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| date | Tue, 04 Jan 2011 10:02:07 +0100 |
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<HTML> <BODY BGCOLOR="white"> <PRE> <FONT color="green">001</FONT> /*<a name="line.1"></a> <FONT color="green">002</FONT> * Licensed to the Apache Software Foundation (ASF) under one or more<a name="line.2"></a> <FONT color="green">003</FONT> * contributor license agreements. See the NOTICE file distributed with<a name="line.3"></a> <FONT color="green">004</FONT> * this work for additional information regarding copyright ownership.<a name="line.4"></a> <FONT color="green">005</FONT> * The ASF licenses this file to You under the Apache License, Version 2.0<a name="line.5"></a> <FONT color="green">006</FONT> * (the "License"); you may not use this file except in compliance with<a name="line.6"></a> <FONT color="green">007</FONT> * the License. You may obtain a copy of the License at<a name="line.7"></a> <FONT color="green">008</FONT> *<a name="line.8"></a> <FONT color="green">009</FONT> * http://www.apache.org/licenses/LICENSE-2.0<a name="line.9"></a> <FONT color="green">010</FONT> *<a name="line.10"></a> <FONT color="green">011</FONT> * Unless required by applicable law or agreed to in writing, software<a name="line.11"></a> <FONT color="green">012</FONT> * distributed under the License is distributed on an "AS IS" BASIS,<a name="line.12"></a> <FONT color="green">013</FONT> * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.<a name="line.13"></a> <FONT color="green">014</FONT> * See the License for the specific language governing permissions and<a name="line.14"></a> <FONT color="green">015</FONT> * limitations under the License.<a name="line.15"></a> <FONT color="green">016</FONT> */<a name="line.16"></a> <FONT color="green">017</FONT> package org.apache.commons.math.util;<a name="line.17"></a> <FONT color="green">018</FONT> <a name="line.18"></a> <FONT color="green">019</FONT> import org.apache.commons.math.ConvergenceException;<a name="line.19"></a> <FONT color="green">020</FONT> import org.apache.commons.math.MathException;<a name="line.20"></a> <FONT color="green">021</FONT> import org.apache.commons.math.MaxIterationsExceededException;<a name="line.21"></a> <FONT color="green">022</FONT> <a name="line.22"></a> <FONT color="green">023</FONT> /**<a name="line.23"></a> <FONT color="green">024</FONT> * Provides a generic means to evaluate continued fractions. Subclasses simply<a name="line.24"></a> <FONT color="green">025</FONT> * provided the a and b coefficients to evaluate the continued fraction.<a name="line.25"></a> <FONT color="green">026</FONT> *<a name="line.26"></a> <FONT color="green">027</FONT> * <p><a name="line.27"></a> <FONT color="green">028</FONT> * References:<a name="line.28"></a> <FONT color="green">029</FONT> * <ul><a name="line.29"></a> <FONT color="green">030</FONT> * <li><a href="http://mathworld.wolfram.com/ContinuedFraction.html"><a name="line.30"></a> <FONT color="green">031</FONT> * Continued Fraction</a></li><a name="line.31"></a> <FONT color="green">032</FONT> * </ul><a name="line.32"></a> <FONT color="green">033</FONT> * </p><a name="line.33"></a> <FONT color="green">034</FONT> *<a name="line.34"></a> <FONT color="green">035</FONT> * @version $Revision: 920558 $ $Date: 2010-03-08 17:57:32 -0500 (Mon, 08 Mar 2010) $<a name="line.35"></a> <FONT color="green">036</FONT> */<a name="line.36"></a> <FONT color="green">037</FONT> public abstract class ContinuedFraction {<a name="line.37"></a> <FONT color="green">038</FONT> <a name="line.38"></a> <FONT color="green">039</FONT> /** Maximum allowed numerical error. */<a name="line.39"></a> <FONT color="green">040</FONT> private static final double DEFAULT_EPSILON = 10e-9;<a name="line.40"></a> <FONT color="green">041</FONT> <a name="line.41"></a> <FONT color="green">042</FONT> /**<a name="line.42"></a> <FONT color="green">043</FONT> * Default constructor.<a name="line.43"></a> <FONT color="green">044</FONT> */<a name="line.44"></a> <FONT color="green">045</FONT> protected ContinuedFraction() {<a name="line.45"></a> <FONT color="green">046</FONT> super();<a name="line.46"></a> <FONT color="green">047</FONT> }<a name="line.47"></a> <FONT color="green">048</FONT> <a name="line.48"></a> <FONT color="green">049</FONT> /**<a name="line.49"></a> <FONT color="green">050</FONT> * Access the n-th a coefficient of the continued fraction. Since a can be<a name="line.50"></a> <FONT color="green">051</FONT> * a function of the evaluation point, x, that is passed in as well.