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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.ode.nonstiff;<a name="line.18"></a> <FONT color="green">019</FONT> <a name="line.19"></a> <FONT color="green">020</FONT> import java.util.Arrays;<a name="line.20"></a> <FONT color="green">021</FONT> import java.util.HashMap;<a name="line.21"></a> <FONT color="green">022</FONT> import java.util.Map;<a name="line.22"></a> <FONT color="green">023</FONT> <a name="line.23"></a> <FONT color="green">024</FONT> import org.apache.commons.math.fraction.BigFraction;<a name="line.24"></a> <FONT color="green">025</FONT> import org.apache.commons.math.linear.Array2DRowFieldMatrix;<a name="line.25"></a> <FONT color="green">026</FONT> import org.apache.commons.math.linear.Array2DRowRealMatrix;<a name="line.26"></a> <FONT color="green">027</FONT> import org.apache.commons.math.linear.DefaultFieldMatrixChangingVisitor;<a name="line.27"></a> <FONT color="green">028</FONT> import org.apache.commons.math.linear.FieldDecompositionSolver;<a name="line.28"></a> <FONT color="green">029</FONT> import org.apache.commons.math.linear.FieldLUDecompositionImpl;<a name="line.29"></a> <FONT color="green">030</FONT> import org.apache.commons.math.linear.FieldMatrix;<a name="line.30"></a> <FONT color="green">031</FONT> import org.apache.commons.math.linear.MatrixUtils;<a name="line.31"></a> <FONT color="green">032</FONT> <a name="line.32"></a> <FONT color="green">033</FONT> /** Transformer to Nordsieck vectors for Adams integrators.<a name="line.33"></a> <FONT color="green">034</FONT> * <p>This class i used by {@link AdamsBashforthIntegrator Adams-Bashforth} and<a name="line.34"></a> <FONT color="green">035</FONT> * {@link AdamsMoultonIntegrator Adams-Moulton} integrators to convert between<a name="line.35"></a> <FONT color="green">036</FONT> * classical representation with several previous first derivatives and Nordsieck<a name="line.36"></a> <FONT color="green">037</FONT> * representation with higher order scaled derivatives.</p><a name="line.37"></a> <FONT color="green">038</FONT> *<a name="line.38"></a> <FONT color="green">039</FONT> * <p>We define scaled derivatives s<sub>i</sub>(n) at step n as:<a name="line.39"></a> <FONT color="green">040</FONT> * <pre><a name="line.40"></a> <FONT color="green">041</FONT> * s<sub>1</sub>(n) = h y'<sub>n</sub> for first derivative<a name="line.41"></a> <FONT color="green">042</FONT> * s<sub>2</sub>(n) = h<sup>2</sup>/2 y''<sub>n</sub> for second derivative<a name="line.42"></a> <FONT color="green">043</FONT> * s<sub>3</sub>(n) = h<sup>3</sup>/6 y'''<sub>n</sub> for third derivative<a name="line.43"></a> <FONT color="green">044</FONT> * ...<a name="line.44"></a> <FONT color="green">045</FONT> * s<sub>k</sub>(n) = h<sup>k</sup>/k! y(k)<sub>n</sub> for k<sup>th</sup> derivative<a name="line.45"></a> <FONT color="green">046</FONT> * </pre></p><a name="line.46"></a> <FONT color="green">047</FONT> *<a name="line.47"></a> <FONT color="green">048</FONT> * <p>With the previous definition, the classical representation of multistep methods<a name="line.48"></a> <FONT color="green">049</FONT> * uses first derivatives only, i.e. it handles y<sub>n</sub>, s<sub>1</sub>(n) and<a name="line.49"></a> <FONT color="green">050</FONT> * q<sub>n</sub> where q<sub>n</sub> is defined as:<a name="line.50"></a> <FONT color="green">051</FONT> * <pre><a name="line.51"></a> <FONT color="green">052</FONT> * q<sub>n</sub> = [ s<sub>1</sub>(n-1) s<sub>1</sub>(n-2) ... s<sub>1</sub>(n-(k-1)) ]<sup>T</sup><a name="line.52"></a> <FONT color="green">053</FONT> * </pre><a name="line.53"></a> <FONT color="green">054</FONT> * (we omit the k index in the notation for clarity).