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| author | dwinter |
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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> <a name="line.17"></a> <FONT color="green">018</FONT> package org.apache.commons.math.linear;<a name="line.18"></a> <FONT color="green">019</FONT> <a name="line.19"></a> <FONT color="green">020</FONT> import org.apache.commons.math.MathRuntimeException;<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.util.MathUtils;<a name="line.22"></a> <FONT color="green">023</FONT> <a name="line.23"></a> <FONT color="green">024</FONT> /**<a name="line.24"></a> <FONT color="green">025</FONT> * Calculates the eigen decomposition of a real <strong>symmetric</strong><a name="line.25"></a> <FONT color="green">026</FONT> * matrix.<a name="line.26"></a> <FONT color="green">027</FONT> * <p><a name="line.27"></a> <FONT color="green">028</FONT> * The eigen decomposition of matrix A is a set of two matrices: V and D such<a name="line.28"></a> <FONT color="green">029</FONT> * that A = V D V<sup>T</sup>. A, V and D are all m &times; m matrices.<a name="line.29"></a> <FONT color="green">030</FONT> * </p><a name="line.30"></a> <FONT color="green">031</FONT> * <p><a name="line.31"></a> <FONT color="green">032</FONT> * As of 2.0, this class supports only <strong>symmetric</strong> matrices, and<a name="line.32"></a> <FONT color="green">033</FONT> * hence computes only real realEigenvalues. This implies the D matrix returned<a name="line.33"></a> <FONT color="green">034</FONT> * by {@link #getD()} is always diagonal and the imaginary values returned<a name="line.34"></a> <FONT color="green">035</FONT> * {@link #getImagEigenvalue(int)} and {@link #getImagEigenvalues()} are always<a name="line.35"></a> <FONT color="green">036</FONT> * null.<a name="line.36"></a> <FONT color="green">037</FONT> * </p><a name="line.37"></a> <FONT color="green">038</FONT> * <p><a name="line.38"></a> <FONT color="green">039</FONT> * When called with a {@link RealMatrix} argument, this implementation only uses<a name="line.39"></a> <FONT color="green">040</FONT> * the upper part of the matrix, the part below the diagonal is not accessed at<a name="line.40"></a> <FONT color="green">041</FONT> * all.<a name="line.41"></a> <FONT color="green">042</FONT> * </p><a name="line.42"></a> <FONT color="green">043</FONT> * <p><a name="line.43"></a> <FONT color="green">044</FONT> * This implementation is based on the paper by A. Drubrulle, R.S. Martin and<a name="line.44"></a> <FONT color="green">045</FONT> * J.H. Wilkinson 'The Implicit QL Algorithm' in Wilksinson and Reinsch (1971)<a name="line.45"></a> <FONT color="green">046</FONT> * Handbook for automatic computation, vol. 2, Linear algebra, Springer-Verlag,<a name="line.46"></a> <FONT color="green">047</FONT> * New-York<a name="line.47"></a> <FONT color="green">048</FONT> * </p><a name="line.48"></a> <FONT color="green">049</FONT> * @version $Revision: 912413 $ $Date: 2010-02-21 16:46:12 -0500 (Sun, 21 Feb 2010) $<a name="line.49"></a> <FONT color="green">050</FONT> * @since 2.0<a name="line.50"></a> <FONT color="green">051</FONT> */<a name="line.51"></a> <FONT color="green">052</FONT> public class EigenDecompositionImpl implements EigenDecomposition {<a name="line.52"></a> <FONT color="green">053</FONT> <a name="line.53"></a> <FONT color="green">054</FONT> /** Maximum number of iterations accepted in the implicit QL transformation */<a name="line.54"></a> <FONT color="green">055</FONT> private byte maxIter = 30;<a name="line.55"></a> <FONT color="green">056</FONT> <a name="line.56"></a> <FONT color="green">057</FONT> /** Main diagonal of the tridiagonal matrix. */<a name="line.57"></a> <FONT color="green">058</FONT> private double[] main;<a name="line.58"></a> <FONT color="green">059</FONT> <a name="line.59"></a> <FONT color="green">060</FONT> /** Secondary diagonal of the tridiagonal matrix. */<a name="line.60"></a> <FONT color="green">061</FONT> private double[] secondary;<a name="line.61"></a> <FONT color="green">062</FONT> <a name="line.62"></a> <FONT color="green">063</FONT> /**<a name="line.63"></a> <FONT color="green">064</FONT> * Transformer to tridiagonal (may be null if matrix is already<a name="line.64"></a> <FONT color="green">065</FONT> * tridiagonal).<a name="line.65"></a> <FONT color="green">066</FONT> */<a name="line.66"></a> <FONT color="green">067</FONT> private TriDiagonalTransformer transformer;<a name="line.67"></a> <FONT color="green">068</FONT> <a name="line.68"></a> <FONT color="green">069</FONT> /** Real part of the realEigenvalues. */<a name="line.69"></a> <FONT color="green">070</FONT> private double[] realEigenvalues;<a name="line.70"></a> <FONT color="green">071</FONT> <a name="line.71"></a> <FONT color="green">072</FONT> /** Imaginary part of the realEigenvalues. */<a name="line.72"></a> <FONT color="green">073</FONT> private double[] imagEigenvalues;<a name="line.73"></a> <FONT color="green">074</FONT> <a name="line.74"></a> <FONT color="green">075</FONT> /** Eigenvectors. */<a name="line.75"></a> <FONT color="green">076</FONT> private ArrayRealVector[] eigenvectors;<a name="line.76"></a> <FONT color="green">077</FONT> <a name="line.77"></a> <FONT color="green">078</FONT> /** Cached value of V. */<a name="line.78"></a> <FONT color="green">079</FONT> private RealMatrix cachedV;<a name="line.79"></a> <FONT color="green">080</FONT> <a name="line.80"></a> <FONT color="green">081</FONT> /** Cached value of D. */<a name="line.81"></a> <FONT color="green">082</FONT> private RealMatrix cachedD;<a name="line.82"></a> <FONT color="green">083</FONT> <a name="line.83"></a> <FONT color="green">084</FONT> /** Cached value of Vt. */<a name="line.84"></a> <FONT color="green">085</FONT> private RealMatrix cachedVt;<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> * Calculates the eigen decomposition of the given symmetric matrix.