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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    *      https://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  /*
19   * This is not the original file distributed by the Apache Software Foundation
20   * It has been modified by the Hipparchus project
21   */
22  
23  package org.hipparchus.ode.nonstiff;
24  
25  import org.hipparchus.Field;
26  import org.hipparchus.CalculusFieldElement;
27  import org.hipparchus.exception.MathIllegalArgumentException;
28  import org.hipparchus.linear.Array2DRowFieldMatrix;
29  import org.hipparchus.linear.FieldMatrix;
30  import org.hipparchus.ode.FieldEquationsMapper;
31  import org.hipparchus.ode.FieldODEStateAndDerivative;
32  import org.hipparchus.util.FastMath;
33  
34  
35  /**
36   * This class implements explicit Adams-Bashforth integrators for Ordinary
37   * Differential Equations.
38   *
39   * <p>Adams-Bashforth methods (in fact due to Adams alone) are explicit
40   * multistep ODE solvers. This implementation is a variation of the classical
41   * one: it uses adaptive stepsize to implement error control, whereas
42   * classical implementations are fixed step size. The value of state vector
43   * at step n+1 is a simple combination of the value at step n and of the
44   * derivatives at steps n, n-1, n-2 ... Depending on the number k of previous
45   * steps one wants to use for computing the next value, different formulas
46   * are available:</p>
47   * <ul>
48   *   <li>k = 1: y<sub>n+1</sub> = y<sub>n</sub> + h y'<sub>n</sub></li>
49   *   <li>k = 2: y<sub>n+1</sub> = y<sub>n</sub> + h (3y'<sub>n</sub>-y'<sub>n-1</sub>)/2</li>
50   *   <li>k = 3: y<sub>n+1</sub> = y<sub>n</sub> + h (23y'<sub>n</sub>-16y'<sub>n-1</sub>+5y'<sub>n-2</sub>)/12</li>
51   *   <li>k = 4: y<sub>n+1</sub> = y<sub>n</sub> + h (55y'<sub>n</sub>-59y'<sub>n-1</sub>+37y'<sub>n-2</sub>-9y'<sub>n-3</sub>)/24</li>
52   *   <li>...</li>
53   * </ul>
54   *
55   * <p>A k-steps Adams-Bashforth method is of order k.</p>
56   *
57   * <p> There must be sufficient time for the {@link #setStarterIntegrator(org.hipparchus.ode.FieldODEIntegrator)
58   * starter integrator} to take several steps between the the last reset event, and the end
59   * of integration, otherwise an exception may be thrown during integration. The user can
60   * adjust the end date of integration, or the step size of the starter integrator to
61   * ensure a sufficient number of steps can be completed before the end of integration.
62   * </p>
63   *
64   * <p><strong>Implementation details</strong></p>
65   *
66   * <p>We define scaled derivatives s<sub>i</sub>(n) at step n as:
67   * \[
68   *   \left\{\begin{align}
69   *   s_1(n) &amp;= h y'_n \text{ for first derivative}\\
70   *   s_2(n) &amp;= \frac{h^2}{2} y_n'' \text{ for second derivative}\\
71   *   s_3(n) &amp;= \frac{h^3}{6} y_n''' \text{ for third derivative}\\
72   *   &amp;\cdots\\
73   *   s_k(n) &amp;= \frac{h^k}{k!} y_n^{(k)} \text{ for } k^\mathrm{th} \text{ derivative}
74   *   \end{align}\right.
75   * \]</p>
76   *
77   * <p>The definitions above use the classical representation with several previous first
78   * derivatives. Lets define
79   * \[
80   *   q_n = [ s_1(n-1) s_1(n-2) \ldots s_1(n-(k-1)) ]^T
81   * \]
82   * (we omit the k index in the notation for clarity). With these definitions,
83   * Adams-Bashforth methods can be written:
84   * \[
85   *   \left\{\begin{align}
86   *   k = 1: &amp; y_{n+1} = y_n +               s_1(n) \\
87   *   k = 2: &amp; y_{n+1} = y_n + \frac{3}{2}   s_1(n) + [ \frac{-1}{2} ] q_n \\
88   *   k = 3: &amp; y_{n+1} = y_n + \frac{23}{12} s_1(n) + [ \frac{-16}{12} \frac{5}{12} ] q_n \\
89   *   k = 4: &amp; y_{n+1} = y_n + \frac{55}{24} s_1(n) + [ \frac{-59}{24} \frac{37}{24} \frac{-9}{24} ] q_n \\
90   *          &amp; \cdots
91   *   \end{align}\right.