</p><a name="line.54"></a> <FONT color="green">055</FONT> *<a name="line.55"></a> <FONT color="green">056</FONT> * <p>Another possible representation uses the Nordsieck vector with<a name="line.56"></a> <FONT color="green">057</FONT> * higher degrees scaled derivatives all taken at the same step, i.e it handles y<sub>n</sub>,<a name="line.57"></a> <FONT color="green">058</FONT> * s<sub>1</sub>(n) and r<sub>n</sub>) where r<sub>n</sub> is defined as:<a name="line.58"></a> <FONT color="green">059</FONT> * <pre><a name="line.59"></a> <FONT color="green">060</FONT> * r<sub>n</sub> = [ s<sub>2</sub>(n), s<sub>3</sub>(n) ... s<sub>k</sub>(n) ]<sup>T</sup><a name="line.60"></a> <FONT color="green">061</FONT> * </pre><a name="line.61"></a> <FONT color="green">062</FONT> * (here again we omit the k index in the notation for clarity)<a name="line.62"></a> <FONT color="green">063</FONT> * </p><a name="line.63"></a> <FONT color="green">064</FONT> *<a name="line.64"></a> <FONT color="green">065</FONT> * <p>Taylor series formulas show that for any index offset i, s<sub>1</sub>(n-i) can be<a name="line.65"></a> <FONT color="green">066</FONT> * computed from s<sub>1</sub>(n), s<sub>2</sub>(n) ... s<sub>k</sub>(n), the formula being exact<a name="line.66"></a> <FONT color="green">067</FONT> * for degree k polynomials.<a name="line.67"></a> <FONT color="green">068</FONT> * <pre><a name="line.68"></a> <FONT color="green">069</FONT> * s<sub>1</sub>(n-i) = s<sub>1</sub>(n) + &sum;<sub>j</sub> j (-i)<sup>j-1</sup> s<sub>j</sub>(n)<a name="line.69"></a> <FONT color="green">070</FONT> * </pre><a name="line.70"></a> <FONT color="green">071</FONT> * The previous formula can be used with several values for i to compute the transform between<a name="line.71"></a> <FONT color="green">072</FONT> * classical representation and Nordsieck vector at step end. The transform between r<sub>n</sub><a name="line.72"></a> <FONT color="green">073</FONT> * and q<sub>n</sub> resulting from the Taylor series formulas above is:<a name="line.73"></a> <FONT color="green">074</FONT> * <pre><a name="line.74"></a> <FONT color="green">075</FONT> * q<sub>n</sub> = s<sub>1</sub>(n) u + P r<sub>n</sub><a name="line.75"></a> <FONT color="green">076</FONT> * </pre><a name="line.76"></a> <FONT color="green">077</FONT> * where u is the [ 1 1 ... 1 ]<sup>T</sup> vector and P is the (k-1)&times;(k-1) matrix built<a name="line.77"></a> <FONT color="green">078</FONT> * with the j (-i)<sup>j-1</sup> terms:<a name="line.78"></a> <FONT color="green">079</FONT> * <pre><a name="line.79"></a> <FONT color="green">080</FONT> * [ -2 3 -4 5 ... ]<a name="line.80"></a> <FONT color="green">081</FONT> * [ -4 12 -32 80 ... ]<a name="line.81"></a> <FONT color="green">082</FONT> * P = [ -6 27 -108 405 ... ]<a name="line.82"></a> <FONT color="green">083</FONT> * [ -8 48 -256 1280 ... ]<a name="line.83"></a> <FONT color="green">084</FONT> * [ ... ]<a name="line.84"></a> <FONT color="green">085</FONT> * </pre></p><a name="line.85"></a> <FONT color="green">086</FONT> *<a name="line.86"></a> <FONT color="green">087</FONT> * <p>Changing -i into +i in the formula above can be used to compute a similar transform between<a name="line.87"></a> <FONT color="green">088</FONT> * classical representation and Nordsieck vector at step start. The resulting matrix is simply<a name="line.88"></a> <FONT color="green">089</FONT> * the absolute value of matrix P.