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
18package org.apache.commons.math.ode.nonstiff;
19
20import org.apache.commons.math.util.FastMath;
21
22
23/**
24 * This class implements the 8(5,3) Dormand-Prince integrator for Ordinary
25 * Differential Equations.
26 *
27 * <p>This integrator is an embedded Runge-Kutta integrator
28 * of order 8(5,3) used in local extrapolation mode (i.e. the solution
29 * is computed using the high order formula) with stepsize control
30 * (and automatic step initialization) and continuous output. This
31 * method uses 12 functions evaluations per step for integration and 4
32 * evaluations for interpolation. However, since the first
33 * interpolation evaluation is the same as the first integration
34 * evaluation of the next step, we have included it in the integrator
35 * rather than in the interpolator and specified the method was an
36 * <i>fsal</i>. Hence, despite we have 13 stages here, the cost is
37 * really 12 evaluations per step even if no interpolation is done,
38 * and the overcost of interpolation is only 3 evaluations.</p>
39 *
40 * <p>This method is based on an 8(6) method by Dormand and Prince
41 * (i.e. order 8 for the integration and order 6 for error estimation)
42 * modified by Hairer and Wanner to use a 5th order error estimator
43 * with 3rd order correction. This modification was introduced because
44 * the original method failed in some cases (wrong steps can be
45 * accepted when step size is too large, for example in the
46 * Brusselator problem) and also had <i>severe difficulties when
47 * applied to problems with discontinuities</i>. This modification is
48 * explained in the second edition of the first volume (Nonstiff
49 * Problems) of the reference book by Hairer, Norsett and Wanner:
50 * <i>Solving Ordinary Differential Equations</i> (Springer-Verlag,
51 * ISBN 3-540-56670-8).</p>
52 *
53 * @version $Revision: 990655 $ $Date: 2010-08-29 23:49:40 +0200 (dim. 29 août 2010) $
54 * @since 1.2
55 */
56
57public class DormandPrince853Integrator extends EmbeddedRungeKuttaIntegrator {
58
59  /** Integrator method name. */
60  private static final String METHOD_NAME = "Dormand-Prince 8 (5, 3)";
61
62  /** Time steps Butcher array. */
63  private static final double[] STATIC_C = {
64    (12.0 - 2.0 * FastMath.sqrt(6.0)) / 135.0, (6.0 - FastMath.sqrt(6.0)) / 45.0, (6.0 - FastMath.sqrt(6.0)) / 30.0,
65    (6.0 + FastMath.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,
66    6.0/7.0, 1.0, 1.0
67  };
68
69  /** Internal weights Butcher array. */
70  private static final double[][] STATIC_A = {
71
72    // k2
73    {(12.0 - 2.0 * FastMath.sqrt(6.0)) / 135.0},
74
75    // k3
76    {(6.0 - FastMath.sqrt(6.0)) / 180.0, (6.0 - FastMath.sqrt(6.0)) / 60.0},
77
78    // k4
79    {(6.0 - FastMath.sqrt(6.0)) / 120.0, 0.0, (6.0 - FastMath.sqrt(6.0)) / 40.0},
80
81    // k5
82    {(462.0 + 107.0 * FastMath.sqrt(6.0)) / 3000.0, 0.0,
83     (-402.0 - 197.0 * FastMath.sqrt(6.0)) / 1000.0, (168.0 + 73.0 * FastMath.sqrt(6.0)) / 375.0},
84
85    // k6
86    {1.0 / 27.0, 0.0, 0.0, (16.0 + FastMath.sqrt(6.0)) / 108.0, (16.0 - FastMath.sqrt(6.0)) / 108.0},
87
88    // k7
89    {19.0 / 512.0, 0.0, 0.0, (118.0 + 23.0 * FastMath.sqrt(6.0)) / 1024.0,
90     (118.0 - 23.0 * FastMath.sqrt(6.0)) / 1024.0, -9.0 / 512.0},
91
92    // k8
93    {13772.0 / 371293.0, 0.0, 0.0, (51544.0 + 4784.0 * FastMath.sqrt(6.0)) / 371293.0,
94     (51544.0 - 4784.0 * FastMath.sqrt(6.0)) / 371293.0, -5688.0 / 371293.0, 3072.0 / 371293.0},
95
96    // k9
97    {58656157643.0 / 93983540625.0, 0.0, 0.0,
98     (-1324889724104.0 - 318801444819.0 * FastMath.sqrt(6.0)) / 626556937500.0,
99     (-1324889724104.0 + 318801444819.0 * FastMath.sqrt(6.0)) / 626556937500.0,
100     96044563816.0 / 3480871875.0, 5682451879168.0 / 281950621875.0,
101     -165125654.0 / 3796875.0},
102
103    // k10
104    {8909899.0 / 18653125.0, 0.0, 0.0,
105     (-4521408.0 - 1137963.0 * FastMath.sqrt(6.0)) / 2937500.0,
106     (-4521408.0 + 1137963.0 * FastMath.sqrt(6.0)) / 2937500.0,
