LegendreGaussIntegrator.java
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* contributor license agreements. See the NOTICE file distributed with
* this work for additional information regarding copyright ownership.
* The ASF licenses this file to You under the Apache License, Version 2.0
* (the "License"); you may not use this file except in compliance with
* the License. You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
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package org.apache.commons.math3.analysis.integration;
import org.apache.commons.math3.exception.MathIllegalArgumentException;
import org.apache.commons.math3.exception.MaxCountExceededException;
import org.apache.commons.math3.exception.NotStrictlyPositiveException;
import org.apache.commons.math3.exception.NumberIsTooSmallException;
import org.apache.commons.math3.exception.TooManyEvaluationsException;
import org.apache.commons.math3.exception.util.LocalizedFormats;
import org.apache.commons.math3.util.FastMath;
/**
* Implements the <a href="http://mathworld.wolfram.com/Legendre-GaussQuadrature.html">
* Legendre-Gauss</a> quadrature formula.
* <p>
* 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.
* </p>
* <p>
* 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.
* </p>
* <p>
* The evaluation abscissas x<sub>i</sub> are the roots of the degree n
* Legendre polynomial. The weights a<sub>i</sub> of the quadrature formula
* integrals from -1 to +1 ∫ Li<sup>2</sup> where Li (x) =
* ∏ (x-x<sub>k</sub>)/(x<sub>i</sub>-x<sub>k</sub>) for k != i.
* </p>
* <p>
* @since 1.2
* @deprecated As of 3.1 (to be removed in 4.0). Please use
* {@link IterativeLegendreGaussIntegrator} instead.
*/
@Deprecated
public class LegendreGaussIntegrator extends BaseAbstractUnivariateIntegrator {
/** Abscissas for the 2 points method. */
private static final double[] ABSCISSAS_2 = {
-1.0 / FastMath.sqrt(3.0),
1.0 / FastMath.sqrt(3.0)
};
/** Weights for the 2 points method. */
private static final double[] WEIGHTS_2 = {
1.0,
1.0
};
/** Abscissas for the 3 points method. */
private static final double[] ABSCISSAS_3 = {
-FastMath.sqrt(0.6),
0.0,
FastMath.sqrt(0.6)
};
/** Weights for the 3 points method. */
private static final double[] WEIGHTS_3 = {
5.0 / 9.0,
8.0 / 9.0,
5.0 / 9.0
};
/** Abscissas for the 4 points method. */
private static final double[] ABSCISSAS_4 = {
-FastMath.sqrt((15.0 + 2.0 * FastMath.sqrt(30.0)) / 35.0),
-FastMath.sqrt((15.0 - 2.0 * FastMath.sqrt(30.0)) / 35.0),
FastMath.sqrt((15.0 - 2.0 * FastMath.sqrt(30.0)) / 35.0),
FastMath.sqrt((15.0 + 2.0 * FastMath.sqrt(30.0)) / 35.0)
};
/** Weights for the 4 points method. */
private static final double[] WEIGHTS_4 = {
(90.0 - 5.0 * FastMath.sqrt(30.0)) / 180.0,
(90.0 + 5.0 * FastMath.sqrt(30.0)) / 180.0,
(90.0 + 5.0 * FastMath.sqrt(30.0)) / 180.0,
(90.0 - 5.0 * FastMath.sqrt(30.0)) / 180.0
};
/** Abscissas for the 5 points method. */
private static final double[] ABSCISSAS_5 = {
-FastMath.sqrt((35.0 + 2.0 * FastMath.sqrt(70.0)) / 63.0),
-FastMath.sqrt((35.0 - 2.0 * FastMath.sqrt(70.0)) / 63.0),
0.0,
FastMath.sqrt((35.0 - 2.0 * FastMath.sqrt(70.0)) / 63.0),
FastMath.sqrt((35.0 + 2.0 * FastMath.sqrt(70.0)) / 63.0)
};
/** Weights for the 5 points method. */
private static final double[] WEIGHTS_5 = {
(322.0 - 13.0 * FastMath.sqrt(70.0)) / 900.0,
(322.0 + 13.0 * FastMath.sqrt(70.0)) / 900.0,
128.0 / 225.0,
(322.0 + 13.0 * FastMath.sqrt(70.0)) / 900.0,
(322.0 - 13.0 * FastMath.sqrt(70.0)) / 900.0
};
/** Abscissas for the current method. */
private final double[] abscissas;
/** Weights for the current method. */
private final double[] weights;
/**
* Build a Legendre-Gauss integrator with given accuracies and iterations counts.
