| /* |
| * Licensed to the Apache Software Foundation (ASF) under one or more |
| * contributor license agreements. See the NOTICE file distributed with |
| * this work for additional information regarding copyright ownership. |
| * The ASF licenses this file to You under the Apache License, Version 2.0 |
| * (the "License"); you may not use this file except in compliance with |
| * the License. You may obtain a copy of the License at |
| * |
| * http://www.apache.org/licenses/LICENSE-2.0 |
| * |
| * Unless required by applicable law or agreed to in writing, software |
| * distributed under the License is distributed on an "AS IS" BASIS, |
| * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. |
| * See the License for the specific language governing permissions and |
| * limitations under the License. |
| */ |
| |
| package org.apache.commons.math.ode.nonstiff; |
| |
| import org.apache.commons.math.util.FastMath; |
| |
| |
| /** |
| * This class implements the 8(5,3) Dormand-Prince integrator for Ordinary |
| * Differential Equations. |
| * |
| * <p>This integrator is an embedded Runge-Kutta integrator |
| * of order 8(5,3) used in local extrapolation mode (i.e. the solution |
| * is computed using the high order formula) with stepsize control |
| * (and automatic step initialization) and continuous output. This |
| * method uses 12 functions evaluations per step for integration and 4 |
| * evaluations for interpolation. However, since the first |
| * interpolation evaluation is the same as the first integration |
| * evaluation of the next step, we have included it in the integrator |
| * rather than in the interpolator and specified the method was an |
| * <i>fsal</i>. Hence, despite we have 13 stages here, the cost is |
| * really 12 evaluations per step even if no interpolation is done, |
| * and the overcost of interpolation is only 3 evaluations.</p> |
| * |
| * <p>This method is based on an 8(6) method by Dormand and Prince |
| * (i.e. order 8 for the integration and order 6 for error estimation) |
| * modified by Hairer and Wanner to use a 5th order error estimator |
| * with 3rd order correction. This modification was introduced because |
| * the original method failed in some cases (wrong steps can be |
| * accepted when step size is too large, for example in the |
| * Brusselator problem) and also had <i>severe difficulties when |
| * applied to problems with discontinuities</i>. This modification is |
| * explained in the second edition of the first volume (Nonstiff |
| * Problems) of the reference book by Hairer, Norsett and Wanner: |
| * <i>Solving Ordinary Differential Equations</i> (Springer-Verlag, |
| * ISBN 3-540-56670-8).</p> |
| * |
| * @version $Revision: 990655 $ $Date: 2010-08-29 23:49:40 +0200 (dim. 29 août 2010) $ |
| * @since 1.2 |
| */ |
| |
| public class DormandPrince853Integrator extends EmbeddedRungeKuttaIntegrator { |
| |
| /** Integrator method name. */ |
| private static final String METHOD_NAME = "Dormand-Prince 8 (5, 3)"; |
| |
| /** Time steps Butcher array. */ |
| private static final double[] STATIC_C = { |
| (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, |
| (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, |
| 6.0/7.0, 1.0, 1.0 |
| }; |
| |
| /** Internal weights Butcher array. */ |
| private static final double[][] STATIC_A = { |
| |
| // k2 |
| {(12.0 - 2.0 * FastMath.sqrt(6.0)) / 135.0}, |
| |
| // k3 |
| {(6.0 - FastMath.sqrt(6.0)) / 180.0, (6.0 - FastMath.sqrt(6.0)) / 60.0}, |
| |
| // k4 |
| {(6.0 - FastMath.sqrt(6.0)) / 120.0, 0.0, (6.0 - FastMath.sqrt(6.0)) / 40.0}, |
| |
| // k5 |
| {(462.0 + 107.0 * FastMath.sqrt(6.0)) / 3000.0, 0.0, |
| (-402.0 - 197.0 * FastMath.sqrt(6.0)) / 1000.0, (168.0 + 73.0 * FastMath.sqrt(6.0)) / 375.0}, |
| |
| // k6 |
