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.math3.ode.nonstiff; 019 020import 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 * @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 * FastMath.sqrt(6.0)) / 135.0, (6.0 - FastMath.sqrt(6.0)) / 45.0, (6.0 - FastMath.sqrt(6.0)) / 30.0, 064 (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, 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 * FastMath.sqrt(6.0)) / 135.0}, 073 074 // k3 075 {(6.0 - FastMath.sqrt(6.0)) / 180.0, (6.0 - FastMath.sqrt(6.0)) / 60.0}, 076 077 // k4 078 {(6.0 - FastMath.sqrt(6.0)) / 120.0, 0.0, (6.0 - FastMath.sqrt(6.0)) / 40.0}, 079 080 // k5 081 {(462.0 + 107.0 * FastMath.sqrt(6.0)) / 3000.0, 0.0, 082 (-402.0 - 197.0 * FastMath.sqrt(6.0)) / 1000.0, (168.0 + 73.0 * FastMath.sqrt(6.0)) / 375.0}, 083 084 // k6 085 {1.0 / 27.0, 0.0, 0.0, (16.0 + FastMath.sqrt(6.0)) / 108.0, (16.0 - FastMath.sqrt(6.0)) / 108.0}, 086 087 // k7 088 {19.0 / 512.0, 0.0, 0.0, (118.0 + 23.0 * FastMath.sqrt(6.0)) / 1024.0, 089 (118.0 - 23.0 * FastMath.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 * FastMath.sqrt(6.0)) / 371293.0, 093 (51544.0 - 4784.0 * FastMath.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 * FastMath.sqrt(6.0)) / 626556937500.0, 098 (-1324889724104.0 + 318801444819.0 * FastMath.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 * FastMath.sqrt(6.0)) / 2937500.0, 105 (-4521408.0 + 1137963.0 * FastMath.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 * FastMath.sqrt(6.0)) / 112847.0, 112 (354216.0 - 94326.0 * FastMath.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 * FastMath.sqrt(6.0)) / 551636.0, 120 (-3457480.0 + 960905.0 * FastMath.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 }; 134 135 /** Propagation weights Butcher array. */ 136 private static final double[] STATIC_B = { 137 104257.0/1920240.0, 138 0.0, 139 0.0, 140 0.0, 141 0.0, 142 3399327.0/763840.0, 143 66578432.0/35198415.0, 144 -1674902723.0/288716400.0, 145 54980371265625.0/176692375811392.0, 146 -734375.0/4826304.0, 147 171414593.0/851261400.0, 148 137909.0/3084480.0, 149 0.0 150 }; 151 152 /** First error weights array, element 1. */ 153 private static final double E1_01 = 116092271.0 / 8848465920.0; 154 155 // elements 2 to 5 are zero, so they are neither stored nor used 156 157 /** First error weights array, element 6. */ 158 private static final double E1_06 = -1871647.0 / 1527680.0; 159 160 /** First error weights array, element 7. */ 161 private static final double E1_07 = -69799717.0 / 140793660.0; 162 163 /** First error weights array, element 8. */ 164 private static final double E1_08 = 1230164450203.0 / 739113984000.0; 165 166 /** First error weights array, element 9. */ 167 private static final double E1_09 = -1980813971228885.0 / 5654156025964544.0; 168 169 /** First error weights array, element 10. */ 170 private static final double E1_10 = 464500805.0 / 1389975552.0; 171 172 /** First error weights array, element 11. */ 173 private static final double E1_11 = 1606764981773.0 / 19613062656000.0; 174 175 /** First error weights array, element 12. */ 176 private static final double E1_12 = -137909.0 / 6168960.0; 177 178 179 /** Second error weights array, element 1. */ 180 private static final double E2_01 = -364463.0 / 1920240.0; 181 182 // elements 2 to 5 are zero, so they are neither stored nor used 183 184 /** Second error weights array, element 6. */ 185 private static final double E2_06 = 3399327.0 / 763840.0; 186 187 /** Second error weights array, element 7. */ 188 private static final double E2_07 = 66578432.0 / 35198415.0; 189 190 /** Second error weights array, element 8. */ 191 private static final double E2_08 = -1674902723.0 / 288716400.0; 192 193 /** Second error weights array, element 9. */ 194 private static final double E2_09 = -74684743568175.0 / 176692375811392.0; 195 196 /** Second error weights array, element 10. */ 197 private static final double E2_10 = -734375.0 / 4826304.0; 198 199 /** Second error weights array, element 11. */ 200 private static final double E2_11 = 171414593.0 / 851261400.0; 201 202 /** Second error weights array, element 12. */ 203 private static final double E2_12 = 69869.0 / 3084480.0; 204 205 /** Simple constructor. 