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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.MathRuntimeException;<a name="line.21"></a> <FONT color="green">022</FONT> import org.apache.commons.math.MaxIterationsExceededException;<a name="line.22"></a> <FONT color="green">023</FONT> import org.apache.commons.math.analysis.UnivariateRealFunction;<a name="line.23"></a> <FONT color="green">024</FONT> import org.apache.commons.math.analysis.polynomials.PolynomialFunction;<a name="line.24"></a> <FONT color="green">025</FONT> import org.apache.commons.math.complex.Complex;<a name="line.25"></a> <FONT color="green">026</FONT> <a name="line.26"></a> <FONT color="green">027</FONT> /**<a name="line.27"></a> <FONT color="green">028</FONT> * Implements the <a href="http://mathworld.wolfram.com/LaguerresMethod.html"><a name="line.28"></a> <FONT color="green">029</FONT> * Laguerre's Method</a> for root finding of real coefficient polynomials.<a name="line.29"></a> <FONT color="green">030</FONT> * For reference, see <b>A First Course in Numerical Analysis</b>,<a name="line.30"></a> <FONT color="green">031</FONT> * ISBN 048641454X, chapter 8.<a name="line.31"></a> <FONT color="green">032</FONT> * <p><a name="line.32"></a> <FONT color="green">033</FONT> * Laguerre's method is global in the sense that it can start with any initial<a name="line.33"></a> <FONT color="green">034</FONT> * approximation and be able to solve all roots from that point.</p><a name="line.34"></a> <FONT color="green">035</FONT> *<a name="line.35"></a> <FONT color="green">036</FONT> * @version $Revision: 922708 $ $Date: 2010-03-13 20:15:47 -0500 (Sat, 13 Mar 2010) $<a name="line.36"></a> <FONT color="green">037</FONT> * @since 1.2<a name="line.37"></a> <FONT color="green">038</FONT> */<a name="line.38"></a> <FONT color="green">039</FONT> public class LaguerreSolver extends UnivariateRealSolverImpl {<a name="line.39"></a> <FONT color="green">040</FONT> <a name="line.40"></a> <FONT color="green">041</FONT> /** Message for non-polynomial function. */<a name="line.41"></a> <FONT color="green">042</FONT> private static final String NON_POLYNOMIAL_FUNCTION_MESSAGE =<a name="line.42"></a> <FONT color="green">043</FONT> "function is not polynomial";<a name="line.43"></a> <FONT color="green">044</FONT> <a name="line.44"></a> <FONT color="green">045</FONT> /** Message for non-positive degree. */<a name="line.45"></a> <FONT color="green">046</FONT> private static final String NON_POSITIVE_DEGREE_MESSAGE =<a name="line.46"></a> <FONT color="green">047</FONT> "polynomial degree must be positive: degree={0}";<a name="line.47"></a> <FONT color="green">048</FONT> <a name="line.48"></a> <FONT color="green">049</FONT> /** polynomial function to solve.<a name="line.49"></a> <FONT color="green">050</FONT> * @deprecated as of 2.0 the function is not stored anymore in the instance<a name="line.50"></a> <FONT color="green">051</FONT> */<a name="line.51"></a> <FONT color="green">052</FONT> @Deprecated<a name="line.52"></a> <FONT color="green">053</FONT> private final PolynomialFunction p;<a name="line.53"></a> <FONT color="green">054</FONT> <a name="line.54"></a> <FONT color="green">055</FONT> /**<a name="line.55"></a> <FONT color="green">056</FONT> * Construct a solver for the given function.<a name="line.56"></a> <FONT color="green">057</FONT> *<a name="line.57"></a> <FONT color="green">058</FONT> * @param f function to solve<a name="line.58"></a> <FONT color="green">059</FONT> * @throws IllegalArgumentException if function is not polynomial<a name="line.59"></a> <FONT color="green">060</FONT> * @deprecated as of 2.0 the function to solve is passed as an argument<a name="line.60"></a> <FONT color="green">061</FONT> * to the {@link #solve(UnivariateRealFunction, double, double)} or<a name="line.61"></a> <FONT color="green">062</FONT> * {@link UnivariateRealSolverImpl#solve(UnivariateRealFunction, double, double, double)}<a name="line.62"></a> <FONT color="green">063</FONT> * method.