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 java.util.Arrays;
26
27 import org.hipparchus.Field;
28 import org.hipparchus.CalculusFieldElement;
29 import org.hipparchus.exception.MathIllegalArgumentException;
30 import org.hipparchus.exception.MathIllegalStateException;
31 import org.hipparchus.linear.Array2DRowFieldMatrix;
32 import org.hipparchus.linear.FieldMatrix;
33 import org.hipparchus.linear.FieldMatrixPreservingVisitor;
34 import org.hipparchus.ode.FieldEquationsMapper;
35 import org.hipparchus.ode.FieldODEStateAndDerivative;
36 import org.hipparchus.ode.LocalizedODEFormats;
37 import org.hipparchus.util.MathArrays;
38 import org.hipparchus.util.MathUtils;
39
40
41 /**
42 * This class implements implicit Adams-Moulton integrators for Ordinary
43 * Differential Equations.
44 *
45 * <p>Adams-Moulton methods (in fact due to Adams alone) are implicit
46 * multistep ODE solvers. This implementation is a variation of the classical
47 * one: it uses adaptive stepsize to implement error control, whereas
48 * classical implementations are fixed step size. The value of state vector
49 * at step n+1 is a simple combination of the value at step n and of the
50 * derivatives at steps n+1, n, n-1 ... Since y'<sub>n+1</sub> is needed to
51 * compute y<sub>n+1</sub>, another method must be used to compute a first
52 * estimate of y<sub>n+1</sub>, then compute y'<sub>n+1</sub>, then compute
53 * a final estimate of y<sub>n+1</sub> using the following formulas. Depending
54 * on the number k of previous steps one wants to use for computing the next
55 * value, different formulas are available for the final estimate:</p>
56 * <ul>
57 * <li>k = 1: y<sub>n+1</sub> = y<sub>n</sub> + h y'<sub>n+1</sub></li>
58 * <li>k = 2: y<sub>n+1</sub> = y<sub>n</sub> + h (y'<sub>n+1</sub>+y'<sub>n</sub>)/2</li>
59 * <li>k = 3: y<sub>n+1</sub> = y<sub>n</sub> + h (5y'<sub>n+1</sub>+8y'<sub>n</sub>-y'<sub>n-1</sub>)/12</li>
60 * <li>k = 4: y<sub>n+1</sub> = y<sub>n</sub> + h (9y'<sub>n+1</sub>+19y'<sub>n</sub>-5y'<sub>n-1</sub>+y'<sub>n-2</sub>)/24</li>
61 * <li>...</li>
62 * </ul>
63 *
64 * <p>A k-steps Adams-Moulton method is of order k+1.</p>
65 *
66 * <p> There must be sufficient time for the {@link #setStarterIntegrator(org.hipparchus.ode.FieldODEIntegrator)
67 * starter integrator} to take several steps between the the last reset event, and the end
68 * of integration, otherwise an exception may be thrown during integration. The user can
69 * adjust the end date of integration, or the step size of the starter integrator to
70 * ensure a sufficient number of steps can be completed before the end of integration.
71 * </p>
72 *
73 * <p><strong>Implementation details</strong></p>
74 *
75 * <p>We define scaled derivatives s<sub>i</sub>(n) at step n as:
76 * \[
77 * \left\{\begin{align}
78 * s_1(n) &= h y'_n \text{ for first derivative}\\
79 * s_2(n) &= \frac{h^2}{2} y_n'' \text{ for second derivative}\\
80 * s_3(n) &= \frac{h^3}{6} y_n''' \text{ for third derivative}\\
81 * &\cdots\\
82 * s_k(n) &= \frac{h^k}{k!} y_n^{(k)} \text{ for } k^\mathrm{th} \text{ derivative}
83 * \end{align}\right.