<a name="line.88"></a> <FONT color="green">089</FONT> * @param matrix The <strong>symmetric</strong> matrix to decompose.<a name="line.89"></a> <FONT color="green">090</FONT> * @param splitTolerance dummy parameter, present for backward compatibility only.<a name="line.90"></a> <FONT color="green">091</FONT> * @exception InvalidMatrixException (wrapping a<a name="line.91"></a> <FONT color="green">092</FONT> * {@link org.apache.commons.math.ConvergenceException} if algorithm<a name="line.92"></a> <FONT color="green">093</FONT> * fails to converge<a name="line.93"></a> <FONT color="green">094</FONT> */<a name="line.94"></a> <FONT color="green">095</FONT> public EigenDecompositionImpl(final RealMatrix matrix,final double splitTolerance)<a name="line.95"></a> <FONT color="green">096</FONT> throws InvalidMatrixException {<a name="line.96"></a> <FONT color="green">097</FONT> if (isSymmetric(matrix)) {<a name="line.97"></a> <FONT color="green">098</FONT> transformToTridiagonal(matrix);<a name="line.98"></a> <FONT color="green">099</FONT> findEigenVectors(transformer.getQ().getData());<a name="line.99"></a> <FONT color="green">100</FONT> } else {<a name="line.100"></a> <FONT color="green">101</FONT> // as of 2.0, non-symmetric matrices (i.e. complex eigenvalues) are<a name="line.101"></a> <FONT color="green">102</FONT> // NOT supported<a name="line.102"></a> <FONT color="green">103</FONT> // see issue https://issues.apache.org/jira/browse/MATH-235<a name="line.103"></a> <FONT color="green">104</FONT> throw new InvalidMatrixException(<a name="line.104"></a> <FONT color="green">105</FONT> "eigen decomposition of assymetric matrices not supported yet");<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> * Calculates the eigen decomposition of the symmetric tridiagonal<a name="line.110"></a> <FONT color="green">111</FONT> * matrix. The Householder matrix is assumed to be the identity matrix.<a name="line.111"></a> <FONT color="green">112</FONT> * @param main Main diagonal of the symmetric triadiagonal form<a name="line.112"></a> <FONT color="green">113</FONT> * @param secondary Secondary of the tridiagonal form<a name="line.113"></a> <FONT color="green">114</FONT> * @param splitTolerance dummy parameter, present for backward compatibility only.<a name="line.114"></a> <FONT color="green">115</FONT> * @exception InvalidMatrixException (wrapping a<a name="line.115"></a> <FONT color="green">116</FONT> * {@link org.apache.commons.math.ConvergenceException} if algorithm<a name="line.116"></a> <FONT color="green">117</FONT> * fails to converge<a name="line.117"></a> <FONT color="green">118</FONT> */<a name="line.118"></a> <FONT color="green">119</FONT> public EigenDecompositionImpl(final double[] main,final double[] secondary,<a name="line.119"></a> <FONT color="green">120</FONT> final double splitTolerance)<a name="line.120"></a> <FONT color="green">121</FONT> throws InvalidMatrixException {<a name="line.121"></a> <FONT color="green">122</FONT> this.main = main.clone();<a name="line.122"></a> <FONT color="green">123</FONT> this.secondary = secondary.clone();<a name="line.123"></a> <FONT color="green">124</FONT> transformer = null;<a name="line.124"></a> <FONT color="green">125</FONT> final int size=main.length;<a name="line.125"></a> <FONT color="green">126</FONT> double[][] z = new double[size][size];<a name="line.126"></a> <FONT color="green">127</FONT> for (int i=0;i<size;i++) {<a name="line.127"></a> <FONT color="green">128</FONT> z[i][i]=1.0;<a name="line.128"></a> <FONT color="green">129</FONT> }<a name="line.129"></a> <FONT color="green">130</FONT> findEigenVectors(z);<a name="line.130"></a> <FONT color="green">131</FONT> }<a name="line.131"></a> <FONT color="green">132</FONT> <a name="line.132"></a> <FONT color="green">133</FONT> /**<a name="line.133"></a> <FONT color="green">134</FONT> * Check if a matrix is symmetric.<a name="line.134"></a> <FONT color="green">135</FONT> * @param matrix<a name="line.135"></a> <FONT color="green">136</FONT> * matrix to check<a name="line.136"></a> <FONT color="green">137</FONT> * @return true if matrix is symmetric<a name="line.137"></a> <FONT color="green">138</FONT> */<a name="line.138"></a> <FONT color="green">139</FONT> private boolean isSymmetric(final RealMatrix matrix) {<a name="line.139"></a> <FONT color="green">140</FONT> final int rows = matrix.getRowDimension();<a name="line.140"></a> <FONT color="green">141</FONT> final int columns = matrix.getColumnDimension();<a name="line.141"></a> <FONT color="green">142</FONT> final double eps = 10 * rows * columns * MathUtils.EPSILON;<a name="line.142"></a> <FONT color="green">143</FONT> for (int i = 0; i < rows; ++i) {<a name="line.143"></a> <FONT color="green">144</FONT> for (int j = i + 1; j < columns; ++j) {<a name="line.144"></a> <FONT color="green">145</FONT> final double mij = matrix.getEntry(i, j);<a name="line.145"></a> <FONT color="green">146</FONT> final double mji = matrix.getEntry(j, i);<a name="line.146"></a> <FONT color="green">147</FONT> if (Math.abs(mij - mji) > (Math.max(Math.abs(mij), Math<a name="line.147"></a> <FONT color="green">148</FONT> .abs(mji)) * eps)) {<a name="line.148"></a> <FONT color="green">149</FONT> return false;<a name="line.149"></a> <FONT color="green">150</FONT> }<a name="line.150"></a> <FONT color="green">151</FONT> }<a name="line.151"></a> <FONT color="green">152</FONT> }<a name="line.152"></a> <FONT color="green">153</FONT> return true;<a name="line.153"></a> <FONT color="green">154</FONT> }<a name="line.154"></a> <FONT color="green">155</FONT> <a name="line.155"></a> <FONT color="green">156</FONT> /** {@inheritDoc} */<a name="line.156"></a> <FONT color="green">157</FONT> public RealMatrix getV() throws InvalidMatrixException {<a name="line.157"></a> <FONT color="green">158</FONT> <a name="line.158"></a> <FONT