<a name="line.63"></a> <FONT color="green">064</FONT> */<a name="line.64"></a> <FONT color="green">065</FONT> @Deprecated<a name="line.65"></a> <FONT color="green">066</FONT> public LaguerreSolver(UnivariateRealFunction f) throws<a name="line.66"></a> <FONT color="green">067</FONT> IllegalArgumentException {<a name="line.67"></a> <FONT color="green">068</FONT> super(f, 100, 1E-6);<a name="line.68"></a> <FONT color="green">069</FONT> if (f instanceof PolynomialFunction) {<a name="line.69"></a> <FONT color="green">070</FONT> p = (PolynomialFunction) f;<a name="line.70"></a> <FONT color="green">071</FONT> } else {<a name="line.71"></a> <FONT color="green">072</FONT> throw MathRuntimeException.createIllegalArgumentException(NON_POLYNOMIAL_FUNCTION_MESSAGE);<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> /**<a name="line.76"></a> <FONT color="green">077</FONT> * Construct a solver.<a name="line.77"></a> <FONT color="green">078</FONT> */<a name="line.78"></a> <FONT color="green">079</FONT> public LaguerreSolver() {<a name="line.79"></a> <FONT color="green">080</FONT> super(100, 1E-6);<a name="line.80"></a> <FONT color="green">081</FONT> p = null;<a name="line.81"></a> <FONT color="green">082</FONT> }<a name="line.82"></a> <FONT color="green">083</FONT> <a name="line.83"></a> <FONT color="green">084</FONT> /**<a name="line.84"></a> <FONT color="green">085</FONT> * Returns a copy of the polynomial function.<a name="line.85"></a> <FONT color="green">086</FONT> *<a name="line.86"></a> <FONT color="green">087</FONT> * @return a fresh copy of the polynomial function<a name="line.87"></a> <FONT color="green">088</FONT> * @deprecated as of 2.0 the function is not stored anymore within the instance.<a name="line.88"></a> <FONT color="green">089</FONT> */<a name="line.89"></a> <FONT color="green">090</FONT> @Deprecated<a name="line.90"></a> <FONT color="green">091</FONT> public PolynomialFunction getPolynomialFunction() {<a name="line.91"></a> <FONT color="green">092</FONT> return new PolynomialFunction(p.getCoefficients());<a name="line.92"></a> <FONT color="green">093</FONT> }<a name="line.93"></a> <FONT color="green">094</FONT> <a name="line.94"></a> <FONT color="green">095</FONT> /** {@inheritDoc} */<a name="line.95"></a> <FONT color="green">096</FONT> @Deprecated<a name="line.96"></a> <FONT color="green">097</FONT> public double solve(final double min, final double max)<a name="line.97"></a> <FONT color="green">098</FONT> throws ConvergenceException, FunctionEvaluationException {<a name="line.98"></a> <FONT color="green">099</FONT> return solve(p, min, max);<a name="line.99"></a> <FONT color="green">100</FONT> }<a name="line.100"></a> <FONT color="green">101</FONT> <a name="line.101"></a> <FONT color="green">102</FONT> /** {@inheritDoc} */<a name="line.102"></a> <FONT color="green">103</FONT> @Deprecated<a name="line.103"></a> <FONT color="green">104</FONT> public double solve(final double min, final double max, final double initial)<a name="line.104"></a> <FONT color="green">105</FONT> throws ConvergenceException, FunctionEvaluationException {<a name="line.105"></a> <FONT color="green">106</FONT> return solve(p, min, max, initial);<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 with initial value.<a name="line.110"></a> <FONT color="green">111</FONT> * <p><a name="line.111"></a> <FONT color="green">112</FONT> * Requires bracketing condition.