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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.analysis.solvers;<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.FunctionEvaluationException;<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> import org.apache.commons.math.analysis.UnivariateRealFunction;<a name="line.22"></a> <FONT color="green">023</FONT> import org.apache.commons.math.util.MathUtils;<a name="line.23"></a> <FONT color="green">024</FONT> <a name="line.24"></a> <FONT color="green">025</FONT> /**<a name="line.25"></a> <FONT color="green">026</FONT> * Implements the <a href="http://mathworld.wolfram.com/MullersMethod.html"><a name="line.26"></a> <FONT color="green">027</FONT> * Muller's Method</a> for root finding of real univariate functions. For<a name="line.27"></a> <FONT color="green">028</FONT> * reference, see <b>Elementary Numerical Analysis</b>, ISBN 0070124477,<a name="line.28"></a> <FONT color="green">029</FONT> * chapter 3.<a name="line.29"></a> <FONT color="green">030</FONT> * <p><a name="line.30"></a> <FONT color="green">031</FONT> * Muller's method applies to both real and complex functions, but here we<a name="line.31"></a> <FONT color="green">032</FONT> * restrict ourselves to real functions. Methods solve() and solve2() find<a name="line.32"></a> <FONT color="green">033</FONT> * real zeros, using different ways to bypass complex arithmetics.</p><a name="line.33"></a> <FONT color="green">034</FONT> *<a name="line.34"></a> <FONT color="green">035</FONT> * @version $Revision: 825919 $ $Date: 2009-10-16 10:51:55 -0400 (Fri, 16 Oct 2009) $<a name="line.35"></a> <FONT color="green">036</FONT> * @since 1.2<a name="line.36"></a> <FONT color="green">037</FONT> */<a name="line.37"></a> <FONT color="green">038</FONT> public class MullerSolver extends UnivariateRealSolverImpl {<a name="line.38"></a> <FONT color="green">039</FONT> <a name="line.39"></a> <FONT color="green">040</FONT> /**<a name="line.40"></a> <FONT color="green">041</FONT> * Construct a solver for the given function.<a name="line.41"></a> <FONT color="green">042</FONT> *<a name="line.42"></a> <FONT color="green">043</FONT> * @param f function to solve<a name="line.43"></a> <FONT color="green">044</FONT> * @deprecated as of 2.0 the function to solve is passed as an argument<a name="line.44"></a> <FONT color="green">045</FONT> * to the {@link #solve(UnivariateRealFunction, double, double)} or<a name="line.45"></a> <FONT color="green">046</FONT> * {@link UnivariateRealSolverImpl#solve(UnivariateRealFunction, double, double, double)}<a name="line.46"></a> <FONT color="green">047</FONT> * method.<a name="line.47"></a> <FONT color="green">048</FONT> */<a name="line.48"></a> <FONT color="green">049</FONT> @Deprecated<a name="line.49"></a> <FONT color="green">050</FONT> public MullerSolver(UnivariateRealFunction f) {<a name="line.50"></a> <FONT color="green">051</FONT> super(f, 100, 1E-6);<a name="line.51"></a> <FONT color="green">052</FONT> }<a name="line.52"></a> <FONT color="green">053</FONT> <a name="line.53"></a> <FONT color="green">054</FONT> /**<a name="line.54"></a> <FONT color="green">055</FONT> * Construct a solver.