color="green">159</FONT> if (cachedV == null) {<a name="line.159"></a> <FONT color="green">160</FONT> final int m = eigenvectors.length;<a name="line.160"></a> <FONT color="green">161</FONT> cachedV = MatrixUtils.createRealMatrix(m, m);<a name="line.161"></a> <FONT color="green">162</FONT> for (int k = 0; k < m; ++k) {<a name="line.162"></a> <FONT color="green">163</FONT> cachedV.setColumnVector(k, eigenvectors[k]);<a name="line.163"></a> <FONT color="green">164</FONT> }<a name="line.164"></a> <FONT color="green">165</FONT> }<a name="line.165"></a> <FONT color="green">166</FONT> // return the cached matrix<a name="line.166"></a> <FONT color="green">167</FONT> return cachedV;<a name="line.167"></a> <FONT color="green">168</FONT> <a name="line.168"></a> <FONT color="green">169</FONT> }<a name="line.169"></a> <FONT color="green">170</FONT> <a name="line.170"></a> <FONT color="green">171</FONT> /** {@inheritDoc} */<a name="line.171"></a> <FONT color="green">172</FONT> public RealMatrix getD() throws InvalidMatrixException {<a name="line.172"></a> <FONT color="green">173</FONT> if (cachedD == null) {<a name="line.173"></a> <FONT color="green">174</FONT> // cache the matrix for subsequent calls<a name="line.174"></a> <FONT color="green">175</FONT> cachedD = MatrixUtils.createRealDiagonalMatrix(realEigenvalues);<a name="line.175"></a> <FONT color="green">176</FONT> }<a name="line.176"></a> <FONT color="green">177</FONT> return cachedD;<a name="line.177"></a> <FONT color="green">178</FONT> }<a name="line.178"></a> <FONT color="green">179</FONT> <a name="line.179"></a> <FONT color="green">180</FONT> /** {@inheritDoc} */<a name="line.180"></a> <FONT color="green">181</FONT> public RealMatrix getVT() throws InvalidMatrixException {<a name="line.181"></a> <FONT color="green">182</FONT> <a name="line.182"></a> <FONT color="green">183</FONT> if (cachedVt == null) {<a name="line.183"></a> <FONT color="green">184</FONT> final int m = eigenvectors.length;<a name="line.184"></a> <FONT color="green">185</FONT> cachedVt = MatrixUtils.createRealMatrix(m, m);<a name="line.185"></a> <FONT color="green">186</FONT> for (int k = 0; k < m; ++k) {<a name="line.186"></a> <FONT color="green">187</FONT> cachedVt.setRowVector(k, eigenvectors[k]);<a name="line.187"></a> <FONT color="green">188</FONT> }<a name="line.188"></a> <FONT color="green">189</FONT> <a name="line.189"></a> <FONT color="green">190</FONT> }<a name="line.190"></a> <FONT color="green">191</FONT> <a name="line.191"></a> <FONT color="green">192</FONT> // return the cached matrix<a name="line.192"></a> <FONT color="green">193</FONT> return cachedVt;<a name="line.193"></a> <FONT color="green">194</FONT> }<a name="line.194"></a> <FONT color="green">195</FONT> <a name="line.195"></a> <FONT color="green">196</FONT> /** {@inheritDoc} */<a name="line.196"></a> <FONT color="green">197</FONT> public double[] getRealEigenvalues() throws InvalidMatrixException {<a name="line.197"></a> <FONT color="green">198</FONT> return realEigenvalues.clone();<a name="line.198"></a> <FONT color="green">199</FONT> }<a name="line.199"></a> <FONT color="green">200</FONT> <a name="line.200"></a> <FONT color="green">201</FONT> /** {@inheritDoc} */<a name="line.201"></a> <FONT color="green">202</FONT> public double getRealEigenvalue(final int i) throws InvalidMatrixException,<a name="line.202"></a> <FONT color="green">203</FONT> ArrayIndexOutOfBoundsException {<a name="line.203"></a> <FONT color="green">204</FONT> return realEigenvalues[i];<a name="line.204"></a> <FONT color="green">205</FONT> }<a name="line.205"></a> <FONT color="green">206</FONT> <a name="line.206"></a> <FONT color="green">207</FONT> /** {@inheritDoc} */<a name="line.207"></a> <FONT color="green">208</FONT> public double[] getImagEigenvalues() throws InvalidMatrixException {<a name="line.208"></a> <FONT color="green">209</FONT> return imagEigenvalues.clone();<a name="line.209"></a> <FONT color="green">210</FONT> }<a name="line.210"></a> <FONT color="green">211</FONT> <a name="line.211"></a> <FONT color="green">212</FONT> /** {@inheritDoc} */<a name="line.212"></a> <FONT color="green">213</FONT> public double getImagEigenvalue(final int i) throws InvalidMatrixException,<a name="line.213"></a> <FONT color="green">214</FONT> ArrayIndexOutOfBoundsException {<a name="line.214"></a> <FONT color="green">215</FONT> return imagEigenvalues[i];<a name="line.215"></a> <FONT color="green">216</FONT> }<a name="line.216"></a> <FONT color="green">217</FONT> <a name="line.217"></a> <FONT color="green">218</FONT> /** {@inheritDoc} */<a name="line.218"></a> <FONT color="green">219</FONT> public RealVector getEigenvector(final int i)<a name="line.219"></a> <FONT color="green">220</FONT> throws InvalidMatrixException, ArrayIndexOutOfBoundsException {<a name="line.220"></a> <FONT color="green">221</FONT> return eigenvectors[i].copy();<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> * Return the determinant of the matrix<a name="line.225"></a> <FONT color="green">226</FONT> * @return determinant of the matrix<a name="line.226"></a> <FONT color="green">227</FONT> */<a name="line.227"></a> <FONT color="green">228</FONT> public double getDeterminant() {<a name="line.228"></a> <FONT color="green">229</FONT> double determinant = 1;<a name="line.229"></a> <FONT color="green">230</FONT> for (double lambda : realEigenvalues) {<a name="line.230"></a> <FONT color="green">231</FONT> determinant *= lambda;<a name="line.231"></a> <FONT color="green">232</FONT> }<a name="line.232"></a> <FONT color="green">233</FONT> return determinant;<a name="line.233"></a> <FONT color="green">234</FONT> }<a name="line.234"></a> <FONT color="green">235</FONT> <a name="line.235"></a> <FONT color="green">236</FONT> /** {@inheritDoc} */<a name="line.236"></a> <FONT color="green">237</FONT> public DecompositionSolver getSolver() {<a name="line.237"></a> <FONT