jidt/java/source/infodynamics/utils/commonsmath3/util/ContinuedFraction.java

/*
 *  Java Information Dynamics Toolkit (JIDT)
 *  Copyright (C) 2012, Joseph T. Lizier
 *  
 *  This program is free software: you can redistribute it and/or modify
 *  it under the terms of the GNU General Public License as published by
 *  the Free Software Foundation, either version 3 of the License, or
 *  (at your option) any later version.
 *  
 *  This program is distributed in the hope that it will be useful,
 *  but WITHOUT ANY WARRANTY; without even the implied warranty of
 *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
 *  GNU General Public License for more details.
 *  
 *  You should have received a copy of the GNU General Public License
 *  along with this program.  If not, see <http://www.gnu.org/licenses/>.
*/

/*
 * This class was originally distributed as part of the Apache Commons
 *  Math3 library, under the Apache License Version 2.0, which is 
 *  copied below. This Apache 2 software is now included as a derivative
 *  work in the GPLv3 licensed JIDT project, as per:
 *  http://www.apache.org/licenses/GPL-compatibility.html
 *  
 * The original Apache source code has been modified as follows:
 * -- We have modified package names to sit inside the JIDT structure.
 */

/*
 * 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 infodynamics.utils.commonsmath3.util;

import infodynamics.utils.commonsmath3.exception.ConvergenceException;
import infodynamics.utils.commonsmath3.exception.MaxCountExceededException;
import infodynamics.utils.commonsmath3.exception.util.LocalizedFormats;

/**
 * Provides a generic means to evaluate continued fractions.  Subclasses simply
 * provided the a and b coefficients to evaluate the continued fraction.
 *
 * <p>
 * References:
 * <ul>
 * <li><a href="http://mathworld.wolfram.com/ContinuedFraction.html">
 * Continued Fraction</a></li>
 * </ul>
 * </p>
 *
 * @version $Id$
 */
public abstract class ContinuedFraction {
    /** Maximum allowed numerical error. */
    private static final double DEFAULT_EPSILON = 10e-9;

    /**
     * Default constructor.
     */
    protected ContinuedFraction() {
        super();
    }

    /**
     * Access the n-th a coefficient of the continued fraction.  Since a can be
     * a function of the evaluation point, x, that is passed in as well.
     * @param n the coefficient index to retrieve.
     * @param x the evaluation point.
     * @return the n-th a coefficient.
     */
    protected abstract double getA(int n, double x);

    /**
     * Access the n-th b coefficient of the continued fraction.  Since b can be
     * a function of the evaluation point, x, that is passed in as well.
     * @param n the coefficient index to retrieve.
     * @param x the evaluation point.
     * @return the n-th b coefficient.
     */
    protected abstract double getB(int n, double x);

    /**
     * Evaluates the continued fraction at the value x.
     * @param x the evaluation point.
     * @return the value of the continued fraction evaluated at x.
     * @throws ConvergenceException if the algorithm fails to converge.
     */
    public double evaluate(double x) throws ConvergenceException {
        return evaluate(x, DEFAULT_EPSILON, Integer.MAX_VALUE);
    }

    /**
     * Evaluates the continued fraction at the value x.
     * @param x the evaluation point.
     * @param epsilon maximum error allowed.
     * @return the value of the continued fraction evaluated at x.
     * @throws ConvergenceException if the algorithm fails to converge.
     */
    public double evaluate(double x, double epsilon) throws ConvergenceException {
        return evaluate(x, epsilon, Integer.MAX_VALUE);
    }

    /**
     * Evaluates the continued fraction at the value x.
     * @param x the evaluation point.
     * @param maxIterations maximum number of convergents
     * @return the value of the continued fraction evaluated at x.
     * @throws ConvergenceException if the algorithm fails to converge.
     * @throws MaxCountExceededException if maximal number of iterations is reached
     */
    public double evaluate(double x, int maxIterations)
        throws ConvergenceException, MaxCountExceededException {
        return evaluate(x, DEFAULT_EPSILON, maxIterations);
    }

    /**
     * Evaluates the continued fraction at the value x.
     * <p>
     * The implementation of this method is based on the modified Lentz algorithm as described
     * on page 18 ff. in:
     * <ul>
     *   <li>
     *   I. J. Thompson,  A. R. Barnett. "Coulomb and Bessel Functions of Complex Arguments and Order."
     *   <a target="_blank" href="http://www.fresco.org.uk/papers/Thompson-JCP64p490.pdf">
     *   http://www.fresco.org.uk/papers/Thompson-JCP64p490.pdf</a>
     *   </li>
     * </ul>
     * <b>Note:</b> the implementation uses the terms a<sub>i</sub> and b<sub>i</sub> as defined in
     * <a href="http://mathworld.wolfram.com/ContinuedFraction.html">Continued Fraction @ MathWorld</a>.
     * </p>
     *
     * @param x the evaluation point.
     * @param epsilon maximum error allowed.
     * @param maxIterations maximum number of convergents
     * @return the value of the continued fraction evaluated at x.
     * @throws ConvergenceException if the algorithm fails to converge.
     * @throws MaxCountExceededException if maximal number of iterations is reached
     */
    public double evaluate(double x, double epsilon, int maxIterations)
        throws ConvergenceException, MaxCountExceededException {
        final double small = 1e-50;
        double hPrev = getA(0, x);

        // use the value of small as epsilon criteria for zero checks
        if (Precision.equals(hPrev, 0.0, small)) {
            hPrev = small;
        }

        int n = 1;
        double dPrev = 0.0;
        double cPrev = hPrev;
        double hN = hPrev;

        while (n < maxIterations) {
            final double a = getA(n, x);
            final double b = getB(n, x);

            double dN = a + b * dPrev;
            if (Precision.equals(dN, 0.0, small)) {
                dN = small;
            }
            double cN = a + b / cPrev;
            if (Precision.equals(cN, 0.0, small)) {
                cN = small;
            }

            dN = 1 / dN;
            final double deltaN = cN * dN;
            hN = hPrev * deltaN;

            if (Double.isInfinite(hN)) {
                throw new ConvergenceException(LocalizedFormats.CONTINUED_FRACTION_INFINITY_DIVERGENCE,
                                               x);
            }
            if (Double.isNaN(hN)) {
                throw new ConvergenceException(LocalizedFormats.CONTINUED_FRACTION_NAN_DIVERGENCE,
                                               x);
            }

            if (FastMath.abs(deltaN - 1.0) < epsilon) {
                break;
            }

            dPrev = dN;
            cPrev = cN;
            hPrev = hN;
            n++;
        }

        if (n >= maxIterations) {
            throw new MaxCountExceededException(LocalizedFormats.NON_CONVERGENT_CONTINUED_FRACTION,
                                                maxIterations, x);
        }

        return hN;
    }

}