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1 <HTML>
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2 <BODY BGCOLOR="white">
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3 <PRE>
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4 <FONT color="green">001</FONT> /*<a name="line.1"></a>
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5 <FONT color="green">002</FONT> * Licensed to the Apache Software Foundation (ASF) under one or more<a name="line.2"></a>
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6 <FONT color="green">003</FONT> * contributor license agreements. See the NOTICE file distributed with<a name="line.3"></a>
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7 <FONT color="green">004</FONT> * this work for additional information regarding copyright ownership.<a name="line.4"></a>
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8 <FONT color="green">005</FONT> * The ASF licenses this file to You under the Apache License, Version 2.0<a name="line.5"></a>
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9 <FONT color="green">006</FONT> * (the "License"); you may not use this file except in compliance with<a name="line.6"></a>
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10 <FONT color="green">007</FONT> * the License. You may obtain a copy of the License at<a name="line.7"></a>
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11 <FONT color="green">008</FONT> *<a name="line.8"></a>
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12 <FONT color="green">009</FONT> * http://www.apache.org/licenses/LICENSE-2.0<a name="line.9"></a>
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13 <FONT color="green">010</FONT> *<a name="line.10"></a>
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14 <FONT color="green">011</FONT> * Unless required by applicable law or agreed to in writing, software<a name="line.11"></a>
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15 <FONT color="green">012</FONT> * distributed under the License is distributed on an "AS IS" BASIS,<a name="line.12"></a>
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16 <FONT color="green">013</FONT> * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.<a name="line.13"></a>
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17 <FONT color="green">014</FONT> * See the License for the specific language governing permissions and<a name="line.14"></a>
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18 <FONT color="green">015</FONT> * limitations under the License.<a name="line.15"></a>
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19 <FONT color="green">016</FONT> */<a name="line.16"></a>
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20 <FONT color="green">017</FONT> <a name="line.17"></a>
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21 <FONT color="green">018</FONT> package org.apache.commons.math.ode.nonstiff;<a name="line.18"></a>
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22 <FONT color="green">019</FONT> <a name="line.19"></a>
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23 <FONT color="green">020</FONT> <a name="line.20"></a>
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24 <FONT color="green">021</FONT> /**<a name="line.21"></a>
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25 <FONT color="green">022</FONT> * This class implements the 5(4) Dormand-Prince integrator for Ordinary<a name="line.22"></a>
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26 <FONT color="green">023</FONT> * Differential Equations.<a name="line.23"></a>
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27 <FONT color="green">024</FONT> <a name="line.24"></a>
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28 <FONT color="green">025</FONT> * <p>This integrator is an embedded Runge-Kutta integrator<a name="line.25"></a>
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29 <FONT color="green">026</FONT> * of order 5(4) used in local extrapolation mode (i.e. the solution<a name="line.26"></a>
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30 <FONT color="green">027</FONT> * is computed using the high order formula) with stepsize control<a name="line.27"></a>
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31 <FONT color="green">028</FONT> * (and automatic step initialization) and continuous output. This<a name="line.28"></a>
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32 <FONT color="green">029</FONT> * method uses 7 functions evaluations per step. However, since this<a name="line.29"></a>
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33 <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>
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34 <FONT color="green">031</FONT> * the first evaluation of the next step and hence can be avoided. So<a name="line.31"></a>
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35 <FONT color="green">032</FONT> * the cost is really 6 functions evaluations per step.</p><a name="line.32"></a>
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36 <FONT color="green">033</FONT> *<a name="line.33"></a>
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37 <FONT color="green">034</FONT> * <p>This method has been published (whithout the continuous output<a name="line.34"></a>
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38 <FONT color="green">035</FONT> * that was added by Shampine in 1986) in the following article :<a name="line.35"></a>
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39 <FONT color="green">036</FONT> * <pre><a name="line.36"></a>