<a name="line.55"></a> <FONT color="green">056</FONT> */<a name="line.56"></a> <FONT color="green">057</FONT> public MullerSolver() {<a name="line.57"></a> <FONT color="green">058</FONT> super(100, 1E-6);<a name="line.58"></a> <FONT color="green">059</FONT> }<a name="line.59"></a> <FONT color="green">060</FONT> <a name="line.60"></a> <FONT color="green">061</FONT> /** {@inheritDoc} */<a name="line.61"></a> <FONT color="green">062</FONT> @Deprecated<a name="line.62"></a> <FONT color="green">063</FONT> public double solve(final double min, final double max)<a name="line.63"></a> <FONT color="green">064</FONT> throws ConvergenceException, FunctionEvaluationException {<a name="line.64"></a> <FONT color="green">065</FONT> return solve(f, min, max);<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> /** {@inheritDoc} */<a name="line.68"></a> <FONT color="green">069</FONT> @Deprecated<a name="line.69"></a> <FONT color="green">070</FONT> public double solve(final double min, final double max, final double initial)<a name="line.70"></a> <FONT color="green">071</FONT> throws ConvergenceException, FunctionEvaluationException {<a name="line.71"></a> <FONT color="green">072</FONT> return solve(f, min, max, initial);<a name="line.72"></a> <FONT color="green">073</FONT> }<a name="line.73"></a> <FONT color="green">074</FONT> <a name="line.74"></a> <FONT color="green">075</FONT> /**<a name="line.75"></a> <FONT color="green">076</FONT> * Find a real root in the given interval with initial value.<a name="line.76"></a> <FONT color="green">077</FONT> * <p><a name="line.77"></a> <FONT color="green">078</FONT> * Requires bracketing condition.</p><a name="line.78"></a> <FONT color="green">079</FONT> *<a name="line.79"></a> <FONT color="green">080</FONT> * @param f the function to solve<a name="line.80"></a> <FONT color="green">081</FONT> * @param min the lower bound for the interval<a name="line.81"></a> <FONT color="green">082</FONT> * @param max the upper bound for the interval<a name="line.82"></a> <FONT color="green">083</FONT> * @param initial the start value to use<a name="line.83"></a> <FONT color="green">084</FONT> * @return the point at which the function value is zero<a name="line.84"></a> <FONT color="green">085</FONT> * @throws MaxIterationsExceededException if the maximum iteration count is exceeded<a name="line.85"></a> <FONT color="green">086</FONT> * or the solver detects convergence problems otherwise<a name="line.86"></a> <FONT color="green">087</FONT> * @throws FunctionEvaluationException if an error occurs evaluating the<a name="line.87"></a> <FONT color="green">088</FONT> * function<a name="line.88"></a> <FONT color="green">089</FONT> * @throws IllegalArgumentException if any parameters are invalid<a name="line.89"></a> <FONT color="green">090</FONT> */<a name="line.90"></a> <FONT color="green">091</FONT> public double solve(final UnivariateRealFunction f,<a name="line.91"></a> <FONT color="green">092</FONT> final double min, final double max, final double initial)<a name="line.92"></a> <FONT color="green">093</FONT> throws MaxIterationsExceededException, FunctionEvaluationException {<a name="line.93"></a> <FONT color="green">094</FONT> <a name="line.94"></a> <FONT color="green">095</FONT> // check for zeros before verifying bracketing<a name="line.95"></a> <FONT color="green">096</FONT> if (f.value(min) == 0.0) { return min; }<a name="line.96"></a> <FONT color="green">097</FONT> if (f.value(max) == 0.0) { return max; }<a name="line.97"></a> <FONT color="green">098</FONT> if (f.value(initial) == 0.0) { return initial; }<a name="line.98"></a> <FONT color="green">099</FONT> <a name="line.99"></a> <FONT color="green">100</FONT> verifyBracketing(min, max, f);<a name="line.100"></a> <FONT color="green">101</FONT> verifySequence(min, initial, max);<a name="line.101"></a> <FONT color="green">102</FONT> if (isBracketing(min, initial, f)) {<a name="line.102"></a> <FONT color="green">103</FONT> return solve(f, min, initial);<a name="line.103"></a> <FONT color="green">104</FONT> } else {<a name="line.104"></a> <FONT color="green">105</FONT> return solve(f, initial, max);<a name="line.105"></a> <FONT color="green">106</FONT> }<a name="line.106"></a> <FONT color="green">107</FONT> }<a name="line.107"></a> <FONT color="green">108</FONT> <a name="line.108"></a> <FONT color="green">109</FONT> /**<a name="line.109"></a> <FONT color="green">110</FONT> * Find a real root in the given interval.