<a name="line.51"></a> <FONT color="green">052</FONT> * @param n the coefficient index to retrieve.<a name="line.52"></a> <FONT color="green">053</FONT> * @param x the evaluation point.<a name="line.53"></a> <FONT color="green">054</FONT> * @return the n-th a coefficient.<a name="line.54"></a> <FONT color="green">055</FONT> */<a name="line.55"></a> <FONT color="green">056</FONT> protected abstract double getA(int n, double x);<a name="line.56"></a> <FONT color="green">057</FONT> <a name="line.57"></a> <FONT color="green">058</FONT> /**<a name="line.58"></a> <FONT color="green">059</FONT> * Access the n-th b coefficient of the continued fraction. Since b can be<a name="line.59"></a> <FONT color="green">060</FONT> * a function of the evaluation point, x, that is passed in as well.<a name="line.60"></a> <FONT color="green">061</FONT> * @param n the coefficient index to retrieve.<a name="line.61"></a> <FONT color="green">062</FONT> * @param x the evaluation point.<a name="line.62"></a> <FONT color="green">063</FONT> * @return the n-th b coefficient.<a name="line.63"></a> <FONT color="green">064</FONT> */<a name="line.64"></a> <FONT color="green">065</FONT> protected abstract double getB(int n, double x);<a name="line.65"></a> <FONT color="green">066</FONT> <a name="line.66"></a> <FONT color="green">067</FONT> /**<a name="line.67"></a> <FONT color="green">068</FONT> * Evaluates the continued fraction at the value x.<a name="line.68"></a> <FONT color="green">069</FONT> * @param x the evaluation point.<a name="line.69"></a> <FONT color="green">070</FONT> * @return the value of the continued fraction evaluated at x.<a name="line.70"></a> <FONT color="green">071</FONT> * @throws MathException if the algorithm fails to converge.<a name="line.71"></a> <FONT color="green">072</FONT> */<a name="line.72"></a> <FONT color="green">073</FONT> public double evaluate(double x) throws MathException {<a name="line.73"></a> <FONT color="green">074</FONT> return evaluate(x, DEFAULT_EPSILON, Integer.MAX_VALUE);<a name="line.74"></a> <FONT color="green">075</FONT> }<a name="line.75"></a> <FONT color="green">076</FONT> <a name="line.76"></a> <FONT color="green">077</FONT> /**<a name="line.77"></a> <FONT color="green">078</FONT> * Evaluates the continued fraction at the value x.<a name="line.78"></a> <FONT color="green">079</FONT> * @param x the evaluation point.<a name="line.79"></a> <FONT color="green">080</FONT> * @param epsilon maximum error allowed.<a name="line.80"></a> <FONT color="green">081</FONT> * @return the value of the continued fraction evaluated at x.<a name="line.81"></a> <FONT color="green">082</FONT> * @throws MathException if the algorithm fails to converge.<a name="line.82"></a> <FONT color="green">083</FONT> */<a name="line.83"></a> <FONT color="green">084</FONT> public double evaluate(double x, double epsilon) throws MathException {<a name="line.84"></a> <FONT color="green">085</FONT> return evaluate(x, epsilon, Integer.MAX_VALUE);<a name="line.85"></a> <FONT color="green">086</FONT> }<a name="line.86"></a> <FONT color="green">087</FONT> <a name="line.87"></a> <FONT color="green">088</FONT> /**<a name="line.88"></a> <FONT color="green">089</FONT> * Evaluates the continued fraction at the value x.<a name="line.89"></a> <FONT color="green">090</FONT> * @param x the evaluation point.<a name="line.90"></a> <FONT color="green">091</FONT> * @param maxIterations maximum number of convergents<a name="line.91"></a> <FONT color="green">092</FONT> * @return the value of the continued fraction evaluated at x.<a name="line.92"></a> <FONT color="green">093</FONT> * @throws MathException if the algorithm fails to converge.<a name="line.93"></a> <FONT color="green">094</FONT> */<a name="line.94"></a> <FONT color="green">095</FONT> public double evaluate(double x, int maxIterations) throws MathException {<a name="line.95"></a> <FONT color="green">096</FONT> return evaluate(x, DEFAULT_EPSILON, maxIterations);<a name="line.96"></a> <FONT color="green">097</FONT> }<a name="line.97"></a> <FONT color="green">098</FONT> <a name="line.98"></a> <FONT color="green">099</FONT> /**<a name="line.99"></a> <FONT color="green">100</FONT> * <p><a name="line.100"></a> <FONT color="green">101</FONT> * Evaluates the continued fraction at the value x.