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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> import java.util.Arrays;<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> import org.apache.commons.math.linear.Array2DRowRealMatrix;<a name="line.22"></a>
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26 <FONT color="green">023</FONT> import org.apache.commons.math.linear.MatrixVisitorException;<a name="line.23"></a>
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27 <FONT color="green">024</FONT> import org.apache.commons.math.linear.RealMatrixPreservingVisitor;<a name="line.24"></a>
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28 <FONT color="green">025</FONT> import org.apache.commons.math.ode.DerivativeException;<a name="line.25"></a>
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29 <FONT color="green">026</FONT> import org.apache.commons.math.ode.FirstOrderDifferentialEquations;<a name="line.26"></a>
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30 <FONT color="green">027</FONT> import org.apache.commons.math.ode.IntegratorException;<a name="line.27"></a>
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31 <FONT color="green">028</FONT> import org.apache.commons.math.ode.events.CombinedEventsManager;<a name="line.28"></a>
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32 <FONT color="green">029</FONT> import org.apache.commons.math.ode.sampling.NordsieckStepInterpolator;<a name="line.29"></a>
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33 <FONT color="green">030</FONT> import org.apache.commons.math.ode.sampling.StepHandler;<a name="line.30"></a>
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34 <FONT color="green">031</FONT> <a name="line.31"></a>
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35 <FONT color="green">032</FONT> <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> * This class implements implicit Adams-Moulton integrators for Ordinary<a name="line.34"></a>
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38 <FONT color="green">035</FONT> * Differential Equations.<a name="line.35"></a>
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39 <FONT color="green">036</FONT> *<a name="line.36"></a>
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40 <FONT color="green">037</FONT> * <p>Adams-Moulton methods (in fact due to Adams alone) are implicit<a name="line.37"></a>
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41 <FONT color="green">038</FONT> * multistep ODE solvers. This implementation is a variation of the classical<a name="line.38"></a>
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42 <FONT color="green">039</FONT> * one: it uses adaptive stepsize to implement error control, whereas<a name="line.39"></a>
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43 <FONT color="green">040</FONT> * classical implementations are fixed step size. The value of state vector<a name="line.40"></a>
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44 <FONT color="green">041</FONT> * at step n+1 is a simple combination of the value at step n and of the<a name="line.41"></a>
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45 <FONT color="green">042</FONT> * derivatives at steps n+1, n, n-1 ... Since y'<sub>n+1</sub> is needed to<a name="line.42"></a>
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46 <FONT color="green">043</FONT> * compute y<sub>n+1</sub>,another method must be used to compute a first<a name="line.43"></a>
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47 <FONT color="green">044</FONT> * estimate of y<sub>n+1</sub>, then compute y'<sub>n+1</sub>, then compute<a name="line.44"></a>
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48 <FONT color="green">045</FONT> * a final estimate of y<sub>n+1</sub> using the following formulas. Depending<a name="line.45"></a>
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49 <FONT color="green">046</FONT> * on the number k of previous steps one wants to use for computing the next<a name="line.46"></a>
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50 <FONT color="green">047</FONT> * value, different formulas are available for the final estimate:</p><a name="line.47"></a>
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51 <FONT color="green">048</FONT> * <ul><a name="line.48"></a>
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52 <FONT color="green">049</FONT> * <li>k = 1: y<sub>n+1</sub> = y<sub>n</sub> + h y'<sub>n+1</sub></li><a name="line.49"></a>
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53 <FONT color="green">050</FONT> * <li>k = 2: y<sub>n+1</sub> = y<sub>n</sub> + h (y'<sub>n+1</sub>+y'<sub>n</sub>)/2</li><a name="line.50"></a>
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54 <FONT color="green">051</FONT> * <li>k = 3: y<sub>n+1</sub> = y<sub>n</sub> + h (5y'<sub>n+1</sub>+8y'<sub>n</sub>-y'<sub>n-1</sub>)/12</li><a name="line.51"></a>
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55 <FONT color="green">052</FONT> * <li>k = 4: y<sub>n+1</sub> = y<sub>n</sub> + h (9y'<sub>n+1</sub>+19y'<sub>n</sub>-5y'<sub>n-1</sub>+y'<sub>n-2</sub>)/24</li><a name="line.52"></a>
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56 <FONT color="green">053</FONT> * <li>...</li><a name="line.53"></a>
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57 <FONT color="green">054</FONT> * </ul><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> * <p>A k-steps Adams-Moulton method is of order k+1.</p><a name="line.56"></a>
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60 <FONT color="green">057</FONT> *<a name="line.57"></a>
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61 <FONT color="green">058</FONT> * <h3>Implementation details</h3><a name="line.58"></a>
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62 <FONT color="green">059</FONT> *<a name="line.59"></a>
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63 <FONT color="green">060</FONT> * <p>We define scaled derivatives s<sub>i</sub>(n) at step n as:<a name="line.60"></a>
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64 <FONT color="green">061</FONT> * <pre><a name="line.61"></a>
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65 <FONT color="green">062</FONT> * s<sub>1</sub>(n) = h y'<sub>n</sub> for first derivative<a name="line.62"></a>
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66 <FONT color="green">063</FONT> * s<sub>2</sub>(n) = h<sup>2</sup>/2 y''<sub>n</sub> for second derivative<a name="line.63"></a>
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67 <FONT color="green">064</FONT> * s<sub>3</sub>(n) = h<sup>3</sup>/6 y'''<sub>n</sub> for third derivative<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> * s<sub>k</sub>(n) = h<sup>k</sup>/k! y(k)<sub>n</sub> for k<sup>th</sup> derivative<a name="line.66"></a>
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70 <FONT color="green">067</FONT> * </pre></p><a name="line.67"></a>
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71 <FONT color="green">068</FONT> *<a name="line.68"></a>
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72 <FONT color="green">069</FONT> * <p>The definitions above use the classical representation with several previous first<a name="line.69"></a>
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73 <FONT color="green">070</FONT> * derivatives. Lets define<a name="line.70"></a>
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74 <FONT color="green">071</FONT> * <pre><a name="line.71"></a>
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75 <FONT color="green">072</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.72"></a>