<a name="line.110"></a> <FONT color="green">111</FONT> * <p><a name="line.111"></a> <FONT color="green">112</FONT> * Original Muller's method would have function evaluation at complex point.<a name="line.112"></a> <FONT color="green">113</FONT> * Since our f(x) is real, we have to find ways to avoid that. Bracketing<a name="line.113"></a> <FONT color="green">114</FONT> * condition is one way to go: by requiring bracketing in every iteration,<a name="line.114"></a> <FONT color="green">115</FONT> * the newly computed approximation is guaranteed to be real.</p><a name="line.115"></a> <FONT color="green">116</FONT> * <p><a name="line.116"></a> <FONT color="green">117</FONT> * Normally Muller's method converges quadratically in the vicinity of a<a name="line.117"></a> <FONT color="green">118</FONT> * zero, however it may be very slow in regions far away from zeros. For<a name="line.118"></a> <FONT color="green">119</FONT> * example, f(x) = exp(x) - 1, min = -50, max = 100. In such case we use<a name="line.119"></a> <FONT color="green">120</FONT> * bisection as a safety backup if it performs very poorly.</p><a name="line.120"></a> <FONT color="green">121</FONT> * <p><a name="line.121"></a> <FONT color="green">122</FONT> * The formulas here use divided differences directly.</p><a name="line.122"></a> <FONT color="green">123</FONT> *<a name="line.123"></a> <FONT color="green">124</FONT> * @param f the function to solve<a name="line.124"></a> <FONT color="green">125</FONT> * @param min the lower bound for the interval<a name="line.125"></a> <FONT color="green">126</FONT> * @param max the upper bound for the interval<a name="line.126"></a> <FONT color="green">127</FONT> * @return the point at which the function value is zero<a name="line.127"></a> <FONT color="green">128</FONT> * @throws MaxIterationsExceededException if the maximum iteration count is exceeded<a name="line.128"></a> <FONT color="green">129</FONT> * or the solver detects convergence problems otherwise<a name="line.129"></a> <FONT color="green">130</FONT> * @throws FunctionEvaluationException if an error occurs evaluating the<a name="line.130"></a> <FONT color="green">131</FONT> * function<a name="line.131"></a> <FONT color="green">132</FONT> * @throws IllegalArgumentException if any parameters are invalid<a name="line.132"></a> <FONT color="green">133</FONT> */<a name="line.133"></a> <FONT color="green">134</FONT> public double solve(final UnivariateRealFunction f,<a name="line.134"></a> <FONT color="green">135</FONT> final double min, final double max)<a name="line.135"></a> <FONT color="green">136</FONT> throws MaxIterationsExceededException, FunctionEvaluationException {<a name="line.136"></a> <FONT color="green">137</FONT> <a name="line.137"></a> <FONT color="green">138</FONT> // [x0, x2] is the bracketing interval in each iteration<a name="line.138"></a> <FONT color="green">139</FONT> // x1 is the last approximation and an interpolation point in (x0, x2)<a name="line.139"></a> <FONT color="green">140</FONT> // x is the new root approximation and new x1 for next round<a name="line.140"></a> <FONT color="green">141</FONT> // d01, d12, d012 are divided differences<a name="line.141"></a> <FONT color="green">142</FONT> <a name="line.142"></a> <FONT color="green">143</FONT> double x0 = min;<a