<a name="line.101"></a> <FONT color="green">102</FONT> * </p><a name="line.102"></a> <FONT color="green">103</FONT> *<a name="line.103"></a> <FONT color="green">104</FONT> * <p><a name="line.104"></a> <FONT color="green">105</FONT> * The implementation of this method is based on equations 14-17 of:<a name="line.105"></a> <FONT color="green">106</FONT> * <ul><a name="line.106"></a> <FONT color="green">107</FONT> * <li><a name="line.107"></a> <FONT color="green">108</FONT> * Eric W. Weisstein. "Continued Fraction." From MathWorld--A Wolfram Web<a name="line.108"></a> <FONT color="green">109</FONT> * Resource. <a target="_blank"<a name="line.109"></a> <FONT color="green">110</FONT> * href="http://mathworld.wolfram.com/ContinuedFraction.html"><a name="line.110"></a> <FONT color="green">111</FONT> * http://mathworld.wolfram.com/ContinuedFraction.html</a><a name="line.111"></a> <FONT color="green">112</FONT> * </li><a name="line.112"></a> <FONT color="green">113</FONT> * </ul><a name="line.113"></a> <FONT color="green">114</FONT> * The recurrence relationship defined in those equations can result in<a name="line.114"></a> <FONT color="green">115</FONT> * very large intermediate results which can result in numerical overflow.<a name="line.115"></a> <FONT color="green">116</FONT> * As a means to combat these overflow conditions, the intermediate results<a name="line.116"></a> <FONT color="green">117</FONT> * are scaled whenever they threaten to become numerically unstable.</p><a name="line.117"></a> <FONT color="green">118</FONT> *<a name="line.118"></a> <FONT color="green">119</FONT> * @param x the evaluation point.<a name="line.119"></a> <FONT color="green">120</FONT> * @param epsilon maximum error allowed.<a name="line.120"></a> <FONT color="green">121</FONT> * @param maxIterations maximum number of convergents<a name="line.121"></a> <FONT color="green">122</FONT> * @return the value of the continued fraction evaluated at x.<a name="line.122"></a> <FONT color="green">123</FONT> * @throws MathException if the algorithm fails to converge.<a name="line.123"></a> <FONT color="green">124</FONT> */<a name="line.124"></a> <FONT color="green">125</FONT> public double evaluate(double x, double epsilon, int maxIterations)<a name="line.125"></a> <FONT color="green">126</FONT> throws MathException<a name="line.126"></a> <FONT color="green">127</FONT> {<a name="line.127"></a> <FONT color="green">128</FONT> double p0 = 1.0;<a name="line.128"></a> <FONT color="green">129</FONT> double p1 = getA(0, x);<a name="line.129"></a> <FONT color="green">130</FONT> double q0 = 0.0;<a name="line.130"></a> <FONT color="green">131</FONT> double q1 = 1.0;<a name="line.131"></a> <FONT color="green">132</FONT> double c = p1 / q1;<a name="line.132"></a> <FONT color="green">133</FONT> int n = 0;<a name="line.133"></a> <FONT color="green">134</FONT> double relativeError = Double.MAX_VALUE;<a name="line.134"></a> <FONT color="green">135</FONT> while (n < maxIterations && relativeError > epsilon) {<a name="line.135"></a> <FONT color="green">136</FONT> ++n;<a name="line.136"></a> <FONT color="green">137</FONT> double a = getA(n, x);<a name="line.137"></a> <FONT color="green">138</FONT> double b = getB(n, x);<a name="line.138"></a> <FONT color="green">139</FONT> double p2 = a * p1 + b * p0;<a name="line.139"></a> <FONT color="green">140</FONT> double q2 = a * q1 + b * q0;<a name="line.140"></a> <FONT color="green">141</FONT> boolean infinite = false;<a name="line.141"></a> <FONT color="green">142</FONT> if (Double.isInfinite(p2) || Double.isInfinite(q2)) {<a name="line.142"></a> <FONT color="green">143</FONT> /*<a name="line.143"></a> <FONT color="green">144</FONT> * Need to scale. Try successive powers of the larger of a or b<a name="line.144"></a> <FONT color="green">145</FONT> * up to 5th power. Throw ConvergenceException if one or both<a name="line.145"></a> <FONT color="green">146</FONT> * of p2, q2 still overflow.