ContinuedFraction.java

/*
 * 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.util;
package jmathlib.toolbox.specfun._private;

import java.io.Serializable;

//import org.apache.commons.math.ConvergenceException;
//import org.apache.commons.math.MathException;

/**
 * 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 $Revision: 1.1 $ $Date: 2007/01/05 08:55:48 $
 */
public abstract class ContinuedFraction implements Serializable {
    
    /** Serialization UID */
    private static final long serialVersionUID = 1768555336266158242L;
    
    /** 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 MathException if the algorithm fails to converge.
     */
    public double evaluate(double x) throws Exception {
        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 MathException if the algorithm fails to converge.
     */
    public double evaluate(double x, double epsilon) throws Exception {
        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 MathException if the algorithm fails to converge.
     */
    public double evaluate(double x, int maxIterations) throws Exception {
        return evaluate(x, DEFAULT_EPSILON, maxIterations);
    }

    /**
     * <p>
     * Evaluates the continued fraction at the value x.
     * </p>
     * 
     * <p>
     * The implementation of this method is based on equations 14-17 of:
     * <ul>
     * <li>
     *   Eric W. Weisstein. "Continued Fraction." From MathWorld--A Wolfram Web
     *   Resource. <a target="_blank"
     *   href="http://mathworld.wolfram.com/ContinuedFraction.html">
     *   http://mathworld.wolfram.com/ContinuedFraction.html</a>
     * </li>
     * </ul>
     * The recurrence relationship defined in those equations can result in
     * very large intermediate results which can result in numerical overflow.
     * As a means to combat these overflow conditions, the intermediate results
     * are scaled whenever they threaten to become numerically unstable.
     *   
     * @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 MathException if the algorithm fails to converge.
     */
    public double evaluate(double x, double epsilon, int maxIterations)
        throws Exception
    {
        double p0 = 1.0;
        double p1 = getA(0, x);
        double q0 = 0.0;
        double q1 = 1.0;
        double c = p1 / q1;
        int n = 0;
        double relativeError = Double.MAX_VALUE;
        while (n < maxIterations && relativeError > epsilon) {
            ++n;
            double a = getA(n, x);
            double b = getB(n, x);
            double p2 = a * p1 + b * p0;
            double q2 = a * q1 + b * q0;
            if (Double.isInfinite(p2) || Double.isInfinite(q2)) {
                // need to scale
                if (a != 0.0) {
                    p2 = p1 + (b / a * p0);
                    q2 = q1 + (b / a * q0);
                } else if (b != 0) {
                    p2 = (a / b * p1) + p0;
                    q2 = (a / b * q1) + q0;
                } else {
                    // can not scale an convergent is unbounded.
                    throw new ConvergenceException(
                        "Continued fraction convergents diverged to +/- " +
                        "infinity.");
                }
            }
            double r = p2 / q2;
            relativeError = Math.abs(r / c - 1.0);
                
            // prepare for next iteration
            c = p2 / q2;
            p0 = p1;
            p1 = p2;
            q0 = q1;
            q1 = q2;
        }

        if (n >= maxIterations) {
            throw new ConvergenceException(
                "Continued fraction convergents failed to converge.");
        }

        return c;
    }
}