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76 <FONT color="green">073</FONT> * </pre><a name="line.73"></a>
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77 <FONT color="green">074</FONT> * (we omit the k index in the notation for clarity). With these definitions,<a name="line.74"></a>
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78 <FONT color="green">075</FONT> * Adams-Moulton methods can be written:<a name="line.75"></a>
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79 <FONT color="green">076</FONT> * <ul><a name="line.76"></a>
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80 <FONT color="green">077</FONT> * <li>k = 1: y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n+1)</li><a name="line.77"></a>
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81 <FONT color="green">078</FONT> * <li>k = 2: y<sub>n+1</sub> = y<sub>n</sub> + 1/2 s<sub>1</sub>(n+1) + [ 1/2 ] q<sub>n+1</sub></li><a name="line.78"></a>
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82 <FONT color="green">079</FONT> * <li>k = 3: y<sub>n+1</sub> = y<sub>n</sub> + 5/12 s<sub>1</sub>(n+1) + [ 8/12 -1/12 ] q<sub>n+1</sub></li><a name="line.79"></a>
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83 <FONT color="green">080</FONT> * <li>k = 4: y<sub>n+1</sub> = y<sub>n</sub> + 9/24 s<sub>1</sub>(n+1) + [ 19/24 -5/24 1/24 ] q<sub>n+1</sub></li><a name="line.80"></a>
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84 <FONT color="green">081</FONT> * <li>...</li><a name="line.81"></a>
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85 <FONT color="green">082</FONT> * </ul></p><a name="line.82"></a>
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86 <FONT color="green">083</FONT> *<a name="line.83"></a>
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87 <FONT color="green">084</FONT> * <p>Instead of using the classical representation with first derivatives only (y<sub>n</sub>,<a name="line.84"></a>
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88 <FONT color="green">085</FONT> * s<sub>1</sub>(n+1) and q<sub>n+1</sub>), our implementation uses the Nordsieck vector with<a name="line.85"></a>
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89 <FONT color="green">086</FONT> * higher degrees scaled derivatives all taken at the same step (y<sub>n</sub>, s<sub>1</sub>(n)<a name="line.86"></a>
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90 <FONT color="green">087</FONT> * and r<sub>n</sub>) where r<sub>n</sub> is defined as:<a name="line.87"></a>
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91 <FONT color="green">088</FONT> * <pre><a name="line.88"></a>
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92 <FONT color="green">089</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.89"></a>
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93 <FONT color="green">090</FONT> * </pre><a name="line.90"></a>
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94 <FONT color="green">091</FONT> * (here again we omit the k index in the notation for clarity)<a name="line.91"></a>
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95 <FONT color="green">092</FONT> * </p><a name="line.92"></a>
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96 <FONT color="green">093</FONT> *<a name="line.93"></a>
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97 <FONT color="green">094</FONT> * <p>Taylor series formulas show that for any index offset i, s<sub>1</sub>(n-i) can be<a name="line.94"></a>
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98 <FONT color="green">095</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.95"></a>
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99 <FONT color="green">096</FONT> * for degree k polynomials.<a name="line.96"></a>
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100 <FONT color="green">097</FONT> * <pre><a name="line.97"></a>
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101 <FONT color="green">098</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.98"></a>
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102 <FONT color="green">099</FONT> * </pre><a name="line.99"></a>
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103 <FONT color="green">100</FONT> * The previous formula can be used with several values for i to compute the transform between<a name="line.100"></a>
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104 <FONT color="green">101</FONT> * classical representation and Nordsieck vector. The transform between r<sub>n</sub><a name="line.101"></a>
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105 <FONT color="green">102</FONT> * and q<sub>n</sub> resulting from the Taylor series formulas above is:<a name="line.102"></a>
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106 <FONT color="green">103</FONT> * <pre><a name="line.103"></a>
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107 <FONT color="green">104</FONT> * q<sub>n</sub> = s<sub>1</sub>(n) u + P r<sub>n</sub><a name="line.104"></a>
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108 <FONT color="green">105</FONT> * </pre><a name="line.105"></a>
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109 <FONT color="green">106</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.106"></a>
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110 <FONT color="green">107</FONT> * with the j (-i)<sup>j-1</sup> terms:<a name="line.107"></a>
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111 <FONT color="green">108</FONT> * <pre><a name="line.108"></a>
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112 <FONT color="green">109</FONT> * [ -2 3 -4 5 ... ]<a name="line.109"></a>
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113 <FONT color="green">110</FONT> * [ -4 12 -32 80 ... ]<a name="line.110"></a>
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114 <FONT color="green">111</FONT> * P = [ -6 27 -108 405 ... ]<a name="line.111"></a>
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115 <FONT color="green">112</FONT> * [ -8 48 -256 1280 ... ]<a name="line.112"></a>
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116 <FONT color="green">113</FONT> * [ ... ]<a name="line.113"></a>
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117 <FONT color="green">114</FONT> * </pre></p><a name="line.114"></a>
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118 <FONT color="green">115</FONT> *<a name="line.115"></a>
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119 <FONT color="green">116</FONT> * <p>Using the Nordsieck vector has several advantages:<a name="line.116"></a>
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120 <FONT color="green">117</FONT> * <ul><a name="line.117"></a>
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121 <FONT color="green">118</FONT> * <li>it greatly simplifies step interpolation as the interpolator mainly applies<a name="line.118"></a>
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122 <FONT color="green">119</FONT> * Taylor series formulas,</li><a name="line.119"></a>
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123 <FONT color="green">120</FONT> * <li>it simplifies step changes that occur when discrete events that truncate<a name="line.120"></a>
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124 <FONT color="green">121</FONT> * the step are triggered,</li><a name="line.121"></a>
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125 <FONT color="green">122</FONT> * <li>it allows to extend the methods in order to support adaptive stepsize.</li><a name="line.122"></a>
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126 <FONT color="green">123</FONT> * </ul></p><a name="line.123"></a>
