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Diffstat (limited to 'src/main/java/org/apache/commons/math3/ode/nonstiff/DormandPrince853Integrator.java')
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diff --git a/src/main/java/org/apache/commons/math3/ode/nonstiff/DormandPrince853Integrator.java b/src/main/java/org/apache/commons/math3/ode/nonstiff/DormandPrince853Integrator.java new file mode 100644 index 0000000..895cb88 --- /dev/null +++ b/src/main/java/org/apache/commons/math3/ode/nonstiff/DormandPrince853Integrator.java @@ -0,0 +1,286 @@ +/* + * 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> + * + * @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); + + } + +} |