206 * Build an eighth order Dormand-Prince integrator with the given step bounds 207 * @param minStep minimal step (sign is irrelevant, regardless of 208 * integration direction, forward or backward), the last step can 209 * be smaller than this 210 * @param maxStep maximal step (sign is irrelevant, regardless of 211 * integration direction, forward or backward), the last step can 212 * be smaller than this 213 * @param scalAbsoluteTolerance allowed absolute error 214 * @param scalRelativeTolerance allowed relative error 215 */ 216 public DormandPrince853Integrator(final double minStep, final double maxStep, 217 final double scalAbsoluteTolerance, 218 final double scalRelativeTolerance) { 219 super(METHOD_NAME, true, STATIC_C, STATIC_A, STATIC_B, 220 new DormandPrince853StepInterpolator(), 221 minStep, maxStep, scalAbsoluteTolerance, scalRelativeTolerance); 222 } 223 224 /** Simple constructor. 225 * Build an eighth order Dormand-Prince integrator with the given step bounds 226 * @param minStep minimal step (sign is irrelevant, regardless of 227 * integration direction, forward or backward), the last step can 228 * be smaller than this 229 * @param maxStep maximal step (sign is irrelevant, regardless of 230 * integration direction, forward or backward), the last step can 231 * be smaller than this 232 * @param vecAbsoluteTolerance allowed absolute error 233 * @param vecRelativeTolerance allowed relative error 234 */ 235 public DormandPrince853Integrator(final double minStep, final double maxStep, 236 final double[] vecAbsoluteTolerance, 237 final double[] vecRelativeTolerance) { 238 super(METHOD_NAME, true, STATIC_C, STATIC_A, STATIC_B, 239 new DormandPrince853StepInterpolator(), 240 minStep, maxStep, vecAbsoluteTolerance, vecRelativeTolerance); 241 } 242 243 /** {@inheritDoc} */ 244 @Override 245 public int getOrder() { 246 return 8; 247 } 248 249 /** {@inheritDoc} */ 250 @Override 251 protected double estimateError(final double[][] yDotK, 252 final double[] y0, final double[] y1, 253 final double h) { 254 double error1 = 0; 255 double error2 = 0; 256 257 for (int j = 0; j < mainSetDimension; ++j) { 258 final double errSum1 = E1_01 * yDotK[0][j] + E1_06 * yDotK[5][j] + 259 E1_07 * yDotK[6][j] + E1_08 * yDotK[7][j] + 260 E1_09 * yDotK[8][j] + E1_10 * yDotK[9][j] + 261 E1_11 * yDotK[10][j] + E1_12 * yDotK[11][j]; 262 final double errSum2 = E2_01 * yDotK[0][j] + E2_06 * yDotK[5][j] + 263 E2_07 * yDotK[6][j] + E2_08 * yDotK[7][j] + 264 E2_09 * yDotK[8][j] + E2_10 * yDotK[9][j] + 265 E2_11 * yDotK[10][j] + E2_12 * yDotK[11][j]; 266 267 final double yScale = FastMath.max(FastMath.abs(y0[j]), FastMath.abs(y1[j])); 268 final double tol = (vecAbsoluteTolerance == null) ? 269 (scalAbsoluteTolerance + scalRelativeTolerance * yScale) : 270 (vecAbsoluteTolerance[j] + vecRelativeTolerance[j] * yScale); 271 final double ratio1 = errSum1 / tol; 272 error1 += ratio1 * ratio1; 273 final double ratio2 = errSum2 / tol; 274 error2 += ratio2 * ratio2; 275 } 276 277 double den = error1 + 0.01 * error2; 278 if (den <= 0.0) { 279 den = 1.0; 280 } 281 282 return FastMath.abs(h) * error1 / FastMath.sqrt(mainSetDimension * den); 283 284 } 285 286}