1 /*
2 * Licensed to the Apache Software Foundation (ASF) under one or more
3 * contributor license agreements. See the NOTICE file distributed with
4 * this work for additional information regarding copyright ownership.
5 * The ASF licenses this file to You under the Apache License, Version 2.0
6 * (the "License"); you may not use this file except in compliance with
7 * the License. You may obtain a copy of the License at
8 *
9 * http://www.apache.org/licenses/LICENSE-2.0
10 *
11 * Unless required by applicable law or agreed to in writing, software
12 * distributed under the License is distributed on an "AS IS" BASIS,
13 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
14 * See the License for the specific language governing permissions and
15 * limitations under the License.
16 */
17
18 package org.apache.commons.math4.legacy.ode.nonstiff;
19
20 import org.apache.commons.math4.legacy.ode.sampling.StepInterpolator;
21 import org.apache.commons.math4.core.jdkmath.JdkMath;
22
23 /**
24 * This class represents an interpolator over the last step during an
25 * ODE integration for the 6th order Luther integrator.
26 *
27 * <p>This interpolator computes dense output inside the last
28 * step computed. The interpolation equation is consistent with the
29 * integration scheme.</p>
30 *
31 * @see LutherIntegrator
32 * @since 3.3
33 */
34
35 class LutherStepInterpolator extends RungeKuttaStepInterpolator {
36
37 /** Serializable version identifier. */
38 private static final long serialVersionUID = 20140416L;
39
40 /** Square root. */
41 private static final double Q = JdkMath.sqrt(21);
42
43 /** Simple constructor.
44 * This constructor builds an instance that is not usable yet, the
45 * {@link
46 * org.apache.commons.math4.legacy.ode.sampling.AbstractStepInterpolator#reinitialize}
47 * method should be called before using the instance in order to
48 * initialize the internal arrays. This constructor is used only
49 * in order to delay the initialization in some cases. The {@link
50 * RungeKuttaIntegrator} class uses the prototyping design pattern
51 * to create the step interpolators by cloning an uninitialized model
52 * and later initializing the copy.
53 */
54 // CHECKSTYLE: stop RedundantModifier
55 // the public modifier here is needed for serialization
56 public LutherStepInterpolator() {
57 }
58 // CHECKSTYLE: resume RedundantModifier
59
60 /** Copy constructor.
61 * @param interpolator interpolator to copy from. The copy is a deep
62 * copy: its arrays are separated from the original arrays of the
63 * instance
64 */
65 LutherStepInterpolator(final LutherStepInterpolator interpolator) {
66 super(interpolator);
67 }
68
69 /** {@inheritDoc} */
70 @Override
71 protected StepInterpolator doCopy() {
72 return new LutherStepInterpolator(this);
73 }
74
75
76 /** {@inheritDoc} */
77 @Override
78 protected void computeInterpolatedStateAndDerivatives(final double theta,
79 final double oneMinusThetaH) {
80
81 // the coefficients below have been computed by solving the
82 // order conditions from a theorem from Butcher (1963), using
83 // the method explained in Folkmar Bornemann paper "Runge-Kutta
84 // Methods, Trees, and Maple", Center of Mathematical Sciences, Munich
85 // University of Technology, February 9, 2001
86 //<http://wwwzenger.informatik.tu-muenchen.de/selcuk/sjam012101.html>
87
88 // the method is implemented in the rkcheck tool
89 // <https://www.spaceroots.org/software/rkcheck/index.html>.
