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1   /*
2    * Licensed to the Apache Software Foundation (ASF) under one or more
3    * contributor license agreements.  See the NOTICE file distributed with
4    * this work for additional information regarding copyright ownership.
5    * The ASF licenses this file to You under the Apache License, Version 2.0
6    * (the "License"); you may not use this file except in compliance with
7    * the License.  You may obtain a copy of the License at
8    *
9    *      http://www.apache.org/licenses/LICENSE-2.0
10   *
11   * Unless required by applicable law or agreed to in writing, software
12   * distributed under the License is distributed on an "AS IS" BASIS,
13   * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
14   * See the License for the specific language governing permissions and
15   * limitations under the License.
16   */
17  
18  package org.apache.commons.math.ode.nonstiff;
19  
20  
21  /**
22   * This class implements the 8(5,3) Dormand-Prince integrator for Ordinary
23   * Differential Equations.
24   *
25   * <p>This integrator is an embedded Runge-Kutta integrator
26   * of order 8(5,3) used in local extrapolation mode (i.e. the solution
27   * is computed using the high order formula) with stepsize control
28   * (and automatic step initialization) and continuous output. This
29   * method uses 12 functions evaluations per step for integration and 4
30   * evaluations for interpolation. However, since the first
31   * interpolation evaluation is the same as the first integration
32   * evaluation of the next step, we have included it in the integrator
33   * rather than in the interpolator and specified the method was an
34   * <i>fsal</i>. Hence, despite we have 13 stages here, the cost is
35   * really 12 evaluations per step even if no interpolation is done,
36   * and the overcost of interpolation is only 3 evaluations.</p>
37   *
38   * <p>This method is based on an 8(6) method by Dormand and Prince
39   * (i.e. order 8 for the integration and order 6 for error estimation)
40   * modified by Hairer and Wanner to use a 5th order error estimator
41   * with 3rd order correction. This modification was introduced because
42   * the original method failed in some cases (wrong steps can be
43   * accepted when step size is too large, for example in the
44   * Brusselator problem) and also had <i>severe difficulties when
45   * applied to problems with discontinuities</i>. This modification is
46   * explained in the second edition of the first volume (Nonstiff
47   * Problems) of the reference book by Hairer, Norsett and Wanner:
48   * <i>Solving Ordinary Differential Equations</i> (Springer-Verlag,
49   * ISBN 3-540-56670-8).</p>
50   *
51   * @version $Revision: 786881 $ $Date: 2009-06-20 14:53:08 -0400 (Sat, 20 Jun 2009) $
52   * @since 1.2
53   */
54  
55  public class DormandPrince853Integrator extends EmbeddedRungeKuttaIntegrator {
56  
57    /** Integrator method name. */
58    private static final String METHOD_NAME = "Dormand-Prince 8 (5, 3)";
59  
60    /** Time steps Butcher array. */
61    private static final double[] staticC = {
62      (12.0 - 2.0 * Math.sqrt(6.0)) / 135.0, (6.0 - Math.sqrt(6.0)) / 45.0, (6.0 - Math.sqrt(6.0)) / 30.0,
63      (6.0 + Math.sqrt(6.0)) / 30.0, 1.0/3.0, 1.0/4.0, 4.0/13.0, 127.0/195.0, 3.0/5.0,
64      6.0/7.0, 1.0, 1.0
65    };
66  
67    /** Internal weights Butcher array. */
68    private static final double[][] staticA = {
69  
70      // k2
71      {(12.0 - 2.0 * Math.sqrt(6.0)) / 135.0},
72  
73      // k3
74      {(6.0 - Math.sqrt(6.0)) / 180.0, (6.0 - Math.sqrt(6.0)) / 60.0},
75  
76      // k4
77      {(6.0 - Math.sqrt(6.0)) / 120.0, 0.0, (6.0 - Math.sqrt(6.0)) / 40.0},
78  
79      // k5
80      {(462.0 + 107.0 * Math.sqrt(6.0)) / 3000.0, 0.0,
81       (-402.0 - 197.0 * Math.sqrt(6.0)) / 1000.0, (168.0 + 73.0 * Math.sqrt(6.0)) / 375.0},
82  
83      // k6
84      {1.0 / 27.0, 0.0, 0.0, (16.0 + Math.sqrt(6.0)) / 108.0, (16.0 - Math.sqrt(6.0)) / 108.0},
85  
86      // k7
87      {19.0 / 512.0, 0.0, 0.0, (118.0 + 23.0 * Math.sqrt(6.0)) / 1024.0,
88       (118.0 - 23.0 * Math.sqrt(6.0)) / 1024.0, -9.0 / 512.0},
89  
90      // k8
91      {13772.0 / 371293.0, 0.0, 0.0, (51544.0 + 4784.0 * Math.sqrt(6.0)) / 371293.0,
92       (51544.0 - 4784.0 * Math.sqrt(6.0)) / 371293.0, -5688.0 / 371293.0, 3072.0 / 371293.0},
93  
94      // k9
95      {58656157643.0 / 93983540625.0, 0.0, 0.0,
96       (-1324889724104.0 - 318801444819.0 * Math.sqrt(6.0)) / 626556937500.0,