</p><a name="line.89"></a> <FONT color="green">090</FONT> *<a name="line.90"></a> <FONT color="green">091</FONT> * <p>For {@link AdamsBashforthIntegrator Adams-Bashforth} method, the Nordsieck vector<a name="line.91"></a> <FONT color="green">092</FONT> * at step n+1 is computed from the Nordsieck vector at step n as follows:<a name="line.92"></a> <FONT color="green">093</FONT> * <ul><a name="line.93"></a> <FONT color="green">094</FONT> * <li>y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n) + u<sup>T</sup> r<sub>n</sub></li><a name="line.94"></a> <FONT color="green">095</FONT> * <li>s<sub>1</sub>(n+1) = h f(t<sub>n+1</sub>, y<sub>n+1</sub>)</li><a name="line.95"></a> <FONT color="green">096</FONT> * <li>r<sub>n+1</sub> = (s<sub>1</sub>(n) - s<sub>1</sub>(n+1)) P<sup>-1</sup> u + P<sup>-1</sup> A P r<sub>n</sub></li><a name="line.96"></a> <FONT color="green">097</FONT> * </ul><a name="line.97"></a> <FONT color="green">098</FONT> * where A is a rows shifting matrix (the lower left part is an identity matrix):<a name="line.98"></a> <FONT color="green">099</FONT> * <pre><a name="line.99"></a> <FONT color="green">100</FONT> * [ 0 0 ... 0 0 | 0 ]<a name="line.100"></a> <FONT color="green">101</FONT> * [ ---------------+---]<a name="line.101"></a> <FONT color="green">102</FONT> * [ 1 0 ... 0 0 | 0 ]<a name="line.102"></a> <FONT color="green">103</FONT> * A = [ 0 1 ... 0 0 | 0 ]<a name="line.103"></a> <FONT color="green">104</FONT> * [ ... | 0 ]<a name="line.104"></a> <FONT color="green">105</FONT> * [ 0 0 ... 1 0 | 0 ]<a name="line.105"></a> <FONT color="green">106</FONT> * [ 0 0 ... 0 1 | 0 ]<a name="line.106"></a> <FONT color="green">107</FONT> * </pre></p><a name="line.107"></a> <FONT color="green">108</FONT> *<a name="line.108"></a> <FONT color="green">109</FONT> * <p>For {@link AdamsMoultonIntegrator Adams-Moulton} method, the predicted Nordsieck vector<a name="line.109"></a> <FONT color="green">110</FONT> * at step n+1 is computed from the Nordsieck vector at step n as follows:<a name="line.110"></a> <FONT color="green">111</FONT> * <ul><a name="line.111"></a> <FONT color="green">112</FONT> * <li>Y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n) + u<sup>T</sup> r<sub>n</sub></li><a name="line.112"></a> <FONT color="green">113</FONT> * <li>S<sub>1</sub>(n+1) = h f(t<sub>n+1</sub>, Y<sub>n+1</sub>)</li><a name="line.113"></a> <FONT color="green">114</FONT> * <li>R<sub>n+1</sub> = (s<sub>1</sub>(n) - s<sub>1</sub>(n+1)) P<sup>-1</sup> u + P<sup>-1</sup> A P r<sub>n</sub></li><a name="line.114"></a> <FONT color="green">115</FONT> * </ul><a name="line.115"></a> <FONT color="green">116</FONT> * From this predicted vector, the corrected vector is computed as follows:<a name="line.116"></a> <FONT color="green">117</FONT> * <ul><a name="line.117"></a> <FONT color="green">118</FONT> * <li>y<sub>n+1</sub> = y<sub>n</sub> + S<sub>1</sub>(n+1) + [ -1 +1 -1 +1 ... &plusmn;1 ] r<sub>n+1</sub></li><a name="line.118"></a> <FONT color="green">119</FONT> * <li>s<sub>1</sub>(n+1) = h f(t<sub>n+1</sub>, y<sub>n+1</sub>)</li><a name="line.119"></a> <FONT color="green">120</FONT> * <li>r<sub>n+1</sub> = R<sub>n+1</sub> + (s<sub>1</sub>(n+1) - S<sub>1</sub>(n+1)) P<sup>-1</sup> u</li><a name="line.120"></a> <FONT color="green">121</FONT> * </ul><a name="line.121"></a> <FONT color="green">122</FONT> * where the upper case Y<sub>n+1</sub>, S<sub>1</sub>(n+1) and R<sub>n+1</sub> represent the<a name="line.122"></a> <FONT color="green">123</FONT> * predicted states whereas the lower case y<sub>n+1</sub>, s<sub>n+1</sub> and r<sub>n+1</sub><a name="line.123"></a> <FONT color="green">124</FONT> * represent the corrected states.</p><a name="line.124"></a> <FONT color="green">125</FONT> *<a name="line.125"></a> <FONT color="green">126</FONT> * <p>We observe that both methods use similar update formulas. In both cases a P<sup>-1</sup>u<a name="line.126"></a> <FONT color="green">127</FONT> * vector and a P<sup>-1</sup> A P matrix are used that do not depend on the state,<a name="line.127"></a> <FONT color="green">128</FONT> * they only depend on k. This class handles these transformations.