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/* * 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 5(4) Dormand-Prince integrator for Ordinary * Differential Equations. * <p>This integrator is an embedded Runge-Kutta integrator * of order 5(4) 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 7 functions evaluations per step. However, since this * is an <i>fsal</i>, the last evaluation of one step is the same as * the first evaluation of the next step and hence can be avoided. So * the cost is really 6 functions evaluations per step.</p> * * <p>This method has been published (whithout the continuous output * that was added by Shampine in 1986) in the following article : * <pre> * A family of embedded Runge-Kutta formulae * J. R. Dormand and P. J. Prince * Journal of Computational and Applied Mathematics * volume 6, no 1, 1980, pp. 19-26 * </pre></p> * * @version $Id: DormandPrince54Integrator.java 1416643 2012-12-03 19:37:14Z tn $ * @since 1.2 */ public class DormandPrince54Integrator extends EmbeddedRungeKuttaIntegrator { /** Integrator method name. */ private static final String METHOD_NAME = "Dormand-Prince 5(4)"; /** Time steps Butcher array. */ private static final double[] STATIC_C = { 1.0 / 5.0, 3.0 / 10.0, 4.0 / 5.0, 8.0 / 9.0, 1.0, 1.0 }; /** Internal weights Butcher array. */ private static final double[][] STATIC_A = { { 1.0 / 5.0 }, { 3.0 / 40.0, 9.0 / 40.0 }, { 44.0 / 45.0, -56.0 / 15.0, 32.0 / 9.0 }, { 19372.0 / 6561.0, -25360.0 / 2187.0, 64448.0 / 6561.0, -212.0 / 729.0 }, { 9017.0 / 3168.0, -355.0 / 33.0, 46732.0 / 5247.0, 49.0 / 176.0, -5103.0 / 18656.0 }, { 35.0 / 384.0, 0.0, 500.0 / 1113.0, 125.0 / 192.0, -2187.0 / 6784.0, 11.0 / 84.0 } }; /** Propagation weights Butcher array. */ private static final double[] STATIC_B = { 35.0 / 384.0, 0.0, 500.0 / 1113.0, 125.0 / 192.0, -2187.0 / 6784.0, 11.0 / 84.0, 0.0 }; /** Error array, element 1. */ private static final double E1 = 71.0 / 57600.0; // element 2 is zero, so it is neither stored nor used /** Error array, element 3. */ private static final double E3 = -71.0 / 16695.0; /** Error array, element 4. */ private static final double E4 = 71.0 / 1920.0; /** Error array, element 5. */ private static final double E5 = -17253.0 / 339200.0; /** Error array, element 6. */ private static final double E6 = 22.0 / 525.0; /** Error array, element 7. */ private static final double E7 = -1.0 / 40.0; /** Simple constructor. * Build a fifth 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 DormandPrince54Integrator(final double minStep, final double maxStep, final double scalAbsoluteTolerance, final double scalRelativeTolerance) { super(METHOD_NAME, true, STATIC_C, STATIC_A, STATIC_B, new DormandPrince54StepInterpolator(), minStep, maxStep, scalAbsoluteTolerance, scalRelativeTolerance); } /** Simple constructor. * Build a fifth 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 DormandPrince54Integrator(final double minStep, final double maxStep, final double[] vecAbsoluteTolerance, final double[] vecRelativeTolerance) { super(METHOD_NAME, true, STATIC_C, STATIC_A, STATIC_B, new DormandPrince54StepInterpolator(), minStep, maxStep, vecAbsoluteTolerance, vecRelativeTolerance); } /** {@inheritDoc} */ @Override public int getOrder() { return 5; } /** {@inheritDoc} */ @Override protected double estimateError(final double[][] yDotK, final double[] y0, final double[] y1, final double h) { double error = 0; for (int j = 0; j < mainSetDimension; ++j) { final double errSum = E1 * yDotK[0][j] + E3 * yDotK[2][j] + E4 * yDotK[3][j] + E5 * yDotK[4][j] + E6 * yDotK[5][j] + E7 * yDotK[6][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 ratio = h * errSum / tol; error += ratio * ratio; } return FastMath.sqrt(error / mainSetDimension); } }