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.math.ode.nonstiff;
019    
020    
021    /**
022     * This class implements the 5(4) Dormand-Prince integrator for Ordinary
023     * Differential Equations.
024    
025     * <p>This integrator is an embedded Runge-Kutta integrator
026     * of order 5(4) used in local extrapolation mode (i.e. the solution
027     * is computed using the high order formula) with stepsize control
028     * (and automatic step initialization) and continuous output. This
029     * method uses 7 functions evaluations per step. However, since this
030     * is an <i>fsal</i>, the last evaluation of one step is the same as
031     * the first evaluation of the next step and hence can be avoided. So
032     * the cost is really 6 functions evaluations per step.</p>
033     *
034     * <p>This method has been published (whithout the continuous output
035     * that was added by Shampine in 1986) in the following article :
036     * <pre>
037     *  A family of embedded Runge-Kutta formulae
038     *  J. R. Dormand and P. J. Prince
039     *  Journal of Computational and Applied Mathematics
040     *  volume 6, no 1, 1980, pp. 19-26
041     * </pre></p>
042     *
043     * @version $Revision: 810196 $ $Date: 2009-09-01 15:47:46 -0400 (Tue, 01 Sep 2009) $
044     * @since 1.2
045     */
046    
047    public class DormandPrince54Integrator extends EmbeddedRungeKuttaIntegrator {
048    
049      /** Integrator method name. */
050      private static final String METHOD_NAME = "Dormand-Prince 5(4)";
051    
052      /** Time steps Butcher array. */
053      private static final double[] STATIC_C = {
054        1.0/5.0, 3.0/10.0, 4.0/5.0, 8.0/9.0, 1.0, 1.0
055      };
056    
057      /** Internal weights Butcher array. */
058      private static final double[][] STATIC_A = {
059        {1.0/5.0},
060        {3.0/40.0, 9.0/40.0},
061        {44.0/45.0, -56.0/15.0, 32.0/9.0},
062        {19372.0/6561.0, -25360.0/2187.0, 64448.0/6561.0,  -212.0/729.0},
063        {9017.0/3168.0, -355.0/33.0, 46732.0/5247.0, 49.0/176.0, -5103.0/18656.0},
064        {35.0/384.0, 0.0, 500.0/1113.0, 125.0/192.0, -2187.0/6784.0, 11.0/84.0}
065      };
066    
067      /** Propagation weights Butcher array. */
068      private static final double[] STATIC_B = {
069        35.0/384.0, 0.0, 500.0/1113.0, 125.0/192.0, -2187.0/6784.0, 11.0/84.0, 0.0
070      };
071    
072      /** Error array, element 1. */
073      private static final double E1 =     71.0 / 57600.0;
074    
075      // element 2 is zero, so it is neither stored nor used
076    
077      /** Error array, element 3. */
078      private static final double E3 =    -71.0 / 16695.0;
079    
080      /** Error array, element 4. */
081      private static final double E4 =     71.0 / 1920.0;
082    
083      /** Error array, element 5. */
084      private static final double E5 = -17253.0 / 339200.0;
085    
086      /** Error array, element 6. */
087      private static final double E6 =     22.0 / 525.0;
088    
089      /** Error array, element 7. */
090      private static final double E7 =     -1.0 / 40.0;
091    
092      /** Simple constructor.
093       * Build a fifth order Dormand-Prince integrator with the given step bounds
094       * @param minStep minimal step (must be positive even for backward
095       * integration), the last step can be smaller than this
096       * @param maxStep maximal step (must be positive even for backward
097       * integration)
098       * @param scalAbsoluteTolerance allowed absolute error
099       * @param scalRelativeTolerance allowed relative error
100       */
101      public DormandPrince54Integrator(final double minStep, final double maxStep,
102                                       final double scalAbsoluteTolerance,
103                                       final double scalRelativeTolerance) {
104        super(METHOD_NAME, true, STATIC_C, STATIC_A, STATIC_B, new DormandPrince54StepInterpolator(),
105              minStep, maxStep, scalAbsoluteTolerance, scalRelativeTolerance);
106      }
107    
108      /** Simple constructor.
109       * Build a fifth order Dormand-Prince integrator with the given step bounds
110       * @param minStep minimal step (must be positive even for backward
111       * integration), the last step can be smaller than this
112       * @param maxStep maximal step (must be positive even for backward
113       * integration)
114       * @param vecAbsoluteTolerance allowed absolute error
115       * @param vecRelativeTolerance allowed relative error
116       */
117      public DormandPrince54Integrator(final double minStep, final double maxStep,
118                                       final double[] vecAbsoluteTolerance,
119                                       final double[] vecRelativeTolerance) {
120        super(METHOD_NAME, true, STATIC_C, STATIC_A, STATIC_B, new DormandPrince54StepInterpolator(),
121              minStep, maxStep, vecAbsoluteTolerance, vecRelativeTolerance);
122      }
123    
124      /** {@inheritDoc} */
125      @Override
126      public int getOrder() {
127        return 5;
128      }
129    
130      /** {@inheritDoc} */
131      @Override
132      protected double estimateError(final double[][] yDotK,
133                                     final double[] y0, final double[] y1,
134                                     final double h) {
135    
136        double error = 0;
137    
138        for (int j = 0; j < y0.length; ++j) {
139            final double errSum = E1 * yDotK[0][j] +  E3 * yDotK[2][j] +
140                                  E4 * yDotK[3][j] +  E5 * yDotK[4][j] +
141                                  E6 * yDotK[5][j] +  E7 * yDotK[6][j];
142    
143            final double yScale = Math.max(Math.abs(y0[j]), Math.abs(y1[j]));
144            final double tol = (vecAbsoluteTolerance == null) ?
145                               (scalAbsoluteTolerance + scalRelativeTolerance * yScale) :
146                                   (vecAbsoluteTolerance[j] + vecRelativeTolerance[j] * yScale);
147            final double ratio  = h * errSum / tol;
148            error += ratio * ratio;
149    
150        }
151    
152        return Math.sqrt(error / y0.length);
153    
154      }
155    
156    }