</p><a name="line.112"></a> <FONT color="green">113</FONT> *<a name="line.113"></a> <FONT color="green">114</FONT> * @param f function to solve (must be polynomial)<a name="line.114"></a> <FONT color="green">115</FONT> * @param min the lower bound for the interval<a name="line.115"></a> <FONT color="green">116</FONT> * @param max the upper bound for the interval<a name="line.116"></a> <FONT color="green">117</FONT> * @param initial the start value to use<a name="line.117"></a> <FONT color="green">118</FONT> * @return the point at which the function value is zero<a name="line.118"></a> <FONT color="green">119</FONT> * @throws ConvergenceException if the maximum iteration count is exceeded<a name="line.119"></a> <FONT color="green">120</FONT> * or the solver detects convergence problems otherwise<a name="line.120"></a> <FONT color="green">121</FONT> * @throws FunctionEvaluationException if an error occurs evaluating the<a name="line.121"></a> <FONT color="green">122</FONT> * function<a name="line.122"></a> <FONT color="green">123</FONT> * @throws IllegalArgumentException if any parameters are invalid<a name="line.123"></a> <FONT color="green">124</FONT> */<a name="line.124"></a> <FONT color="green">125</FONT> public double solve(final UnivariateRealFunction f,<a name="line.125"></a> <FONT color="green">126</FONT> final double min, final double max, final double initial)<a name="line.126"></a> <FONT color="green">127</FONT> throws ConvergenceException, FunctionEvaluationException {<a name="line.127"></a> <FONT color="green">128</FONT> <a name="line.128"></a> <FONT color="green">129</FONT> // check for zeros before verifying bracketing<a name="line.129"></a> <FONT color="green">130</FONT> if (f.value(min) == 0.0) {<a name="line.130"></a> <FONT color="green">131</FONT> return min;<a name="line.131"></a> <FONT color="green">132</FONT> }<a name="line.132"></a> <FONT color="green">133</FONT> if (f.value(max) == 0.0) {<a name="line.133"></a> <FONT color="green">134</FONT> return max;<a name="line.134"></a> <FONT color="green">135</FONT> }<a name="line.135"></a> <FONT color="green">136</FONT> if (f.value(initial) == 0.0) {<a name="line.136"></a> <FONT color="green">137</FONT> return initial;<a name="line.137"></a> <FONT color="green">138</FONT> }<a name="line.138"></a> <FONT color="green">139</FONT> <a name="line.139"></a> <FONT color="green">140</FONT> verifyBracketing(min, max, f);<a name="line.140"></a> <FONT color="green">141</FONT> verifySequence(min, initial, max);<a name="line.141"></a> <FONT color="green">142</FONT> if (isBracketing(min, initial, f)) {<a name="line.142"></a> <FONT color="green">143</FONT> return solve(f, min, initial);<a name="line.143"></a> <FONT color="green">144</FONT> } else {<a name="line.144"></a> <FONT color="green">145</FONT> return solve(f, initial, max);<a name="line.145"></a> <FONT color="green">146</FONT> }<a name="line.146"></a> <FONT color="green">147</FONT> <a name="line.147"></a> <FONT color="green">148</FONT> }<a name="line.148"></a> <FONT color="green">149</FONT> <a name="line.149"></a> <FONT color="green">150</FONT> /**<a name="line.150"></a> <FONT color="green">151</FONT> * Find a real root in the given interval.<a name="line.151"></a> <FONT color="green">152</FONT> * <p><a name="line.152"></a> <FONT color="green">153</FONT> * Despite the bracketing condition, the root returned by solve(Complex[],<a name="line.153"></a> <FONT color="green">154</FONT> * Complex) may not be a real zero inside [min, max]. For example,<a name="line.154"></a> <FONT color="green">155</FONT> * p(x) = x^3 + 1, min = -2, max = 2, initial = 0. We can either try<a name="line.155"></a> <FONT color="green">156</FONT> * another initial value, or, as we did here, call solveAll() to obtain<a name="line.156"></a> <FONT color="green">157</FONT> * all roots and pick up the one that we're looking for.