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127 <FONT color="green">124</FONT> *<a name="line.124"></a>
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128 <FONT color="green">125</FONT> * <p>The predicted Nordsieck vector at step n+1 is computed from the Nordsieck vector at step<a name="line.125"></a>
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129 <FONT color="green">126</FONT> * n as follows:<a name="line.126"></a>
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130 <FONT color="green">127</FONT> * <ul><a name="line.127"></a>
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131 <FONT color="green">128</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.128"></a>
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132 <FONT color="green">129</FONT> * <li>S<sub>1</sub>(n+1) = h f(t<sub>n+1</sub>, Y<sub>n+1</sub>)</li><a name="line.129"></a>
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133 <FONT color="green">130</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.130"></a>
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134 <FONT color="green">131</FONT> * </ul><a name="line.131"></a>
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135 <FONT color="green">132</FONT> * where A is a rows shifting matrix (the lower left part is an identity matrix):<a name="line.132"></a>
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136 <FONT color="green">133</FONT> * <pre><a name="line.133"></a>
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137 <FONT color="green">134</FONT> * [ 0 0 ... 0 0 | 0 ]<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> * [ 1 0 ... 0 0 | 0 ]<a name="line.136"></a>
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140 <FONT color="green">137</FONT> * A = [ 0 1 ... 0 0 | 0 ]<a name="line.137"></a>
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141 <FONT color="green">138</FONT> * [ ... | 0 ]<a name="line.138"></a>
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142 <FONT color="green">139</FONT> * [ 0 0 ... 1 0 | 0 ]<a name="line.139"></a>
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143 <FONT color="green">140</FONT> * [ 0 0 ... 0 1 | 0 ]<a name="line.140"></a>
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144 <FONT color="green">141</FONT> * </pre><a name="line.141"></a>
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145 <FONT color="green">142</FONT> * From this predicted vector, the corrected vector is computed as follows:<a name="line.142"></a>
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146 <FONT color="green">143</FONT> * <ul><a name="line.143"></a>
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147 <FONT color="green">144</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.144"></a>
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148 <FONT color="green">145</FONT> * <li>s<sub>1</sub>(n+1) = h f(t<sub>n+1</sub>, y<sub>n+1</sub>)</li><a name="line.145"></a>
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149 <FONT color="green">146</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.146"></a>
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150 <FONT color="green">147</FONT> * </ul><a name="line.147"></a>
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151 <FONT color="green">148</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.148"></a>
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152 <FONT color="green">149</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.149"></a>
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153 <FONT color="green">150</FONT> * represent the corrected states.</p><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> * <p>The P<sup>-1</sup>u vector and the P<sup>-1</sup> A P matrix do not depend on the state,<a name="line.152"></a>
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156 <FONT color="green">153</FONT> * they only depend on k and therefore are precomputed once for all.</p><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> * @version $Revision: 927202 $ $Date: 2010-03-24 18:11:51 -0400 (Wed, 24 Mar 2010) $<a name="line.155"></a>
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159 <FONT color="green">156</FONT> * @since 2.0<a name="line.156"></a>
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160 <FONT color="green">157</FONT> */<a name="line.157"></a>
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161 <FONT color="green">158</FONT> public class AdamsMoultonIntegrator extends AdamsIntegrator {<a name="line.158"></a>
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162 <FONT color="green">159</FONT> <a name="line.159"></a>
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163 <FONT color="green">160</FONT> /**<a name="line.160"></a>
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164 <FONT color="green">161</FONT> * Build an Adams-Moulton integrator with the given order and error control parameters.<a name="line.161"></a>
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165 <FONT color="green">162</FONT> * @param nSteps number of steps of the method excluding the one being computed<a name="line.162"></a>
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166 <FONT color="green">163</FONT> * @param minStep minimal step (must be positive even for backward<a name="line.163"></a>
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167 <FONT color="green">164</FONT> * integration), the last step can be smaller than this<a name="line.164"></a>
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168 <FONT color="green">165</FONT> * @param maxStep maximal step (must be positive even for backward<a name="line.165"></a>
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169 <FONT color="green">166</FONT> * integration)<a name="line.166"></a>
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170 <FONT color="green">167</FONT> * @param scalAbsoluteTolerance allowed absolute error<a name="line.167"></a>
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171 <FONT color="green">168</FONT> * @param scalRelativeTolerance allowed relative error<a name="line.168"></a>
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172 <FONT color="green">169</FONT> * @exception IllegalArgumentException if order is 1 or less<a name="line.169"></a>
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173 <FONT color="green">170</FONT> */<a name="line.170"></a>
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174 <FONT color="green">171</FONT> public AdamsMoultonIntegrator(final int nSteps,<a name="line.171"></a>
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175 <FONT color="green">172</FONT> final double minStep, final double maxStep,<a name="line.172"></a>
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176 <FONT color="green">173</FONT> final double scalAbsoluteTolerance,<a name="line.173"></a>
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177 <FONT color="green">174</FONT> final double scalRelativeTolerance)<a name="line.174"></a>
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178 <FONT color="green">175</FONT> throws IllegalArgumentException {<a name="line.175"></a>
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179 <FONT color="green">176</FONT> super("Adams-Moulton", nSteps, nSteps + 1, minStep, maxStep,<a name="line.176"></a>
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180 <FONT color="green">177</FONT> scalAbsoluteTolerance, scalRelativeTolerance);<a name="line.177"></a>
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181 <FONT color="green">178</FONT> }<a name="line.178"></a>