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40 <FONT color="green">037</FONT> * A family of embedded Runge-Kutta formulae<a name="line.37"></a>
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41 <FONT color="green">038</FONT> * J. R. Dormand and P. J. Prince<a name="line.38"></a>
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42 <FONT color="green">039</FONT> * Journal of Computational and Applied Mathematics<a name="line.39"></a>
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43 <FONT color="green">040</FONT> * volume 6, no 1, 1980, pp. 19-26<a name="line.40"></a>
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44 <FONT color="green">041</FONT> * </pre></p><a name="line.41"></a>
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45 <FONT color="green">042</FONT> *<a name="line.42"></a>
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46 <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>
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47 <FONT color="green">044</FONT> * @since 1.2<a name="line.44"></a>
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48 <FONT color="green">045</FONT> */<a name="line.45"></a>
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49 <FONT color="green">046</FONT> <a name="line.46"></a>
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50 <FONT color="green">047</FONT> public class DormandPrince54Integrator extends EmbeddedRungeKuttaIntegrator {<a name="line.47"></a>
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51 <FONT color="green">048</FONT> <a name="line.48"></a>
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52 <FONT color="green">049</FONT> /** Integrator method name. */<a name="line.49"></a>
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53 <FONT color="green">050</FONT> private static final String METHOD_NAME = "Dormand-Prince 5(4)";<a name="line.50"></a>
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54 <FONT color="green">051</FONT> <a name="line.51"></a>
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55 <FONT color="green">052</FONT> /** Time steps Butcher array. */<a name="line.52"></a>
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56 <FONT color="green">053</FONT> private static final double[] STATIC_C = {<a name="line.53"></a>
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57 <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>
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58 <FONT color="green">055</FONT> };<a name="line.55"></a>
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59 <FONT color="green">056</FONT> <a name="line.56"></a>
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60 <FONT color="green">057</FONT> /** Internal weights Butcher array. */<a name="line.57"></a>
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61 <FONT color="green">058</FONT> private static final double[][] STATIC_A = {<a name="line.58"></a>
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62 <FONT color="green">059</FONT> {1.0/5.0},<a name="line.59"></a>
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63 <FONT color="green">060</FONT> {3.0/40.0, 9.0/40.0},<a name="line.60"></a>
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64 <FONT color="green">061</FONT> {44.0/45.0, -56.0/15.0, 32.0/9.0},<a name="line.61"></a>
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65 <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>
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66 <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>
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67 <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>
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68 <FONT color="green">065</FONT> };<a name="line.65"></a>
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69 <FONT color="green">066</FONT> <a name="line.66"></a>
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70 <FONT color="green">067</FONT> /** Propagation weights Butcher array. */<a name="line.67"></a>
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71 <FONT color="green">068</FONT> private static final double[] STATIC_B = {<a name="line.68"></a>
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72 <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>
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73 <FONT color="green">070</FONT> };<a name="line.70"></a>
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74 <FONT color="green">071</FONT> <a name="line.71"></a>
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75 <FONT color="green">072</FONT> /** Error array, element 1. */<a name="line.72"></a>
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76 <FONT color="green">073</FONT> private static final double E1 = 71.0 / 57600.0;<a name="line.73"></a>
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77 <FONT color="green">074</FONT> <a name="line.74"></a>
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78 <FONT color="green">075</FONT> // element 2 is zero, so it is neither stored nor used<a name="line.75"></a>
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79 <FONT color="green">076</FONT> <a name="line.76"></a>
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80 <FONT color="green">077</FONT> /** Error array, element 3. */<a name="line.77"></a>
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81 <FONT color="green">078</FONT> private static final double E3 = -71.0 / 16695.0;<a name="line.78"></a>
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82 <FONT color="green">079</FONT> <a name="line.79"></a>
