/*
* 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.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.
*
* <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. */
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 (Math.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;
}
}