| {1.0 / 27.0, 0.0, 0.0, (16.0 + FastMath.sqrt(6.0)) / 108.0, (16.0 - FastMath.sqrt(6.0)) / 108.0}, |
| |
| // k7 |
| {19.0 / 512.0, 0.0, 0.0, (118.0 + 23.0 * FastMath.sqrt(6.0)) / 1024.0, |
| (118.0 - 23.0 * FastMath.sqrt(6.0)) / 1024.0, -9.0 / 512.0}, |
| |
| // k8 |
| {13772.0 / 371293.0, 0.0, 0.0, (51544.0 + 4784.0 * FastMath.sqrt(6.0)) / 371293.0, |
| (51544.0 - 4784.0 * FastMath.sqrt(6.0)) / 371293.0, -5688.0 / 371293.0, 3072.0 / 371293.0}, |
| |
| // k9 |
| {58656157643.0 / 93983540625.0, 0.0, 0.0, |
| (-1324889724104.0 - 318801444819.0 * FastMath.sqrt(6.0)) / 626556937500.0, |
| (-1324889724104.0 + 318801444819.0 * FastMath.sqrt(6.0)) / 626556937500.0, |
| 96044563816.0 / 3480871875.0, 5682451879168.0 / 281950621875.0, |
| -165125654.0 / 3796875.0}, |
| |
| // k10 |
| {8909899.0 / 18653125.0, 0.0, 0.0, |
| (-4521408.0 - 1137963.0 * FastMath.sqrt(6.0)) / 2937500.0, |
| (-4521408.0 + 1137963.0 * FastMath.sqrt(6.0)) / 2937500.0, |
| 96663078.0 / 4553125.0, 2107245056.0 / 137915625.0, |
| -4913652016.0 / 147609375.0, -78894270.0 / 3880452869.0}, |
| |
| // k11 |
| {-20401265806.0 / 21769653311.0, 0.0, 0.0, |
| (354216.0 + 94326.0 * FastMath.sqrt(6.0)) / 112847.0, |
| (354216.0 - 94326.0 * FastMath.sqrt(6.0)) / 112847.0, |
| -43306765128.0 / 5313852383.0, -20866708358144.0 / 1126708119789.0, |
| 14886003438020.0 / 654632330667.0, 35290686222309375.0 / 14152473387134411.0, |
| -1477884375.0 / 485066827.0}, |
| |
| // k12 |
| {39815761.0 / 17514443.0, 0.0, 0.0, |
| (-3457480.0 - 960905.0 * FastMath.sqrt(6.0)) / 551636.0, |
| (-3457480.0 + 960905.0 * FastMath.sqrt(6.0)) / 551636.0, |
| -844554132.0 / 47026969.0, 8444996352.0 / 302158619.0, |
| -2509602342.0 / 877790785.0, -28388795297996250.0 / 3199510091356783.0, |
| 226716250.0 / 18341897.0, 1371316744.0 / 2131383595.0}, |
| |
| // k13 should be for interpolation only, but since it is the same |
| // stage as the first evaluation of the next step, we perform it |
| // here at no cost by specifying this is an fsal method |
| {104257.0/1920240.0, 0.0, 0.0, 0.0, 0.0, 3399327.0/763840.0, |
| 66578432.0/35198415.0, -1674902723.0/288716400.0, |
| 54980371265625.0/176692375811392.0, -734375.0/4826304.0, |
| 171414593.0/851261400.0, 137909.0/3084480.0} |
| |
| }; |
| |
| /** Propagation weights Butcher array. */ |
| private static final double[] STATIC_B = { |
| 104257.0/1920240.0, |
| 0.0, |
| 0.0, |
| 0.0, |
| 0.0, |
| 3399327.0/763840.0, |
| 66578432.0/35198415.0, |
| -1674902723.0/288716400.0, |
| 54980371265625.0/176692375811392.0, |
| -734375.0/4826304.0, |
| 171414593.0/851261400.0, |
| 137909.0/3084480.0, |
| 0.0 |
| }; |
| |
| /** First error weights array, element 1. */ |
| private static final double E1_01 = 116092271.0 / 8848465920.0; |
| |
| // elements 2 to 5 are zero, so they are neither stored nor used |
| |
| /** First error weights array, element 6. */ |
| private static final double E1_06 = -1871647.0 / 1527680.0; |
| |
| /** First error weights array, element 7. */ |
| private static final double E1_07 = -69799717.0 / 140793660.0; |
| |
| /** First error weights array, element 8. */ |
| private static final double E1_08 = 1230164450203.0 / 739113984000.0; |
| |
| /** First error weights array, element 9. */ |
| private static final double E1_09 = -1980813971228885.0 / 5654156025964544.0; |
| |
| /** First error weights array, element 10. */ |
| private static final double E1_10 = 464500805.0 / 1389975552.0; |
| |
| /** First error weights array, element 11. */ |
| private static final double E1_11 = 1606764981773.0 / 19613062656000.0; |
| |
| /** First error weights array, element 12. */ |
| private static final double E1_12 = -137909.0 / 6168960.0; |
| |
| |
| /** Second error weights array, element 1. */ |
| private static final double E2_01 = -364463.0 / 1920240.0; |
| |
| // elements 2 to 5 are zero, so they are neither stored nor used |
| |