</p><a name="line.128"></a> <FONT color="green">129</FONT> *<a name="line.129"></a> <FONT color="green">130</FONT> * @version $Revision: 810196 $ $Date: 2009-09-01 15:47:46 -0400 (Tue, 01 Sep 2009) $<a name="line.130"></a> <FONT color="green">131</FONT> * @since 2.0<a name="line.131"></a> <FONT color="green">132</FONT> */<a name="line.132"></a> <FONT color="green">133</FONT> public class AdamsNordsieckTransformer {<a name="line.133"></a> <FONT color="green">134</FONT> <a name="line.134"></a> <FONT color="green">135</FONT> /** Cache for already computed coefficients. */<a name="line.135"></a> <FONT color="green">136</FONT> private static final Map<Integer, AdamsNordsieckTransformer> CACHE =<a name="line.136"></a> <FONT color="green">137</FONT> new HashMap<Integer, AdamsNordsieckTransformer>();<a name="line.137"></a> <FONT color="green">138</FONT> <a name="line.138"></a> <FONT color="green">139</FONT> /** Initialization matrix for the higher order derivatives wrt y'', y''' ... */<a name="line.139"></a> <FONT color="green">140</FONT> private final Array2DRowRealMatrix initialization;<a name="line.140"></a> <FONT color="green">141</FONT> <a name="line.141"></a> <FONT color="green">142</FONT> /** Update matrix for the higher order derivatives h<sup>2</sup>/2y'', h<sup>3</sup>/6 y''' ... */<a name="line.142"></a> <FONT color="green">143</FONT> private final Array2DRowRealMatrix update;<a name="line.143"></a> <FONT color="green">144</FONT> <a name="line.144"></a> <FONT color="green">145</FONT> /** Update coefficients of the higher order derivatives wrt y'. */<a name="line.145"></a> <FONT color="green">146</FONT> private final double[] c1;<a name="line.146"></a> <FONT color="green">147</FONT> <a name="line.147"></a> <FONT color="green">148</FONT> /** Simple constructor.<a name="line.148"></a> <FONT color="green">149</FONT> * @param nSteps number of steps of the multistep method<a name="line.149"></a> <FONT color="green">150</FONT> * (excluding the one being computed)<a name="line.150"></a> <FONT color="green">151</FONT> */<a name="line.151"></a> <FONT color="green">152</FONT> private AdamsNordsieckTransformer(final int nSteps) {<a name="line.152"></a> <FONT color="green">153</FONT> <a name="line.153"></a> <FONT color="green">154</FONT> // compute exact coefficients<a name="line.154"></a> <FONT color="green">155</FONT> FieldMatrix<BigFraction> bigP = buildP(nSteps);<a name="line.155"></a> <FONT color="green">156</FONT> FieldDecompositionSolver<BigFraction> pSolver =<a name="line.156"></a> <FONT color="green">157</FONT> new FieldLUDecompositionImpl<BigFraction>(bigP).getSolver();<a name="line.157"></a> <FONT color="green">158</FONT> <a name="line.158"></a> <FONT color="green">159</FONT> BigFraction[] u = new BigFraction[nSteps];<a name="line.159"></a> <FONT color="green">160</FONT> Arrays.fill(u, BigFraction.ONE);<a name="line.160"></a> <FONT color="green">161</FONT> BigFraction[] bigC1 = pSolver.solve(u);<a name="line.161"></a> <FONT color="green">162</FONT> <a name="line.162"></a> <FONT color="green">163</FONT> // update coefficients are computed by combining transform from<a name="line.163"></a> <FONT color="green">164</FONT> // Nordsieck to multistep, then shifting rows to represent step advance<a name="line.164"></a> <FONT color="green">165</FONT> // then applying inverse transform<a name="line.165"></a> <FONT color="green">166</FONT> BigFraction[][] shiftedP = bigP.getData();<a name="line.166"></a> <FONT color="green">167</FONT> for (int i = shiftedP.length - 1; i > 0; --i) {<a name="line.167"></a> <FONT color="green">168</FONT> // shift rows<a name="line.168"></a> <FONT color="green">169</FONT> shiftedP[i] = shiftedP[i - 1];<a name="line.169"></a> <FONT color="green">170</FONT> }<a name="line.170"></a> <FONT color="green">171</FONT> shiftedP[0] = new BigFraction[nSteps];<a name="line.171"></a> <FONT color="green">172</FONT> Arrays.fill(shiftedP[0], BigFraction.ZERO);<a name="line.172"></a> <FONT color="green">173</FONT> FieldMatrix<BigFraction> bigMSupdate =<a name="line.173"></a> <FONT color="green">174</FONT> pSolver.solve(new Array2DRowFieldMatrix<BigFraction>(shiftedP, false));<a name="line.174"></a> <FONT color="green">175</FONT> <a name="line.175"></a> <FONT color="green">176</FONT> // initialization coefficients, computed from a R matrix = abs(P)<a name="line.176"></a> <FONT color="green">177</FONT> bigP.walkInOptimizedOrder(new DefaultFieldMatrixChangingVisitor<BigFraction>(BigFraction.ZERO) {<a name="line.177"></a> <FONT color="green">178</FONT> /** {@inheritDoc} */<a name="line.178"></a> <FONT color="green">179</FONT> @Override<a name="line.179"></a> <FONT color="green">180</FONT> public BigFraction visit(int row, int column, BigFraction value) {<a name="line.180"></a> <FONT color="green">181</FONT> return ((column & 0x1) == 0x1) ? value : value.negate();<a name="line.181"></a> <FONT color="green">182</FONT> }<a name="line.182"></a> <FONT color="green">183</FONT> });<a name="line.183"></a> <FONT color="green">184</FONT> FieldMatrix<BigFraction> bigRInverse =<a name="line.184"></a> <FONT color="green">185</FONT> new FieldLUDecompositionImpl<BigFraction>(bigP).getSolver().getInverse();<a name="line.185"></a> <FONT color="green">186</FONT> <a name="line.186"></a> <FONT color="green">187</FONT> // convert coefficients to double<a name="line.187"></a> <FONT color="green">188</FONT> initialization = MatrixUtils.bigFractionMatrixToRealMatrix(bigRInverse);<a name="line.188"></a> <FONT color="green">189</FONT> update = MatrixUtils.bigFractionMatrixToRealMatrix(bigMSupdate);<a name="line.189"></a> <FONT color="green">190</FONT> c1 = new double[nSteps];<a name="line.190"></a> <FONT color="green">191</FONT> for (int i = 0; i < nSteps; ++i) {<a name="line.191"></a> <FONT color="green">192</FONT> c1[i] = bigC1[i].doubleValue();<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> }<a name="line.195"></a> <FONT color="green">196</FONT> <a name="line.196"></a> <FONT color="green">197</FONT> /** Get the Nordsieck transformer for a given number of steps.<a name="line.197"></a> <FONT color="green">198</FONT> * @param nSteps number of steps of the multistep method<a name="line.198"></a> <FONT color="green">199</FONT> * (excluding the one being computed)<a name="line.199"></a> <FONT color="green">200</FONT> * @return Nordsieck transformer for the specified number of steps<a name="line.200"></a> <FONT color="green">201</FONT> */<a name="line.201"></a> <FONT color="green">202</FONT> public static AdamsNordsieckTransformer getInstance(final int nSteps) {<a name="line.202"></a> <FONT color="green">203</FONT> synchronized(CACHE) {<a name="line.203"></a> <FONT color="green">204</FONT> AdamsNordsieckTransformer t = CACHE.get(nSteps);<a name="line.204"></a> <FONT color="green">205</FONT> if (t == null) {<a name="line.205"></a> <FONT color="green">206</FONT> t = new AdamsNordsieckTransformer(nSteps);<a name="line.206"></a> <FONT color="green">207</FONT> CACHE.put(nSteps, t);<a name="line.207"></a> <FONT color="green">208</FONT> }<a name="line.208"></a> <FONT color="green">209</FONT> return t;<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> <a name="line.212"></a> <FONT color="green">213</FONT> /** Get the number of steps of the method<a name="line.213"></a> <FONT color="green">214</FONT> * (excluding the one being computed).<a name="line.214"></a> <FONT color="green">215</FONT> * @return number of steps of the method<a name="line.215"></a> <FONT color="green">216</FONT> * (excluding the one being computed)<a name="line.216"></a> <FONT color="green">217</FONT> */<a name="line.217"></a> <FONT color="green">218</FONT> public int getNSteps() {<a name="line.218"></a> <FONT color="green">219</FONT> return c1.length;<a name="line.219"></a> <FONT color="green">220</FONT> }<a name="line.220"></a> <FONT color="green">221</FONT> <a name="line.221"></a> <FONT color="green">222</FONT> /** Build the P matrix.