90 // Running it for order 5 gives the following order conditions
91 // for an interpolator:
92 // order 1 conditions
93 // \sum_{i=1}^{i=s}\left(b_{i} \right) =1
94 // order 2 conditions
95 // \sum_{i=1}^{i=s}\left(b_{i} c_{i}\right) = \frac{\theta}{2}
96 // order 3 conditions
97 // \sum_{i=2}^{i=s}\left(b_{i} \sum_{j=1}^{j=i-1}{\left(a_{i,j} c_{j} \right)}\right) = \frac{\theta^{2}}{6}
98 // \sum_{i=1}^{i=s}\left(b_{i} c_{i}^{2}\right) = \frac{\theta^{2}}{3}
99 // order 4 conditions
100 // \sum_{i=3}^{i=s}\left(b_{i} \sum_{j=2}^{j=i-1}{\left(a_{i,j} \sum_{k=1}^{k=j-1}{\left(a_{j,k} c_{k} \right)} \right)}\right) = \frac{\theta^{3}}{24}
101 // \sum_{i=2}^{i=s}\left(b_{i} \sum_{j=1}^{j=i-1}{\left(a_{i,j} c_{j}^{2} \right)}\right) = \frac{\theta^{3}}{12}
102 // \sum_{i=2}^{i=s}\left(b_{i} c_{i}\sum_{j=1}^{j=i-1}{\left(a_{i,j} c_{j} \right)}\right) = \frac{\theta^{3}}{8}
103 // \sum_{i=1}^{i=s}\left(b_{i} c_{i}^{3}\right) = \frac{\theta^{3}}{4}
104 // order 5 conditions
105 // \sum_{i=4}^{i=s}\left(b_{i} \sum_{j=3}^{j=i-1}{\left(a_{i,j} \sum_{k=2}^{k=j-1}{\left(a_{j,k} \sum_{l=1}^{l=k-1}{\left(a_{k,l} c_{l} \right)} \right)} \right)}\right) = \frac{\theta^{4}}{120}
106 // \sum_{i=3}^{i=s}\left(b_{i} \sum_{j=2}^{j=i-1}{\left(a_{i,j} \sum_{k=1}^{k=j-1}{\left(a_{j,k} c_{k}^{2} \right)} \right)}\right) = \frac{\theta^{4}}{60}
107 // \sum_{i=3}^{i=s}\left(b_{i} \sum_{j=2}^{j=i-1}{\left(a_{i,j} c_{j}\sum_{k=1}^{k=j-1}{\left(a_{j,k} c_{k} \right)} \right)}\right) = \frac{\theta^{4}}{40}
108 // \sum_{i=2}^{i=s}\left(b_{i} \sum_{j=1}^{j=i-1}{\left(a_{i,j} c_{j}^{3} \right)}\right) = \frac{\theta^{4}}{20}
109 // \sum_{i=3}^{i=s}\left(b_{i} c_{i}\sum_{j=2}^{j=i-1}{\left(a_{i,j} \sum_{k=1}^{k=j-1}{\left(a_{j,k} c_{k} \right)} \right)}\right) = \frac{\theta^{4}}{30}
110 // \sum_{i=2}^{i=s}\left(b_{i} c_{i}\sum_{j=1}^{j=i-1}{\left(a_{i,j} c_{j}^{2} \right)}\right) = \frac{\theta^{4}}{15}
111 // \sum_{i=2}^{i=s}\left(b_{i} \left(\sum_{j=1}^{j=i-1}{\left(a_{i,j} c_{j} \right)} \right)^{2}\right) = \frac{\theta^{4}}{20}
112 // \sum_{i=2}^{i=s}\left(b_{i} c_{i}^{2}\sum_{j=1}^{j=i-1}{\left(a_{i,j} c_{j} \right)}\right) = \frac{\theta^{4}}{10}
113 // \sum_{i=1}^{i=s}\left(b_{i} c_{i}^{4}\right) = \frac{\theta^{4}}{5}
114
115 // The a_{j,k} and c_{k} are given by the integrator Butcher arrays. What remains to solve
116 // are the b_i for the interpolator. They are found by solving the above equations.
117 // For a given interpolator, some equations are redundant, so in our case when we select
118 // all equations from order 1 to 4, we still don't have enough independent equations
119 // to solve from b_1 to b_7. We need to also select one equation from order 5. Here,
120 // we selected the last equation. It appears this choice implied at least the last 3 equations
121 // are fulfilled, but some of the former ones are not, so the resulting interpolator is order 5.
122 // At the end, we get the b_i as polynomials in theta.