name="line.143"></a> <FONT color="green">144</FONT> double y0 = f.value(x0);<a name="line.144"></a> <FONT color="green">145</FONT> double x2 = max;<a name="line.145"></a> <FONT color="green">146</FONT> double y2 = f.value(x2);<a name="line.146"></a> <FONT color="green">147</FONT> double x1 = 0.5 * (x0 + x2);<a name="line.147"></a> <FONT color="green">148</FONT> double y1 = f.value(x1);<a name="line.148"></a> <FONT color="green">149</FONT> <a name="line.149"></a> <FONT color="green">150</FONT> // check for zeros before verifying bracketing<a name="line.150"></a> <FONT color="green">151</FONT> if (y0 == 0.0) {<a name="line.151"></a> <FONT color="green">152</FONT> return min;<a name="line.152"></a> <FONT color="green">153</FONT> }<a name="line.153"></a> <FONT color="green">154</FONT> if (y2 == 0.0) {<a name="line.154"></a> <FONT color="green">155</FONT> return max;<a name="line.155"></a> <FONT color="green">156</FONT> }<a name="line.156"></a> <FONT color="green">157</FONT> verifyBracketing(min, max, f);<a name="line.157"></a> <FONT color="green">158</FONT> <a name="line.158"></a> <FONT color="green">159</FONT> double oldx = Double.POSITIVE_INFINITY;<a name="line.159"></a> <FONT color="green">160</FONT> for (int i = 1; i <= maximalIterationCount; ++i) {<a name="line.160"></a> <FONT color="green">161</FONT> // Muller's method employs quadratic interpolation through<a name="line.161"></a> <FONT color="green">162</FONT> // x0, x1, x2 and x is the zero of the interpolating parabola.<a name="line.162"></a> <FONT color="green">163</FONT> // Due to bracketing condition, this parabola must have two<a name="line.163"></a> <FONT color="green">164</FONT> // real roots and we choose one in [x0, x2] to be x.<a name="line.164"></a> <FONT color="green">165</FONT> final double d01 = (y1 - y0) / (x1 - x0);<a name="line.165"></a> <FONT color="green">166</FONT> final double d12 = (y2 - y1) / (x2 - x1);<a name="line.166"></a> <FONT color="green">167</FONT> final double d012 = (d12 - d01) / (x2 - x0);<a name="line.167"></a> <FONT color="green">168</FONT> final double c1 = d01 + (x1 - x0) * d012;<a name="line.168"></a> <FONT color="green">169</FONT> final double delta = c1 * c1 - 4 * y1 * d012;<a name="line.169"></a> <FONT color="green">170</FONT> final double xplus = x1 + (-2.0 * y1) / (c1 + Math.sqrt(delta));<a name="line.170"></a> <FONT color="green">171</FONT> final double xminus = x1 + (-2.0 * y1) / (c1 - Math.sqrt(delta));<a name="line.171"></a> <FONT color="green">172</FONT> // xplus and xminus are two roots of parabola and at least<a name="line.172"></a> <FONT color="green">173</FONT> // one of them should lie in (x0, x2)<a name="line.173"></a> <FONT color="green">174</FONT> final double x = isSequence(x0, xplus, x2) ? xplus : xminus;<a name="line.174"></a> <FONT color="green">175</FONT> final double y = f.value(x);<a name="line.175"></a> <FONT color="green">176</FONT> <a name="line.176"></a> <FONT color="green">177</FONT> // check for convergence<a name="line.177"></a> <FONT color="green">178</FONT> final double tolerance = Math.max(relativeAccuracy * Math.abs(x), absoluteAccuracy);<a name="line.178"></a> <FONT color="green">179</FONT> if (Math.abs(x - oldx) <= tolerance) {<a name="line.179"></a> <FONT color="green">180</FONT> setResult(x, i);<a name="line.180"></a> <FONT color="green">181</FONT> return result;<a name="line.181"></a> <FONT color="green">182</FONT> }<a name="line.182"></a> <FONT color="green">183</FONT> if (Math.abs(y) <= functionValueAccuracy) {<a name="line.183"></a> <FONT color="green">184</FONT> setResult(x, i);<a name="line.184"></a> <FONT color="green">185</FONT> return result;<a name="line.185"></a> <FONT color="green">186</FONT> }<a name="line.186"></a> <FONT color="green">187</FONT> <a name="line.187"></a> <FONT color="green">188</FONT> // Bisect if convergence is too slow. Bisection would waste<a name="line.188"></a> <FONT color="green">189</FONT> // our calculation of x, hopefully it won't happen often.