92   * \]
93   *
94   * <p>Instead of using the classical representation with first derivatives only (y<sub>n</sub>,
95   * s<sub>1</sub>(n) and q<sub>n</sub>), our implementation uses the Nordsieck vector with
96   * higher degrees scaled derivatives all taken at the same step (y<sub>n</sub>, s<sub>1</sub>(n)
97   * and r<sub>n</sub>) where r<sub>n</sub> is defined as:
98   * \[
99   * r_n = [ s_2(n), s_3(n) \ldots s_k(n) ]^T
100  * \]
101  * (here again we omit the k index in the notation for clarity)
102  * </p>
103  *
104  * <p>Taylor series formulas show that for any index offset i, s<sub>1</sub>(n-i) can be
105  * computed from s<sub>1</sub>(n), s<sub>2</sub>(n) ... s<sub>k</sub>(n), the formula being exact
106  * for degree k polynomials.
107  * \[
108  * s_1(n-i) = s_1(n) + \sum_{j\gt 0} (j+1) (-i)^j s_{j+1}(n)
109  * \]
110  * The previous formula can be used with several values for i to compute the transform between
111  * classical representation and Nordsieck vector. The transform between r<sub>n</sub>
112  * and q<sub>n</sub> resulting from the Taylor series formulas above is:
113  * \[
114  * q_n = s_1(n) u + P r_n
115  * \]
116  * where u is the [ 1 1 ... 1 ]<sup>T</sup> vector and P is the (k-1)&times;(k-1) matrix built
117  * with the \((j+1) (-i)^j\) terms with i being the row number starting from 1 and j being
118  * the column number starting from 1:
119  * \[
120  *   P=\begin{bmatrix}
121  *   -2  &amp;  3 &amp;   -4 &amp;    5 &amp; \ldots \\
122  *   -4  &amp; 12 &amp;  -32 &amp;   80 &amp; \ldots \\
123  *   -6  &amp; 27 &amp; -108 &amp;  405 &amp; \ldots \\
124  *   -8  &amp; 48 &amp; -256 &amp; 1280 &amp; \ldots \\
125  *       &amp;    &amp;  \ldots\\
126  *    \end{bmatrix}
127  * \]
128  *
129  * <p>Using the Nordsieck vector has several advantages:</p>
130  * <ul>
131  *   <li>it greatly simplifies step interpolation as the interpolator mainly applies
132  *   Taylor series formulas,</li>
133  *   <li>it simplifies step changes that occur when discrete events that truncate
134  *   the step are triggered,</li>
135  *   <li>it allows to extend the methods in order to support adaptive stepsize.</li>
136  * </ul>
137  *
138  * <p>The Nordsieck vector at step n+1 is computed from the Nordsieck vector at step n as follows:
139  * <ul>
140  *   <li>y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n) + u<sup>T</sup> r<sub>n</sub></li>
141  *   <li>s<sub>1</sub>(n+1) = h f(t<sub>n+1</sub>, y<sub>n+1</sub>)</li>
142  *   <li>r<sub>n+1</sub> = (s<sub>1</sub>(n) - s<sub>1</sub>(n+1)) P<sup>-1</sup> u + P<sup>-1</sup> A P r<sub>n</sub></li>
143  * </ul>
144  * <p>where A is a rows shifting matrix (the lower left part is an identity matrix):</p>
145  * <pre>
146  *        [ 0 0   ...  0 0 | 0 ]
147  *        [ ---------------+---]
148  *        [ 1 0   ...  0 0 | 0 ]
149  *    A = [ 0 1   ...  0 0 | 0 ]
150  *        [       ...      | 0 ]
151  *        [ 0 0   ...  1 0 | 0 ]
152  *        [ 0 0   ...  0 1 | 0 ]
153  * </pre>
154  *
155  * <p>The P<sup>-1</sup>u vector and the P<sup>-1</sup> A P matrix do not depend on the state,
156  * they only depend on k and therefore are precomputed once for all.</p>
157  *
158  * @param <T> the type of the field elements
159  */
160 public class AdamsBashforthFieldIntegrator<T extends CalculusFieldElement<T>> extends AdamsFieldIntegrator<T> {
161 
162     /** Integrator method name. */
163     private static final String METHOD_NAME = "Adams-Bashforth";
164 
165     /**
166      * Build an Adams-Bashforth integrator with the given order and step control parameters.
