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/*
* 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.math3.analysis.integration.gauss;
import org.apache.commons.math3.exception.DimensionMismatchException;
import org.apache.commons.math3.util.Pair;
/**
* Factory that creates Gauss-type quadrature rule using Legendre polynomials.
* In this implementation, the lower and upper bounds of the natural interval
* of integration are -1 and 1, respectively.
* The Legendre polynomials are evaluated using the recurrence relation
* presented in <a href="http://en.wikipedia.org/wiki/Abramowitz_and_Stegun">
* Abramowitz and Stegun, 1964</a>.
*
* @since 3.1
*/
public class LegendreRuleFactory extends BaseRuleFactory<Double> {
/** {@inheritDoc} */
@Override
protected Pair<Double[], Double[]> computeRule(int numberOfPoints)
throws DimensionMismatchException {
if (numberOfPoints == 1) {
// Break recursion.
return new Pair<Double[], Double[]>(new Double[] { 0d },
new Double[] { 2d });
}
// Get previous rule.
// If it has not been computed yet it will trigger a recursive call
// to this method.
final Double[] previousPoints = getRuleInternal(numberOfPoints - 1).getFirst();
// Compute next rule.
final Double[] points = new Double[numberOfPoints];
final Double[] weights = new Double[numberOfPoints];
// Find i-th root of P[n+1] by bracketing.
final int iMax = numberOfPoints / 2;
for (int i = 0; i < iMax; i++) {
// Lower-bound of the interval.
double a = (i == 0) ? -1 : previousPoints[i - 1].doubleValue();
// Upper-bound of the interval.
double b = (iMax == 1) ? 1 : previousPoints[i].doubleValue();
// P[j-1](a)
double pma = 1;
// P[j](a)
double pa = a;
// P[j-1](b)
double pmb = 1;
// P[j](b)
double pb = b;
for (int j = 1; j < numberOfPoints; j++) {
final int two_j_p_1 = 2 * j + 1;
final int j_p_1 = j + 1;
// P[j+1](a)
final double ppa = (two_j_p_1 * a * pa - j * pma) / j_p_1;
// P[j+1](b)
final double ppb = (two_j_p_1 * b * pb - j * pmb) / j_p_1;
pma = pa;
pa = ppa;
pmb = pb;
pb = ppb;
}
// Now pa = P[n+1](a), and pma = P[n](a) (same holds for b).
// Middle of the interval.
double c = 0.5 * (a + b);
// P[j-1](c)
double pmc = 1;
// P[j](c)
double pc = c;
boolean done = false;
while (!done) {
done = b - a <= Math.ulp(c);
pmc = 1;
pc = c;
for (int j = 1; j < numberOfPoints; j++) {
// P[j+1](c)
final double ppc = ((2 * j + 1) * c * pc - j * pmc) / (j + 1);
pmc = pc;
pc = ppc;
}
// Now pc = P[n+1](c) and pmc = P[n](c).
if (!done) {
if (pa * pc <= 0) {
b = c;
pmb = pmc;
pb = pc;
} else {
a = c;
pma = pmc;
pa = pc;
}
c = 0.5 * (a + b);
}
}
final double d = numberOfPoints * (pmc - c * pc);
final double w = 2 * (1 - c * c) / (d * d);
points[i] = c;
weights[i] = w;
final int idx = numberOfPoints - i - 1;
points[idx] = -c;
weights[idx] = w;
}
// If "numberOfPoints" is odd, 0 is a root.
// Note: as written, the test for oddness will work for negative
// integers too (although it is not necessary here), preventing
// a FindBugs warning.
if (numberOfPoints % 2 != 0) {
double pmc = 1;
for (int j = 1; j < numberOfPoints; j += 2) {
pmc = -j * pmc / (j + 1);
}
final double d = numberOfPoints * pmc;
final double w = 2 / (d * d);
points[iMax] = 0d;
weights[iMax] = w;
}
return new Pair<Double[], Double[]>(points, weights);
}
}
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