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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> <a name="line.20"></a> <FONT color="green">021</FONT> /**<a name="line.21"></a> <FONT color="green">022</FONT> * This class implements the 5(4) Dormand-Prince integrator for Ordinary<a name="line.22"></a> <FONT color="green">023</FONT> * Differential Equations.<a name="line.23"></a> <FONT color="green">024</FONT> <a name="line.24"></a> <FONT color="green">025</FONT> * <p>This integrator is an embedded Runge-Kutta integrator<a name="line.25"></a> <FONT color="green">026</FONT> * of order 5(4) used in local extrapolation mode (i.e. the solution<a name="line.26"></a> <FONT color="green">027</FONT> * is computed using the high order formula) with stepsize control<a name="line.27"></a> <FONT color="green">028</FONT> * (and automatic step initialization) and continuous output. This<a name="line.28"></a> <FONT color="green">029</FONT> * method uses 7 functions evaluations per step. However, since this<a name="line.29"></a> <FONT color="green">030</FONT> * is an <i>fsal</i>, the last evaluation of one step is the same as<a name="line.30"></a> <FONT color="green">031</FONT> * the first evaluation of the next step and hence can be avoided. So<a name="line.31"></a> <FONT color="green">032</FONT> * the cost is really 6 functions evaluations per step.</p><a name="line.32"></a> <FONT color="green">033</FONT> *<a name="line.33"></a> <FONT color="green">034</FONT> * <p>This method has been published (whithout the continuous output<a name="line.34"></a> <FONT color="green">035</FONT> * that was added by Shampine in 1986) in the following article :<a name="line.35"></a> <FONT color="green">036</FONT> * <pre><a name="line.36"></a> <FONT color="green">037</FONT> * A family of embedded Runge-Kutta formulae<a name="line.37"></a> <FONT color="green">038</FONT> * J. R. Dormand and P. J. Prince<a name="line.38"></a> <FONT color="green">039</FONT> * Journal of Computational and Applied Mathematics<a name="line.39"></a> <FONT color="green">040</FONT> * volume 6, no 1, 1980, pp. 19-26<a name="line.40"></a> <FONT color="green">041</FONT> * </pre></p><a name="line.41"></a> <FONT color="green">042</FONT> *<a name="line.42"></a> <FONT color="green">043</FONT> * @version $Revision: 810196 $ $Date: 2009-09-01 15:47:46 -0400 (Tue, 01 Sep 2009) $<a name="line.43"></a> <FONT color="green">044</FONT> * @since 1.2<a name="line.44"></a> <FONT color="green">045</FONT> */<a name="line.45"></a> <FONT color="green">046</FONT> <a name="line.46"></a> <FONT color="green">047</FONT> public class DormandPrince54Integrator extends EmbeddedRungeKuttaIntegrator {<a name="line.47"></a> <FONT color="green">048</FONT> <a name="line.48"></a> <FONT color="green">049</FONT> /** Integrator method name. */<a name="line.49"></a> <FONT color="green">050</FONT> private static final String METHOD_NAME = "Dormand-Prince 5(4)";<a name="line.50"></a> <FONT color="green">051</FONT> <a name="line.51"></a> <FONT color="green">052</FONT> /** Time steps Butcher array. */<a name="line.52"></a> <FONT color="green">053</FONT> private static final double[] STATIC_C = {<a name="line.53"></a> <FONT color="green">054</FONT> 1.0/5.0, 3.0/10.0, 4.0/5.0, 8.0/9.0, 1.0, 1.0<a name="line.54"></a> <FONT color="green">055</FONT> };<a name="line.55"></a> <FONT color="green">056</FONT> <a name="line.56"></a> <FONT color="green">057</FONT> /** Internal weights Butcher array. */<a name="line.57"></a> <FONT color="green">058</FONT> private static final double[][] STATIC_A = {<a name="line.58"></a> <FONT color="green">059</FONT> {1.0/5.0},<a name="line.59"></a> <FONT color="green">060</FONT> {3.0/40.0, 9.0/40.0},<a name="line.60"></a> <FONT color="green">061</FONT> {44.0/45.0, -56.0/15.0, 32.0/9.0},<a name="line.61"></a> <FONT color="green">062</FONT> {19372.0/6561.0, -25360.0/2187.0, 64448.0/6561.0, -212.0/729.0},<a