<a name="line.189"></a> <FONT color="green">190</FONT> // the real number equality test x == x1 is intentional and<a name="line.190"></a> <FONT color="green">191</FONT> // completes the proximity tests above it<a name="line.191"></a> <FONT color="green">192</FONT> boolean bisect = (x < x1 && (x1 - x0) > 0.95 * (x2 - x0)) ||<a name="line.192"></a> <FONT color="green">193</FONT> (x > x1 && (x2 - x1) > 0.95 * (x2 - x0)) ||<a name="line.193"></a> <FONT color="green">194</FONT> (x == x1);<a name="line.194"></a> <FONT color="green">195</FONT> // prepare the new bracketing interval for next iteration<a name="line.195"></a> <FONT color="green">196</FONT> if (!bisect) {<a name="line.196"></a> <FONT color="green">197</FONT> x0 = x < x1 ? x0 : x1;<a name="line.197"></a> <FONT color="green">198</FONT> y0 = x < x1 ? y0 : y1;<a name="line.198"></a> <FONT color="green">199</FONT> x2 = x > x1 ? x2 : x1;<a name="line.199"></a> <FONT color="green">200</FONT> y2 = x > x1 ? y2 : y1;<a name="line.200"></a> <FONT color="green">201</FONT> x1 = x; y1 = y;<a name="line.201"></a> <FONT color="green">202</FONT> oldx = x;<a name="line.202"></a> <FONT color="green">203</FONT> } else {<a name="line.203"></a> <FONT color="green">204</FONT> double xm = 0.5 * (x0 + x2);<a name="line.204"></a> <FONT color="green">205</FONT> double ym = f.value(xm);<a name="line.205"></a> <FONT color="green">206</FONT> if (MathUtils.sign(y0) + MathUtils.sign(ym) == 0.0) {<a name="line.206"></a> <FONT color="green">207</FONT> x2 = xm; y2 = ym;<a name="line.207"></a> <FONT color="green">208</FONT> } else {<a name="line.208"></a> <FONT color="green">209</FONT> x0 = xm; y0 = ym;<a name="line.209"></a> <FONT color="green">210</FONT> }<a name="line.210"></a> <FONT color="green">211</FONT> x1 = 0.5 * (x0 + x2);<a name="line.211"></a> <FONT color="green">212</FONT> y1 = f.value(x1);<a name="line.212"></a> <FONT color="green">213</FONT> oldx = Double.POSITIVE_INFINITY;<a name="line.213"></a> <FONT color="green">214</FONT> }<a name="line.214"></a> <FONT color="green">215</FONT> }<a name="line.215"></a> <FONT color="green">216</FONT> throw new MaxIterationsExceededException(maximalIterationCount);<a name="line.216"></a> <FONT color="green">217</FONT> }<a name="line.217"></a> <FONT color="green">218</FONT> <a name="line.218"></a> <FONT color="green">219</FONT> /**<a name="line.219"></a> <FONT color="green">220</FONT> * Find a real root in the given interval.<a name="line.220"></a> <FONT color="green">221</FONT> * <p><a name="line.221"></a> <FONT color="green">222</FONT> * solve2() differs from solve() in the way it avoids complex operations.<a name="line.222"></a> <FONT color="green">223</FONT> * Except for the initial [min, max], solve2() does not require bracketing<a name="line.223"></a> <FONT color="green">224</FONT> * condition, e.g. f(x0), f(x1), f(x2) can have the same sign. If complex<a name="line.224"></a> <FONT color="green">225</FONT> * number arises in the computation, we simply use its modulus as real<a name="line.225"></a> <FONT color="green">226</FONT> * approximation.