/*
* Licensed to the Apache Software Foundation (ASF) under one or more
* 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
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*/
package org.apache.commons.math.analysis.integration;
import org.apache.commons.math.ConvergenceException;
import org.apache.commons.math.MathRuntimeException;
import org.apache.commons.math.analysis.UnivariateRealFunction;
import org.apache.commons.math.exception.MaxCountExceededException;
import org.apache.commons.math.exception.MathUserException;
import org.apache.commons.math.exception.util.LocalizedFormats;
import org.apache.commons.math.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 functions evaluations. A
* Legendre-Gauss integrator using an n-points quadrature formula can
* integrate exactly 2n-1 degree polynomialss.
* </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>
* @version $Id: LegendreGaussIntegrator.java 1131229 2011-06-03 20:49:25Z luc $
* @since 1.2
*/
public class LegendreGaussIntegrator extends UnivariateRealIntegratorImpl {
/** 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.
* @param n number of points desired (must be between 2 and 5 inclusive)
* @param defaultMaximalIterationCount maximum number of iterations
* @exception IllegalArgumentException if the number of points is not
* in the supported range
*/
public LegendreGaussIntegrator(final int n, final int defaultMaximalIterationCount)
throws IllegalArgumentException {
super(defaultMaximalIterationCount);
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 MathRuntimeException.createIllegalArgumentException(
LocalizedFormats.N_POINTS_GAUSS_LEGENDRE_INTEGRATOR_NOT_SUPPORTED,
n, 2, 5);
}
}
/** {@inheritDoc} */
public double integrate(final UnivariateRealFunction f, final double min, final double max)
throws ConvergenceException, MathUserException, IllegalArgumentException {
clearResult();
verifyInterval(min, max);
verifyIterationCount();
// compute first estimate with a single step
double oldt = stage(f, min, max, 1);
int n = 2;
for (int i = 0; i < maximalIterationCount; ++i) {
// improve integral with a larger number of steps
final double t = stage(f, min, max, n);
// estimate error
final double delta = FastMath.abs(t - oldt);
final double limit =
FastMath.max(absoluteAccuracy,
relativeAccuracy * (FastMath.abs(oldt) + FastMath.abs(t)) * 0.5);
// check convergence
if ((i + 1 >= minimalIterationCount) && (delta <= limit)) {
setResult(t, i);
return result;
}
// 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;
}
throw new MaxCountExceededException(maximalIterationCount);
}
/**
* Compute the n-th stage integral.
* @param f the integrand function
* @param min the lower bound for the interval
* @param max the upper bound for the interval
* @param n number of steps
* @return the value of n-th stage integral
* @throws MathUserException if an error occurs evaluating the
* function
*/
private double stage(final UnivariateRealFunction f,
final double min, final double max, final int n)
throws MathUserException {
// set up the step for the current stage
final double step = (max - min) / n;
final double halfStep = step / 2.0;
// integrate over all elementary steps
double midPoint = min + halfStep;
double sum = 0.0;
for (int i = 0; i < n; ++i) {
for (int j = 0; j < abscissas.length; ++j) {
sum += weights[j] * f.value(midPoint + halfStep * abscissas[j]);
}
midPoint += step;
}
return halfStep * sum;
}
}