001 /* 002 * Licensed to the Apache Software Foundation (ASF) under one or more 003 * contributor license agreements. See the NOTICE file distributed with 004 * this work for additional information regarding copyright ownership. 005 * The ASF licenses this file to You under the Apache License, Version 2.0 006 * (the "License"); you may not use this file except in compliance with 007 * the License. You may obtain a copy of the License at 008 * 009 * http://www.apache.org/licenses/LICENSE-2.0 010 * 011 * Unless required by applicable law or agreed to in writing, software 012 * distributed under the License is distributed on an "AS IS" BASIS, 013 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. 014 * See the License for the specific language governing permissions and 015 * limitations under the License. 016 */ 017 package org.apache.commons.math.util; 018 019 import org.apache.commons.math.ConvergenceException; 020 import org.apache.commons.math.MathException; 021 import org.apache.commons.math.MaxIterationsExceededException; 022 import org.apache.commons.math.exception.util.LocalizedFormats; 023 024 /** 025 * Provides a generic means to evaluate continued fractions. Subclasses simply 026 * provided the a and b coefficients to evaluate the continued fraction. 027 * 028 * <p> 029 * References: 030 * <ul> 031 * <li><a href="http://mathworld.wolfram.com/ContinuedFraction.html"> 032 * Continued Fraction</a></li> 033 * </ul> 034 * </p> 035 * 036 * @version $Revision: 990655 $ $Date: 2010-08-29 23:49:40 +0200 (dim. 29 ao??t 2010) $ 037 */ 038 public abstract class ContinuedFraction { 039 040 /** Maximum allowed numerical error. */ 041 private static final double DEFAULT_EPSILON = 10e-9; 042 043 /** 044 * Default constructor. 045 */ 046 protected ContinuedFraction() { 047 super(); 048 } 049 050 /** 051 * Access the n-th a coefficient of the continued fraction. Since a can be 052 * a function of the evaluation point, x, that is passed in as well. 053 * @param n the coefficient index to retrieve. 054 * @param x the evaluation point. 055 * @return the n-th a coefficient. 056 */ 057 protected abstract double getA(int n, double x); 058 059 /** 060 * Access the n-th b coefficient of the continued fraction. Since b can be 061 * a function of the evaluation point, x, that is passed in as well. 062 * @param n the coefficient index to retrieve. 063 * @param x the evaluation point. 064 * @return the n-th b coefficient. 065 */ 066 protected abstract double getB(int n, double x); 067 068 /** 069 * Evaluates the continued fraction at the value x. 070 * @param x the evaluation point. 071 * @return the value of the continued fraction evaluated at x. 072 * @throws MathException if the algorithm fails to converge. 073 */ 074 public double evaluate(double x) throws MathException { 075 return evaluate(x, DEFAULT_EPSILON, Integer.MAX_VALUE); 076 } 077 078 /** 079 * Evaluates the continued fraction at the value x. 080 * @param x the evaluation point. 081 * @param epsilon maximum error allowed. 082 * @return the value of the continued fraction evaluated at x. 083 * @throws MathException if the algorithm fails to converge. 084 */ 085 public double evaluate(double x, double epsilon) throws MathException { 086 return evaluate(x, epsilon, Integer.MAX_VALUE); 087 } 088 089 /** 090 * Evaluates the continued fraction at the value x. 091 * @param x the evaluation point. 092 * @param maxIterations maximum number of convergents 093 * @return the value of the continued fraction evaluated at x. 094 * @throws MathException if the algorithm fails to converge. 095 */ 096 public double evaluate(double x, int maxIterations) throws MathException { 097 return evaluate(x, DEFAULT_EPSILON, maxIterations); 098 } 099 100 /** 101 * <p> 102 * Evaluates the continued fraction at the value x. 103 * </p> 104 * 105 * <p> 106 * The implementation of this method is based on equations 14-17 of: 107 * <ul> 108 * <li> 109 * Eric W. Weisstein. "Continued Fraction." From MathWorld--A Wolfram Web 110 * Resource. <a target="_blank" 111 * href="http://mathworld.wolfram.com/ContinuedFraction.html"> 112 * http://mathworld.wolfram.com/ContinuedFraction.html</a> 113 * </li> 114 * </ul> 115 * The recurrence relationship defined in those equations can result in 116 * very large intermediate results which can result in numerical overflow. 117 * As a means to combat these overflow conditions, the intermediate results 118 * are scaled whenever they threaten to become numerically unstable.</p> 119 * 120 * @param x the evaluation point. 121 * @param epsilon maximum error allowed. 122 * @param maxIterations maximum number of convergents 123 * @return the value of the continued fraction evaluated at x. 124 * @throws MathException if the algorithm fails to converge. 125 */ 126 public double evaluate(double x, double epsilon, int maxIterations) 127 throws MathException 128 { 129 double p0 = 1.0; 130 double p1 = getA(0, x); 131 double q0 = 0.0; 132 double q1 = 1.0; 133 double c = p1 / q1; 134 int n = 0; 135 double relativeError = Double.MAX_VALUE; 136 while (n < maxIterations && relativeError > epsilon) { 137 ++n; 138 double a = getA(n, x); 139 double b = getB(n, x); 140 double p2 = a * p1 + b * p0; 141 double q2 = a * q1 + b * q0; 142 boolean infinite = false; 143 if (Double.isInfinite(p2) || Double.isInfinite(q2)) { 144 /* 145 * Need to scale. Try successive powers of the larger of a or b 146 * up to 5th power. Throw ConvergenceException if one or both 147 * of p2, q2 still overflow. 148 */ 149 double scaleFactor = 1d; 150 double lastScaleFactor = 1d; 151 final int maxPower = 5; 152 final double scale = FastMath.max(a,b); 153 if (scale <= 0) { // Can't scale 154 throw new ConvergenceException( 155 LocalizedFormats.CONTINUED_FRACTION_INFINITY_DIVERGENCE, 156 x); 157 } 158 infinite = true; 159 for (int i = 0; i < maxPower; i++) { 160 lastScaleFactor = scaleFactor; 161 scaleFactor *= scale; 162 if (a != 0.0 && a > b) { 163 p2 = p1 / lastScaleFactor + (b / scaleFactor * p0); 164 q2 = q1 / lastScaleFactor + (b / scaleFactor * q0); 165 } else if (b != 0) { 166 p2 = (a / scaleFactor * p1) + p0 / lastScaleFactor; 167 q2 = (a / scaleFactor * q1) + q0 / lastScaleFactor; 168 } 169 infinite = Double.isInfinite(p2) || Double.isInfinite(q2); 170 if (!infinite) { 171 break; 172 } 173 } 174 } 175 176 if (infinite) { 177 // Scaling failed 178 throw new ConvergenceException( 179 LocalizedFormats.CONTINUED_FRACTION_INFINITY_DIVERGENCE, 180 x); 181 } 182 183 double r = p2 / q2; 184 185 if (Double.isNaN(r)) { 186 throw new ConvergenceException( 187 LocalizedFormats.CONTINUED_FRACTION_NAN_DIVERGENCE, 188 x); 189 } 190 relativeError = FastMath.abs(r / c - 1.0); 191 192 // prepare for next iteration 193 c = p2 / q2; 194 p0 = p1; 195 p1 = p2; 196 q0 = q1; 197 q1 = q2; 198 } 199 200 if (n >= maxIterations) { 201 throw new MaxIterationsExceededException(maxIterations, 202 LocalizedFormats.NON_CONVERGENT_CONTINUED_FRACTION, 203 x); 204 } 205 206 return c; 207 } 208 }