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.math.ode.nonstiff;
019
020
021 /**
022 * This class implements the 8(5,3) Dormand-Prince integrator for Ordinary
023 * Differential Equations.
024 *
025 * <p>This integrator is an embedded Runge-Kutta integrator
026 * of order 8(5,3) used in local extrapolation mode (i.e. the solution
027 * is computed using the high order formula) with stepsize control
028 * (and automatic step initialization) and continuous output. This
029 * method uses 12 functions evaluations per step for integration and 4
030 * evaluations for interpolation. However, since the first
031 * interpolation evaluation is the same as the first integration
032 * evaluation of the next step, we have included it in the integrator
033 * rather than in the interpolator and specified the method was an
034 * <i>fsal</i>. Hence, despite we have 13 stages here, the cost is
035 * really 12 evaluations per step even if no interpolation is done,
036 * and the overcost of interpolation is only 3 evaluations.</p>
037 *
038 * <p>This method is based on an 8(6) method by Dormand and Prince
039 * (i.e. order 8 for the integration and order 6 for error estimation)
040 * modified by Hairer and Wanner to use a 5th order error estimator
041 * with 3rd order correction. This modification was introduced because
042 * the original method failed in some cases (wrong steps can be
043 * accepted when step size is too large, for example in the
044 * Brusselator problem) and also had <i>severe difficulties when
045 * applied to problems with discontinuities</i>. This modification is
046 * explained in the second edition of the first volume (Nonstiff
047 * Problems) of the reference book by Hairer, Norsett and Wanner:
048 * <i>Solving Ordinary Differential Equations</i> (Springer-Verlag,
049 * ISBN 3-540-56670-8).</p>
050 *
051 * @version $Revision: 810196 $ $Date: 2009-09-01 15:47:46 -0400 (Tue, 01 Sep 2009) $
052 * @since 1.2
053 */
054
055 public class DormandPrince853Integrator extends EmbeddedRungeKuttaIntegrator {
056
057 /** Integrator method name. */
058 private static final String METHOD_NAME = "Dormand-Prince 8 (5, 3)";
059
060 /** Time steps Butcher array. */
061 private static final double[] STATIC_C = {
062 (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,
063 (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,
064 6.0/7.0, 1.0, 1.0
065 };
066
067 /** Internal weights Butcher array. */
068 private static final double[][] STATIC_A = {
069
070 // k2
071 {(12.0 - 2.0 * Math.sqrt(6.0)) / 135.0},
072
073 // k3
074 {(6.0 - Math.sqrt(6.0)) / 180.0, (6.0 - Math.sqrt(6.0)) / 60.0},
075
076 // k4
077 {(6.0 - Math.sqrt(6.0)) / 120.0, 0.0, (6.0 - Math.sqrt(6.0)) / 40.0},
078
079 // k5
080 {(462.0 + 107.0 * Math.sqrt(6.0)) / 3000.0, 0.0,
081 (-402.0 - 197.0 * Math.sqrt(6.0)) / 1000.0, (168.0 + 73.0 * Math.sqrt(6.0)) / 375.0},
082
083 // k6
084 {1.0 / 27.0, 0.0, 0.0, (16.0 + Math.sqrt(6.0)) / 108.0, (16.0 - Math.sqrt(6.0)) / 108.0},
085
086 // k7
087 {19.0 / 512.0, 0.0, 0.0, (118.0 + 23.0 * Math.sqrt(6.0)) / 1024.0,
088 (118.0 - 23.0 * Math.sqrt(6.0)) / 1024.0, -9.0 / 512.0},
089
090 // k8
091 {13772.0 / 371293.0, 0.0, 0.0, (51544.0 + 4784.0 * Math.sqrt(6.0)) / 371293.0,
092 (51544.0 - 4784.0 * Math.sqrt(6.0)) / 371293.0, -5688.0 / 371293.0, 3072.0 / 371293.0},
093
094 // k9
095 {58656157643.0 / 93983540625.0, 0.0, 0.0,
096 (-1324889724104.0 - 318801444819.0 * Math.sqrt(6.0)) / 626556937500.0,
097 (-1324889724104.0 + 318801444819.0 * Math.sqrt(6.0)) / 626556937500.0,
098 96044563816.0 / 3480871875.0, 5682451879168.0 / 281950621875.0,
099 -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[] 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 (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, STATIC_C, STATIC_A, STATIC_B,
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, STATIC_C, STATIC_A, STATIC_B,
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 }