97       (-1324889724104.0 + 318801444819.0 * Math.sqrt(6.0)) / 626556937500.0,
98       96044563816.0 / 3480871875.0, 5682451879168.0 / 281950621875.0,
99       -165125654.0 / 3796875.0},
100 
101     // k10
102     {8909899.0 / 18653125.0, 0.0, 0.0,
103      (-4521408.0 - 1137963.0 * Math.sqrt(6.0)) / 2937500.0,
104      (-4521408.0 + 1137963.0 * Math.sqrt(6.0)) / 2937500.0,
105      96663078.0 / 4553125.0, 2107245056.0 / 137915625.0,
106      -4913652016.0 / 147609375.0, -78894270.0 / 3880452869.0},
107 
108     // k11
109     {-20401265806.0 / 21769653311.0, 0.0, 0.0,
110      (354216.0 + 94326.0 * Math.sqrt(6.0)) / 112847.0,
111      (354216.0 - 94326.0 * Math.sqrt(6.0)) / 112847.0,
112      -43306765128.0 / 5313852383.0, -20866708358144.0 / 1126708119789.0,
113      14886003438020.0 / 654632330667.0, 35290686222309375.0 / 14152473387134411.0,
114      -1477884375.0 / 485066827.0},
115 
116     // k12
117     {39815761.0 / 17514443.0, 0.0, 0.0,
118      (-3457480.0 - 960905.0 * Math.sqrt(6.0)) / 551636.0,
119      (-3457480.0 + 960905.0 * Math.sqrt(6.0)) / 551636.0,
120      -844554132.0 / 47026969.0, 8444996352.0 / 302158619.0,
121      -2509602342.0 / 877790785.0, -28388795297996250.0 / 3199510091356783.0,
122      226716250.0 / 18341897.0, 1371316744.0 / 2131383595.0},
123 
124     // k13 should be for interpolation only, but since it is the same
125     // stage as the first evaluation of the next step, we perform it
126     // here at no cost by specifying this is an fsal method
127     {104257.0/1920240.0, 0.0, 0.0, 0.0, 0.0, 3399327.0/763840.0,
128      66578432.0/35198415.0, -1674902723.0/288716400.0,
129      54980371265625.0/176692375811392.0, -734375.0/4826304.0,
130      171414593.0/851261400.0, 137909.0/3084480.0}
131 
132   };
133 
134   /** Propagation weights Butcher array. */
135   private static final double[] staticB = {
136       104257.0/1920240.0,
137       0.0,
138       0.0,
139       0.0,
140       0.0,
141       3399327.0/763840.0,
142       66578432.0/35198415.0,
143       -1674902723.0/288716400.0,
144       54980371265625.0/176692375811392.0,
145       -734375.0/4826304.0,
146       171414593.0/851261400.0,
147       137909.0/3084480.0,
148       0.0
149   };
150 
151   /** First error weights array, element 1. */
152   private static final double e1_01 =         116092271.0 / 8848465920.0;
153 
154   // elements 2 to 5 are zero, so they are neither stored nor used
155 
156   /** First error weights array, element 6. */
157   private static final double e1_06 =          -1871647.0 / 1527680.0;
158 
159   /** First error weights array, element 7. */
160   private static final double e1_07 =         -69799717.0 / 140793660.0;
161 
162   /** First error weights array, element 8. */
163   private static final double e1_08 =     1230164450203.0 / 739113984000.0;
164 
165   /** First error weights array, element 9. */
166   private static final double e1_09 = -1980813971228885.0 / 5654156025964544.0;
167 
168   /** First error weights array, element 10. */
169   private static final double e1_10 =         464500805.0 / 1389975552.0;
170 
171   /** First error weights array, element 11. */
172   private static final double e1_11 =     1606764981773.0 / 19613062656000.0;
173 
174   /** First error weights array, element 12. */
175   private static final double e1_12 =           -137909.0 / 6168960.0;
176 
177 
178   /** Second error weights array, element 1. */
179   private static final double e2_01 =           -364463.0 / 1920240.0;
180 
181   // elements 2 to 5 are zero, so they are neither stored nor used
182 
183   /** Second error weights array, element 6. */
184   private static final double e2_06 =           3399327.0 / 763840.0;
185 
186   /** Second error weights array, element 7. */
187   private static final double e2_07 =          66578432.0 / 35198415.0;
188 
189   /** Second error weights array, element 8. */
190   private static final double e2_08 =       -1674902723.0 / 288716400.0;
191 
192   /** Second error weights array, element 9. */
193   private static final double e2_09 =   -74684743568175.0 / 176692375811392.0;
194 
195   /** Second error weights array, element 10. */
196   private static final double e2_10 =           -734375.0 / 4826304.0;
197 
198   /** Second error weights array, element 11. */
199   private static final double e2_11 =         171414593.0 / 851261400.0;
200 
201   /** Second error weights array, element 12. */
202   private static final double e2_12 =             69869.0 / 3084480.0;
203 
204   /** Simple constructor.
