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 }