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}