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182 <FONT color="green">179</FONT> <a name="line.179"></a>
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183 <FONT color="green">180</FONT> /**<a name="line.180"></a>
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184 <FONT color="green">181</FONT> * Build an Adams-Moulton integrator with the given order and error control parameters.<a name="line.181"></a>
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185 <FONT color="green">182</FONT> * @param nSteps number of steps of the method excluding the one being computed<a name="line.182"></a>
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186 <FONT color="green">183</FONT> * @param minStep minimal step (must be positive even for backward<a name="line.183"></a>
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187 <FONT color="green">184</FONT> * integration), the last step can be smaller than this<a name="line.184"></a>
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188 <FONT color="green">185</FONT> * @param maxStep maximal step (must be positive even for backward<a name="line.185"></a>
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189 <FONT color="green">186</FONT> * integration)<a name="line.186"></a>
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190 <FONT color="green">187</FONT> * @param vecAbsoluteTolerance allowed absolute error<a name="line.187"></a>
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191 <FONT color="green">188</FONT> * @param vecRelativeTolerance allowed relative error<a name="line.188"></a>
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192 <FONT color="green">189</FONT> * @exception IllegalArgumentException if order is 1 or less<a name="line.189"></a>
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193 <FONT color="green">190</FONT> */<a name="line.190"></a>
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194 <FONT color="green">191</FONT> public AdamsMoultonIntegrator(final int nSteps,<a name="line.191"></a>
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195 <FONT color="green">192</FONT> final double minStep, final double maxStep,<a name="line.192"></a>
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196 <FONT color="green">193</FONT> final double[] vecAbsoluteTolerance,<a name="line.193"></a>
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197 <FONT color="green">194</FONT> final double[] vecRelativeTolerance)<a name="line.194"></a>
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198 <FONT color="green">195</FONT> throws IllegalArgumentException {<a name="line.195"></a>
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199 <FONT color="green">196</FONT> super("Adams-Moulton", nSteps, nSteps + 1, minStep, maxStep,<a name="line.196"></a>
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200 <FONT color="green">197</FONT> vecAbsoluteTolerance, vecRelativeTolerance);<a name="line.197"></a>
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201 <FONT color="green">198</FONT> }<a name="line.198"></a>
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202 <FONT color="green">199</FONT> <a name="line.199"></a>
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203 <FONT color="green">200</FONT> <a name="line.200"></a>
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204 <FONT color="green">201</FONT> /** {@inheritDoc} */<a name="line.201"></a>
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205 <FONT color="green">202</FONT> @Override<a name="line.202"></a>
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206 <FONT color="green">203</FONT> public double integrate(final FirstOrderDifferentialEquations equations,<a name="line.203"></a>
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207 <FONT color="green">204</FONT> final double t0, final double[] y0,<a name="line.204"></a>
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208 <FONT color="green">205</FONT> final double t, final double[] y)<a name="line.205"></a>
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209 <FONT color="green">206</FONT> throws DerivativeException, IntegratorException {<a name="line.206"></a>
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210 <FONT color="green">207</FONT> <a name="line.207"></a>
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211 <FONT color="green">208</FONT> final int n = y0.length;<a name="line.208"></a>
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212 <FONT color="green">209</FONT> sanityChecks(equations, t0, y0, t, y);<a name="line.209"></a>
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213 <FONT color="green">210</FONT> setEquations(equations);<a name="line.210"></a>
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214 <FONT color="green">211</FONT> resetEvaluations();<a name="line.211"></a>
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215 <FONT color="green">212</FONT> final boolean forward = t > t0;<a name="line.212"></a>
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216 <FONT color="green">213</FONT> <a name="line.213"></a>
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217 <FONT color="green">214</FONT> // initialize working arrays<a name="line.214"></a>
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218 <FONT color="green">215</FONT> if (y != y0) {<a name="line.215"></a>
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219 <FONT color="green">216</FONT> System.arraycopy(y0, 0, y, 0, n);<a name="line.216"></a>
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220 <FONT color="green">217</FONT> }<a name="line.217"></a>
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221 <FONT color="green">218</FONT> final double[] yDot = new double[y0.length];<a name="line.218"></a>
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222 <FONT color="green">219</FONT> final double[] yTmp = new double[y0.length];<a name="line.219"></a>
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223 <FONT color="green">220</FONT> <a name="line.220"></a>
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224 <FONT color="green">221</FONT> // set up two interpolators sharing the integrator arrays<a name="line.221"></a>
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225 <FONT color="green">222</FONT> final NordsieckStepInterpolator interpolator = new NordsieckStepInterpolator();<a name="line.222"></a>
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226 <FONT color="green">223</FONT> interpolator.reinitialize(y, forward);<a name="line.223"></a>
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227 <FONT color="green">224</FONT> final NordsieckStepInterpolator interpolatorTmp = new NordsieckStepInterpolator();<a name="line.224"></a>
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228 <FONT color="green">225</FONT> interpolatorTmp.reinitialize(yTmp, forward);<a name="line.225"></a>
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229 <FONT color="green">226</FONT> <a name="line.226"></a>
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230 <FONT color="green">227</FONT> // set up integration control objects<a name="line.227"></a>
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231 <FONT color="green">228</FONT> for (StepHandler handler : stepHandlers) {<a name="line.228"></a>
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232 <FONT color="green">229</FONT> handler.reset();<a name="line.229"></a>
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233 <FONT color="green">230</FONT> }<a name="line.230"></a>
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234 <FONT color="green">231</FONT> CombinedEventsManager manager = addEndTimeChecker(t0, t, eventsHandlersManager);<a name="line.231"></a>