107     96663078.0 / 4553125.0, 2107245056.0 / 137915625.0,
108     -4913652016.0 / 147609375.0, -78894270.0 / 3880452869.0},
109
110    // k11
111    {-20401265806.0 / 21769653311.0, 0.0, 0.0,
112     (354216.0 + 94326.0 * FastMath.sqrt(6.0)) / 112847.0,
113     (354216.0 - 94326.0 * FastMath.sqrt(6.0)) / 112847.0,
114     -43306765128.0 / 5313852383.0, -20866708358144.0 / 1126708119789.0,
115     14886003438020.0 / 654632330667.0, 35290686222309375.0 / 14152473387134411.0,
116     -1477884375.0 / 485066827.0},
117
118    // k12
119    {39815761.0 / 17514443.0, 0.0, 0.0,
120     (-3457480.0 - 960905.0 * FastMath.sqrt(6.0)) / 551636.0,
121     (-3457480.0 + 960905.0 * FastMath.sqrt(6.0)) / 551636.0,
122     -844554132.0 / 47026969.0, 8444996352.0 / 302158619.0,
123     -2509602342.0 / 877790785.0, -28388795297996250.0 / 3199510091356783.0,
124     226716250.0 / 18341897.0, 1371316744.0 / 2131383595.0},
125
126    // k13 should be for interpolation only, but since it is the same
127    // stage as the first evaluation of the next step, we perform it
128    // here at no cost by specifying this is an fsal method
129    {104257.0/1920240.0, 0.0, 0.0, 0.0, 0.0, 3399327.0/763840.0,
130     66578432.0/35198415.0, -1674902723.0/288716400.0,
131     54980371265625.0/176692375811392.0, -734375.0/4826304.0,
132     171414593.0/851261400.0, 137909.0/3084480.0}
133
134  };
135
136  /** Propagation weights Butcher array. */
137  private static final double[] STATIC_B = {
138      104257.0/1920240.0,
139      0.0,
140      0.0,
141      0.0,
142      0.0,
143      3399327.0/763840.0,
144      66578432.0/35198415.0,
145      -1674902723.0/288716400.0,
146      54980371265625.0/176692375811392.0,
147      -734375.0/4826304.0,
148      171414593.0/851261400.0,
149      137909.0/3084480.0,
150      0.0
151  };
152
153  /** First error weights array, element 1. */
154  private static final double E1_01 =         116092271.0 / 8848465920.0;
155
156  // elements 2 to 5 are zero, so they are neither stored nor used
157
158  /** First error weights array, element 6. */
159  private static final double E1_06 =          -1871647.0 / 1527680.0;
160
161  /** First error weights array, element 7. */
162  private static final double E1_07 =         -69799717.0 / 140793660.0;
163
164  /** First error weights array, element 8. */
165  private static final double E1_08 =     1230164450203.0 / 739113984000.0;
166
167  /** First error weights array, element 9. */
168  private static final double E1_09 = -1980813971228885.0 / 5654156025964544.0;
169
170  /** First error weights array, element 10. */
171  private static final double E1_10 =         464500805.0 / 1389975552.0;
172
173  /** First error weights array, element 11. */
174  private static final double E1_11 =     1606764981773.0 / 19613062656000.0;
175
176  /** First error weights array, element 12. */
177  private static final double E1_12 =           -137909.0 / 6168960.0;
178
179
180  /** Second error weights array, element 1. */
181  private static final double E2_01 =           -364463.0 / 1920240.0;
182
183  // elements 2 to 5 are zero, so they are neither stored nor used
184
185  /** Second error weights array, element 6. */
186  private static final double E2_06 =           3399327.0 / 763840.0;
187
188  /** Second error weights array, element 7. */
189  private static final double E2_07 =          66578432.0 / 35198415.0;
190
191  /** Second error weights array, element 8. */
192  private static final double E2_08 =       -1674902723.0 / 288716400.0;
193
194  /** Second error weights array, element 9. */
195  private static final double E2_09 =   -74684743568175.0 / 176692375811392.0;
196
197  /** Second error weights array, element 10. */
198  private static final double E2_10 =           -734375.0 / 4826304.0;
199
200  /** Second error weights array, element 11. */
201  private static final double E2_11 =         171414593.0 / 851261400.0;
202
203  /** Second error weights array, element 12. */
204  private static final double E2_12 =             69869.0 / 3084480.0;
205
206  /** Simple constructor.
