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    
018    package org.apache.commons.math3.ode.nonstiff;
019    
020    import org.apache.commons.math3.util.FastMath;
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     * @version $Id: DormandPrince853Integrator.java 1416643 2012-12-03 19:37:14Z tn $
054     * @since 1.2
055     */
056    
057    public class DormandPrince853Integrator extends EmbeddedRungeKuttaIntegrator {
058    
059      /** Integrator method name. */
060      private static final String METHOD_NAME = "Dormand-Prince 8 (5, 3)";
061    
062      /** Time steps Butcher array. */
063      private static final double[] STATIC_C = {
064        (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,
065        (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,
066        6.0/7.0, 1.0, 1.0
067      };
068    
069      /** Internal weights Butcher array. */
070      private static final double[][] STATIC_A = {
071    
072        // k2
073        {(12.0 - 2.0 * FastMath.sqrt(6.0)) / 135.0},
074    
075        // k3
076        {(6.0 - FastMath.sqrt(6.0)) / 180.0, (6.0 - FastMath.sqrt(6.0)) / 60.0},
077    
078        // k4
079        {(6.0 - FastMath.sqrt(6.0)) / 120.0, 0.0, (6.0 - FastMath.sqrt(6.0)) / 40.0},
080    
081        // k5
082        {(462.0 + 107.0 * FastMath.sqrt(6.0)) / 3000.0, 0.0,
083         (-402.0 - 197.0 * FastMath.sqrt(6.0)) / 1000.0, (168.0 + 73.0 * FastMath.sqrt(6.0)) / 375.0},
084    
085        // k6
086        {1.0 / 27.0, 0.0, 0.0, (16.0 + FastMath.sqrt(6.0)) / 108.0, (16.0 - FastMath.sqrt(6.0)) / 108.0},
087    
088        // k7
089        {19.0 / 512.0, 0.0, 0.0, (118.0 + 23.0 * FastMath.sqrt(6.0)) / 1024.0,
090         (118.0 - 23.0 * FastMath.sqrt(6.0)) / 1024.0, -9.0 / 512.0},
091    
092        // k8
093        {13772.0 / 371293.0, 0.0, 0.0, (51544.0 + 4784.0 * FastMath.sqrt(6.0)) / 371293.0,
094         (51544.0 - 4784.0 * FastMath.sqrt(6.0)) / 371293.0, -5688.0 / 371293.0, 3072.0 / 371293.0},
095    
096        // k9
097        {58656157643.0 / 93983540625.0, 0.0, 0.0,
098         (-1324889724104.0 - 318801444819.0 * FastMath.sqrt(6.0)) / 626556937500.0,
099         (-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 (sign is irrelevant, regardless of
209       * integration direction, forward or backward), the last step can
210       * be smaller than this
211       * @param maxStep maximal step (sign is irrelevant, regardless of
212       * integration direction, forward or backward), the last step can
213       * be smaller than this
214       * @param scalAbsoluteTolerance allowed absolute error
215       * @param scalRelativeTolerance allowed relative error
216       */
217      public DormandPrince853Integrator(final double minStep, final double maxStep,
218                                        final double scalAbsoluteTolerance,
219                                        final double scalRelativeTolerance) {
220        super(METHOD_NAME, true, STATIC_C, STATIC_A, STATIC_B,
221              new DormandPrince853StepInterpolator(),
222              minStep, maxStep, scalAbsoluteTolerance, scalRelativeTolerance);
223      }
224    
225      /** Simple constructor.
226       * Build an eighth order Dormand-Prince integrator with the given step bounds
227       * @param minStep minimal step (sign is irrelevant, regardless of
228       * integration direction, forward or backward), the last step can
229       * be smaller than this
230       * @param maxStep maximal step (sign is irrelevant, regardless of
231       * integration direction, forward or backward), the last step can
232       * be smaller than this
233       * @param vecAbsoluteTolerance allowed absolute error
234       * @param vecRelativeTolerance allowed relative error
235       */
236      public DormandPrince853Integrator(final double minStep, final double maxStep,
237                                        final double[] vecAbsoluteTolerance,
238                                        final double[] vecRelativeTolerance) {
239        super(METHOD_NAME, true, STATIC_C, STATIC_A, STATIC_B,
240              new DormandPrince853StepInterpolator(),
241              minStep, maxStep, vecAbsoluteTolerance, vecRelativeTolerance);
242      }
243    
244      /** {@inheritDoc} */
245      @Override
246      public int getOrder() {
247        return 8;
248      }
249    
250      /** {@inheritDoc} */
251      @Override
252      protected double estimateError(final double[][] yDotK,
253                                     final double[] y0, final double[] y1,
254                                     final double h) {
255        double error1 = 0;
256        double error2 = 0;
257    
258        for (int j = 0; j < mainSetDimension; ++j) {
259          final double errSum1 = E1_01 * yDotK[0][j]  + E1_06 * yDotK[5][j] +
260                                 E1_07 * yDotK[6][j]  + E1_08 * yDotK[7][j] +
261                                 E1_09 * yDotK[8][j]  + E1_10 * yDotK[9][j] +
262                                 E1_11 * yDotK[10][j] + E1_12 * yDotK[11][j];
263          final double errSum2 = E2_01 * yDotK[0][j]  + E2_06 * yDotK[5][j] +
264                                 E2_07 * yDotK[6][j]  + E2_08 * yDotK[7][j] +
265                                 E2_09 * yDotK[8][j]  + E2_10 * yDotK[9][j] +
266                                 E2_11 * yDotK[10][j] + E2_12 * yDotK[11][j];
267    
268          final double yScale = FastMath.max(FastMath.abs(y0[j]), FastMath.abs(y1[j]));
269          final double tol = (vecAbsoluteTolerance == null) ?
270                             (scalAbsoluteTolerance + scalRelativeTolerance * yScale) :
271                             (vecAbsoluteTolerance[j] + vecRelativeTolerance[j] * yScale);
272          final double ratio1  = errSum1 / tol;
273          error1        += ratio1 * ratio1;
274          final double ratio2  = errSum2 / tol;
275          error2        += ratio2 * ratio2;
276        }
277    
278        double den = error1 + 0.01 * error2;
279        if (den <= 0.0) {
280          den = 1.0;
281        }
282    
283        return FastMath.abs(h) * error1 / FastMath.sqrt(mainSetDimension * den);
284    
285      }
286    
287    }