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83 <FONT color="green">080</FONT> /** Error array, element 4. */<a name="line.80"></a>
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84 <FONT color="green">081</FONT> private static final double E4 = 71.0 / 1920.0;<a name="line.81"></a>
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85 <FONT color="green">082</FONT> <a name="line.82"></a>
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86 <FONT color="green">083</FONT> /** Error array, element 5. */<a name="line.83"></a>
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87 <FONT color="green">084</FONT> private static final double E5 = -17253.0 / 339200.0;<a name="line.84"></a>
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88 <FONT color="green">085</FONT> <a name="line.85"></a>
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89 <FONT color="green">086</FONT> /** Error array, element 6. */<a name="line.86"></a>
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90 <FONT color="green">087</FONT> private static final double E6 = 22.0 / 525.0;<a name="line.87"></a>
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91 <FONT color="green">088</FONT> <a name="line.88"></a>
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92 <FONT color="green">089</FONT> /** Error array, element 7. */<a name="line.89"></a>
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93 <FONT color="green">090</FONT> private static final double E7 = -1.0 / 40.0;<a name="line.90"></a>
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94 <FONT color="green">091</FONT> <a name="line.91"></a>
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95 <FONT color="green">092</FONT> /** Simple constructor.<a name="line.92"></a>
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96 <FONT color="green">093</FONT> * Build a fifth order Dormand-Prince integrator with the given step bounds<a name="line.93"></a>
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97 <FONT color="green">094</FONT> * @param minStep minimal step (must be positive even for backward<a name="line.94"></a>
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98 <FONT color="green">095</FONT> * integration), the last step can be smaller than this<a name="line.95"></a>
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99 <FONT color="green">096</FONT> * @param maxStep maximal step (must be positive even for backward<a name="line.96"></a>
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100 <FONT color="green">097</FONT> * integration)<a name="line.97"></a>
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101 <FONT color="green">098</FONT> * @param scalAbsoluteTolerance allowed absolute error<a name="line.98"></a>
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102 <FONT color="green">099</FONT> * @param scalRelativeTolerance allowed relative error<a name="line.99"></a>
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103 <FONT color="green">100</FONT> */<a name="line.100"></a>
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104 <FONT color="green">101</FONT> public DormandPrince54Integrator(final double minStep, final double maxStep,<a name="line.101"></a>
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105 <FONT color="green">102</FONT> final double scalAbsoluteTolerance,<a name="line.102"></a>
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106 <FONT color="green">103</FONT> final double scalRelativeTolerance) {<a name="line.103"></a>
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107 <FONT color="green">104</FONT> super(METHOD_NAME, true, STATIC_C, STATIC_A, STATIC_B, new DormandPrince54StepInterpolator(),<a name="line.104"></a>
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108 <FONT color="green">105</FONT> minStep, maxStep, scalAbsoluteTolerance, scalRelativeTolerance);<a name="line.105"></a>
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109 <FONT color="green">106</FONT> }<a name="line.106"></a>
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110 <FONT color="green">107</FONT> <a name="line.107"></a>
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111 <FONT color="green">108</FONT> /** Simple constructor.<a name="line.108"></a>
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112 <FONT color="green">109</FONT> * Build a fifth order Dormand-Prince integrator with the given step bounds<a name="line.109"></a>
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113 <FONT color="green">110</FONT> * @param minStep minimal step (must be positive even for backward<a name="line.110"></a>
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114 <FONT color="green">111</FONT> * integration), the last step can be smaller than this<a name="line.111"></a>
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115 <FONT color="green">112</FONT> * @param maxStep maximal step (must be positive even for backward<a name="line.112"></a>
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116 <FONT color="green">113</FONT> * integration)<a name="line.113"></a>
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117 <FONT color="green">114</FONT> * @param vecAbsoluteTolerance allowed absolute error<a name="line.114"></a>
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118 <FONT color="green">115</FONT> * @param vecRelativeTolerance allowed relative error<a name="line.115"></a>
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119 <FONT color="green">116</FONT> */<a name="line.116"></a>
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120 <FONT color="green">117</FONT> public DormandPrince54Integrator(final double minStep, final double maxStep,<a name="line.117"></a>
