001/*
002 * Licensed to the Apache Software Foundation (ASF) under one or more
003 * contributor license agreements.  See the NOTICE file distributed with
004 * this work for additional information regarding copyright ownership.
005 * The ASF licenses this file to You under the Apache License, Version 2.0
006 * (the "License"); you may not use this file except in compliance with
007 * the License.  You may obtain a copy of the License at
008 *
009 *      http://www.apache.org/licenses/LICENSE-2.0
010 *
011 * Unless required by applicable law or agreed to in writing, software
012 * distributed under the License is distributed on an "AS IS" BASIS,
013 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
014 * See the License for the specific language governing permissions and
015 * limitations under the License.
016 */
017
018package org.apache.commons.math4.legacy.ode.nonstiff;
019
020import org.apache.commons.math4.core.jdkmath.JdkMath;
021
022
023/**
024 * This class implements the 8(5,3) Dormand-Prince integrator for Ordinary
025 * Differential Equations.
026 *
027 * <p>This integrator is an embedded Runge-Kutta integrator
028 * of order 8(5,3) used in local extrapolation mode (i.e. the solution
029 * is computed using the high order formula) with stepsize control
030 * (and automatic step initialization) and continuous output. This
031 * method uses 12 functions evaluations per step for integration and 4
032 * evaluations for interpolation. However, since the first
033 * interpolation evaluation is the same as the first integration
034 * evaluation of the next step, we have included it in the integrator
035 * rather than in the interpolator and specified the method was an
036 * <i>fsal</i>. Hence, despite we have 13 stages here, the cost is
037 * really 12 evaluations per step even if no interpolation is done,
038 * and the overcost of interpolation is only 3 evaluations.</p>
039 *
040 * <p>This method is based on an 8(6) method by Dormand and Prince
041 * (i.e. order 8 for the integration and order 6 for error estimation)
042 * modified by Hairer and Wanner to use a 5th order error estimator
043 * with 3rd order correction. This modification was introduced because
044 * the original method failed in some cases (wrong steps can be
045 * accepted when step size is too large, for example in the
046 * Brusselator problem) and also had <i>severe difficulties when
047 * applied to problems with discontinuities</i>. This modification is
048 * explained in the second edition of the first volume (Nonstiff
049 * Problems) of the reference book by Hairer, Norsett and Wanner:
050 * <i>Solving Ordinary Differential Equations</i> (Springer-Verlag,
051 * ISBN 3-540-56670-8).</p>
052 *
053 * @since 1.2
054 */
055
056public class DormandPrince853Integrator extends EmbeddedRungeKuttaIntegrator {
057
058  /** Integrator method name. */
059  private static final String METHOD_NAME = "Dormand-Prince 8 (5, 3)";
060
061  /** Time steps Butcher array. */
062  private static final double[] STATIC_C = {
063    (12.0 - 2.0 * JdkMath.sqrt(6.0)) / 135.0, (6.0 - JdkMath.sqrt(6.0)) / 45.0, (6.0 - JdkMath.sqrt(6.0)) / 30.0,
064    (6.0 + JdkMath.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,
065    6.0/7.0, 1.0, 1.0
066  };
067
068  /** Internal weights Butcher array. */
069  private static final double[][] STATIC_A = {
070
071    // k2
072    {(12.0 - 2.0 * JdkMath.sqrt(6.0)) / 135.0},
073
074    // k3
075    {(6.0 - JdkMath.sqrt(6.0)) / 180.0, (6.0 - JdkMath.sqrt(6.0)) / 60.0},
076
077    // k4
078    {(6.0 - JdkMath.sqrt(6.0)) / 120.0, 0.0, (6.0 - JdkMath.sqrt(6.0)) / 40.0},
079
080    // k5
081    {(462.0 + 107.0 * JdkMath.sqrt(6.0)) / 3000.0, 0.0,
082     (-402.0 - 197.0 * JdkMath.sqrt(6.0)) / 1000.0, (168.0 + 73.0 * JdkMath.sqrt(6.0)) / 375.0},
083
084    // k6
085    {1.0 / 27.0, 0.0, 0.0, (16.0 + JdkMath.sqrt(6.0)) / 108.0, (16.0 - JdkMath.sqrt(6.0)) / 108.0},
086
087    // k7
088    {19.0 / 512.0, 0.0, 0.0, (118.0 + 23.0 * JdkMath.sqrt(6.0)) / 1024.0,
089     (118.0 - 23.0 * JdkMath.sqrt(6.0)) / 1024.0, -9.0 / 512.0},
090
091    // k8
092    {13772.0 / 371293.0, 0.0, 0.0, (51544.0 + 4784.0 * JdkMath.sqrt(6.0)) / 371293.0,
093     (51544.0 - 4784.0 * JdkMath.sqrt(6.0)) / 371293.0, -5688.0 / 371293.0, 3072.0 / 371293.0},
094
095    // k9
096    {58656157643.0 / 93983540625.0, 0.0, 0.0,
097     (-1324889724104.0 - 318801444819.0 * JdkMath.sqrt(6.0)) / 626556937500.0,
098     (-1324889724104.0 + 318801444819.0 * JdkMath.sqrt(6.0)) / 626556937500.0,
099     96044563816.0 / 3480871875.0, 5682451879168.0 / 281950621875.0,
100     -165125654.0 / 3796875.0},
101
102    // k10
103    {8909899.0 / 18653125.0, 0.0, 0.0,