84 * \]</p>
85 *
86 * <p>The definitions above use the classical representation with several previous first
87 * derivatives. Lets define
88 * \[
89 * q_n = [ s_1(n-1) s_1(n-2) \ldots s_1(n-(k-1)) ]^T
90 * \]
91 * (we omit the k index in the notation for clarity). With these definitions,
92 * Adams-Moulton methods can be written:</p>
93 * <ul>
94 * <li>k = 1: y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n+1)</li>
95 * <li>k = 2: y<sub>n+1</sub> = y<sub>n</sub> + 1/2 s<sub>1</sub>(n+1) + [ 1/2 ] q<sub>n+1</sub></li>
96 * <li>k = 3: y<sub>n+1</sub> = y<sub>n</sub> + 5/12 s<sub>1</sub>(n+1) + [ 8/12 -1/12 ] q<sub>n+1</sub></li>
97 * <li>k = 4: y<sub>n+1</sub> = y<sub>n</sub> + 9/24 s<sub>1</sub>(n+1) + [ 19/24 -5/24 1/24 ] q<sub>n+1</sub></li>
98 * <li>...</li>
99 * </ul>
100 *
101 * <p>Instead of using the classical representation with first derivatives only (y<sub>n</sub>,
102 * s<sub>1</sub>(n+1) and q<sub>n+1</sub>), our implementation uses the Nordsieck vector with
103 * higher degrees scaled derivatives all taken at the same step (y<sub>n</sub>, s<sub>1</sub>(n)
104 * and r<sub>n</sub>) where r<sub>n</sub> is defined as:
105 * \[
106 * r_n = [ s_2(n), s_3(n) \ldots s_k(n) ]^T
107 * \]
108 * (here again we omit the k index in the notation for clarity)
109 * </p>
110 *
111 * <p>Taylor series formulas show that for any index offset i, s<sub>1</sub>(n-i) can be
112 * computed from s<sub>1</sub>(n), s<sub>2</sub>(n) ... s<sub>k</sub>(n), the formula being exact
113 * for degree k polynomials.
114 * \[
115 * s_1(n-i) = s_1(n) + \sum_{j\gt 0} (j+1) (-i)^j s_{j+1}(n)
116 * \]
117 * The previous formula can be used with several values for i to compute the transform between
118 * classical representation and Nordsieck vector. The transform between r<sub>n</sub>
119 * and q<sub>n</sub> resulting from the Taylor series formulas above is:
120 * \[
121 * q_n = s_1(n) u + P r_n
122 * \]
123 * where u is the [ 1 1 ... 1 ]<sup>T</sup> vector and P is the (k-1)×(k-1) matrix built
124 * with the \((j+1) (-i)^j\) terms with i being the row number starting from 1 and j being
125 * the column number starting from 1:
126 * \[
127 * P=\begin{bmatrix}
128 * -2 & 3 & -4 & 5 & \ldots \\
129 * -4 & 12 & -32 & 80 & \ldots \\
130 * -6 & 27 & -108 & 405 & \ldots \\
131 * -8 & 48 & -256 & 1280 & \ldots \\
132 * & & \ldots\\
133 * \end{bmatrix}
134 * \]
135 *
136 * <p>Using the Nordsieck vector has several advantages:</p>
137 * <ul>
138 * <li>it greatly simplifies step interpolation as the interpolator mainly applies
139 * Taylor series formulas,</li>
140 * <li>it simplifies step changes that occur when discrete events that truncate
141 * the step are triggered,</li>
142 * <li>it allows to extend the methods in order to support adaptive stepsize.</li>
143 * </ul>
144 *
145 * <p>The predicted Nordsieck vector at step n+1 is computed from the Nordsieck vector at step
146 * n as follows:
147 * <ul>
148 * <li>Y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n) + u<sup>T</sup> r<sub>n</sub></li>
149 * <li>S<sub>1</sub>(n+1) = h f(t<sub>n+1</sub>, Y<sub>n+1</sub>)</li>
150 * <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>
151 * </ul>
152 * where A is a rows shifting matrix (the lower left part is an identity matrix):
153 * <pre>
154 * [ 0 0 ... 0 0 | 0 ]
155 * [ ---------------+---]
156 * [ 1 0 ... 0 0 | 0 ]
157 * A = [ 0 1 ... 0 0 | 0 ]
158 * [ ... | 0 ]
159 * [ 0 0 ... 1 0 | 0 ]
160 * [ 0 0 ... 0 1 | 0 ]
161 * </pre>
162 * From this predicted vector, the corrected vector is computed as follows:
163 * <ul>
164 * <li>y<sub>n+1</sub> = y<sub>n</sub> + S<sub>1</sub>(n+1) + [ -1 +1 -1 +1 ... ±1 ] r<sub>n+1</sub></li>