* @param n number of points desired (must be between 2 and 5 inclusive)
* @param relativeAccuracy relative accuracy of the result
* @param absoluteAccuracy absolute accuracy of the result
* @param minimalIterationCount minimum number of iterations
* @param maximalIterationCount maximum number of iterations
* @exception MathIllegalArgumentException if number of points is out of [2; 5]
* @exception NotStrictlyPositiveException if minimal number of iterations
* is not strictly positive
* @exception NumberIsTooSmallException if maximal number of iterations
* is lesser than or equal to the minimal number of iterations
*/
public LegendreGaussIntegrator(final int n,
final double relativeAccuracy,
final double absoluteAccuracy,
final int minimalIterationCount,
final int maximalIterationCount)
throws MathIllegalArgumentException, NotStrictlyPositiveException, NumberIsTooSmallException {
super(relativeAccuracy, absoluteAccuracy, minimalIterationCount, maximalIterationCount);
switch(n) {
case 2 :
abscissas = ABSCISSAS_2;
weights = WEIGHTS_2;
break;
case 3 :
abscissas = ABSCISSAS_3;
weights = WEIGHTS_3;
break;
case 4 :
abscissas = ABSCISSAS_4;
weights = WEIGHTS_4;
break;
case 5 :
abscissas = ABSCISSAS_5;
weights = WEIGHTS_5;
break;
default :
throw new MathIllegalArgumentException(
LocalizedFormats.N_POINTS_GAUSS_LEGENDRE_INTEGRATOR_NOT_SUPPORTED,
n, 2, 5);
}
}
/**
* Build a Legendre-Gauss integrator with given accuracies.
* @param n number of points desired (must be between 2 and 5 inclusive)
* @param relativeAccuracy relative accuracy of the result
* @param absoluteAccuracy absolute accuracy of the result
* @exception MathIllegalArgumentException if number of points is out of [2; 5]
*/
public LegendreGaussIntegrator(final int n,
final double relativeAccuracy,
final double absoluteAccuracy)
throws MathIllegalArgumentException {
this(n, relativeAccuracy, absoluteAccuracy,
DEFAULT_MIN_ITERATIONS_COUNT, DEFAULT_MAX_ITERATIONS_COUNT);
}
/**
* Build a Legendre-Gauss integrator with given iteration counts.
* @param n number of points desired (must be between 2 and 5 inclusive)
* @param minimalIterationCount minimum number of iterations
* @param maximalIterationCount maximum number of iterations
* @exception MathIllegalArgumentException if number of points is out of [2; 5]
* @exception NotStrictlyPositiveException if minimal number of iterations
* is not strictly positive
* @exception NumberIsTooSmallException if maximal number of iterations
* is lesser than or equal to the minimal number of iterations
*/
public LegendreGaussIntegrator(final int n,
final int minimalIterationCount,
final int maximalIterationCount)
throws MathIllegalArgumentException {
this(n, DEFAULT_RELATIVE_ACCURACY, DEFAULT_ABSOLUTE_ACCURACY,
minimalIterationCount, maximalIterationCount);
}
/** {@inheritDoc} */
@Override
protected double doIntegrate()
throws MathIllegalArgumentException, TooManyEvaluationsException, MaxCountExceededException {
// compute first estimate with a single step
double oldt = stage(1);
int n = 2;
while (true) {
// improve integral with a larger number of steps
final double t = stage(n);
// estimate error
final double delta = FastMath.abs(t - oldt);
final double limit =
FastMath.max(getAbsoluteAccuracy(),
getRelativeAccuracy() * (FastMath.abs(oldt) + FastMath.abs(t)) * 0.5);
// check convergence
if ((iterations.getCount() + 1 >= getMinimalIterationCount()) && (delta <= limit)) {
return t;
}
// prepare next iteration
double ratio = FastMath.min(4, FastMath.pow(delta / limit, 0.5 / abscissas.length));
n = FastMath.max((int) (ratio * n), n + 1);
oldt = t;
iterations.incrementCount();
}
}
/**
* Compute the n-th stage integral.
* @param n number of steps
* @return the value of n-th stage integral
* @throws TooManyEvaluationsException if the maximum number of evaluations
* is exceeded.
*/
private double stage(final int n)
throws TooManyEvaluationsException {
// set up the step for the current stage
final double step = (getMax() - getMin()) / n;
final double halfStep = step / 2.0;
// integrate over all elementary steps
double midPoint = getMin() + halfStep;
double sum = 0.0;
for (int i = 0; i < n; ++i) {
for (int j = 0; j < abscissas.length; ++j) {
sum += weights[j] * computeObjectiveValue(midPoint + halfStep * abscissas[j]);
}
midPoint += step;
}
return halfStep * sum;
}
}