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Commons Math example source code file (LegendreGaussIntegrator.java)

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

Java - Commons Math tags/keywords

abscissas_3, abscissas_4, abscissas_5, convergenceexception, functionevaluationexception, functionevaluationexception, illegalargumentexception, illegalargumentexception, legendregaussintegrator, univariaterealfunction, weights_2, weights_3, weights_4, weights_4

The Commons Math LegendreGaussIntegrator.java 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.math.analysis.integration;

import org.apache.commons.math.ConvergenceException;
import org.apache.commons.math.FunctionEvaluationException;
import org.apache.commons.math.MathRuntimeException;
import org.apache.commons.math.MaxIterationsExceededException;
import org.apache.commons.math.analysis.UnivariateRealFunction;

/**
 * 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 are the roots of the degree n
 * Legendre polynomial. The weights a<sub>i of the quadrature formula
 * integrals from -1 to +1 ∫ Li<sup>2 where Li (x) =
 * ∏ (x-x<sub>k)/(xi-xk) for k != i.
 * </p>
 * <p>
 * @version $Revision: 811685 $ $Date: 2009-09-05 13:36:48 -0400 (Sat, 05 Sep 2009) $
 * @since 1.2
 */

public class LegendreGaussIntegrator extends UnivariateRealIntegratorImpl {

    /** Abscissas for the 2 points method. */
    private static final double[] ABSCISSAS_2 = {
        -1.0 / Math.sqrt(3.0),
         1.0 / Math.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 = {
        -Math.sqrt(0.6),
         0.0,
         Math.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 = {
        -Math.sqrt((15.0 + 2.0 * Math.sqrt(30.0)) / 35.0),
        -Math.sqrt((15.0 - 2.0 * Math.sqrt(30.0)) / 35.0),
         Math.sqrt((15.0 - 2.0 * Math.sqrt(30.0)) / 35.0),
         Math.sqrt((15.0 + 2.0 * Math.sqrt(30.0)) / 35.0)
    };

    /** Weights for the 4 points method. */
    private static final double[] WEIGHTS_4 = {
        (90.0 - 5.0 * Math.sqrt(30.0)) / 180.0,
        (90.0 + 5.0 * Math.sqrt(30.0)) / 180.0,
        (90.0 + 5.0 * Math.sqrt(30.0)) / 180.0,
        (90.0 - 5.0 * Math.sqrt(30.0)) / 180.0
    };

    /** Abscissas for the 5 points method. */
    private static final double[] ABSCISSAS_5 = {
        -Math.sqrt((35.0 + 2.0 * Math.sqrt(70.0)) / 63.0),
        -Math.sqrt((35.0 - 2.0 * Math.sqrt(70.0)) / 63.0),
         0.0,
         Math.sqrt((35.0 - 2.0 * Math.sqrt(70.0)) / 63.0),
         Math.sqrt((35.0 + 2.0 * Math.sqrt(70.0)) / 63.0)
    };

    /** Weights for the 5 points method. */
    private static final double[] WEIGHTS_5 = {
        (322.0 - 13.0 * Math.sqrt(70.0)) / 900.0,
        (322.0 + 13.0 * Math.sqrt(70.0)) / 900.0,
        128.0 / 225.0,
        (322.0 + 13.0 * Math.sqrt(70.0)) / 900.0,
        (322.0 - 13.0 * Math.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(
                    "{0} points Legendre-Gauss integrator not supported, " +
                    "number of points must be in the {1}-{2} range",
                    n, 2, 5);
        }

    }

    /** {@inheritDoc} */
    @Deprecated
    public double integrate(final double min, final double max)
        throws ConvergenceException,  FunctionEvaluationException, IllegalArgumentException {
        return integrate(f, min, max);
    }

    /** {@inheritDoc} */
    public double integrate(final UnivariateRealFunction f,
            final double min, final double max)
        throws ConvergenceException,  FunctionEvaluationException, 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 = Math.abs(t - oldt);
            final double limit =
                Math.max(absoluteAccuracy,
                         relativeAccuracy * (Math.abs(oldt) + Math.abs(t)) * 0.5);

            // check convergence
            if ((i + 1 >= minimalIterationCount) && (delta <= limit)) {
                setResult(t, i);
                return result;
            }

            // prepare next iteration
            double ratio = Math.min(4, Math.pow(delta / limit, 0.5 / abscissas.length));
            n = Math.max((int) (ratio * n), n + 1);
            oldt = t;

        }

        throw new MaxIterationsExceededException(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 FunctionEvaluationException if an error occurs evaluating the
     * function
     */
    private double stage(final UnivariateRealFunction f,
                         final double min, final double max, final int n)
        throws FunctionEvaluationException {

        // 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;

    }

}

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