104     (-4521408.0 - 1137963.0 * JdkMath.sqrt(6.0)) / 2937500.0,
105     (-4521408.0 + 1137963.0 * JdkMath.sqrt(6.0)) / 2937500.0,
106     96663078.0 / 4553125.0, 2107245056.0 / 137915625.0,
107     -4913652016.0 / 147609375.0, -78894270.0 / 3880452869.0},
108
109    // k11
110    {-20401265806.0 / 21769653311.0, 0.0, 0.0,
111     (354216.0 + 94326.0 * JdkMath.sqrt(6.0)) / 112847.0,
112     (354216.0 - 94326.0 * JdkMath.sqrt(6.0)) / 112847.0,
113     -43306765128.0 / 5313852383.0, -20866708358144.0 / 1126708119789.0,
114     14886003438020.0 / 654632330667.0, 35290686222309375.0 / 14152473387134411.0,
115     -1477884375.0 / 485066827.0},
116
117    // k12
118    {39815761.0 / 17514443.0, 0.0, 0.0,
119     (-3457480.0 - 960905.0 * JdkMath.sqrt(6.0)) / 551636.0,
120     (-3457480.0 + 960905.0 * JdkMath.sqrt(6.0)) / 551636.0,
121     -844554132.0 / 47026969.0, 8444996352.0 / 302158619.0,
122     -2509602342.0 / 877790785.0, -28388795297996250.0 / 3199510091356783.0,
123     226716250.0 / 18341897.0, 1371316744.0 / 2131383595.0},
124
125    // k13 should be for interpolation only, but since it is the same
126    // stage as the first evaluation of the next step, we perform it
127    // here at no cost by specifying this is an fsal method
128    {104257.0/1920240.0, 0.0, 0.0, 0.0, 0.0, 3399327.0/763840.0,
129     66578432.0/35198415.0, -1674902723.0/288716400.0,
130     54980371265625.0/176692375811392.0, -734375.0/4826304.0,
131     171414593.0/851261400.0, 137909.0/3084480.0}
132  };
133
134  /** Propagation weights Butcher array. */
135  private static final double[] STATIC_B = {
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 (sign is irrelevant, regardless of
207   * integration direction, forward or backward), the last step can
208   * be smaller than this
209   * @param maxStep maximal step (sign is irrelevant, regardless of
210   * integration direction, forward or backward), the last step can
211   * be smaller than this
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 (sign is irrelevant, regardless of
226   * integration direction, forward or backward), the last step can
227   * be smaller than this
228   * @param maxStep maximal step (sign is irrelevant, regardless of
229   * integration direction, forward or backward), the last step can
230   * be smaller than this
231   * @param vecAbsoluteTolerance allowed absolute error
232   * @param vecRelativeTolerance allowed relative error
233   */
234  public DormandPrince853Integrator(final double minStep, final double maxStep,
235                                    final double[] vecAbsoluteTolerance,
236                                    final double[] vecRelativeTolerance) {
237    super(METHOD_NAME, true, STATIC_C, STATIC_A, STATIC_B,
238          new DormandPrince853StepInterpolator(),
239          minStep, maxStep, vecAbsoluteTolerance, vecRelativeTolerance);
240  }
241
242  /** {@inheritDoc} */
243  @Override
244  public int getOrder() {
245    return 8;
246  }
247
248  /** {@inheritDoc} */
249  @Override
250  protected double estimateError(final double[][] yDotK,
251                                 final double[] y0, final double[] y1,
252                                 final double h) {
253    double error1 = 0;
254    double error2 = 0;
255
256    for (int j = 0; j < mainSetDimension; ++j) {
257      final double errSum1 = E1_01 * yDotK[0][j]  + E1_06 * yDotK[5][j] +
258                             E1_07 * yDotK[6][j]  + E1_08 * yDotK[7][j] +
259                             E1_09 * yDotK[8][j]  + E1_10 * yDotK[9][j] +
260                             E1_11 * yDotK[10][j] + E1_12 * yDotK[11][j];
261      final double errSum2 = E2_01 * yDotK[0][j]  + E2_06 * yDotK[5][j] +
262                             E2_07 * yDotK[6][j]  + E2_08 * yDotK[7][j] +
263                             E2_09 * yDotK[8][j]  + E2_10 * yDotK[9][j] +
264                             E2_11 * yDotK[10][j] + E2_12 * yDotK[11][j];
265
266      final double yScale = JdkMath.max(JdkMath.abs(y0[j]), JdkMath.abs(y1[j]));
267      final double tol = (vecAbsoluteTolerance == null) ?
268                         (scalAbsoluteTolerance + scalRelativeTolerance * yScale) :
269                         (vecAbsoluteTolerance[j] + vecRelativeTolerance[j] * yScale);
270      final double ratio1  = errSum1 / tol;
271      error1        += ratio1 * ratio1;
272      final double ratio2  = errSum2 / tol;
273      error2        += ratio2 * ratio2;
274    }
275
276    double den = error1 + 0.01 * error2;
277    if (den <= 0.0) {
278      den = 1.0;
279    }
280
281    return JdkMath.abs(h) * error1 / JdkMath.sqrt(mainSetDimension * den);
282  }
283}