207   * Build an eighth order Dormand-Prince integrator with the given step bounds
208   * @param minStep minimal step (must be positive even for backward
209   * integration), the last step can be smaller than this
210   * @param maxStep maximal step (must be positive even for backward
211   * integration)
212   * @param scalAbsoluteTolerance allowed absolute error
213   * @param scalRelativeTolerance allowed relative error
214   */
215  public DormandPrince853Integrator(final double minStep, final double maxStep,
216                                    final double scalAbsoluteTolerance,
217                                    final double scalRelativeTolerance) {
218    super(METHOD_NAME, true, STATIC_C, STATIC_A, STATIC_B,
219          new DormandPrince853StepInterpolator(),
220          minStep, maxStep, scalAbsoluteTolerance, scalRelativeTolerance);
221  }
222
223  /** Simple constructor.
224   * Build an eighth order Dormand-Prince integrator with the given step bounds
225   * @param minStep minimal step (must be positive even for backward
226   * integration), the last step can be smaller than this
227   * @param maxStep maximal step (must be positive even for backward
228   * integration)
229   * @param vecAbsoluteTolerance allowed absolute error
230   * @param vecRelativeTolerance allowed relative error
231   */
232  public DormandPrince853Integrator(final double minStep, final double maxStep,
233                                    final double[] vecAbsoluteTolerance,
234                                    final double[] vecRelativeTolerance) {
235    super(METHOD_NAME, true, STATIC_C, STATIC_A, STATIC_B,
236          new DormandPrince853StepInterpolator(),
237          minStep, maxStep, vecAbsoluteTolerance, vecRelativeTolerance);
238  }
239
240  /** {@inheritDoc} */
241  @Override
242  public int getOrder() {
243    return 8;
244  }
245
246  /** {@inheritDoc} */
247  @Override
248  protected double estimateError(final double[][] yDotK,
249                                 final double[] y0, final double[] y1,
250                                 final double h) {
251    double error1 = 0;
252    double error2 = 0;
253
254    for (int j = 0; j < mainSetDimension; ++j) {
255      final double errSum1 = E1_01 * yDotK[0][j]  + E1_06 * yDotK[5][j] +
256                             E1_07 * yDotK[6][j]  + E1_08 * yDotK[7][j] +
257                             E1_09 * yDotK[8][j]  + E1_10 * yDotK[9][j] +
258                             E1_11 * yDotK[10][j] + E1_12 * yDotK[11][j];
259      final double errSum2 = E2_01 * yDotK[0][j]  + E2_06 * yDotK[5][j] +
260                             E2_07 * yDotK[6][j]  + E2_08 * yDotK[7][j] +
261                             E2_09 * yDotK[8][j]  + E2_10 * yDotK[9][j] +
262                             E2_11 * yDotK[10][j] + E2_12 * yDotK[11][j];
263
264      final double yScale = FastMath.max(FastMath.abs(y0[j]), FastMath.abs(y1[j]));
265      final double tol = (vecAbsoluteTolerance == null) ?
266                         (scalAbsoluteTolerance + scalRelativeTolerance * yScale) :
267                         (vecAbsoluteTolerance[j] + vecRelativeTolerance[j] * yScale);
268      final double ratio1  = errSum1 / tol;
269      error1        += ratio1 * ratio1;
270      final double ratio2  = errSum2 / tol;
271      error2        += ratio2 * ratio2;
272    }
273
274    double den = error1 + 0.01 * error2;
275    if (den <= 0.0) {
276      den = 1.0;
277    }
278
279    return FastMath.abs(h) * error1 / FastMath.sqrt(mainSetDimension * den);
280
281  }
282
283}
284