    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
    018    package org.apache.commons.math3.ode.nonstiff;
    019
    020    import org.apache.commons.math3.util.FastMath;
    021
    022
    023    /**
    024     * This class implements the 5(4) Dormand-Prince integrator for Ordinary
    025     * Differential Equations.
    026
    027     * <p>This integrator is an embedded Runge-Kutta integrator
    028     * of order 5(4) 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 7 functions evaluations per step. However, since this
    032     * is an <i>fsal</i>, the last evaluation of one step is the same as
    033     * the first evaluation of the next step and hence can be avoided. So
    034     * the cost is really 6 functions evaluations per step.</p>
    035     *
    036     * <p>This method has been published (whithout the continuous output
    037     * that was added by Shampine in 1986) in the following article :
    038     * <pre>
    039     *  A family of embedded Runge-Kutta formulae
    040     *  J. R. Dormand and P. J. Prince
    041     *  Journal of Computational and Applied Mathematics
    042     *  volume 6, no 1, 1980, pp. 19-26
    043     * </pre></p>
    044     *
    045     * @version $Id: DormandPrince54Integrator.java 1416643 2012-12-03 19:37:14Z tn $
    046     * @since 1.2
    047     */
    048
    049    public class DormandPrince54Integrator extends EmbeddedRungeKuttaIntegrator {
    050
    051      /** Integrator method name. */
    052      private static final String METHOD_NAME = "Dormand-Prince 5(4)";
    053
    054      /** Time steps Butcher array. */
    055      private static final double[] STATIC_C = {
    056        1.0/5.0, 3.0/10.0, 4.0/5.0, 8.0/9.0, 1.0, 1.0
    057      };
    058
    059      /** Internal weights Butcher array. */
    060      private static final double[][] STATIC_A = {
    061        {1.0/5.0},
    062        {3.0/40.0, 9.0/40.0},
    063        {44.0/45.0, -56.0/15.0, 32.0/9.0},
    064        {19372.0/6561.0, -25360.0/2187.0, 64448.0/6561.0,  -212.0/729.0},
    065        {9017.0/3168.0, -355.0/33.0, 46732.0/5247.0, 49.0/176.0, -5103.0/18656.0},
    066        {35.0/384.0, 0.0, 500.0/1113.0, 125.0/192.0, -2187.0/6784.0, 11.0/84.0}
    067      };
    068
    069      /** Propagation weights Butcher array. */
    070      private static final double[] STATIC_B = {
    071        35.0/384.0, 0.0, 500.0/1113.0, 125.0/192.0, -2187.0/6784.0, 11.0/84.0, 0.0
    072      };
    073
    074      /** Error array, element 1. */
    075      private static final double E1 =     71.0 / 57600.0;
    076
    077      // element 2 is zero, so it is neither stored nor used
    078
    079      /** Error array, element 3. */
    080      private static final double E3 =    -71.0 / 16695.0;
    081
    082      /** Error array, element 4. */
    083      private static final double E4 =     71.0 / 1920.0;
    084
    085      /** Error array, element 5. */
    086      private static final double E5 = -17253.0 / 339200.0;
    087
    088      /** Error array, element 6. */
    089      private static final double E6 =     22.0 / 525.0;
    090
    091      /** Error array, element 7. */
    092      private static final double E7 =     -1.0 / 40.0;
    093
    094      /** Simple constructor.
    095       * Build a fifth order Dormand-Prince integrator with the given step bounds
    096       * @param minStep minimal step (sign is irrelevant, regardless of
    097       * integration direction, forward or backward), the last step can
    098       * be smaller than this
    099       * @param maxStep maximal step (sign is irrelevant, regardless of
    100       * integration direction, forward or backward), the last step can
    101       * be smaller than this
    102       * @param scalAbsoluteTolerance allowed absolute error
    103       * @param scalRelativeTolerance allowed relative error
    104       */
    105      public DormandPrince54Integrator(final double minStep, final double maxStep,
    106                                       final double scalAbsoluteTolerance,
    107                                       final double scalRelativeTolerance) {
    108        super(METHOD_NAME, true, STATIC_C, STATIC_A, STATIC_B, new DormandPrince54StepInterpolator(),
    109              minStep, maxStep, scalAbsoluteTolerance, scalRelativeTolerance);
    110      }
    111
    112      /** Simple constructor.
    113       * Build a fifth order Dormand-Prince integrator with the given step bounds
    114       * @param minStep minimal step (sign is irrelevant, regardless of
    115       * integration direction, forward or backward), the last step can
    116       * be smaller than this
    117       * @param maxStep maximal step (sign is irrelevant, regardless of
    118       * integration direction, forward or backward), the last step can
    119       * be smaller than this
    120       * @param vecAbsoluteTolerance allowed absolute error
    121       * @param vecRelativeTolerance allowed relative error
    122       */
    123      public DormandPrince54Integrator(final double minStep, final double maxStep,
    124                                       final double[] vecAbsoluteTolerance,
    125                                       final double[] vecRelativeTolerance) {
    126        super(METHOD_NAME, true, STATIC_C, STATIC_A, STATIC_B, new DormandPrince54StepInterpolator(),
    127              minStep, maxStep, vecAbsoluteTolerance, vecRelativeTolerance);
    128      }
    129
    130      /** {@inheritDoc} */
    131      @Override
    132      public int getOrder() {
    133        return 5;
    134      }
    135
    136      /** {@inheritDoc} */
    137      @Override
    138      protected double estimateError(final double[][] yDotK,
    139                                     final double[] y0, final double[] y1,
    140                                     final double h) {
    141
    142        double error = 0;
    143
    144        for (int j = 0; j < mainSetDimension; ++j) {
    145            final double errSum = E1 * yDotK[0][j] +  E3 * yDotK[2][j] +
    146                                  E4 * yDotK[3][j] +  E5 * yDotK[4][j] +
    147                                  E6 * yDotK[5][j] +  E7 * yDotK[6][j];
    148
    149            final double yScale = FastMath.max(FastMath.abs(y0[j]), FastMath.abs(y1[j]));
    150            final double tol = (vecAbsoluteTolerance == null) ?
    151                               (scalAbsoluteTolerance + scalRelativeTolerance * yScale) :
    152                                   (vecAbsoluteTolerance[j] + vecRelativeTolerance[j] * yScale);
    153            final double ratio  = h * errSum / tol;
    154            error += ratio * ratio;
    155
    156        }
    157
    158        return FastMath.sqrt(error / mainSetDimension);
    159
    160      }
    161
    162    }