</p><a name="line.157"></a> <FONT color="green">158</FONT> *<a name="line.158"></a> <FONT color="green">159</FONT> * @param f the function to solve<a name="line.159"></a> <FONT color="green">160</FONT> * @param min the lower bound for the interval<a name="line.160"></a> <FONT color="green">161</FONT> * @param max the upper bound for the interval<a name="line.161"></a> <FONT color="green">162</FONT> * @return the point at which the function value is zero<a name="line.162"></a> <FONT color="green">163</FONT> * @throws ConvergenceException if the maximum iteration count is exceeded<a name="line.163"></a> <FONT color="green">164</FONT> * or the solver detects convergence problems otherwise<a name="line.164"></a> <FONT color="green">165</FONT> * @throws FunctionEvaluationException if an error occurs evaluating the<a name="line.165"></a> <FONT color="green">166</FONT> * function<a name="line.166"></a> <FONT color="green">167</FONT> * @throws IllegalArgumentException if any parameters are invalid<a name="line.167"></a> <FONT color="green">168</FONT> */<a name="line.168"></a> <FONT color="green">169</FONT> public double solve(final UnivariateRealFunction f,<a name="line.169"></a> <FONT color="green">170</FONT> final double min, final double max)<a name="line.170"></a> <FONT color="green">171</FONT> throws ConvergenceException, FunctionEvaluationException {<a name="line.171"></a> <FONT color="green">172</FONT> <a name="line.172"></a> <FONT color="green">173</FONT> // check function type<a name="line.173"></a> <FONT color="green">174</FONT> if (!(f instanceof PolynomialFunction)) {<a name="line.174"></a> <FONT color="green">175</FONT> throw MathRuntimeException.createIllegalArgumentException(NON_POLYNOMIAL_FUNCTION_MESSAGE);<a name="line.175"></a> <FONT color="green">176</FONT> }<a name="line.176"></a> <FONT color="green">177</FONT> <a name="line.177"></a> <FONT color="green">178</FONT> // check for zeros before verifying bracketing<a name="line.178"></a> <FONT color="green">179</FONT> if (f.value(min) == 0.0) { return min; }<a name="line.179"></a> <FONT color="green">180</FONT> if (f.value(max) == 0.0) { return max; }<a name="line.180"></a> <FONT color="green">181</FONT> verifyBracketing(min, max, f);<a name="line.181"></a> <FONT color="green">182</FONT> <a name="line.182"></a> <FONT color="green">183</FONT> double coefficients[] = ((PolynomialFunction) f).getCoefficients();<a name="line.183"></a> <FONT color="green">184</FONT> Complex c[] = new Complex[coefficients.length];<a name="line.184"></a> <FONT color="green">185</FONT> for (int i = 0; i < coefficients.length; i++) {<a name="line.185"></a> <FONT color="green">186</FONT> c[i] = new Complex(coefficients[i], 0.0);<a name="line.186"></a> <FONT color="green">187</FONT> }<a name="line.187"></a> <FONT color="green">188</FONT> Complex initial = new Complex(0.5 * (min + max), 0.0);<a name="line.188"></a> <FONT color="green">189</FONT> Complex z = solve(c, initial);<a name="line.189"></a> <FONT color="green">190</FONT> if (isRootOK(min, max, z)) {<a name="line.190"></a> <FONT color="green">191</FONT> setResult(z.getReal(), iterationCount);<a name="line.191"></a> <FONT color="green">192</FONT> return result;<a name="line.192"></a> <FONT color="green">193</FONT> }<a name="line.193"></a> <FONT color="green">194</FONT> <a name="line.194"></a> <FONT color="green">195</FONT> // solve all roots and select the one we're seeking<a name="line.195"></a> <FONT color="green">196</FONT> Complex[] root = solveAll(c, initial);<a name="line.196"></a> <FONT color="green">197</FONT> for (int i = 0; i < root.length; i++) {<a name="line.197"></a> <FONT color="green">198</FONT> if (isRootOK(min, max, root[i])) {<a name="line.198"></a> <FONT color="green">199</FONT> setResult(root[i].getReal(), iterationCount);<a name="line.199"></a> <FONT color="green">200</FONT> return result;<a name="line.200"></a> <FONT color="green">201</FONT> }<a name="line.201"></a> <FONT color="green">202</FONT> }<a name="line.202"></a> <FONT color="green">203</FONT> <a name="line.203"></a> <FONT color="green">204</FONT> // should never happen<a name="line.204"></a> <FONT color="green">205</FONT> throw new ConvergenceException();<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> <FONT color="green">208</FONT> /**<a name="line.208"></a> <FONT color="green">209</FONT> * Returns true iff the given complex root is actually a real zero<a name="line.209"></a> <FONT color="green">210</FONT> * in the given interval, within the solver tolerance level.