205    * Build an eighth order Dormand-Prince integrator with the given step bounds
206    * @param minStep minimal step (must be positive even for backward
207    * integration), the last step can be smaller than this
208    * @param maxStep maximal step (must be positive even for backward
209    * integration)
210    * @param scalAbsoluteTolerance allowed absolute error
211    * @param scalRelativeTolerance allowed relative error
212    */
213   public DormandPrince853Integrator(final double minStep, final double maxStep,
214                                     final double scalAbsoluteTolerance,
215                                     final double scalRelativeTolerance) {
216     super(METHOD_NAME, true, staticC, staticA, staticB,
217           new DormandPrince853StepInterpolator(),
218           minStep, maxStep, scalAbsoluteTolerance, scalRelativeTolerance);
219   }
220 
221   /** Simple constructor.
222    * Build an eighth order Dormand-Prince integrator with the given step bounds
223    * @param minStep minimal step (must be positive even for backward
224    * integration), the last step can be smaller than this
225    * @param maxStep maximal step (must be positive even for backward
226    * integration)
227    * @param vecAbsoluteTolerance allowed absolute error
228    * @param vecRelativeTolerance allowed relative error
229    */
230   public DormandPrince853Integrator(final double minStep, final double maxStep,
231                                     final double[] vecAbsoluteTolerance,
232                                     final double[] vecRelativeTolerance) {
233     super(METHOD_NAME, true, staticC, staticA, staticB,
234           new DormandPrince853StepInterpolator(),
235           minStep, maxStep, vecAbsoluteTolerance, vecRelativeTolerance);
236   }
237 
238   /** {@inheritDoc} */
239   @Override
240   public int getOrder() {
241     return 8;
242   }
243 
244   /** {@inheritDoc} */
245   @Override
246   protected double estimateError(final double[][] yDotK,
247                                  final double[] y0, final double[] y1,
248                                  final double h) {
249     double error1 = 0;
250     double error2 = 0;
251 
252     for (int j = 0; j < y0.length; ++j) {
253       final double errSum1 = e1_01 * yDotK[0][j]  + e1_06 * yDotK[5][j] +
254                              e1_07 * yDotK[6][j]  + e1_08 * yDotK[7][j] +
255                              e1_09 * yDotK[8][j]  + e1_10 * yDotK[9][j] +
256                              e1_11 * yDotK[10][j] + e1_12 * yDotK[11][j];
257       final double errSum2 = e2_01 * yDotK[0][j]  + e2_06 * yDotK[5][j] +
258                              e2_07 * yDotK[6][j]  + e2_08 * yDotK[7][j] +
259                              e2_09 * yDotK[8][j]  + e2_10 * yDotK[9][j] +
260                              e2_11 * yDotK[10][j] + e2_12 * yDotK[11][j];
261 
262       final double yScale = Math.max(Math.abs(y0[j]), Math.abs(y1[j]));
263       final double tol = (vecAbsoluteTolerance == null) ?
264                          (scalAbsoluteTolerance + scalRelativeTolerance * yScale) :
265                          (vecAbsoluteTolerance[j] + vecRelativeTolerance[j] * yScale);
266       final double ratio1  = errSum1 / tol;
267       error1        += ratio1 * ratio1;
268       final double ratio2  = errSum2 / tol;
269       error2        += ratio2 * ratio2;
270     }
271 
272     double den = error1 + 0.01 * error2;
273     if (den <= 0.0) {
274       den = 1.0;
275     }
276 
277     return Math.abs(h) * error1 / Math.sqrt(y0.length * den);
278 
279   }
280 
281 }