyEvaluationsException:: if the maximum number of evaluations is exceeded.

public Class LegendreGaussIntegrator

Field Summary

Constructor Summary

Method Summary

Field Detail

Constructor Detail

Method Detail