<a name="line.210"></a> <FONT color="green">211</FONT> *<a name="line.211"></a> <FONT color="green">212</FONT> * @param min the lower bound for the interval<a name="line.212"></a> <FONT color="green">213</FONT> * @param max the upper bound for the interval<a name="line.213"></a> <FONT color="green">214</FONT> * @param z the complex root<a name="line.214"></a> <FONT color="green">215</FONT> * @return true iff z is the sought-after real zero<a name="line.215"></a> <FONT color="green">216</FONT> */<a name="line.216"></a> <FONT color="green">217</FONT> protected boolean isRootOK(double min, double max, Complex z) {<a name="line.217"></a> <FONT color="green">218</FONT> double tolerance = Math.max(relativeAccuracy * z.abs(), absoluteAccuracy);<a name="line.218"></a> <FONT color="green">219</FONT> return (isSequence(min, z.getReal(), max)) &&<a name="line.219"></a> <FONT color="green">220</FONT> (Math.abs(z.getImaginary()) <= tolerance ||<a name="line.220"></a> <FONT color="green">221</FONT> z.abs() <= functionValueAccuracy);<a name="line.221"></a> <FONT color="green">222</FONT> }<a name="line.222"></a> <FONT color="green">223</FONT> <a name="line.223"></a> <FONT color="green">224</FONT> /**<a name="line.224"></a> <FONT color="green">225</FONT> * Find all complex roots for the polynomial with the given coefficients,<a name="line.225"></a> <FONT color="green">226</FONT> * starting from the given initial value.<a name="line.226"></a> <FONT color="green">227</FONT> *<a name="line.227"></a> <FONT color="green">228</FONT> * @param coefficients the polynomial coefficients array<a name="line.228"></a> <FONT color="green">229</FONT> * @param initial the start value to use<a name="line.229"></a> <FONT color="green">230</FONT> * @return the point at which the function value is zero<a name="line.230"></a> <FONT color="green">231</FONT> * @throws ConvergenceException if the maximum iteration count is exceeded<a name="line.231"></a> <FONT color="green">232</FONT> * or the solver detects convergence problems otherwise<a name="line.232"></a> <FONT color="green">233</FONT> * @throws FunctionEvaluationException if an error occurs evaluating the<a name="line.233"></a> <FONT color="green">234</FONT> * function<a name="line.234"></a> <FONT color="green">235</FONT> * @throws IllegalArgumentException if any parameters are invalid<a name="line.235"></a> <FONT color="green">236</FONT> */<a name="line.236"></a> <FONT color="green">237</FONT> public Complex[] solveAll(double coefficients[], double initial) throws<a name="line.237"></a> <FONT color="green">238</FONT> ConvergenceException, FunctionEvaluationException {<a name="line.238"></a> <FONT color="green">239</FONT> <a name="line.239"></a> <FONT color="green">240</FONT> Complex c[] = new Complex[coefficients.length];<a name="line.240"></a> <FONT color="green">241</FONT> Complex z = new Complex(initial, 0.0);<a name="line.241"></a> <FONT color="green">242</FONT> for (int i = 0; i < c.length; i++) {<a name="line.242"></a> <FONT color="green">243</FONT> c[i] = new Complex(coefficients[i], 0.0);<a name="line.243"></a> <FONT color="green">244</FONT> }<a name="line.244"></a> <FONT color="green">245</FONT> return solveAll(c, z);<a name="line.245"></a> <FONT color="green">246</FONT> }<a name="line.246"></a> <FONT color="green">247</FONT> <a name="line.247"></a> <FONT color="green">248</FONT> /**<a name="line.248"></a> <FONT color="green">249</FONT> * Find all complex roots for the polynomial with the given coefficients,<a name="line.249"></a> <FONT color="green">250</FONT> * starting from the given initial value.