123
124 final double coeffDot1 = 1 + theta * ( -54 / 5.0 + theta * ( 36 + theta * ( -47 + theta * 21)));
125 final double coeffDot2 = 0;
126 final double coeffDot3 = theta * (-208 / 15.0 + theta * ( 320 / 3.0 + theta * (-608 / 3.0 + theta * 112)));
127 final double coeffDot4 = theta * ( 324 / 25.0 + theta * ( -486 / 5.0 + theta * ( 972 / 5.0 + theta * -567 / 5.0)));
128 final double coeffDot5 = theta * ((833 + 343 * Q) / 150.0 + theta * ((-637 - 357 * Q) / 30.0 + theta * ((392 + 287 * Q) / 15.0 + theta * (-49 - 49 * Q) / 5.0)));
129 final double coeffDot6 = theta * ((833 - 343 * Q) / 150.0 + theta * ((-637 + 357 * Q) / 30.0 + theta * ((392 - 287 * Q) / 15.0 + theta * (-49 + 49 * Q) / 5.0)));
130 final double coeffDot7 = theta * ( 3 / 5.0 + theta * ( -3 + theta * 3));
131
132 if (previousState != null && theta <= 0.5) {
133
134 final double coeff1 = 1 + theta * ( -27 / 5.0 + theta * ( 12 + theta * ( -47 / 4.0 + theta * 21 / 5.0)));
135 final double coeff2 = 0;
136 final double coeff3 = theta * (-104 / 15.0 + theta * ( 320 / 9.0 + theta * (-152 / 3.0 + theta * 112 / 5.0)));
137 final double coeff4 = theta * ( 162 / 25.0 + theta * ( -162 / 5.0 + theta * ( 243 / 5.0 + theta * -567 / 25.0)));
138 final double coeff5 = theta * ((833 + 343 * Q) / 300.0 + theta * ((-637 - 357 * Q) / 90.0 + theta * ((392 + 287 * Q) / 60.0 + theta * (-49 - 49 * Q) / 25.0)));
139 final double coeff6 = theta * ((833 - 343 * Q) / 300.0 + theta * ((-637 + 357 * Q) / 90.0 + theta * ((392 - 287 * Q) / 60.0 + theta * (-49 + 49 * Q) / 25.0)));
140 final double coeff7 = theta * ( 3 / 10.0 + theta * ( -1 + theta * ( 3 / 4.0)));
141 for (int i = 0; i < interpolatedState.length; ++i) {
142 final double yDot1 = yDotK[0][i];
143 final double yDot2 = yDotK[1][i];
144 final double yDot3 = yDotK[2][i];
145 final double yDot4 = yDotK[3][i];
146 final double yDot5 = yDotK[4][i];
147 final double yDot6 = yDotK[5][i];
148 final double yDot7 = yDotK[6][i];
149 interpolatedState[i] = previousState[i] +
150 theta * h * (coeff1 * yDot1 + coeff2 * yDot2 + coeff3 * yDot3 +
151 coeff4 * yDot4 + coeff5 * yDot5 + coeff6 * yDot6 + coeff7 * yDot7);
152 interpolatedDerivatives[i] = coeffDot1 * yDot1 + coeffDot2 * yDot2 + coeffDot3 * yDot3 +
153 coeffDot4 * yDot4 + coeffDot5 * yDot5 + coeffDot6 * yDot6 + coeffDot7 * yDot7;
154 }
155 } else {
156
157 final double coeff1 = -1 / 20.0 + theta * ( 19 / 20.0 + theta * ( -89 / 20.0 + theta * ( 151 / 20.0 + theta * -21 / 5.0)));
158 final double coeff2 = 0;
159 final double coeff3 = -16 / 45.0 + theta * ( -16 / 45.0 + theta * ( -328 / 45.0 + theta * ( 424 / 15.0 + theta * -112 / 5.0)));
160 final double coeff4 = theta * ( theta * ( 162 / 25.0 + theta * ( -648 / 25.0 + theta * 567 / 25.0)));
161 final double coeff5 = -49 / 180.0 + theta * ( -49 / 180.0 + theta * ((2254 + 1029 * Q) / 900.0 + theta * ((-1372 - 847 * Q) / 300.0 + theta * ( 49 + 49 * Q) / 25.0)));
162 final double coeff6 = -49 / 180.0 + theta * ( -49 / 180.0 + theta * ((2254 - 1029 * Q) / 900.0 + theta * ((-1372 + 847 * Q) / 300.0 + theta * ( 49 - 49 * Q) / 25.0)));
163 final double coeff7 = -1 / 20.0 + theta * ( -1 / 20.0 + theta * ( 1 / 4.0 + theta * ( -3 / 4.0)));
164 for (int i = 0; i < interpolatedState.length; ++i) {
165 final double yDot1 = yDotK[0][i];
166 final double yDot2 = yDotK[1][i];
167 final double yDot3 = yDotK[2][i];
168 final double yDot4 = yDotK[3][i];
169 final double yDot5 = yDotK[4][i];
170 final double yDot6 = yDotK[5][i];
171 final double yDot7 = yDotK[6][i];
172 interpolatedState[i] = currentState[i] +
173 oneMinusThetaH * (coeff1 * yDot1 + coeff2 * yDot2 + coeff3 * yDot3 +
174 coeff4 * yDot4 + coeff5 * yDot5 + coeff6 * yDot6 + coeff7 * yDot7);
175 interpolatedDerivatives[i] = coeffDot1 * yDot1 + coeffDot2 * yDot2 + coeffDot3 * yDot3 +
176 coeffDot4 * yDot4 + coeffDot5 * yDot5 + coeffDot6 * yDot6 + coeffDot7 * yDot7;
177 }
178 }
179 }
180 }