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 }