<a name="line.250"></a> <FONT color="green">251</FONT> *<a name="line.251"></a> <FONT color="green">252</FONT> * @param coefficients the polynomial coefficients array<a name="line.252"></a> <FONT color="green">253</FONT> * @param initial the start value to use<a name="line.253"></a> <FONT color="green">254</FONT> * @return the point at which the function value is zero<a name="line.254"></a> <FONT color="green">255</FONT> * @throws MaxIterationsExceededException if the maximum iteration count is exceeded<a name="line.255"></a> <FONT color="green">256</FONT> * or the solver detects convergence problems otherwise<a name="line.256"></a> <FONT color="green">257</FONT> * @throws FunctionEvaluationException if an error occurs evaluating the<a name="line.257"></a> <FONT color="green">258</FONT> * function<a name="line.258"></a> <FONT color="green">259</FONT> * @throws IllegalArgumentException if any parameters are invalid<a name="line.259"></a> <FONT color="green">260</FONT> */<a name="line.260"></a> <FONT color="green">261</FONT> public Complex[] solveAll(Complex coefficients[], Complex initial) throws<a name="line.261"></a> <FONT color="green">262</FONT> MaxIterationsExceededException, FunctionEvaluationException {<a name="line.262"></a> <FONT color="green">263</FONT> <a name="line.263"></a> <FONT color="green">264</FONT> int n = coefficients.length - 1;<a name="line.264"></a> <FONT color="green">265</FONT> int iterationCount = 0;<a name="line.265"></a> <FONT color="green">266</FONT> if (n < 1) {<a name="line.266"></a> <FONT color="green">267</FONT> throw MathRuntimeException.createIllegalArgumentException(<a name="line.267"></a> <FONT color="green">268</FONT> NON_POSITIVE_DEGREE_MESSAGE, n);<a name="line.268"></a> <FONT color="green">269</FONT> }<a name="line.269"></a> <FONT color="green">270</FONT> Complex c[] = new Complex[n+1]; // coefficients for deflated polynomial<a name="line.270"></a> <FONT color="green">271</FONT> for (int i = 0; i <= n; i++) {<a name="line.271"></a> <FONT color="green">272</FONT> c[i] = coefficients[i];<a name="line.272"></a> <FONT color="green">273</FONT> }<a name="line.273"></a> <FONT color="green">274</FONT> <a name="line.274"></a> <FONT color="green">275</FONT> // solve individual root successively<a name="line.275"></a> <FONT color="green">276</FONT> Complex root[] = new Complex[n];<a name="line.276"></a> <FONT color="green">277</FONT> for (int i = 0; i < n; i++) {<a name="line.277"></a> <FONT color="green">278</FONT> Complex subarray[] = new Complex[n-i+1];<a name="line.278"></a> <FONT color="green">279</FONT> System.arraycopy(c, 0, subarray, 0, subarray.length);<a name="line.279"></a> <FONT color="green">280</FONT> root[i] = solve(subarray, initial);<a name="line.280"></a> <FONT color="green">281</FONT> // polynomial deflation using synthetic division<a name="line.281"></a> <FONT color="green">282</FONT> Complex newc = c[n-i];<a name="line.282"></a> <FONT color="green">283</FONT> Complex oldc = null;<a name="line.283"></a> <FONT color="green">284</FONT> for (int j = n-i-1; j >= 0; j--) {<a name="line.284"></a> <FONT color="green">285</FONT> oldc = c[j];<a name="line.285"></a> <FONT color="green">286</FONT> c[j] = newc;<a name="line.286"></a> <FONT color="green">287</FONT> newc = oldc.add(newc.multiply(root[i]));<a name="line.287"></a> <FONT color="green">288</FONT> }<a name="line.288"></a> <FONT color="green">289</FONT> iterationCount += this.iterationCount;<a name="line.289"></a> <FONT color="green">290</FONT> }<a name="line.290"></a> <FONT color="green">291</FONT> <a name="line.291"></a> <FONT color="green">292</FONT> resultComputed = true;<a name="line.292"></a> <FONT color="green">293</FONT> this.iterationCount = iterationCount;<a name="line.293"></a> <FONT color="green">294</FONT> return root;<a name="line.294"></a> <FONT color="green">295</FONT> }<a name="line.295"></a> <FONT color="green">296</FONT> <a name="line.296"></a> <FONT color="green">297</FONT> /**<a name="line.297"></a> <FONT color="green">298</FONT> * Find a complex root for the polynomial with the given coefficients,<a name="line.298"></a> <FONT color="green">299</FONT> * starting from the given initial value.