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235 <FONT color="green">232</FONT> <a name="line.232"></a>
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236 <FONT color="green">233</FONT> <a name="line.233"></a>
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237 <FONT color="green">234</FONT> // compute the initial Nordsieck vector using the configured starter integrator<a name="line.234"></a>
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238 <FONT color="green">235</FONT> start(t0, y, t);<a name="line.235"></a>
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239 <FONT color="green">236</FONT> interpolator.reinitialize(stepStart, stepSize, scaled, nordsieck);<a name="line.236"></a>
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240 <FONT color="green">237</FONT> interpolator.storeTime(stepStart);<a name="line.237"></a>
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241 <FONT color="green">238</FONT> <a name="line.238"></a>
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242 <FONT color="green">239</FONT> double hNew = stepSize;<a name="line.239"></a>
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243 <FONT color="green">240</FONT> interpolator.rescale(hNew);<a name="line.240"></a>
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244 <FONT color="green">241</FONT> <a name="line.241"></a>
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245 <FONT color="green">242</FONT> boolean lastStep = false;<a name="line.242"></a>
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246 <FONT color="green">243</FONT> while (!lastStep) {<a name="line.243"></a>
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247 <FONT color="green">244</FONT> <a name="line.244"></a>
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248 <FONT color="green">245</FONT> // shift all data<a name="line.245"></a>
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249 <FONT color="green">246</FONT> interpolator.shift();<a name="line.246"></a>
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250 <FONT color="green">247</FONT> <a name="line.247"></a>
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251 <FONT color="green">248</FONT> double error = 0;<a name="line.248"></a>
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252 <FONT color="green">249</FONT> for (boolean loop = true; loop;) {<a name="line.249"></a>
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253 <FONT color="green">250</FONT> <a name="line.250"></a>
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254 <FONT color="green">251</FONT> stepSize = hNew;<a name="line.251"></a>
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255 <FONT color="green">252</FONT> <a name="line.252"></a>
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256 <FONT color="green">253</FONT> // predict a first estimate of the state at step end (P in the PECE sequence)<a name="line.253"></a>
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257 <FONT color="green">254</FONT> final double stepEnd = stepStart + stepSize;<a name="line.254"></a>
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258 <FONT color="green">255</FONT> interpolator.setInterpolatedTime(stepEnd);<a name="line.255"></a>
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259 <FONT color="green">256</FONT> System.arraycopy(interpolator.getInterpolatedState(), 0, yTmp, 0, y0.length);<a name="line.256"></a>
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260 <FONT color="green">257</FONT> <a name="line.257"></a>
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261 <FONT color="green">258</FONT> // evaluate a first estimate of the derivative (first E in the PECE sequence)<a name="line.258"></a>
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262 <FONT color="green">259</FONT> computeDerivatives(stepEnd, yTmp, yDot);<a name="line.259"></a>
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263 <FONT color="green">260</FONT> <a name="line.260"></a>
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264 <FONT color="green">261</FONT> // update Nordsieck vector<a name="line.261"></a>
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265 <FONT color="green">262</FONT> final double[] predictedScaled = new double[y0.length];<a name="line.262"></a>
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266 <FONT color="green">263</FONT> for (int j = 0; j < y0.length; ++j) {<a name="line.263"></a>
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267 <FONT color="green">264</FONT> predictedScaled[j] = stepSize * yDot[j];<a name="line.264"></a>
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268 <FONT color="green">265</FONT> }<a name="line.265"></a>
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269 <FONT color="green">266</FONT> final Array2DRowRealMatrix nordsieckTmp = updateHighOrderDerivativesPhase1(nordsieck);<a name="line.266"></a>
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270 <FONT color="green">267</FONT> updateHighOrderDerivativesPhase2(scaled, predictedScaled, nordsieckTmp);<a name="line.267"></a>
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271 <FONT color="green">268</FONT> <a name="line.268"></a>
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272 <FONT color="green">269</FONT> // apply correction (C in the PECE sequence)<a name="line.269"></a>
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273 <FONT color="green">270</FONT> error = nordsieckTmp.walkInOptimizedOrder(new Corrector(y, predictedScaled, yTmp));<a name="line.270"></a>
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274 <FONT color="green">271</FONT> <a name="line.271"></a>
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275 <FONT color="green">272</FONT> if (error <= 1.0) {<a name="line.272"></a>
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276 <FONT color="green">273</FONT> <a name="line.273"></a>
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277 <FONT color="green">274</FONT> // evaluate a final estimate of the derivative (second E in the PECE sequence)<a name="line.274"></a>
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278 <FONT color="green">275</FONT> computeDerivatives(stepEnd, yTmp, yDot);<a name="line.275"></a>
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279 <FONT color="green">276</FONT> <a name="line.276"></a>
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280 <FONT color="green">277</FONT> // update Nordsieck vector<a name="line.277"></a>
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281 <FONT color="green">278</FONT> final double[] correctedScaled = new double[y0.length];<a name="line.278"></a>
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282 <FONT color="green">279</FONT> for (int j = 0; j < y0.length; ++j) {<a name="line.279"></a>
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283 <FONT color="green">280</FONT> correctedScaled[j] = stepSize * yDot[j];<a name="line.280"></a>
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284 <FONT color="green">281</FONT> }<a name="line.281"></a>
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285 <FONT color="green">282</FONT> updateHighOrderDerivativesPhase2(predictedScaled, correctedScaled, nordsieckTmp);<a name="line.282"></a>
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286 <FONT color="green">283</FONT> <a name="line.283"></a>
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287 <FONT color="green">284</FONT> // discrete events handling<a name="line.284"></a>
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288 <FONT color="green">285</FONT> interpolatorTmp.reinitialize(stepEnd, stepSize, correctedScaled, nordsieckTmp);<a name="line.285"></a>