<a name="line.146"></a> <FONT color="green">147</FONT> */<a name="line.147"></a> <FONT color="green">148</FONT> double scaleFactor = 1d;<a name="line.148"></a> <FONT color="green">149</FONT> double lastScaleFactor = 1d;<a name="line.149"></a> <FONT color="green">150</FONT> final int maxPower = 5;<a name="line.150"></a> <FONT color="green">151</FONT> final double scale = Math.max(a,b);<a name="line.151"></a> <FONT color="green">152</FONT> if (scale <= 0) { // Can't scale<a name="line.152"></a> <FONT color="green">153</FONT> throw new ConvergenceException(<a name="line.153"></a> <FONT color="green">154</FONT> "Continued fraction convergents diverged to +/- infinity for value {0}",<a name="line.154"></a> <FONT color="green">155</FONT> x);<a name="line.155"></a> <FONT color="green">156</FONT> }<a name="line.156"></a> <FONT color="green">157</FONT> infinite = true;<a name="line.157"></a> <FONT color="green">158</FONT> for (int i = 0; i < maxPower; i++) {<a name="line.158"></a> <FONT color="green">159</FONT> lastScaleFactor = scaleFactor;<a name="line.159"></a> <FONT color="green">160</FONT> scaleFactor *= scale;<a name="line.160"></a> <FONT color="green">161</FONT> if (a != 0.0 && a > b) {<a name="line.161"></a> <FONT color="green">162</FONT> p2 = p1 / lastScaleFactor + (b / scaleFactor * p0);<a name="line.162"></a> <FONT color="green">163</FONT> q2 = q1 / lastScaleFactor + (b / scaleFactor * q0);<a name="line.163"></a> <FONT color="green">164</FONT> } else if (b != 0) {<a name="line.164"></a> <FONT color="green">165</FONT> p2 = (a / scaleFactor * p1) + p0 / lastScaleFactor;<a name="line.165"></a> <FONT color="green">166</FONT> q2 = (a / scaleFactor * q1) + q0 / lastScaleFactor;<a name="line.166"></a> <FONT color="green">167</FONT> }<a name="line.167"></a> <FONT color="green">168</FONT> infinite = Double.isInfinite(p2) || Double.isInfinite(q2);<a name="line.168"></a> <FONT color="green">169</FONT> if (!infinite) {<a name="line.169"></a> <FONT color="green">170</FONT> break;<a name="line.170"></a> <FONT color="green">171</FONT> }<a name="line.171"></a> <FONT color="green">172</FONT> }<a name="line.172"></a> <FONT color="green">173</FONT> }<a name="line.173"></a> <FONT color="green">174</FONT> <a name="line.174"></a> <FONT color="green">175</FONT> if (infinite) {<a name="line.175"></a> <FONT color="green">176</FONT> // Scaling failed<a name="line.176"></a> <FONT color="green">177</FONT> throw new ConvergenceException(<a name="line.177"></a> <FONT color="green">178</FONT> "Continued fraction convergents diverged to +/- infinity for value {0}",<a name="line.178"></a> <FONT color="green">179</FONT> x);<a name="line.179"></a> <FONT color="green">180</FONT> }<a name="line.180"></a> <FONT color="green">181</FONT> <a name="line.181"></a> <FONT color="green">182</FONT> double r = p2 / q2;<a name="line.182"></a> <FONT color="green">183</FONT> <a name="line.183"></a> <FONT color="green">184</FONT> if (Double.isNaN(r)) {<a name="line.184"></a> <FONT color="green">185</FONT> throw new ConvergenceException(<a name="line.185"></a> <FONT color="green">186</FONT> "Continued fraction diverged to NaN for value {0}",<a name="line.186"></a> <FONT color="green">187</FONT> x);<a name="line.187"></a> <FONT color="green">188</FONT> }<a name="line.188"></a> <FONT color="green">189</FONT> relativeError = Math.abs(r / c - 1.0);<a name="line.189"></a> <FONT color="green">190</FONT> <a name="line.190"></a> <FONT color="green">191</FONT> // prepare for next iteration<a name="line.191"></a> <FONT color="green">192</FONT> c = p2 / q2;<a name="line.192"></a> <FONT color="green">193</FONT> p0 = p1;<a name="line.193"></a> <FONT color="green">194</FONT> p1 = p2;<a name="line.194"></a> <FONT color="green">195</FONT> q0 = q1;<a name="line.195"></a> <FONT color="green">196</FONT> q1 = q2;<a name="line.196"></a> <FONT color="green">197</FONT> }<a name="line.197"></a> <FONT color="green">198</FONT> <a name="line.198"></a> <FONT color="green">199</FONT> if (n >= maxIterations) {<a name="line.199"></a> <FONT color="green">200</FONT> throw new MaxIterationsExceededException(maxIterations,<a name="line.200"></a> <FONT color="green">201</FONT> "Continued fraction convergents failed to converge for value {0}",<a name="line.201"></a> <FONT color="green">202</FONT> x);<a name="line.202"></a> <FONT color="green">203</FONT> }<a name="line.203"></a> <FONT color="green">204</FONT> <a name="line.204"></a> <FONT color="green">205</FONT> return c;<a name="line.205"></a> <FONT color="green">206</FONT> }<a name="line.206"></a> <FONT color="green">207</FONT> }<a name="line.207"></a> </PRE> </BODY> </HTML>