<a name="line.299"></a> <FONT color="green">300</FONT> *<a name="line.300"></a> <FONT color="green">301</FONT> * @param coefficients the polynomial coefficients array<a name="line.301"></a> <FONT color="green">302</FONT> * @param initial the start value to use<a name="line.302"></a> <FONT color="green">303</FONT> * @return the point at which the function value is zero<a name="line.303"></a> <FONT color="green">304</FONT> * @throws MaxIterationsExceededException if the maximum iteration count is exceeded<a name="line.304"></a> <FONT color="green">305</FONT> * or the solver detects convergence problems otherwise<a name="line.305"></a> <FONT color="green">306</FONT> * @throws FunctionEvaluationException if an error occurs evaluating the<a name="line.306"></a> <FONT color="green">307</FONT> * function<a name="line.307"></a> <FONT color="green">308</FONT> * @throws IllegalArgumentException if any parameters are invalid<a name="line.308"></a> <FONT color="green">309</FONT> */<a name="line.309"></a> <FONT color="green">310</FONT> public Complex solve(Complex coefficients[], Complex initial) throws<a name="line.310"></a> <FONT color="green">311</FONT> MaxIterationsExceededException, FunctionEvaluationException {<a name="line.311"></a> <FONT color="green">312</FONT> <a name="line.312"></a> <FONT color="green">313</FONT> int n = coefficients.length - 1;<a name="line.313"></a> <FONT color="green">314</FONT> if (n < 1) {<a name="line.314"></a> <FONT color="green">315</FONT> throw MathRuntimeException.createIllegalArgumentException(<a name="line.315"></a> <FONT color="green">316</FONT> NON_POSITIVE_DEGREE_MESSAGE, n);<a name="line.316"></a> <FONT color="green">317</FONT> }<a name="line.317"></a> <FONT color="green">318</FONT> Complex N = new Complex(n, 0.0);<a name="line.318"></a> <FONT color="green">319</FONT> Complex N1 = new Complex(n - 1, 0.0);<a name="line.319"></a> <FONT color="green">320</FONT> <a name="line.320"></a> <FONT color="green">321</FONT> int i = 1;<a name="line.321"></a> <FONT color="green">322</FONT> Complex pv = null;<a name="line.322"></a> <FONT color="green">323</FONT> Complex dv = null;<a name="line.323"></a> <FONT color="green">324</FONT> Complex d2v = null;<a name="line.324"></a> <FONT color="green">325</FONT> Complex G = null;<a name="line.325"></a> <FONT color="green">326</FONT> Complex G2 = null;<a name="line.326"></a> <FONT color="green">327</FONT> Complex H = null;<a name="line.327"></a> <FONT color="green">328</FONT> Complex delta = null;<a name="line.328"></a> <FONT color="green">329</FONT> Complex denominator = null;<a name="line.329"></a> <FONT color="green">330</FONT> Complex z = initial;<a name="line.330"></a> <FONT color="green">331</FONT> Complex oldz = new Complex(Double.POSITIVE_INFINITY, Double.POSITIVE_INFINITY);<a name="line.331"></a> <FONT color="green">332</FONT> while (i <= maximalIterationCount) {<a name="line.332"></a> <FONT color="green">333</FONT> // Compute pv (polynomial value), dv (derivative value), and<a name="line.333"></a> <FONT color="green">334</FONT> // d2v (second derivative value) simultaneously.