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121 <FONT color="green">118</FONT> final double[] vecAbsoluteTolerance,<a name="line.118"></a>
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122 <FONT color="green">119</FONT> final double[] vecRelativeTolerance) {<a name="line.119"></a>
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123 <FONT color="green">120</FONT> super(METHOD_NAME, true, STATIC_C, STATIC_A, STATIC_B, new DormandPrince54StepInterpolator(),<a name="line.120"></a>
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124 <FONT color="green">121</FONT> minStep, maxStep, vecAbsoluteTolerance, vecRelativeTolerance);<a name="line.121"></a>
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125 <FONT color="green">122</FONT> }<a name="line.122"></a>
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126 <FONT color="green">123</FONT> <a name="line.123"></a>
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127 <FONT color="green">124</FONT> /** {@inheritDoc} */<a name="line.124"></a>
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128 <FONT color="green">125</FONT> @Override<a name="line.125"></a>
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129 <FONT color="green">126</FONT> public int getOrder() {<a name="line.126"></a>
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130 <FONT color="green">127</FONT> return 5;<a name="line.127"></a>
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131 <FONT color="green">128</FONT> }<a name="line.128"></a>
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132 <FONT color="green">129</FONT> <a name="line.129"></a>
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133 <FONT color="green">130</FONT> /** {@inheritDoc} */<a name="line.130"></a>
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134 <FONT color="green">131</FONT> @Override<a name="line.131"></a>
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135 <FONT color="green">132</FONT> protected double estimateError(final double[][] yDotK,<a name="line.132"></a>
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136 <FONT color="green">133</FONT> final double[] y0, final double[] y1,<a name="line.133"></a>
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137 <FONT color="green">134</FONT> final double h) {<a name="line.134"></a>
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138 <FONT color="green">135</FONT> <a name="line.135"></a>
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139 <FONT color="green">136</FONT> double error = 0;<a name="line.136"></a>
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140 <FONT color="green">137</FONT> <a name="line.137"></a>
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141 <FONT color="green">138</FONT> for (int j = 0; j < y0.length; ++j) {<a name="line.138"></a>
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142 <FONT color="green">139</FONT> final double errSum = E1 * yDotK[0][j] + E3 * yDotK[2][j] +<a name="line.139"></a>
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143 <FONT color="green">140</FONT> E4 * yDotK[3][j] + E5 * yDotK[4][j] +<a name="line.140"></a>
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144 <FONT color="green">141</FONT> E6 * yDotK[5][j] + E7 * yDotK[6][j];<a name="line.141"></a>
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145 <FONT color="green">142</FONT> <a name="line.142"></a>
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146 <FONT color="green">143</FONT> final double yScale = Math.max(Math.abs(y0[j]), Math.abs(y1[j]));<a name="line.143"></a>
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147 <FONT color="green">144</FONT> final double tol = (vecAbsoluteTolerance == null) ?<a name="line.144"></a>
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148 <FONT color="green">145</FONT> (scalAbsoluteTolerance + scalRelativeTolerance * yScale) :<a name="line.145"></a>
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149 <FONT color="green">146</FONT> (vecAbsoluteTolerance[j] + vecRelativeTolerance[j] * yScale);<a name="line.146"></a>
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150 <FONT color="green">147</FONT> final double ratio = h * errSum / tol;<a name="line.147"></a>
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151 <FONT color="green">148</FONT> error += ratio * ratio;<a name="line.148"></a>
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152 <FONT color="green">149</FONT> <a name="line.149"></a>
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153 <FONT color="green">150</FONT> }<a name="line.150"></a>
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154 <FONT color="green">151</FONT> <a name="line.151"></a>
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155 <FONT color="green">152</FONT> return Math.sqrt(error / y0.length);<a name="line.152"></a>
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156 <FONT color="green">153</FONT> <a name="line.153"></a>
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157 <FONT color="green">154</FONT> }<a name="line.154"></a>
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158 <FONT color="green">155</FONT> <a name="line.155"></a>
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159 <FONT color="green">156</FONT> }<a name="line.156"></a>
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220 </PRE>
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221 </BODY>
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222 </HTML>
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