name="line.62"></a> <FONT color="green">063</FONT> {9017.0/3168.0, -355.0/33.0, 46732.0/5247.0, 49.0/176.0, -5103.0/18656.0},<a name="line.63"></a> <FONT color="green">064</FONT> {35.0/384.0, 0.0, 500.0/1113.0, 125.0/192.0, -2187.0/6784.0, 11.0/84.0}<a name="line.64"></a> <FONT color="green">065</FONT> };<a name="line.65"></a> <FONT color="green">066</FONT> <a name="line.66"></a> <FONT color="green">067</FONT> /** Propagation weights Butcher array. */<a name="line.67"></a> <FONT color="green">068</FONT> private static final double[] STATIC_B = {<a name="line.68"></a> <FONT color="green">069</FONT> 35.0/384.0, 0.0, 500.0/1113.0, 125.0/192.0, -2187.0/6784.0, 11.0/84.0, 0.0<a name="line.69"></a> <FONT color="green">070</FONT> };<a name="line.70"></a> <FONT color="green">071</FONT> <a name="line.71"></a> <FONT color="green">072</FONT> /** Error array, element 1. */<a name="line.72"></a> <FONT color="green">073</FONT> private static final double E1 = 71.0 / 57600.0;<a name="line.73"></a> <FONT color="green">074</FONT> <a name="line.74"></a> <FONT color="green">075</FONT> // element 2 is zero, so it is neither stored nor used<a name="line.75"></a> <FONT color="green">076</FONT> <a name="line.76"></a> <FONT color="green">077</FONT> /** Error array, element 3. */<a name="line.77"></a> <FONT color="green">078</FONT> private static final double E3 = -71.0 / 16695.0;<a name="line.78"></a> <FONT color="green">079</FONT> <a name="line.79"></a> <FONT color="green">080</FONT> /** Error array, element 4. */<a name="line.80"></a> <FONT color="green">081</FONT> private static final double E4 = 71.0 / 1920.0;<a name="line.81"></a> <FONT color="green">082</FONT> <a name="line.82"></a> <FONT color="green">083</FONT> /** Error array, element 5. */<a name="line.83"></a> <FONT color="green">084</FONT> private static final double E5 = -17253.0 / 339200.0;<a name="line.84"></a> <FONT color="green">085</FONT> <a name="line.85"></a> <FONT color="green">086</FONT> /** Error array, element 6. */<a name="line.86"></a> <FONT color="green">087</FONT> private static final double E6 = 22.0 / 525.0;<a name="line.87"></a> <FONT color="green">088</FONT> <a name="line.88"></a> <FONT color="green">089</FONT> /** Error array, element 7. */<a name="line.89"></a> <FONT color="green">090</FONT> private static final double E7 = -1.0 / 40.0;<a name="line.90"></a> <FONT color="green">091</FONT> <a name="line.91"></a> <FONT color="green">092</FONT> /** Simple constructor.<a name="line.92"></a> <FONT color="green">093</FONT> * Build a fifth order Dormand-Prince integrator with the given step bounds<a name="line.93"></a> <FONT color="green">094</FONT> * @param minStep minimal step (must be positive even for backward<a name="line.94"></a> <FONT color="green">095</FONT> * integration), the last step can be smaller than this<a name="line.95"></a> <FONT color="green">096</FONT> * @param maxStep maximal step (must be positive even for backward<a name="line.96"></a> <FONT color="green">097</FONT> * integration)<a name="line.97"></a> <FONT color="green">098</FONT> * @param scalAbsoluteTolerance allowed absolute error<a name="line.98"></a> <FONT color="green">099</FONT> * @param scalRelativeTolerance allowed relative error<a name="line.99"></a> <FONT color="green">100</FONT> */<a name="line.100"></a> <FONT color="green">101</FONT> public DormandPrince54Integrator(final double minStep, final double maxStep,<a name="line.101"></a> <FONT color="green">102</FONT> final double scalAbsoluteTolerance,<a name="line.102"></a> <FONT color="green">103</FONT> final double scalRelativeTolerance) {<a name="line.103"></a> <FONT color="green">104</FONT> super(METHOD_NAME, true, STATIC_C, STATIC_A, STATIC_B, new DormandPrince54StepInterpolator(),<a name="line.104"></a> <FONT color="green">105</FONT> minStep, maxStep, scalAbsoluteTolerance, scalRelativeTolerance);<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> /** Simple constructor.