165 * <li>s<sub>1</sub>(n+1) = h f(t<sub>n+1</sub>, y<sub>n+1</sub>)</li>
166 * <li>r<sub>n+1</sub> = R<sub>n+1</sub> + (s<sub>1</sub>(n+1) - S<sub>1</sub>(n+1)) P<sup>-1</sup> u</li>
167 * </ul>
168 * <p>where the upper case Y<sub>n+1</sub>, S<sub>1</sub>(n+1) and R<sub>n+1</sub> represent the
169 * predicted states whereas the lower case y<sub>n+1</sub>, s<sub>n+1</sub> and r<sub>n+1</sub>
170 * represent the corrected states.</p>
171 *
172 * <p>The P<sup>-1</sup>u vector and the P<sup>-1</sup> A P matrix do not depend on the state,
173 * they only depend on k and therefore are precomputed once for all.</p>
174 *
175 * @param <T> the type of the field elements
176 */
177 public class AdamsMoultonFieldIntegrator<T extends CalculusFieldElement<T>> extends AdamsFieldIntegrator<T> {
178
179 /** Integrator method name. */
180 private static final String METHOD_NAME = "Adams-Moulton";
181
182 /**
183 * Build an Adams-Moulton integrator with the given order and error control parameters.
184 * @param field field to which the time and state vector elements belong
185 * @param nSteps number of steps of the method excluding the one being computed
186 * @param minStep minimal step (sign is irrelevant, regardless of
187 * integration direction, forward or backward), the last step can
188 * be smaller than this
189 * @param maxStep maximal step (sign is irrelevant, regardless of
190 * integration direction, forward or backward), the last step can
191 * be smaller than this
192 * @param scalAbsoluteTolerance allowed absolute error
193 * @param scalRelativeTolerance allowed relative error
194 * @exception MathIllegalArgumentException if order is 1 or less
195 */
196 public AdamsMoultonFieldIntegrator(final Field<T> field, final int nSteps,
197 final double minStep, final double maxStep,
198 final double scalAbsoluteTolerance,
199 final double scalRelativeTolerance)
200 throws MathIllegalArgumentException {
201 super(field, METHOD_NAME, nSteps, nSteps + 1, minStep, maxStep,
202 scalAbsoluteTolerance, scalRelativeTolerance);
203 }
204
205 /**
206 * Build an Adams-Moulton integrator with the given order and error control parameters.
207 * @param field field to which the time and state vector elements belong
208 * @param nSteps number of steps of the method excluding the one being computed
209 * @param minStep minimal step (sign is irrelevant, regardless of
210 * integration direction, forward or backward), the last step can
211 * be smaller than this
212 * @param maxStep maximal step (sign is irrelevant, regardless of
213 * integration direction, forward or backward), the last step can
214 * be smaller than this
215 * @param vecAbsoluteTolerance allowed absolute error
216 * @param vecRelativeTolerance allowed relative error
217 * @exception IllegalArgumentException if order is 1 or less
218 */
219 public AdamsMoultonFieldIntegrator(final Field<T> field, final int nSteps,
220 final double minStep, final double maxStep,
221 final double[] vecAbsoluteTolerance,
222 final double[] vecRelativeTolerance)
223 throws IllegalArgumentException {
224 super(field, METHOD_NAME, nSteps, nSteps + 1, minStep, maxStep,
225 vecAbsoluteTolerance, vecRelativeTolerance);
226 }
227
228 /** {@inheritDoc} */
229 @Override
230 protected double errorEstimation(final T[] previousState, final T predictedTime,
231 final T[] predictedState, final T[] predictedScaled,
232 final FieldMatrix<T> predictedNordsieck) {
233 final double error = predictedNordsieck.walkInOptimizedOrder(new Corrector(previousState, predictedScaled, predictedState)).getReal();
234 if (Double.isNaN(error)) {
235 throw new MathIllegalStateException(LocalizedODEFormats.NAN_APPEARING_DURING_INTEGRATION,
236 predictedTime.getReal());