167      * @param field field to which the time and state vector elements belong
168      * @param nSteps number of steps of the method excluding the one being computed
169      * @param minStep minimal step (sign is irrelevant, regardless of
170      * integration direction, forward or backward), the last step can
171      * be smaller than this
172      * @param maxStep maximal step (sign is irrelevant, regardless of
173      * integration direction, forward or backward), the last step can
174      * be smaller than this
175      * @param scalAbsoluteTolerance allowed absolute error
176      * @param scalRelativeTolerance allowed relative error
177      * @exception MathIllegalArgumentException if order is 1 or less
178      */
179     public AdamsBashforthFieldIntegrator(final Field<T> field, final int nSteps,
180                                          final double minStep, final double maxStep,
181                                          final double scalAbsoluteTolerance,
182                                          final double scalRelativeTolerance)
183         throws MathIllegalArgumentException {
184         super(field, METHOD_NAME, nSteps, nSteps, minStep, maxStep,
185               scalAbsoluteTolerance, scalRelativeTolerance);
186     }
187 
188     /**
189      * Build an Adams-Bashforth integrator with the given order and step control parameters.
190      * @param field field to which the time and state vector elements belong
191      * @param nSteps number of steps of the method excluding the one being computed
192      * @param minStep minimal step (sign is irrelevant, regardless of
193      * integration direction, forward or backward), the last step can
194      * be smaller than this
195      * @param maxStep maximal step (sign is irrelevant, regardless of
196      * integration direction, forward or backward), the last step can
197      * be smaller than this
198      * @param vecAbsoluteTolerance allowed absolute error
199      * @param vecRelativeTolerance allowed relative error
200      * @exception IllegalArgumentException if order is 1 or less
201      */
202     public AdamsBashforthFieldIntegrator(final Field<T> field, final int nSteps,
203                                          final double minStep, final double maxStep,
204                                          final double[] vecAbsoluteTolerance,
205                                          final double[] vecRelativeTolerance)
206         throws IllegalArgumentException {
207         super(field, METHOD_NAME, nSteps, nSteps, minStep, maxStep,
208               vecAbsoluteTolerance, vecRelativeTolerance);
209     }
210 
211     /** {@inheritDoc} */
212     @Override
213     protected double errorEstimation(final T[] previousState, final T predictedTime,
214                                      final T[] predictedState, final T[] predictedScaled,
215                                      final FieldMatrix<T> predictedNordsieck) {
216 
217         final StepsizeHelper helper = getStepSizeHelper();
218         double error = 0;
219         for (int i = 0; i < helper.getMainSetDimension(); ++i) {
220             final double tol = helper.getTolerance(i, FastMath.abs(predictedState[i].getReal()));
221 
222             // apply Taylor formula from high order to low order,
223             // for the sake of numerical accuracy
224             double variation = 0;
225             int sign = predictedNordsieck.getRowDimension() % 2 == 0 ? -1 : 1;
226             for (int k = predictedNordsieck.getRowDimension() - 1; k >= 0; --k) {
227                 variation += sign * predictedNordsieck.getEntry(k, i).getReal();
228                 sign       = -sign;
229             }
230             variation -= predictedScaled[i].getReal();
231 
232             final double ratio  = (predictedState[i].getReal() - previousState[i].getReal() + variation) / tol;
233             error              += ratio * ratio;
234 
235         }
236 
237         return FastMath.sqrt(error / helper.getMainSetDimension());
238 
239     }
240 
241     /** {@inheritDoc} */
242     @Override
243     protected AdamsFieldStateInterpolator<T> finalizeStep(final T stepSize, final T[] predictedY,
244                                                           final T[] predictedScaled, final Array2DRowFieldMatrix<T> predictedNordsieck,
245                                                           final boolean isForward,
246                                                           final FieldODEStateAndDerivative<T> globalPreviousState,
247                                                           final FieldODEStateAndDerivative<T> globalCurrentState,
248                                                           final FieldEquationsMapper<T> equationsMapper) {
249         return new AdamsFieldStateInterpolator<>(getStepSize(), globalCurrentState,
250                                                  predictedScaled, predictedNordsieck, isForward,
251                                                  getStepStart(), globalCurrentState, equationsMapper);
252     }
253 
254 }