color="green">238</FONT> return new Solver(realEigenvalues, imagEigenvalues, eigenvectors);<a name="line.238"></a> <FONT color="green">239</FONT> }<a name="line.239"></a> <FONT color="green">240</FONT> <a name="line.240"></a> <FONT color="green">241</FONT> /** Specialized solver. */<a name="line.241"></a> <FONT color="green">242</FONT> private static class Solver implements DecompositionSolver {<a name="line.242"></a> <FONT color="green">243</FONT> <a name="line.243"></a> <FONT color="green">244</FONT> /** Real part of the realEigenvalues. */<a name="line.244"></a> <FONT color="green">245</FONT> private double[] realEigenvalues;<a name="line.245"></a> <FONT color="green">246</FONT> <a name="line.246"></a> <FONT color="green">247</FONT> /** Imaginary part of the realEigenvalues. */<a name="line.247"></a> <FONT color="green">248</FONT> private double[] imagEigenvalues;<a name="line.248"></a> <FONT color="green">249</FONT> <a name="line.249"></a> <FONT color="green">250</FONT> /** Eigenvectors. */<a name="line.250"></a> <FONT color="green">251</FONT> private final ArrayRealVector[] eigenvectors;<a name="line.251"></a> <FONT color="green">252</FONT> <a name="line.252"></a> <FONT color="green">253</FONT> /**<a name="line.253"></a> <FONT color="green">254</FONT> * Build a solver from decomposed matrix.<a name="line.254"></a> <FONT color="green">255</FONT> * @param realEigenvalues<a name="line.255"></a> <FONT color="green">256</FONT> * real parts of the eigenvalues<a name="line.256"></a> <FONT color="green">257</FONT> * @param imagEigenvalues<a name="line.257"></a> <FONT color="green">258</FONT> * imaginary parts of the eigenvalues<a name="line.258"></a> <FONT color="green">259</FONT> * @param eigenvectors<a name="line.259"></a> <FONT color="green">260</FONT> * eigenvectors<a name="line.260"></a> <FONT color="green">261</FONT> */<a name="line.261"></a> <FONT color="green">262</FONT> private Solver(final double[] realEigenvalues,<a name="line.262"></a> <FONT color="green">263</FONT> final double[] imagEigenvalues,<a name="line.263"></a> <FONT color="green">264</FONT> final ArrayRealVector[] eigenvectors) {<a name="line.264"></a> <FONT color="green">265</FONT> this.realEigenvalues = realEigenvalues;<a name="line.265"></a> <FONT color="green">266</FONT> this.imagEigenvalues = imagEigenvalues;<a name="line.266"></a> <FONT color="green">267</FONT> this.eigenvectors = eigenvectors;<a name="line.267"></a> <FONT color="green">268</FONT> }<a name="line.268"></a> <FONT color="green">269</FONT> <a name="line.269"></a> <FONT color="green">270</FONT> /**<a name="line.270"></a> <FONT color="green">271</FONT> * Solve the linear equation A &times; X = B for symmetric matrices A.<a name="line.271"></a> <FONT color="green">272</FONT> * <p><a name="line.272"></a> <FONT color="green">273</FONT> * This method only find exact linear solutions, i.e. solutions for<a name="line.273"></a> <FONT color="green">274</FONT> * which ||A &times; X - B|| is exactly 0.<a name="line.274"></a> <FONT color="green">275</FONT> * </p><a name="line.275"></a> <FONT color="green">276</FONT> * @param b<a name="line.276"></a> <FONT color="green">277</FONT> * right-hand side of the equation A &times; X = B<a name="line.277"></a> <FONT color="green">278</FONT> * @return a vector X that minimizes the two norm of A &times; X - B<a name="line.278"></a> <FONT color="green">279</FONT> * @exception IllegalArgumentException<a name="line.279"></a> <FONT color="green">280</FONT> * if matrices dimensions don't match<a name="line.280"></a> <FONT color="green">281</FONT> * @exception InvalidMatrixException<a name="line.281"></a> <FONT color="green">282</FONT> * if decomposed matrix is singular<a name="line.282"></a> <FONT color="green">283</FONT> */<a name="line.283"></a> <FONT color="green">284</FONT> public double[] solve(final double[] b)<a name="line.284"></a> <FONT color="green">285</FONT> throws IllegalArgumentException, InvalidMatrixException {<a name="line.285"></a> <FONT color="green">286</FONT> <a name="line.286"></a> <FONT color="green">287</FONT> if (!isNonSingular()) {<a name="line.287"></a> <FONT color="green">288</FONT> throw new SingularMatrixException();<a name="line.288"></a> <FONT color="green">289</FONT> }<a name="line.289"></a> <FONT color="green">290</FONT> <a name="line.290"></a> <FONT color="green">291</FONT> final int m = realEigenvalues.length;<a name="line.291"></a> <FONT color="green">292</FONT> if (b.length != m) {<a name="line.292"></a> <FONT color="green">293</FONT> throw MathRuntimeException.createIllegalArgumentException(<a name="line.293"></a> <FONT color="green">294</FONT> "vector length mismatch: got {0} but expected {1}",<a name="line.294"></a> <FONT color="green">295</FONT> b.length, m);<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> final double[] bp = new double[m];<a name="line.298"></a> <FONT color="green">299</FONT> for (int i = 0; i < m; ++i) {<a name="line.299"></a> <FONT color="green">300</FONT> final ArrayRealVector v = eigenvectors[i];<a name="line.300"></a> <FONT color="green">301</FONT> final double[] vData = v.getDataRef();<a name="line.301"></a> <FONT color="green">302</FONT> final double s = v.dotProduct(b) / realEigenvalues[i];<a name="line.302"></a> <FONT color="green">303</FONT> for (int j = 0; j < m; ++j) {<a name="line.303"></a> <FONT color="green">304</FONT> bp[j] += s * vData[j];<a name="line.304"></a> <FONT color="green">305</FONT> }<a name="line.305"></a> <FONT color="green">306</FONT> }<a name="line.306"></a> <FONT color="green">307</FONT> <a name="line.307"></a> <FONT color="green">308</FONT> return bp;<a name="line.308"></a> <FONT color="green">309</FONT> <a name="line.309"></a> <FONT color="green">310</FONT> }<a name="line.310"></a> <FONT color="green">311</FONT> <a name="line.311"></a> <FONT color="green">312</FONT> /**<a name="line.312"></a> <FONT color="green">313</FONT> * Solve the linear equation A &times; X = B for symmetric matrices A.