</p><a name="line.226"></a> <FONT color="green">227</FONT> * <p><a name="line.227"></a> <FONT color="green">228</FONT> * Because the interval may not be bracketing, bisection alternative is<a name="line.228"></a> <FONT color="green">229</FONT> * not applicable here. However in practice our treatment usually works<a name="line.229"></a> <FONT color="green">230</FONT> * well, especially near real zeros where the imaginary part of complex<a name="line.230"></a> <FONT color="green">231</FONT> * approximation is often negligible.</p><a name="line.231"></a> <FONT color="green">232</FONT> * <p><a name="line.232"></a> <FONT color="green">233</FONT> * The formulas here do not use divided differences directly.</p><a name="line.233"></a> <FONT color="green">234</FONT> *<a name="line.234"></a> <FONT color="green">235</FONT> * @param min the lower bound for the interval<a name="line.235"></a> <FONT color="green">236</FONT> * @param max the upper bound for the interval<a name="line.236"></a> <FONT color="green">237</FONT> * @return the point at which the function value is zero<a name="line.237"></a> <FONT color="green">238</FONT> * @throws MaxIterationsExceededException if the maximum iteration count is exceeded<a name="line.238"></a> <FONT color="green">239</FONT> * or the solver detects convergence problems otherwise<a name="line.239"></a> <FONT color="green">240</FONT> * @throws FunctionEvaluationException if an error occurs evaluating the<a name="line.240"></a> <FONT color="green">241</FONT> * function<a name="line.241"></a> <FONT color="green">242</FONT> * @throws IllegalArgumentException if any parameters are invalid<a name="line.242"></a> <FONT color="green">243</FONT> * @deprecated replaced by {@link #solve2(UnivariateRealFunction, double, double)}<a name="line.243"></a> <FONT color="green">244</FONT> * since 2.0<a name="line.244"></a> <FONT color="green">245</FONT> */<a name="line.245"></a> <FONT color="green">246</FONT> @Deprecated<a name="line.246"></a> <FONT color="green">247</FONT> public double solve2(final double min, final double max)<a name="line.247"></a> <FONT color="green">248</FONT> throws MaxIterationsExceededException, FunctionEvaluationException {<a name="line.248"></a> <FONT color="green">249</FONT> return solve2(f, min, max);<a name="line.249"></a> <FONT color="green">250</FONT> }<a name="line.250"></a> <FONT color="green">251</FONT> <a name="line.251"></a> <FONT color="green">252</FONT> /**<a name="line.252"></a> <FONT color="green">253</FONT> * Find a real root in the given interval.<a name="line.253"></a> <FONT color="green">254</FONT> * <p><a name="line.254"></a> <FONT color="green">255</FONT> * solve2() differs from solve() in the way it avoids complex operations.<a name="line.255"></a> <FONT color="green">256</FONT> * Except for the initial [min, max], solve2() does not require bracketing<a name="line.256"></a> <FONT color="green">257</FONT> * condition, e.g. f(x0), f(x1), f(x2) can have the same sign. If complex<a name="line.257"></a> <FONT color="green">258</FONT> * number arises in the computation, we simply use its modulus as real<a name="line.258"></a> <FONT color="green">259</FONT> * approximation.</p><a name="line.259"></a> <FONT color="green">260</FONT> * <p><a name="line.260"></a> <FONT color="green">261</FONT> * Because the interval may not be bracketing, bisection alternative is<a name="line.261"></a> <FONT color="green">262</FONT> * not applicable here. However in practice our treatment usually works<a name="line.262"></a> <FONT color="green">263</FONT> * well, especially near real zeros where the imaginary part of complex<a name="line.263"></a> <FONT color="green">264</FONT> * approximation is often negligible.</p><a name="line.264"></a> <FONT color="green">265</FONT> * <p><a name="line.265"></a> <FONT color="green">266</FONT> * The formulas here do not use divided differences directly.