| /** Second error weights array, element 6. */ |
| private static final double E2_06 = 3399327.0 / 763840.0; |
| |
| /** Second error weights array, element 7. */ |
| private static final double E2_07 = 66578432.0 / 35198415.0; |
| |
| /** Second error weights array, element 8. */ |
| private static final double E2_08 = -1674902723.0 / 288716400.0; |
| |
| /** Second error weights array, element 9. */ |
| private static final double E2_09 = -74684743568175.0 / 176692375811392.0; |
| |
| /** Second error weights array, element 10. */ |
| private static final double E2_10 = -734375.0 / 4826304.0; |
| |
| /** Second error weights array, element 11. */ |
| private static final double E2_11 = 171414593.0 / 851261400.0; |
| |
| /** Second error weights array, element 12. */ |
| private static final double E2_12 = 69869.0 / 3084480.0; |
| |
| /** Simple constructor. |
| * Build an eighth order Dormand-Prince integrator with the given step bounds |
| * @param minStep minimal step (must be positive even for backward |
| * integration), the last step can be smaller than this |
| * @param maxStep maximal step (must be positive even for backward |
| * integration) |
| * @param scalAbsoluteTolerance allowed absolute error |
| * @param scalRelativeTolerance allowed relative error |
| */ |
| public DormandPrince853Integrator(final double minStep, final double maxStep, |
| final double scalAbsoluteTolerance, |
| final double scalRelativeTolerance) { |
| super(METHOD_NAME, true, STATIC_C, STATIC_A, STATIC_B, |
| new DormandPrince853StepInterpolator(), |
| minStep, maxStep, scalAbsoluteTolerance, scalRelativeTolerance); |
| } |
| |
| /** Simple constructor. |
| * Build an eighth order Dormand-Prince integrator with the given step bounds |
| * @param minStep minimal step (must be positive even for backward |
| * integration), the last step can be smaller than this |
| * @param maxStep maximal step (must be positive even for backward |
| * integration) |
| * @param vecAbsoluteTolerance allowed absolute error |
| * @param vecRelativeTolerance allowed relative error |
| */ |
| public DormandPrince853Integrator(final double minStep, final double maxStep, |
| final double[] vecAbsoluteTolerance, |
| final double[] vecRelativeTolerance) { |
| super(METHOD_NAME, true, STATIC_C, STATIC_A, STATIC_B, |
| new DormandPrince853StepInterpolator(), |
| minStep, maxStep, vecAbsoluteTolerance, vecRelativeTolerance); |
| } |
| |
| /** {@inheritDoc} */ |
| @Override |
| public int getOrder() { |
| return 8; |
| } |
| |
| /** {@inheritDoc} */ |
| @Override |
| protected double estimateError(final double[][] yDotK, |
| final double[] y0, final double[] y1, |
| final double h) { |
| double error1 = 0; |
| double error2 = 0; |
| |
| for (int j = 0; j < mainSetDimension; ++j) { |
| final double errSum1 = E1_01 * yDotK[0][j] + E1_06 * yDotK[5][j] + |
| E1_07 * yDotK[6][j] + E1_08 * yDotK[7][j] + |
| E1_09 * yDotK[8][j] + E1_10 * yDotK[9][j] + |
| E1_11 * yDotK[10][j] + E1_12 * yDotK[11][j]; |
| final double errSum2 = E2_01 * yDotK[0][j] + E2_06 * yDotK[5][j] + |
| E2_07 * yDotK[6][j] + E2_08 * yDotK[7][j] + |
| E2_09 * yDotK[8][j] + E2_10 * yDotK[9][j] + |
| E2_11 * yDotK[10][j] + E2_12 * yDotK[11][j]; |
| |
| final double yScale = FastMath.max(FastMath.abs(y0[j]), FastMath.abs(y1[j])); |
| final double tol = (vecAbsoluteTolerance == null) ? |
| (scalAbsoluteTolerance + scalRelativeTolerance * yScale) : |
| (vecAbsoluteTolerance[j] + vecRelativeTolerance[j] * yScale); |
| final double ratio1 = errSum1 / tol; |
| error1 += ratio1 * ratio1; |
| final double ratio2 = errSum2 / tol; |
| error2 += ratio2 * ratio2; |
| } |
| |
| double den = error1 + 0.01 * error2; |
| if (den <= 0.0) { |
| den = 1.0; |
| } |
| |
| return FastMath.abs(h) * error1 / FastMath.sqrt(mainSetDimension * den); |
| |
| } |
| |
| } |