<a name="line.313"></a> <FONT color="green">314</FONT> * <p><a name="line.314"></a> <FONT color="green">315</FONT> * This method only find exact linear solutions, i.e. solutions for<a name="line.315"></a> <FONT color="green">316</FONT> * which ||A &times; X - B|| is exactly 0.<a name="line.316"></a> <FONT color="green">317</FONT> * </p><a name="line.317"></a> <FONT color="green">318</FONT> * @param b<a name="line.318"></a> <FONT color="green">319</FONT> * right-hand side of the equation A &times; X = B<a name="line.319"></a> <FONT color="green">320</FONT> * @return a vector X that minimizes the two norm of A &times; X - B<a name="line.320"></a> <FONT color="green">321</FONT> * @exception IllegalArgumentException<a name="line.321"></a> <FONT color="green">322</FONT> * if matrices dimensions don't match<a name="line.322"></a> <FONT color="green">323</FONT> * @exception InvalidMatrixException<a name="line.323"></a> <FONT color="green">324</FONT> * if decomposed matrix is singular<a name="line.324"></a> <FONT color="green">325</FONT> */<a name="line.325"></a> <FONT color="green">326</FONT> public RealVector solve(final RealVector b)<a name="line.326"></a> <FONT color="green">327</FONT> throws IllegalArgumentException, InvalidMatrixException {<a name="line.327"></a> <FONT color="green">328</FONT> <a name="line.328"></a> <FONT color="green">329</FONT> if (!isNonSingular()) {<a name="line.329"></a> <FONT color="green">330</FONT> throw new SingularMatrixException();<a name="line.330"></a> <FONT color="green">331</FONT> }<a name="line.331"></a> <FONT color="green">332</FONT> <a name="line.332"></a> <FONT color="green">333</FONT> final int m = realEigenvalues.length;<a name="line.333"></a> <FONT color="green">334</FONT> if (b.getDimension() != m) {<a name="line.334"></a> <FONT color="green">335</FONT> throw MathRuntimeException.createIllegalArgumentException(<a name="line.335"></a> <FONT color="green">336</FONT> "vector length mismatch: got {0} but expected {1}", b<a name="line.336"></a> <FONT color="green">337</FONT> .getDimension(), m);<a name="line.337"></a> <FONT color="green">338</FONT> }<a name="line.338"></a> <FONT color="green">339</FONT> <a name="line.339"></a> <FONT color="green">340</FONT> final double[] bp = new double[m];<a name="line.340"></a> <FONT color="green">341</FONT> for (int i = 0; i < m; ++i) {<a name="line.341"></a> <FONT color="green">342</FONT> final ArrayRealVector v = eigenvectors[i];<a name="line.342"></a> <FONT color="green">343</FONT> final double[] vData = v.getDataRef();<a name="line.343"></a> <FONT color="green">344</FONT> final double s = v.dotProduct(b) / realEigenvalues[i];<a name="line.344"></a> <FONT color="green">345</FONT> for (int j = 0; j < m; ++j) {<a name="line.345"></a> <FONT color="green">346</FONT> bp[j] += s * vData[j];<a name="line.346"></a> <FONT color="green">347</FONT> }<a name="line.347"></a> <FONT color="green">348</FONT> }<a name="line.348"></a> <FONT color="green">349</FONT> <a name="line.349"></a> <FONT color="green">350</FONT> return new ArrayRealVector(bp, false);<a name="line.350"></a> <FONT color="green">351</FONT> <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> <FONT color="green">354</FONT> /**<a name="line.354"></a> <FONT color="green">355</FONT> * Solve the linear equation A &times; X = B for symmetric matrices A.<a name="line.355"></a> <FONT color="green">356</FONT> * <p><a name="line.356"></a> <FONT color="green">357</FONT> * This method only find exact linear solutions, i.e. solutions for<a name="line.357"></a> <FONT color="green">358</FONT> * which ||A &times; X - B|| is exactly 0.<a name="line.358"></a> <FONT color="green">359</FONT> * </p><a name="line.359"></a> <FONT color="green">360</FONT> * @param b<a name="line.360"></a> <FONT color="green">361</FONT> * right-hand side of the equation A &times; X = B<a name="line.361"></a> <FONT color="green">362</FONT> * @return a matrix X that minimizes the two norm of A &times; X - B<a name="line.362"></a> <FONT color="green">363</FONT> * @exception IllegalArgumentException<a name="line.363"></a> <FONT color="green">364</FONT> * if matrices dimensions don't match<a name="line.364"></a> <FONT color="green">365</FONT> * @exception InvalidMatrixException<a name="line.365"></a> <FONT color="green">366</FONT> * if decomposed matrix is singular<a name="line.366"></a> <FONT color="green">367</FONT> */<a name="line.367"></a> <FONT color="green">368</FONT> public RealMatrix solve(final RealMatrix b)<a name="line.368"></a> <FONT color="green">369</FONT> throws IllegalArgumentException, InvalidMatrixException {<a name="line.369"></a> <FONT color="green">370</FONT> <a name="line.370"></a> <FONT color="green">371</FONT> if (!isNonSingular()) {<a name="line.371"></a> <FONT color="green">372</FONT> throw new SingularMatrixException();<a name="line.372"></a> <FONT color="green">373</FONT> }<a name="line.373"></a> <FONT color="green">374</FONT> <a name="line.374"></a> <FONT color="green">375</FONT> final int m = realEigenvalues.length;<a name="line.375"></a> <FONT color="green">376</FONT> if (b.getRowDimension() != m) {<a name="line.376"></a> <FONT color="green">377</FONT> throw MathRuntimeException<a name="line.377"></a> <FONT color="green">378</FONT> .createIllegalArgumentException(<a name="line.378"></a> <FONT color="green">379</FONT> "dimensions mismatch: got {0}x{1} but expected {2}x{3}",<a name="line.379"></a> <FONT color="green">380</FONT> b.getRowDimension(), b.getColumnDimension(), m,<a name="line.380"></a> <FONT color="green">381</FONT> "n");<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> <FONT color="green">384</FONT> final int nColB = b.getColumnDimension();<a name="line.384"></a> <FONT color="green">385</FONT> final double[][] bp = new double[m][nColB];<a name="line.385"></a> <FONT color="green">386</FONT> for (int