<a name="line.222"></a> <FONT color="green">223</FONT> * <p>The P matrix general terms are shifted j (-i)<sup>j-1</sup> terms:<a name="line.223"></a> <FONT color="green">224</FONT> * <pre><a name="line.224"></a> <FONT color="green">225</FONT> * [ -2 3 -4 5 ... ]<a name="line.225"></a> <FONT color="green">226</FONT> * [ -4 12 -32 80 ... ]<a name="line.226"></a> <FONT color="green">227</FONT> * P = [ -6 27 -108 405 ... ]<a name="line.227"></a> <FONT color="green">228</FONT> * [ -8 48 -256 1280 ... ]<a name="line.228"></a> <FONT color="green">229</FONT> * [ ... ]<a name="line.229"></a> <FONT color="green">230</FONT> * </pre></p><a name="line.230"></a> <FONT color="green">231</FONT> * @param nSteps number of steps of the multistep method<a name="line.231"></a> <FONT color="green">232</FONT> * (excluding the one being computed)<a name="line.232"></a> <FONT color="green">233</FONT> * @return P matrix<a name="line.233"></a> <FONT color="green">234</FONT> */<a name="line.234"></a> <FONT color="green">235</FONT> private FieldMatrix<BigFraction> buildP(final int nSteps) {<a name="line.235"></a> <FONT color="green">236</FONT> <a name="line.236"></a> <FONT color="green">237</FONT> final BigFraction[][] pData = new BigFraction[nSteps][nSteps];<a name="line.237"></a> <FONT color="green">238</FONT> <a name="line.238"></a> <FONT color="green">239</FONT> for (int i = 0; i < pData.length; ++i) {<a name="line.239"></a> <FONT color="green">240</FONT> // build the P matrix elements from Taylor series formulas<a name="line.240"></a> <FONT color="green">241</FONT> final BigFraction[] pI = pData[i];<a name="line.241"></a> <FONT color="green">242</FONT> final int factor = -(i + 1);<a name="line.242"></a> <FONT color="green">243</FONT> int aj = factor;<a name="line.243"></a> <FONT color="green">244</FONT> for (int j = 0; j < pI.length; ++j) {<a name="line.244"></a> <FONT color="green">245</FONT> pI[j] = new BigFraction(aj * (j + 2));<a name="line.245"></a> <FONT color="green">246</FONT> aj *= factor;<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> <a name="line.249"></a> <FONT color="green">250</FONT> return new Array2DRowFieldMatrix<BigFraction>(pData, false);<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> <a name="line.253"></a> <FONT color="green">254</FONT> /** Initialize the high order scaled derivatives at step start.<a name="line.254"></a> <FONT color="green">255</FONT> * @param first first scaled derivative at step start<a name="line.255"></a> <FONT color="green">256</FONT> * @param multistep scaled derivatives after step start (hy'1, ..., hy'k-1)<a name="line.256"></a> <FONT color="green">257</FONT> * will be modified<a name="line.257"></a> <FONT color="green">258</FONT> * @return high order derivatives at step start<a name="line.258"></a> <FONT color="green">259</FONT> */<a name="line.259"></a> <FONT color="green">260</FONT> public Array2DRowRealMatrix initializeHighOrderDerivatives(final double[] first,<a name="line.260"></a> <FONT color="green">261</FONT> final double[][] multistep) {<a name="line.261"></a> <FONT color="green">262</FONT> for (int i = 0; i < multistep.length; ++i) {<a name="line.262"></a> <FONT color="green">263</FONT> final double[] msI = multistep[i];<a name="line.263"></a> <FONT color="green">264</FONT> for (int j = 0; j < first.length; ++j) {<a name="line.264"></a> <FONT color="green">265</FONT> msI[j] -= first[j];<a name="line.265"></a> <FONT color="green">266</FONT> }<a name="line.266"></a> <FONT color="green">267</FONT> }<a name="line.267"></a> <FONT color="green">268</FONT> return initialization.multiply(new Array2DRowRealMatrix(multistep, false));<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> /** Update the high order scaled derivatives for Adams integrators (phase 1).