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 * (the "License"); you may not use this file except in compliance with
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 *      http://www.apache.org/licenses/LICENSE-2.0
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 * distributed under the License is distributed on an "AS IS" BASIS,
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package org.apache.commons.math3.util;

import org.apache.commons.math3.exception.ConvergenceException;
import org.apache.commons.math3.exception.MaxCountExceededException;
import org.apache.commons.math3.exception.util.LocalizedFormats;

Provides a generic means to evaluate continued fractions.  Subclasses simply
provided the a and b coefficients to evaluate the continued fraction.

References:


Continued Fraction


/**
 * 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>
 *
 */
public abstract class ContinuedFraction {
    
Maximum allowed numerical error. /** Maximum allowed numerical error. */
    private static final double DEFAULT_EPSILON = 10e-9;

    
Default constructor.
/**
     * 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.
Params:
n – the coefficient index to retrieve.
x – the evaluation point.
Returns:
the n-th a coefficient./**
     * 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.
Params:
n – the coefficient index to retrieve.
x – the evaluation point.
Returns:
the n-th b coefficient./**
     * 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.
Params:
x – the evaluation point.
Throws:
ConvergenceException – if the algorithm fails to converge.
Returns:
the value of the continued fraction evaluated at 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.
Params:
x – the evaluation point.
epsilon – maximum error allowed.
Throws:
ConvergenceException – if the algorithm fails to converge.
Returns:
the value of the continued fraction evaluated at x./**
     * 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.
Params:
x – the evaluation point.
maxIterations – maximum number of convergents
Throws:
ConvergenceException – if the algorithm fails to converge.
MaxCountExceededException – if maximal number of iterations is reached
Returns:
the value of the continued fraction evaluated at x./**
     * 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.

The implementation of this method is based on the modified Lentz algorithm as described
on page 18 ff. in:

  

  I. J. Thompson,  A. R. Barnett. "Coulomb and Bessel Functions of Complex Arguments and Order."
  
  http://www.fresco.org.uk/papers/Thompson-JCP64p490.pdf
  

Note: the implementation uses the terms ai and bi as defined in
Continued Fraction @ MathWorld.

Params:
x – the evaluation point.
epsilon – maximum error allowed.
maxIterations – maximum number of convergents
Throws:
ConvergenceException – if the algorithm fails to converge.
MaxCountExceededException – if maximal number of iterations is reached
Returns:
the value of the continued fraction evaluated at x./**
     * 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;
    }

}