</p><a name="line.266"></a> <FONT color="green">267</FONT> *<a name="line.267"></a> <FONT color="green">268</FONT> * @param f the function to solve<a name="line.268"></a> <FONT color="green">269</FONT> * @param min the lower bound for the interval<a name="line.269"></a> <FONT color="green">270</FONT> * @param max the upper bound for the interval<a name="line.270"></a> <FONT color="green">271</FONT> * @return the point at which the function value is zero<a name="line.271"></a> <FONT color="green">272</FONT> * @throws MaxIterationsExceededException if the maximum iteration count is exceeded<a name="line.272"></a> <FONT color="green">273</FONT> * or the solver detects convergence problems otherwise<a name="line.273"></a> <FONT color="green">274</FONT> * @throws FunctionEvaluationException if an error occurs evaluating the<a name="line.274"></a> <FONT color="green">275</FONT> * function<a name="line.275"></a> <FONT color="green">276</FONT> * @throws IllegalArgumentException if any parameters are invalid<a name="line.276"></a> <FONT color="green">277</FONT> */<a name="line.277"></a> <FONT color="green">278</FONT> public double solve2(final UnivariateRealFunction f,<a name="line.278"></a> <FONT color="green">279</FONT> final double min, final double max)<a name="line.279"></a> <FONT color="green">280</FONT> throws MaxIterationsExceededException, FunctionEvaluationException {<a name="line.280"></a> <FONT color="green">281</FONT> <a name="line.281"></a> <FONT color="green">282</FONT> // x2 is the last root approximation<a name="line.282"></a> <FONT color="green">283</FONT> // x is the new approximation and new x2 for next round<a name="line.283"></a> <FONT color="green">284</FONT> // x0 < x1 < x2 does not hold here<a name="line.284"></a> <FONT color="green">285</FONT> <a name="line.285"></a> <FONT color="green">286</FONT> double x0 = min;<a name="line.286"></a> <FONT color="green">287</FONT> double y0 = f.value(x0);<a name="line.287"></a> <FONT color="green">288</FONT> double x1 = max;<a name="line.288"></a> <FONT color="green">289</FONT> double y1 = f.value(x1);<a name="line.289"></a> <FONT color="green">290</FONT> double x2 = 0.5 * (x0 + x1);<a name="line.290"></a> <FONT color="green">291</FONT> double y2 = f.value(x2);<a name="line.291"></a> <FONT color="green">292</FONT> <a name="line.292"></a> <FONT color="green">293</FONT> // check for zeros before verifying bracketing<a name="line.293"></a> <FONT color="green">294</FONT> if (y0 == 0.0) { return min; }<a name="line.294"></a> <FONT color="green">295</FONT> if (y1 == 0.0) { return max; }<a name="line.295"></a> <FONT color="green">296</FONT> verifyBracketing(min, max, f);<a name="line.296"></a> <FONT color="green">297</FONT> <a name="line.297"></a> <FONT color="green">298</FONT> double oldx = Double.POSITIVE_INFINITY;<a name="line.298"></a> <FONT color="green">299</FONT> for (int i = 1; i <= maximalIterationCount; ++i) {<a name="line.299"></a> <FONT color="green">300</FONT> // quadratic interpolation through x0, x1, x2<a name="line.300"></a> <FONT color="green">301</FONT> final double q = (x2 - x1) / (x1 - x0);<a name="line.301"></a> <FONT color="green">302</FONT> final double a = q * (y2 - (1 + q) * y1 + q * y0);<a name="line.302"></a> <FONT color="green">303</FONT> final double b = (2 * q + 1) * y2 - (1 + q) * (1 + q) * y1 + q * q * y0;<a name="line.303"></a> <FONT color="green">304</FONT> final double c = (1 + q) * y2;<a name="line.304"></a> <FONT color="green">305</FONT> final double delta = b * b - 4 * a * c;<a name="line.305"></a> <FONT color="green">306</FONT> double x;<a name="line.306"></a> <FONT color="green">307</FONT> final double