<a name="line.271"></a> <FONT color="green">272</FONT> * <p>The complete update of high order derivatives has a form similar to:<a name="line.272"></a> <FONT color="green">273</FONT> * <pre><a name="line.273"></a> <FONT color="green">274</FONT> * r<sub>n+1</sub> = (s<sub>1</sub>(n) - s<sub>1</sub>(n+1)) P<sup>-1</sup> u + P<sup>-1</sup> A P r<sub>n</sub><a name="line.274"></a> <FONT color="green">275</FONT> * </pre><a name="line.275"></a> <FONT color="green">276</FONT> * this method computes the P<sup>-1</sup> A P r<sub>n</sub> part.</p><a name="line.276"></a> <FONT color="green">277</FONT> * @param highOrder high order scaled derivatives<a name="line.277"></a> <FONT color="green">278</FONT> * (h<sup>2</sup>/2 y'', ... h<sup>k</sup>/k! y(k))<a name="line.278"></a> <FONT color="green">279</FONT> * @return updated high order derivatives<a name="line.279"></a> <FONT color="green">280</FONT> * @see #updateHighOrderDerivativesPhase2(double[], double[], Array2DRowRealMatrix)<a name="line.280"></a> <FONT color="green">281</FONT> */<a name="line.281"></a> <FONT color="green">282</FONT> public Array2DRowRealMatrix updateHighOrderDerivativesPhase1(final Array2DRowRealMatrix highOrder) {<a name="line.282"></a> <FONT color="green">283</FONT> return update.multiply(highOrder);<a name="line.283"></a> <FONT color="green">284</FONT> }<a name="line.284"></a> <FONT color="green">285</FONT> <a name="line.285"></a> <FONT color="green">286</FONT> /** Update the high order scaled derivatives Adams integrators (phase 2).<a name="line.286"></a> <FONT color="green">287</FONT> * <p>The complete update of high order derivatives has a form similar to:<a name="line.287"></a> <FONT color="green">288</FONT> * <pre><a name="line.288"></a> <FONT color="green">289</FONT> * r<sub>n+1</sub> = (s<sub>1</sub>(n) - s<sub>1</sub>(n+1)) P<sup>-1</sup> u + P<sup>-1</sup> A P r<sub>n</sub><a name="line.289"></a> <FONT color="green">290</FONT> * </pre><a name="line.290"></a> <FONT color="green">291</FONT> * this method computes the (s<sub>1</sub>(n) - s<sub>1</sub>(n+1)) P<sup>-1</sup> u part.</p><a name="line.291"></a> <FONT color="green">292</FONT> * <p>Phase 1 of the update must already have been performed.</p><a name="line.292"></a> <FONT color="green">293</FONT> * @param start first order scaled derivatives at step start<a name="line.293"></a> <FONT color="green">294</FONT> * @param end first order scaled derivatives at step end<a name="line.294"></a> <FONT color="green">295</FONT> * @param highOrder high order scaled derivatives, will be modified<a name="line.295"></a> <FONT color="green">296</FONT> * (h<sup>2</sup>/2 y'', ... h<sup>k</sup>/k! y(k))<a name="line.296"></a> <FONT color="green">297</FONT> * @see #updateHighOrderDerivativesPhase1(Array2DRowRealMatrix)<a name="line.297"></a> <FONT color="green">298</FONT> */<a name="line.298"></a> <FONT color="green">299</FONT> public void updateHighOrderDerivativesPhase2(final double[] start,<a name="line.299"></a> <FONT color="green">300</FONT> final double[] end,<a name="line.300"></a> <FONT color="green">301</FONT> final Array2DRowRealMatrix highOrder) {<a name="line.301"></a> <FONT color="green">302</FONT> final double[][] data = highOrder.getDataRef();<a name="line.302"></a> <FONT color="green">303</FONT> for (int i = 0; i < data.length; ++i) {<a name="line.303"></a> <FONT color="green">304</FONT> final double[] dataI = data[i];<a name="line.304"></a> <FONT color="green">305</FONT> final double c1I = c1[i];<a name="line.305"></a> <FONT color="green">306</FONT> for (int j = 0; j < dataI.length; ++j) {<a name="line.306"></a> <FONT color="green">307</FONT> dataI[j] += c1I * (start[j] - end[j]);<a name="line.307"></a> <FONT color="green">308</FONT> }<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> </PRE> </BODY> </HTML>