k = 0; k < nColB; ++k) {<a name="line.386"></a> <FONT color="green">387</FONT> for (int i = 0; i < m; ++i) {<a name="line.387"></a> <FONT color="green">388</FONT> final ArrayRealVector v = eigenvectors[i];<a name="line.388"></a> <FONT color="green">389</FONT> final double[] vData = v.getDataRef();<a name="line.389"></a> <FONT color="green">390</FONT> double s = 0;<a name="line.390"></a> <FONT color="green">391</FONT> for (int j = 0; j < m; ++j) {<a name="line.391"></a> <FONT color="green">392</FONT> s += v.getEntry(j) * b.getEntry(j, k);<a name="line.392"></a> <FONT color="green">393</FONT> }<a name="line.393"></a> <FONT color="green">394</FONT> s /= realEigenvalues[i];<a name="line.394"></a> <FONT color="green">395</FONT> for (int j = 0; j < m; ++j) {<a name="line.395"></a> <FONT color="green">396</FONT> bp[j][k] += s * vData[j];<a name="line.396"></a> <FONT color="green">397</FONT> }<a name="line.397"></a> <FONT color="green">398</FONT> }<a name="line.398"></a> <FONT color="green">399</FONT> }<a name="line.399"></a> <FONT color="green">400</FONT> <a name="line.400"></a> <FONT color="green">401</FONT> return MatrixUtils.createRealMatrix(bp);<a name="line.401"></a> <FONT color="green">402</FONT> <a name="line.402"></a> <FONT color="green">403</FONT> }<a name="line.403"></a> <FONT color="green">404</FONT> <a name="line.404"></a> <FONT color="green">405</FONT> /**<a name="line.405"></a> <FONT color="green">406</FONT> * Check if the decomposed matrix is non-singular.<a name="line.406"></a> <FONT color="green">407</FONT> * @return true if the decomposed matrix is non-singular<a name="line.407"></a> <FONT color="green">408</FONT> */<a name="line.408"></a> <FONT color="green">409</FONT> public boolean isNonSingular() {<a name="line.409"></a> <FONT color="green">410</FONT> for (int i = 0; i < realEigenvalues.length; ++i) {<a name="line.410"></a> <FONT color="green">411</FONT> if ((realEigenvalues[i] == 0) && (imagEigenvalues[i] == 0)) {<a name="line.411"></a> <FONT color="green">412</FONT> return false;<a name="line.412"></a> <FONT color="green">413</FONT> }<a name="line.413"></a> <FONT color="green">414</FONT> }<a name="line.414"></a> <FONT color="green">415</FONT> return true;<a name="line.415"></a> <FONT color="green">416</FONT> }<a name="line.416"></a> <FONT color="green">417</FONT> <a name="line.417"></a> <FONT color="green">418</FONT> /**<a name="line.418"></a> <FONT color="green">419</FONT> * Get the inverse of the decomposed matrix.<a name="line.419"></a> <FONT color="green">420</FONT> * @return inverse matrix<a name="line.420"></a> <FONT color="green">421</FONT> * @throws InvalidMatrixException<a name="line.421"></a> <FONT color="green">422</FONT> * if decomposed matrix is singular<a name="line.422"></a> <FONT color="green">423</FONT> */<a name="line.423"></a> <FONT color="green">424</FONT> public RealMatrix getInverse() throws InvalidMatrixException {<a name="line.424"></a> <FONT color="green">425</FONT> <a name="line.425"></a> <FONT color="green">426</FONT> if (!isNonSingular()) {<a name="line.426"></a> <FONT color="green">427</FONT> throw new SingularMatrixException();<a name="line.427"></a> <FONT color="green">428</FONT> }<a name="line.428"></a> <FONT color="green">429</FONT> <a name="line.429"></a> <FONT color="green">430</FONT> final int m = realEigenvalues.length;<a name="line.430"></a> <FONT color="green">431</FONT> final double[][] invData = new double[m][m];<a name="line.431"></a> <FONT color="green">432</FONT> <a name="line.432"></a> <FONT color="green">433</FONT> for (int i = 0; i < m; ++i) {<a name="line.433"></a> <FONT color="green">434</FONT> final double[] invI = invData[i];<a name="line.434"></a> <FONT color="green">435</FONT> for (int j = 0; j < m; ++j) {<a name="line.435"></a> <FONT color="green">436</FONT> double invIJ = 0;<a name="line.436"></a> <FONT color="green">437</FONT> for (int k = 0; k < m; ++k) {<a name="line.437"></a> <FONT color="green">438</FONT> final double[] vK = eigenvectors[k].getDataRef();<a name="line.438"></a> <FONT color="green">439</FONT> invIJ += vK[i] * vK[j] / realEigenvalues[k];<a name="line.439"></a> <FONT color="green">440</FONT> }<a name="line.440"></a> <FONT color="green">441</FONT> invI[j] = invIJ;<a name="line.441"></a> <FONT color="green">442</FONT> }<a name="line.442"></a> <FONT color="green">443</FONT> }<a name="line.443"></a> <FONT color="green">444</FONT> return MatrixUtils.createRealMatrix(invData);<a name="line.444"></a> <FONT color="green">445</FONT> <a name="line.445"></a> <FONT color="green">446</FONT> }<a name="line.446"></a> <FONT color="green">447</FONT> <a name="line.447"></a> <FONT color="green">448</FONT> }<a name="line.448"></a> <FONT color="green">449</FONT> <a name="line.449"></a> <FONT color="green">450</FONT> /**<a name="line.450"></a> <FONT color="green">451</FONT> * Transform matrix to tridiagonal.<a name="line.451"></a> <FONT color="green">452</FONT> * @param matrix<a name="line.452"></a> <FONT color="green">453</FONT> * matrix to transform<a name="line.453"></a> <FONT color="green">454</FONT> */<a name="line.454"></a> <FONT color="green">455</FONT> private void transformToTridiagonal(final RealMatrix matrix) {<a name="line.455"></a> <FONT color="green">456</FONT> <a name="line.456"></a> <FONT color="green">457</FONT> // transform the matrix to tridiagonal<a name="line.457"></a> <FONT color="green">458</FONT> transformer = new TriDiagonalTransformer(matrix);<a name="line.458"></a> <FONT color="green">459</FONT> main = transformer.getMainDiagonalRef();<a name="line.459"></a> <FONT color="green">460</FONT> secondary = transformer.getSecondaryDiagonalRef();<a name="line.460"></a> <FONT color="green">461</FONT> <a name="line.461"></a> <FONT color="green">462</FONT> }<a name="line.462"></a> <FONT color="green">463</FONT> <a name="line.463"></a> <FONT color="green">464</FONT> /**<a name="line.464"></a> <FONT color="green">465</FONT> * Find eigenvalues and eigenvectors (Dubrulle et al., 1971)<a name="line.465"></a> <FONT color="green">466</FONT> * @param householderMatrix Householder matrix of the transformation<a name="line.466"></a> <FONT color="green">467</FONT> * to tri-diagonal form.