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289 <FONT color="green">286</FONT> interpolatorTmp.storeTime(stepStart);<a name="line.286"></a>
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290 <FONT color="green">287</FONT> interpolatorTmp.shift();<a name="line.287"></a>
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291 <FONT color="green">288</FONT> interpolatorTmp.storeTime(stepEnd);<a name="line.288"></a>
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292 <FONT color="green">289</FONT> if (manager.evaluateStep(interpolatorTmp)) {<a name="line.289"></a>
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293 <FONT color="green">290</FONT> final double dt = manager.getEventTime() - stepStart;<a name="line.290"></a>
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294 <FONT color="green">291</FONT> if (Math.abs(dt) <= Math.ulp(stepStart)) {<a name="line.291"></a>
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295 <FONT color="green">292</FONT> // we cannot simply truncate the step, reject the current computation<a name="line.292"></a>
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296 <FONT color="green">293</FONT> // and let the loop compute another state with the truncated step.<a name="line.293"></a>
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297 <FONT color="green">294</FONT> // it is so small (much probably exactly 0 due to limited accuracy)<a name="line.294"></a>
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298 <FONT color="green">295</FONT> // that the code above would fail handling it.<a name="line.295"></a>
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299 <FONT color="green">296</FONT> // So we set up an artificial 0 size step by copying states<a name="line.296"></a>
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300 <FONT color="green">297</FONT> interpolator.storeTime(stepStart);<a name="line.297"></a>
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301 <FONT color="green">298</FONT> System.arraycopy(y, 0, yTmp, 0, y0.length);<a name="line.298"></a>
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302 <FONT color="green">299</FONT> hNew = 0;<a name="line.299"></a>
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303 <FONT color="green">300</FONT> stepSize = 0;<a name="line.300"></a>
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304 <FONT color="green">301</FONT> loop = false;<a name="line.301"></a>
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305 <FONT color="green">302</FONT> } else {<a name="line.302"></a>
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306 <FONT color="green">303</FONT> // reject the step to match exactly the next switch time<a name="line.303"></a>
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307 <FONT color="green">304</FONT> hNew = dt;<a name="line.304"></a>
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308 <FONT color="green">305</FONT> interpolator.rescale(hNew);<a name="line.305"></a>
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309 <FONT color="green">306</FONT> }<a name="line.306"></a>
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310 <FONT color="green">307</FONT> } else {<a name="line.307"></a>
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311 <FONT color="green">308</FONT> // accept the step<a name="line.308"></a>
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312 <FONT color="green">309</FONT> scaled = correctedScaled;<a name="line.309"></a>
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313 <FONT color="green">310</FONT> nordsieck = nordsieckTmp;<a name="line.310"></a>
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314 <FONT color="green">311</FONT> interpolator.reinitialize(stepEnd, stepSize, scaled, nordsieck);<a name="line.311"></a>
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315 <FONT color="green">312</FONT> loop = false;<a name="line.312"></a>
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316 <FONT color="green">313</FONT> }<a name="line.313"></a>
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317 <FONT color="green">314</FONT> <a name="line.314"></a>
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318 <FONT color="green">315</FONT> } else {<a name="line.315"></a>
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319 <FONT color="green">316</FONT> // reject the step and attempt to reduce error by stepsize control<a name="line.316"></a>
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320 <FONT color="green">317</FONT> final double factor = computeStepGrowShrinkFactor(error);<a name="line.317"></a>
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321 <FONT color="green">318</FONT> hNew = filterStep(stepSize * factor, forward, false);<a name="line.318"></a>
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322 <FONT color="green">319</FONT> interpolator.rescale(hNew);<a name="line.319"></a>
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323 <FONT color="green">320</FONT> }<a name="line.320"></a>
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324 <FONT color="green">321</FONT> <a name="line.321"></a>
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325 <FONT color="green">322</FONT> }<a name="line.322"></a>
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326 <FONT color="green">323</FONT> <a name="line.323"></a>
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327 <FONT color="green">324</FONT> // the step has been accepted (may have been truncated)<a name="line.324"></a>
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328 <FONT color="green">325</FONT> final double nextStep = stepStart + stepSize;<a name="line.325"></a>
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329 <FONT color="green">326</FONT> System.arraycopy(yTmp, 0, y, 0, n);<a name="line.326"></a>
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330 <FONT color="green">327</FONT> interpolator.storeTime(nextStep);<a name="line.327"></a>
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331 <FONT color="green">328</FONT> manager.stepAccepted(nextStep, y);<a name="line.328"></a>
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332 <FONT color="green">329</FONT> lastStep = manager.stop();<a name="line.329"></a>
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333 <FONT color="green">330</FONT> <a name="line.330"></a>
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334 <FONT color="green">331</FONT> // provide the step data to the step handler<a name="line.331"></a>
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335 <FONT color="green">332</FONT> for (StepHandler handler : stepHandlers) {<a name="line.332"></a>
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336 <FONT color="green">333</FONT> interpolator.setInterpolatedTime(nextStep);<a name="line.333"></a>
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337 <FONT color="green">334</FONT> handler.handleStep(interpolator, lastStep);<a name="line.334"></a>
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338 <FONT color="green">335</FONT> }<a name="line.335"></a>
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339 <FONT color="green">336</FONT> stepStart = nextStep;<a name="line.336"></a>
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340 <FONT color="green">337</FONT> <a name="line.337"></a>
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341 <FONT color="green">338</FONT> if (!lastStep && manager.reset(stepStart, y)) {<a name="line.338"></a>
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342 <FONT color="green">339</FONT> <a name="line.339"></a>