<a name="line.334"></a> <FONT color="green">335</FONT> pv = coefficients[n];<a name="line.335"></a> <FONT color="green">336</FONT> dv = Complex.ZERO;<a name="line.336"></a> <FONT color="green">337</FONT> d2v = Complex.ZERO;<a name="line.337"></a> <FONT color="green">338</FONT> for (int j = n-1; j >= 0; j--) {<a name="line.338"></a> <FONT color="green">339</FONT> d2v = dv.add(z.multiply(d2v));<a name="line.339"></a> <FONT color="green">340</FONT> dv = pv.add(z.multiply(dv));<a name="line.340"></a> <FONT color="green">341</FONT> pv = coefficients[j].add(z.multiply(pv));<a name="line.341"></a> <FONT color="green">342</FONT> }<a name="line.342"></a> <FONT color="green">343</FONT> d2v = d2v.multiply(new Complex(2.0, 0.0));<a name="line.343"></a> <FONT color="green">344</FONT> <a name="line.344"></a> <FONT color="green">345</FONT> // check for convergence<a name="line.345"></a> <FONT color="green">346</FONT> double tolerance = Math.max(relativeAccuracy * z.abs(),<a name="line.346"></a> <FONT color="green">347</FONT> absoluteAccuracy);<a name="line.347"></a> <FONT color="green">348</FONT> if ((z.subtract(oldz)).abs() <= tolerance) {<a name="line.348"></a> <FONT color="green">349</FONT> resultComputed = true;<a name="line.349"></a> <FONT color="green">350</FONT> iterationCount = i;<a name="line.350"></a> <FONT color="green">351</FONT> return z;<a name="line.351"></a> <FONT color="green">352</FONT> }<a name="line.352"></a> <FONT color="green">353</FONT> if (pv.abs() <= functionValueAccuracy) {<a name="line.353"></a> <FONT color="green">354</FONT> resultComputed = true;<a name="line.354"></a> <FONT color="green">355</FONT> iterationCount = i;<a name="line.355"></a> <FONT color="green">356</FONT> return z;<a name="line.356"></a> <FONT color="green">357</FONT> }<a name="line.357"></a> <FONT color="green">358</FONT> <a name="line.358"></a> <FONT color="green">359</FONT> // now pv != 0, calculate the new approximation<a name="line.359"></a> <FONT color="green">360</FONT> G = dv.divide(pv);<a name="line.360"></a> <FONT color="green">361</FONT> G2 = G.multiply(G);<a name="line.361"></a> <FONT color="green">362</FONT> H = G2.subtract(d2v.divide(pv));<a name="line.362"></a> <FONT color="green">363</FONT> delta = N1.multiply((N.multiply(H)).subtract(G2));<a name="line.363"></a> <FONT color="green">364</FONT> // choose a denominator larger in magnitude<a name="line.364"></a> <FONT color="green">365</FONT> Complex deltaSqrt = delta.sqrt();<a name="line.365"></a> <FONT color="green">366</FONT> Complex dplus = G.add(deltaSqrt);<a name="line.366"></a> <FONT color="green">367</FONT> Complex dminus = G.subtract(deltaSqrt);<a name="line.367"></a> <FONT color="green">368</FONT> denominator = dplus.abs() > dminus.abs() ? dplus : dminus;<a name="line.368"></a> <FONT color="green">369</FONT> // Perturb z if denominator is zero, for instance,<a name="line.369"></a> <FONT color="green">370</FONT> // p(x) = x^3 + 1, z = 0.<a name="line.370"></a> <FONT color="green">371</FONT> if (denominator.equals(new Complex(0.0, 0.0))) {<a name="line.371"></a> <FONT color="green">372</FONT> z = z.add(new Complex(absoluteAccuracy, absoluteAccuracy));<a name="line.372"></a> <FONT color="green">373</FONT> oldz = new Complex(Double.POSITIVE_INFINITY,<a name="line.373"></a> <FONT color="green">374</FONT> Double.POSITIVE_INFINITY);<a name="line.374"></a> <FONT color="green">375</FONT> } else {<a name="line.375"></a> <FONT color="green">376</FONT> oldz = z;<a name="line.376"></a> <FONT color="green">377</FONT> z = z.subtract(N.divide(denominator));<a name="line.377"></a> <FONT color="green">378</FONT> }<a name="line.378"></a> <FONT color="green">379</FONT> i++;<a name="line.379"></a> <FONT color="green">380</FONT> }<a name="line.380"></a> <FONT color="green">381</FONT> throw new MaxIterationsExceededException(maximalIterationCount);<a name="line.381"></a> <FONT color="green">382</FONT> }<a name="line.382"></a> <FONT color="green">383</FONT> }<a name="line.383"></a> </PRE> </BODY> </HTML>