denominator;<a name="line.307"></a> <FONT color="green">308</FONT> if (delta >= 0.0) {<a name="line.308"></a> <FONT color="green">309</FONT> // choose a denominator larger in magnitude<a name="line.309"></a> <FONT color="green">310</FONT> double dplus = b + Math.sqrt(delta);<a name="line.310"></a> <FONT color="green">311</FONT> double dminus = b - Math.sqrt(delta);<a name="line.311"></a> <FONT color="green">312</FONT> denominator = Math.abs(dplus) > Math.abs(dminus) ? dplus : dminus;<a name="line.312"></a> <FONT color="green">313</FONT> } else {<a name="line.313"></a> <FONT color="green">314</FONT> // take the modulus of (B +/- Math.sqrt(delta))<a name="line.314"></a> <FONT color="green">315</FONT> denominator = Math.sqrt(b * b - delta);<a name="line.315"></a> <FONT color="green">316</FONT> }<a name="line.316"></a> <FONT color="green">317</FONT> if (denominator != 0) {<a name="line.317"></a> <FONT color="green">318</FONT> x = x2 - 2.0 * c * (x2 - x1) / denominator;<a name="line.318"></a> <FONT color="green">319</FONT> // perturb x if it exactly coincides with x1 or x2<a name="line.319"></a> <FONT color="green">320</FONT> // the equality tests here are intentional<a name="line.320"></a> <FONT color="green">321</FONT> while (x == x1 || x == x2) {<a name="line.321"></a> <FONT color="green">322</FONT> x += absoluteAccuracy;<a name="line.322"></a> <FONT color="green">323</FONT> }<a name="line.323"></a> <FONT color="green">324</FONT> } else {<a name="line.324"></a> <FONT color="green">325</FONT> // extremely rare case, get a random number to skip it<a name="line.325"></a> <FONT color="green">326</FONT> x = min + Math.random() * (max - min);<a name="line.326"></a> <FONT color="green">327</FONT> oldx = Double.POSITIVE_INFINITY;<a name="line.327"></a> <FONT color="green">328</FONT> }<a name="line.328"></a> <FONT color="green">329</FONT> final double y = f.value(x);<a name="line.329"></a> <FONT color="green">330</FONT> <a name="line.330"></a> <FONT color="green">331</FONT> // check for convergence<a name="line.331"></a> <FONT color="green">332</FONT> final double tolerance = Math.max(relativeAccuracy * Math.abs(x), absoluteAccuracy);<a name="line.332"></a> <FONT color="green">333</FONT> if (Math.abs(x - oldx) <= tolerance) {<a name="line.333"></a> <FONT color="green">334</FONT> setResult(x, i);<a name="line.334"></a> <FONT color="green">335</FONT> return result;<a name="line.335"></a> <FONT color="green">336</FONT> }<a name="line.336"></a> <FONT color="green">337</FONT> if (Math.abs(y) <= functionValueAccuracy) {<a name="line.337"></a> <FONT color="green">338</FONT> setResult(x, i);<a name="line.338"></a> <FONT color="green">339</FONT> return result;<a name="line.339"></a> <FONT color="green">340</FONT> }<a name="line.340"></a> <FONT color="green">341</FONT> <a name="line.341"></a> <FONT color="green">342</FONT> // prepare the next iteration<a name="line.342"></a> <FONT color="green">343</FONT> x0 = x1;<a name="line.343"></a> <FONT color="green">344</FONT> y0 = y1;<a name="line.344"></a> <FONT color="green">345</FONT> x1 = x2;<a name="line.345"></a> <FONT color="green">346</FONT> y1 = y2;<a name="line.346"></a> <FONT color="green">347</FONT> x2 = x;<a name="line.347"></a> <FONT color="green">348</FONT> y2 = y;<a name="line.348"></a> <FONT color="green">349</FONT> oldx = x;<a name="line.349"></a> <FONT color="green">350</FONT> }<a name="line.350"></a> <FONT color="green">351</FONT> throw new MaxIterationsExceededException(maximalIterationCount);<a name="line.351"></a> <FONT color="green">352</FONT> }<a name="line.352"></a> <FONT color="green">353</FONT> }<a name="line.353"></a> </PRE> </BODY> </HTML>