<a name="line.108"></a> <FONT color="green">109</FONT> * Build a fifth order Dormand-Prince integrator with the given step bounds<a name="line.109"></a> <FONT color="green">110</FONT> * @param minStep minimal step (must be positive even for backward<a name="line.110"></a> <FONT color="green">111</FONT> * integration), the last step can be smaller than this<a name="line.111"></a> <FONT color="green">112</FONT> * @param maxStep maximal step (must be positive even for backward<a name="line.112"></a> <FONT color="green">113</FONT> * integration)<a name="line.113"></a> <FONT color="green">114</FONT> * @param vecAbsoluteTolerance allowed absolute error<a name="line.114"></a> <FONT color="green">115</FONT> * @param vecRelativeTolerance allowed relative error<a name="line.115"></a> <FONT color="green">116</FONT> */<a name="line.116"></a> <FONT color="green">117</FONT> public DormandPrince54Integrator(final double minStep, final double maxStep,<a name="line.117"></a> <FONT color="green">118</FONT> final double[] vecAbsoluteTolerance,<a name="line.118"></a> <FONT color="green">119</FONT> final double[] vecRelativeTolerance) {<a name="line.119"></a> <FONT color="green">120</FONT> super(METHOD_NAME, true, STATIC_C, STATIC_A, STATIC_B, new DormandPrince54StepInterpolator(),<a name="line.120"></a> <FONT color="green">121</FONT> minStep, maxStep, vecAbsoluteTolerance, vecRelativeTolerance);<a name="line.121"></a> <FONT color="green">122</FONT> }<a name="line.122"></a> <FONT color="green">123</FONT> <a name="line.123"></a> <FONT color="green">124</FONT> /** {@inheritDoc} */<a name="line.124"></a> <FONT color="green">125</FONT> @Override<a name="line.125"></a> <FONT color="green">126</FONT> public int getOrder() {<a name="line.126"></a> <FONT color="green">127</FONT> return 5;<a name="line.127"></a> <FONT color="green">128</FONT> }<a name="line.128"></a> <FONT color="green">129</FONT> <a name="line.129"></a> <FONT color="green">130</FONT> /** {@inheritDoc} */<a name="line.130"></a> <FONT color="green">131</FONT> @Override<a name="line.131"></a> <FONT color="green">132</FONT> protected double estimateError(final double[][] yDotK,<a name="line.132"></a> <FONT color="green">133</FONT> final double[] y0, final double[] y1,<a name="line.133"></a> <FONT color="green">134</FONT> final double h) {<a name="line.134"></a> <FONT color="green">135</FONT> <a name="line.135"></a> <FONT color="green">136</FONT> double error = 0;<a name="line.136"></a> <FONT color="green">137</FONT> <a name="line.137"></a> <FONT color="green">138</FONT> for (int j = 0; j < y0.length; ++j) {<a name="line.138"></a> <FONT color="green">139</FONT> final double errSum = E1 * yDotK[0][j] + E3 * yDotK[2][j] +<a name="line.139"></a> <FONT color="green">140</FONT> E4 * yDotK[3][j] + E5 * yDotK[4][j] +<a name="line.140"></a> <FONT color="green">141</FONT> E6 * yDotK[5][j] + E7 * yDotK[6][j];<a name="line.141"></a> <FONT color="green">142</FONT> <a name="line.142"></a> <FONT color="green">143</FONT> final double yScale = Math.max(Math.abs(y0[j]), Math.abs(y1[j]));<a name="line.143"></a> <FONT color="green">144</FONT> final double tol = (vecAbsoluteTolerance == null) ?<a name="line.144"></a> <FONT color="green">145</FONT> (scalAbsoluteTolerance + scalRelativeTolerance * yScale) :<a name="line.145"></a> <FONT color="green">146</FONT> (vecAbsoluteTolerance[j] + vecRelativeTolerance[j] * yScale);<a name="line.146"></a> <FONT color="green">147</FONT> final double ratio = h * errSum / tol;<a name="line.147"></a> <FONT color="green">148</FONT> error += ratio * ratio;<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> <a name="line.151"></a> <FONT color="green">152</FONT> return Math.sqrt(error / y0.length);<a name="line.152"></a> <FONT color="green">153</FONT> <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> }<a name="line.156"></a> </PRE> </BODY> </HTML>