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Java example source code file (LegendreRuleFactory.java)

This example Java source code file (LegendreRuleFactory.java) is included in the alvinalexander.com "Java Source Code Warehouse" project. The intent of this project is to help you "Learn Java by Example" TM.

Learn more about this Java project at its project page.

Java - Java tags/keywords

baserulefactory, dimensionmismatchexception, double, legendrerulefactory, override, pair

The LegendreRuleFactory.java Java example source code

/*
 * 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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