<a name="line.467"></a> <FONT color="green">468</FONT> */<a name="line.468"></a> <FONT color="green">469</FONT> private void findEigenVectors(double[][] householderMatrix) {<a name="line.469"></a> <FONT color="green">470</FONT> <a name="line.470"></a> <FONT color="green">471</FONT> double[][]z = householderMatrix.clone();<a name="line.471"></a> <FONT color="green">472</FONT> final int n = main.length;<a name="line.472"></a> <FONT color="green">473</FONT> realEigenvalues = new double[n];<a name="line.473"></a> <FONT color="green">474</FONT> imagEigenvalues = new double[n];<a name="line.474"></a> <FONT color="green">475</FONT> double[] e = new double[n];<a name="line.475"></a> <FONT color="green">476</FONT> for (int i = 0; i < n - 1; i++) {<a name="line.476"></a> <FONT color="green">477</FONT> realEigenvalues[i] = main[i];<a name="line.477"></a> <FONT color="green">478</FONT> e[i] = secondary[i];<a name="line.478"></a> <FONT color="green">479</FONT> }<a name="line.479"></a> <FONT color="green">480</FONT> realEigenvalues[n - 1] = main[n - 1];<a name="line.480"></a> <FONT color="green">481</FONT> e[n - 1] = 0.0;<a name="line.481"></a> <FONT color="green">482</FONT> <a name="line.482"></a> <FONT color="green">483</FONT> // Determine the largest main and secondary value in absolute term.<a name="line.483"></a> <FONT color="green">484</FONT> double maxAbsoluteValue=0.0;<a name="line.484"></a> <FONT color="green">485</FONT> for (int i = 0; i < n; i++) {<a name="line.485"></a> <FONT color="green">486</FONT> if (Math.abs(realEigenvalues[i])>maxAbsoluteValue) {<a name="line.486"></a> <FONT color="green">487</FONT> maxAbsoluteValue=Math.abs(realEigenvalues[i]);<a name="line.487"></a> <FONT color="green">488</FONT> }<a name="line.488"></a> <FONT color="green">489</FONT> if (Math.abs(e[i])>maxAbsoluteValue) {<a name="line.489"></a> <FONT color="green">490</FONT> maxAbsoluteValue=Math.abs(e[i]);<a name="line.490"></a> <FONT color="green">491</FONT> }<a name="line.491"></a> <FONT color="green">492</FONT> }<a name="line.492"></a> <FONT color="green">493</FONT> // Make null any main and secondary value too small to be significant<a name="line.493"></a> <FONT color="green">494</FONT> if (maxAbsoluteValue!=0.0) {<a name="line.494"></a> <FONT color="green">495</FONT> for (int i=0; i < n; i++) {<a name="line.495"></a> <FONT color="green">496</FONT> if (Math.abs(realEigenvalues[i])<=MathUtils.EPSILON*maxAbsoluteValue) {<a name="line.496"></a> <FONT color="green">497</FONT> realEigenvalues[i]=0.0;<a name="line.497"></a> <FONT color="green">498</FONT> }<a name="line.498"></a> <FONT color="green">499</FONT> if (Math.abs(e[i])<=MathUtils.EPSILON*maxAbsoluteValue) {<a name="line.499"></a> <FONT color="green">500</FONT> e[i]=0.0;<a name="line.500"></a> <FONT color="green">501</FONT> }<a name="line.501"></a> <FONT color="green">502</FONT> }<a name="line.502"></a> <FONT color="green">503</FONT> }<a name="line.503"></a> <FONT color="green">504</FONT> <a name="line.504"></a> <FONT color="green">505</FONT> for (int j = 0; j < n; j++) {<a name="line.505"></a> <FONT color="green">506</FONT> int its = 0;<a name="line.506"></a> <FONT color="green">507</FONT> int m;<a name="line.507"></a> <FONT color="green">508</FONT> do {<a name="line.508"></a> <FONT color="green">509</FONT> for (m = j; m < n - 1; m++) {<a name="line.509"></a> <FONT color="green">510</FONT> double delta = Math.abs(realEigenvalues[m]) + Math.abs(realEigenvalues[m + 1]);<a name="line.510"></a> <FONT color="green">511</FONT> if (Math.abs(e[m]) + delta == delta) {<a name="line.511"></a> <FONT color="green">512</FONT> break;<a name="line.512"></a> <FONT color="green">513</FONT> }<a name="line.513"></a> <FONT color="green">514</FONT> }<a name="line.514"></a> <FONT color="green">515</FONT> if (m != j) {<a name="line.515"></a> <FONT color="green">516</FONT> if (its == maxIter)<a name="line.516"></a> <FONT color="green">517</FONT> throw new InvalidMatrixException(<a name="line.517"></a> <FONT color="green">518</FONT> new MaxIterationsExceededException(maxIter));<a name="line.518"></a> <FONT color="green">519</FONT> its++;<a name="line.519"></a> <FONT color="green">520</FONT> double q = (realEigenvalues[j + 1] - realEigenvalues[j]) / (2 * e[j]);<a name="line.520"></a> <FONT color="green">521</FONT> double t = Math.sqrt(1 + q * q);<a name="line.521"></a> <FONT color="green">522</FONT> if (q < 0.0) {<a name="line.522"></a> <FONT color="green">523</FONT> q = realEigenvalues[m] - realEigenvalues[j] + e[j] / (q - t);<a name="line.523"></a> <FONT color="green">524</FONT> } else {<a name="line.524"></a> <FONT color="green">525</FONT> q = realEigenvalues[m] - realEigenvalues[j] + e[j] / (q + t);<a name="line.525"></a> <FONT color="green">526</FONT> }<a name="line.526"></a> <FONT color="green">527</FONT> double u = 0.0;<a name="line.527"></a> <FONT color="green">528</FONT> double s = 1.0;<a name="line.528"></a> <FONT color="green">529</FONT> double c = 1.0;<a name="line.529"></a> <FONT color="green">530</FONT> int i;<a name="line.530"></a> <FONT color="green">531</FONT> for (i = m - 1; i >= j; i--) {<a name="line.531"></a> <FONT color="green">532</FONT> double p = s * e[i];<a name="line.532"></a> <FONT color="green">533</FONT> double h = c * e[i];<a name="line.533"></a> <FONT color="green">534</FONT> if (Math.abs(p) >= Math.abs(q)) {<a name="line.534"></a> <FONT color="green">535</FONT> c = q / p;<a name="line.535"></a> <FONT color="green">536</FONT> t = Math.sqrt(c * c + 1.0);<a name="line.536"></a> <FONT color="green">537</FONT> e[i + 1] = p * t;<a name="line.537"></a> <FONT color="green">538</FONT> s = 1.0 / t;<a name="line.538"></a> <FONT color="green">539</FONT> c = c * s;<a name="line.539"></a> <FONT color="green">540</FONT> } else {<a name="line.540"></a> <FONT color="green">541</FONT> s = p / q;<a name="line.541"></a> <FONT color="green">542</FONT> t = Math.sqrt(s * s + 1.0);<a name="line.542"></a> <FONT color="green">543</FONT> e[i + 1] = q * t;<a name="line.543"></a> <FONT color="green">544</FONT> c = 1.0 / t;<a name="line.544"></a> <FONT color="green">545</FONT> s = s * c;<a name="line.545"></a> <FONT color="green">546</FONT> }<a name="line.546"></a> <FONT color="green">547</FONT> if (e[i + 1] == 0.0) {<a name="line.547"></a> <FONT color="green">548</FONT> realEigenvalues[i + 1] -= u;<a name="line.548"></a> <FONT color="green">549</FONT> e[m] = 0.0;<a name="line.549"></a> <FONT color="green">550</FONT> break;<a name="line.550"></a> <FONT color="green">551</FONT> }<a name="line.551"></a> <FONT color="green">552</FONT> q = realEigenvalues[i + 1] - u;<a name="line.552"></a> <FONT color="green">553</FONT> t = (realEigenvalues[i] - q) * s + 2.0 * c * h;<a name="line.553"></a> <FONT color="green">554</FONT> u = s * t;<a name="line.554"></a> <FONT color="green">555</FONT> realEigenvalues[i + 1] = q + u;<a name="line.555"></a> <FONT color="green">556</FONT> q = c * t - h;<a name="line.556"></a> <FONT color="green">557</FONT> for (int ia = 0; ia < n; ia++) {<a name="line.557"></a> <FONT color="green">558</FONT> p = z[ia][i + 1];<a name="line.558"></a> <FONT color="green">559</FONT> z[ia][i + 1] = s * z[ia][i] + c * p;<a name="line.559"></a> <FONT color="green">560</FONT> z[ia][i] = c * z[ia][i] - s * p;<a name="line.560"></a> <FONT color="green">561</FONT> }<a name="line.561"></a> <FONT color="green">562</FONT> }<a name="line.562"></a> <FONT color="green">563</FONT> if (e[i + 1] == 0.0 && i >= j)<a name="line.563"></a> <FONT color="green">564</FONT> continue;<a name="line.564"></a> <FONT color="green">565</FONT> realEigenvalues[j] -= u;<a name="line.565"></a> <FONT color="green">566</FONT> e[j] = q;<a name="line.566"></a> <FONT color="green">567</FONT> e[m] = 0.0;<a name="line.567"></a> <FONT color="green">568</FONT> }<a name="line.568"></a> <FONT color="green">569</FONT> } while (m != j);<a name="line.569"></a> <FONT color="green">570</FONT> }<a name="line.570"></a> <FONT color="green">571</FONT> <a name="line.571"></a> <FONT color="green">572</FONT> //Sort the eigen values (and vectors) in increase order<a name="line.572"></a> <FONT color="green">573</FONT> for (int i = 0; i < n; i++) {<a name="line.573"></a> <FONT color="green">574</FONT> int k = i;<a name="line.574"></a> <FONT color="green">575</FONT> double p = realEigenvalues[i];<a name="line.575"></a> <FONT color="green">576</FONT> for (int j = i + 1; j < n; j++) {<a name="line.576"></a> <FONT color="green">577</FONT> if (realEigenvalues[j] > p) {<a name="line.577"></a> <FONT color="green">578</FONT> k = j;<a name="line.578"></a> <FONT color="green">579</FONT> p = realEigenvalues[j];<a name="line.579"></a> <FONT color="green">580</FONT> }<a name="line.580"></a> <FONT color="green">581</FONT> }<a name="line.581"></a> <FONT color="green">582</FONT> if (k != i) {<a name="line.582"></a> <FONT color="green">583</FONT> realEigenvalues[k] = realEigenvalues[i];<a name="line.583"></a> <FONT color="green">584</FONT> realEigenvalues[i] = p;<a name="line.584"></a> <FONT color="green">585</FONT> for (int j = 0; j < n; j++) {<a name="line.585"></a> <FONT color="green">586</FONT> p = z[j][i];<a name="line.586"></a> <FONT color="green">587</FONT> z[j][i] = z[j][k];<a name="line.587"></a> <FONT color="green">588</FONT> z[j][k] = p;<a name="line.588"></a> <FONT color="green">589</FONT> }<a name="line.589"></a> <FONT color="green">590</FONT> }<a name="line.590"></a> <FONT color="green">591</FONT> }<a name="line.591"></a> <FONT color="green">592</FONT> <a name="line.592"></a> <FONT color="green">593</FONT> // Determine the largest eigen value in absolute term.<a name="line.593"></a> <FONT color="green">594</FONT> maxAbsoluteValue=0.0;<a name="line.594"></a> <FONT color="green">595</FONT> for (int i = 0; i < n; i++) {<a name="line.595"></a> <FONT color="green">596</FONT> if (Math.abs(realEigenvalues[i])>maxAbsoluteValue) {<a name="line.596"></a> <FONT color="green">597</FONT> maxAbsoluteValue=Math.abs(realEigenvalues[i]);<a name="line.597"></a> <FONT color="green">598</FONT> }<a name="line.598"></a> <FONT color="green">599</FONT> }<a name="line.599"></a> <FONT color="green">600</FONT> // Make null any eigen value too small to be significant<a name="line.600"></a> <FONT color="green">601</FONT> if (maxAbsoluteValue!=0.0) {<a name="line.601"></a> <FONT color="green">602</FONT> for (int i=0; i < n; i++) {<a name="line.602"></a> <FONT color="green">603</FONT> if (Math.abs(realEigenvalues[i])<MathUtils.EPSILON*maxAbsoluteValue) {<a name="line.603"></a> <FONT color="green">604</FONT> realEigenvalues[i]=0.0;<a name="line.604"></a> <FONT color="green">605</FONT> }<a name="line.605"></a> <FONT color="green">606</FONT> }<a name="line.606"></a> <FONT color="green">607</FONT> }<a name="line.607"></a> <FONT color="green">608</FONT> eigenvectors = new ArrayRealVector[n];<a name="line.608"></a> <FONT color="green">609</FONT> double[] tmp = new double[n];<a name="line.609"></a> <FONT color="green">610</FONT> for (int i = 0; i < n; i++) {<a name="line.610"></a> <FONT color="green">611</FONT> for (int j = 0; j < n; j++) {<a name="line.611"></a> <FONT color="green">612</FONT> tmp[j] = z[j][i];<a name="line.612"></a> <FONT color="green">613</FONT> }<a name="line.613"></a> <FONT color="green">614</FONT> eigenvectors[i] = new ArrayRealVector(tmp);<a name="line.614"></a> <FONT color="green">615</FONT> }<a name="line.615"></a> <FONT color="green">616</FONT> }<a name="line.616"></a> <FONT color="green">617</FONT> }<a name="line.617"></a> </PRE> </BODY> </HTML>