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343 <FONT color="green">340</FONT> // some events handler has triggered changes that<a name="line.340"></a>
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344 <FONT color="green">341</FONT> // invalidate the derivatives, we need to restart from scratch<a name="line.341"></a>
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345 <FONT color="green">342</FONT> start(stepStart, y, t);<a name="line.342"></a>
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346 <FONT color="green">343</FONT> interpolator.reinitialize(stepStart, stepSize, scaled, nordsieck);<a name="line.343"></a>
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347 <FONT color="green">344</FONT> <a name="line.344"></a>
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348 <FONT color="green">345</FONT> }<a name="line.345"></a>
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349 <FONT color="green">346</FONT> <a name="line.346"></a>
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350 <FONT color="green">347</FONT> if (! lastStep) {<a name="line.347"></a>
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351 <FONT color="green">348</FONT> // in some rare cases we may get here with stepSize = 0, for example<a name="line.348"></a>
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352 <FONT color="green">349</FONT> // when an event occurs at integration start, reducing the first step<a name="line.349"></a>
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353 <FONT color="green">350</FONT> // to zero; we have to reset the step to some safe non zero value<a name="line.350"></a>
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354 <FONT color="green">351</FONT> stepSize = filterStep(stepSize, forward, true);<a name="line.351"></a>
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355 <FONT color="green">352</FONT> <a name="line.352"></a>
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356 <FONT color="green">353</FONT> // stepsize control for next step<a name="line.353"></a>
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357 <FONT color="green">354</FONT> final double factor = computeStepGrowShrinkFactor(error);<a name="line.354"></a>
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358 <FONT color="green">355</FONT> final double scaledH = stepSize * factor;<a name="line.355"></a>
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359 <FONT color="green">356</FONT> final double nextT = stepStart + scaledH;<a name="line.356"></a>
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360 <FONT color="green">357</FONT> final boolean nextIsLast = forward ? (nextT >= t) : (nextT <= t);<a name="line.357"></a>
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361 <FONT color="green">358</FONT> hNew = filterStep(scaledH, forward, nextIsLast);<a name="line.358"></a>
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362 <FONT color="green">359</FONT> interpolator.rescale(hNew);<a name="line.359"></a>
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363 <FONT color="green">360</FONT> }<a name="line.360"></a>
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364 <FONT color="green">361</FONT> <a name="line.361"></a>
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365 <FONT color="green">362</FONT> }<a name="line.362"></a>
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366 <FONT color="green">363</FONT> <a name="line.363"></a>
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367 <FONT color="green">364</FONT> final double stopTime = stepStart;<a name="line.364"></a>
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368 <FONT color="green">365</FONT> stepStart = Double.NaN;<a name="line.365"></a>
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369 <FONT color="green">366</FONT> stepSize = Double.NaN;<a name="line.366"></a>
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370 <FONT color="green">367</FONT> return stopTime;<a name="line.367"></a>
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371 <FONT color="green">368</FONT> <a name="line.368"></a>
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372 <FONT color="green">369</FONT> }<a name="line.369"></a>
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373 <FONT color="green">370</FONT> <a name="line.370"></a>
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374 <FONT color="green">371</FONT> /** Corrector for current state in Adams-Moulton method.<a name="line.371"></a>
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375 <FONT color="green">372</FONT> * <p><a name="line.372"></a>
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376 <FONT color="green">373</FONT> * This visitor implements the Taylor series formula:<a name="line.373"></a>
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377 <FONT color="green">374</FONT> * <pre><a name="line.374"></a>
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378 <FONT color="green">375</FONT> * 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><a name="line.375"></a>
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379 <FONT color="green">376</FONT> * </pre><a name="line.376"></a>
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380 <FONT color="green">377</FONT> * </p><a name="line.377"></a>
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381 <FONT color="green">378</FONT> */<a name="line.378"></a>
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382 <FONT color="green">379</FONT> private class Corrector implements RealMatrixPreservingVisitor {<a name="line.379"></a>
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383 <FONT color="green">380</FONT> <a name="line.380"></a>
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384 <FONT color="green">381</FONT> /** Previous state. */<a name="line.381"></a>
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385 <FONT color="green">382</FONT> private final double[] previous;<a name="line.382"></a>
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386 <FONT color="green">383</FONT> <a name="line.383"></a>
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387 <FONT color="green">384</FONT> /** Current scaled first derivative. */<a name="line.384"></a>
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388 <FONT color="green">385</FONT> private final double[] scaled;<a name="line.385"></a>
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389 <FONT color="green">386</FONT> <a name="line.386"></a>
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390 <FONT color="green">387</FONT> /** Current state before correction. */<a name="line.387"></a>
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391 <FONT color="green">388</FONT> private final double[] before;<a name="line.388"></a>
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392 <FONT color="green">389</FONT> <a name="line.389"></a>
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393 <FONT color="green">390</FONT> /** Current state after correction. */<a name="line.390"></a>
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394 <FONT color="green">391</FONT> private final double[] after;<a name="line.391"></a>
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395 <FONT color="green">392</FONT> <a name="line.392"></a>
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396 <FONT color="green">393</FONT> /** Simple constructor.<a name="line.393"></a>
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397 <FONT color="green">394</FONT> * @param previous previous state<a name="line.394"></a>
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398 <FONT color="green">395</FONT> * @param scaled current scaled first derivative<a name="line.395"></a>
