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 import 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 aot 2010) $ 54 * @since 1.2 55 */ 56 57 public 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
