Java tutorial
/* * 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.math3.ode.nonstiff; import org.apache.commons.math3.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 $Id: DormandPrince853Integrator.java 1416643 2012-12-03 19:37:14Z tn $ * @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 (sign is irrelevant, regardless of * integration direction, forward or backward), the last step can * be smaller than this * @param maxStep maximal step (sign is irrelevant, regardless of * integration direction, forward or backward), the last step can * be smaller than this * @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 (sign is irrelevant, regardless of * integration direction, forward or backward), the last step can * be smaller than this * @param maxStep maximal step (sign is irrelevant, regardless of * integration direction, forward or backward), the last step can * be smaller than this * @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); } }