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
018package org.apache.commons.math3.ode.nonstiff;
019
020import 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 * @since 1.2
046 */
047
048public class DormandPrince54Integrator extends EmbeddedRungeKuttaIntegrator {
049
050  /** Integrator method name. */
051  private static final String METHOD_NAME = "Dormand-Prince 5(4)";
052
053  /** Time steps Butcher array. */
054  private static final double[] STATIC_C = {
055    1.0/5.0, 3.0/10.0, 4.0/5.0, 8.0/9.0, 1.0, 1.0
056  };
057
058  /** Internal weights Butcher array. */
059  private static final double[][] STATIC_A = {
060    {1.0/5.0},
061    {3.0/40.0, 9.0/40.0},
062    {44.0/45.0, -56.0/15.0, 32.0/9.0},
063    {19372.0/6561.0, -25360.0/2187.0, 64448.0/6561.0,  -212.0/729.0},
064    {9017.0/3168.0, -355.0/33.0, 46732.0/5247.0, 49.0/176.0, -5103.0/18656.0},
065    {35.0/384.0, 0.0, 500.0/1113.0, 125.0/192.0, -2187.0/6784.0, 11.0/84.0}
066  };
067
068  /** Propagation weights Butcher array. */
069  private static final double[] STATIC_B = {
070    35.0/384.0, 0.0, 500.0/1113.0, 125.0/192.0, -2187.0/6784.0, 11.0/84.0, 0.0
071  };
072
073  /** Error array, element 1. */
074  private static final double E1 =     71.0 / 57600.0;
075
076  // element 2 is zero, so it is neither stored nor used
077
078  /** Error array, element 3. */
079  private static final double E3 =    -71.0 / 16695.0;
080
081  /** Error array, element 4. */
082  private static final double E4 =     71.0 / 1920.0;
083
084  /** Error array, element 5. */
085  private static final double E5 = -17253.0 / 339200.0;
086
087  /** Error array, element 6. */
088  private static final double E6 =     22.0 / 525.0;
089
090  /** Error array, element 7. */
091  private static final double E7 =     -1.0 / 40.0;
092
093  /** Simple constructor.
094   * Build a fifth order Dormand-Prince integrator with the given step bounds
095   * @param minStep minimal step (sign is irrelevant, regardless of
096   * integration direction, forward or backward), the last step can
097   * be smaller than this
098   * @param maxStep maximal step (sign is irrelevant, regardless of
099   * integration direction, forward or backward), the last step can
100   * be smaller than this
101   * @param scalAbsoluteTolerance allowed absolute error
102   * @param scalRelativeTolerance allowed relative error
103   */
104  public DormandPrince54Integrator(final double minStep, final double maxStep,
105                                   final double scalAbsoluteTolerance,
106                                   final double scalRelativeTolerance) {
107    super(METHOD_NAME, true, STATIC_C, STATIC_A, STATIC_B, new DormandPrince54StepInterpolator(),
108          minStep, maxStep, scalAbsoluteTolerance, scalRelativeTolerance);
109  }
110
111  /** Simple constructor.
112   * Build a fifth order Dormand-Prince integrator with the given step bounds
113   * @param minStep minimal step (sign is irrelevant, regardless of
114   * integration direction, forward or backward), the last step can
115   * be smaller than this
116   * @param maxStep maximal step (sign is irrelevant, regardless of
117   * integration direction, forward or backward), the last step can
118   * be smaller than this
119   * @param vecAbsoluteTolerance allowed absolute error
120   * @param vecRelativeTolerance allowed relative error
121   */
122  public DormandPrince54Integrator(final double minStep, final double maxStep,
123                                   final double[] vecAbsoluteTolerance,
124                                   final double[] vecRelativeTolerance) {
125    super(METHOD_NAME, true, STATIC_C, STATIC_A, STATIC_B, new DormandPrince54StepInterpolator(),
126          minStep, maxStep, vecAbsoluteTolerance, vecRelativeTolerance);
127  }
128
129  /** {@inheritDoc} */
130  @Override
131  public int getOrder() {
132    return 5;
133  }
134
135  /** {@inheritDoc} */
136  @Override
137  protected double estimateError(final double[][] yDotK,
138                                 final double[] y0, final double[] y1,
139                                 final double h) {
140
141    double error = 0;
142
143    for (int j = 0; j < mainSetDimension; ++j) {
144        final double errSum = E1 * yDotK[0][j] +  E3 * yDotK[2][j] +
145                              E4 * yDotK[3][j] +  E5 * yDotK[4][j] +
146                              E6 * yDotK[5][j] +  E7 * yDotK[6][j];
147
148        final double yScale = FastMath.max(FastMath.abs(y0[j]), FastMath.abs(y1[j]));
149        final double tol = (vecAbsoluteTolerance == null) ?
150                           (scalAbsoluteTolerance + scalRelativeTolerance * yScale) :
151                               (vecAbsoluteTolerance[j] + vecRelativeTolerance[j] * yScale);
152        final double ratio  = h * errSum / tol;
153        error += ratio * ratio;
154
155    }
156
157    return FastMath.sqrt(error / mainSetDimension);
158
159  }
160
161}