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399 <FONT color="green">396</FONT> * @param state state to correct (will be overwritten after visit)<a name="line.396"></a>
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400 <FONT color="green">397</FONT> */<a name="line.397"></a>
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401 <FONT color="green">398</FONT> public Corrector(final double[] previous, final double[] scaled, final double[] state) {<a name="line.398"></a>
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402 <FONT color="green">399</FONT> this.previous = previous;<a name="line.399"></a>
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403 <FONT color="green">400</FONT> this.scaled = scaled;<a name="line.400"></a>
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404 <FONT color="green">401</FONT> this.after = state;<a name="line.401"></a>
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405 <FONT color="green">402</FONT> this.before = state.clone();<a name="line.402"></a>
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406 <FONT color="green">403</FONT> }<a name="line.403"></a>
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407 <FONT color="green">404</FONT> <a name="line.404"></a>
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408 <FONT color="green">405</FONT> /** {@inheritDoc} */<a name="line.405"></a>
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409 <FONT color="green">406</FONT> public void start(int rows, int columns,<a name="line.406"></a>
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410 <FONT color="green">407</FONT> int startRow, int endRow, int startColumn, int endColumn) {<a name="line.407"></a>
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411 <FONT color="green">408</FONT> Arrays.fill(after, 0.0);<a name="line.408"></a>
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412 <FONT color="green">409</FONT> }<a name="line.409"></a>
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413 <FONT color="green">410</FONT> <a name="line.410"></a>
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414 <FONT color="green">411</FONT> /** {@inheritDoc} */<a name="line.411"></a>
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415 <FONT color="green">412</FONT> public void visit(int row, int column, double value)<a name="line.412"></a>
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416 <FONT color="green">413</FONT> throws MatrixVisitorException {<a name="line.413"></a>
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417 <FONT color="green">414</FONT> if ((row & 0x1) == 0) {<a name="line.414"></a>
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418 <FONT color="green">415</FONT> after[column] -= value;<a name="line.415"></a>
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419 <FONT color="green">416</FONT> } else {<a name="line.416"></a>
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420 <FONT color="green">417</FONT> after[column] += value;<a name="line.417"></a>
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421 <FONT color="green">418</FONT> }<a name="line.418"></a>
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422 <FONT color="green">419</FONT> }<a name="line.419"></a>
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423 <FONT color="green">420</FONT> <a name="line.420"></a>
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424 <FONT color="green">421</FONT> /**<a name="line.421"></a>
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425 <FONT color="green">422</FONT> * End visiting te Nordsieck vector.<a name="line.422"></a>
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426 <FONT color="green">423</FONT> * <p>The correction is used to control stepsize. So its amplitude is<a name="line.423"></a>
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427 <FONT color="green">424</FONT> * considered to be an error, which must be normalized according to<a name="line.424"></a>
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428 <FONT color="green">425</FONT> * error control settings. If the normalized value is greater than 1,<a name="line.425"></a>
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429 <FONT color="green">426</FONT> * the correction was too large and the step must be rejected.</p><a name="line.426"></a>
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430 <FONT color="green">427</FONT> * @return the normalized correction, if greater than 1, the step<a name="line.427"></a>
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431 <FONT color="green">428</FONT> * must be rejected<a name="line.428"></a>
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432 <FONT color="green">429</FONT> */<a name="line.429"></a>
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433 <FONT color="green">430</FONT> public double end() {<a name="line.430"></a>
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434 <FONT color="green">431</FONT> <a name="line.431"></a>
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435 <FONT color="green">432</FONT> double error = 0;<a name="line.432"></a>
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436 <FONT color="green">433</FONT> for (int i = 0; i < after.length; ++i) {<a name="line.433"></a>
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437 <FONT color="green">434</FONT> after[i] += previous[i] + scaled[i];<a name="line.434"></a>
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438 <FONT color="green">435</FONT> final double yScale = Math.max(Math.abs(previous[i]), Math.abs(after[i]));<a name="line.435"></a>
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439 <FONT color="green">436</FONT> final double tol = (vecAbsoluteTolerance == null) ?<a name="line.436"></a>
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440 <FONT color="green">437</FONT> (scalAbsoluteTolerance + scalRelativeTolerance * yScale) :<a name="line.437"></a>
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441 <FONT color="green">438</FONT> (vecAbsoluteTolerance[i] + vecRelativeTolerance[i] * yScale);<a name="line.438"></a>
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442 <FONT color="green">439</FONT> final double ratio = (after[i] - before[i]) / tol;<a name="line.439"></a>
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443 <FONT color="green">440</FONT> error += ratio * ratio;<a name="line.440"></a>
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444 <FONT color="green">441</FONT> }<a name="line.441"></a>
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445 <FONT color="green">442</FONT> <a name="line.442"></a>
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446 <FONT color="green">443</FONT> return Math.sqrt(error / after.length);<a name="line.443"></a>
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447 <FONT color="green">444</FONT> <a name="line.444"></a>
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448 <FONT color="green">445</FONT> }<a name="line.445"></a>
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449 <FONT color="green">446</FONT> }<a name="line.446"></a>
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450 <FONT color="green">447</FONT> <a name="line.447"></a>
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451 <FONT color="green">448</FONT> }<a name="line.448"></a>
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511
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512 </PRE>
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513 </BODY>
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514 </HTML>
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