237 }
238 return error;
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
250 final T[] correctedYDot = computeDerivatives(globalCurrentState.getTime(), predictedY);
251
252 // update Nordsieck vector
253 final T[] correctedScaled = MathArrays.buildArray(getField(), predictedY.length);
254 for (int j = 0; j < correctedScaled.length; ++j) {
255 correctedScaled[j] = getStepSize().multiply(correctedYDot[j]);
256 }
257 updateHighOrderDerivativesPhase2(predictedScaled, correctedScaled, predictedNordsieck);
258
259 final FieldODEStateAndDerivative<T> updatedStepEnd =
260 equationsMapper.mapStateAndDerivative(globalCurrentState.getTime(), predictedY, correctedYDot);
261 return new AdamsFieldStateInterpolator<>(getStepSize(), updatedStepEnd,
262 correctedScaled, predictedNordsieck, isForward,
263 getStepStart(), updatedStepEnd,
264 equationsMapper);
265
266 }
267
268 /** Corrector for current state in Adams-Moulton method.
269 * <p>
270 * This visitor implements the Taylor series formula:
271 * <pre>
272 * Y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n+1) + [ -1 +1 -1 +1 ... ±1 ] r<sub>n+1</sub>
273 * </pre>
274 * </p>
275 */
276 private class Corrector implements FieldMatrixPreservingVisitor<T> {
277
278 /** Previous state. */
279 private final T[] previous;
280
281 /** Current scaled first derivative. */
282 private final T[] scaled;
283
284 /** Current state before correction. */
285 private final T[] before;
286
287 /** Current state after correction. */
288 private final T[] after;
289
290 /** Simple constructor.
291 * <p>
292 * All arrays will be stored by reference to caller arrays.
293 * </p>
294 * @param previous previous state
295 * @param scaled current scaled first derivative
296 * @param state state to correct (will be overwritten after visit)
297 */
298 Corrector(final T[] previous, final T[] scaled, final T[] state) {
299 this.previous = previous; // NOPMD - array reference storage is intentional and documented here
300 this.scaled = scaled; // NOPMD - array reference storage is intentional and documented here
301 this.after = state; // NOPMD - array reference storage is intentional and documented here
302 this.before = state.clone();
303 }
304
305 /** {@inheritDoc} */
306 @Override
307 public void start(int rows, int columns,
308 int startRow, int endRow, int startColumn, int endColumn) {
309 Arrays.fill(after, getField().getZero());
310 }
311
312 /** {@inheritDoc} */
313 @Override
314 public void visit(int row, int column, T value) {
315 if ((row & 0x1) == 0) {
316 after[column] = after[column].subtract(value);
317 } else {
318 after[column] = after[column].add(value);
319 }
320 }
321
322 /**
323 * End visiting the Nordsieck vector.
324 * <p>The correction is used to control stepsize. So its amplitude is
325 * considered to be an error, which must be normalized according to
326 * error control settings. If the normalized value is greater than 1,
327 * the correction was too large and the step must be rejected.</p>
328 * @return the normalized correction, if greater than 1, the step
329 * must be rejected
330 */
331 @Override
332 public T end() {
333
334 final StepsizeHelper helper = getStepSizeHelper();
335 T error = getField().getZero();
336 for (int i = 0; i < after.length; ++i) {
337 after[i] = after[i].add(previous[i].add(scaled[i]));
338 if (i < helper.getMainSetDimension()) {
339 final T tol = helper.getTolerance(i, MathUtils.max(previous[i].abs(), after[i].abs()));
340 final T ratio = after[i].subtract(before[i]).divide(tol); // (corrected-predicted)/tol
341 error = error.add(ratio.multiply(ratio));
342 }
343 }
344
345 return error.divide(helper.getMainSetDimension()).sqrt();
346
347 }
348 }
349
350 }