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 java.util.Arrays;
21
22 import org.apache.commons.math4.legacy.exception.DimensionMismatchException;
23 import org.apache.commons.math4.legacy.exception.MaxCountExceededException;
24 import org.apache.commons.math4.legacy.exception.NoBracketingException;
25 import org.apache.commons.math4.legacy.exception.NumberIsTooSmallException;
26 import org.apache.commons.math4.legacy.linear.Array2DRowRealMatrix;
27 import org.apache.commons.math4.legacy.linear.RealMatrixPreservingVisitor;
28 import org.apache.commons.math4.legacy.ode.EquationsMapper;
29 import org.apache.commons.math4.legacy.ode.ExpandableStatefulODE;
30 import org.apache.commons.math4.legacy.ode.sampling.NordsieckStepInterpolator;
31 import org.apache.commons.math4.core.jdkmath.JdkMath;
32
33
34 /**
35 * This class implements implicit Adams-Moulton integrators for Ordinary
36 * Differential Equations.
37 *
38 * <p>Adams-Moulton methods (in fact due to Adams alone) are implicit
39 * multistep ODE solvers. This implementation is a variation of the classical
40 * one: it uses adaptive stepsize to implement error control, whereas
41 * classical implementations are fixed step size. The value of state vector
42 * at step n+1 is a simple combination of the value at step n and of the
43 * derivatives at steps n+1, n, n-1 ... Since y'<sub>n+1</sub> is needed to
44 * compute y<sub>n+1</sub>, another method must be used to compute a first
45 * estimate of y<sub>n+1</sub>, then compute y'<sub>n+1</sub>, then compute
46 * a final estimate of y<sub>n+1</sub> using the following formulas. Depending
47 * on the number k of previous steps one wants to use for computing the next
48 * value, different formulas are available for the final estimate:</p>
49 * <ul>
50 * <li>k = 1: y<sub>n+1</sub> = y<sub>n</sub> + h y'<sub>n+1</sub></li>
51 * <li>k = 2: y<sub>n+1</sub> = y<sub>n</sub> + h (y'<sub>n+1</sub>+y'<sub>n</sub>)/2</li>
52 * <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>
53 * <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>
54 * <li>...</li>
55 * </ul>
56 *
57 * <p>A k-steps Adams-Moulton method is of order k+1.</p>
58 *
59 * <p><b>Implementation details</b></p>
60 *
61 * <p>We define scaled derivatives s<sub>i</sub>(n) at step n as:
62 * <div style="white-space: pre"><code>
63 * s<sub>1</sub>(n) = h y'<sub>n</sub> for first derivative
64 * s<sub>2</sub>(n) = h<sup>2</sup>/2 y''<sub>n</sub> for second derivative
65 * s<sub>3</sub>(n) = h<sup>3</sup>/6 y'''<sub>n</sub> for third derivative
66 * ...
67 * s<sub>k</sub>(n) = h<sup>k</sup>/k! y<sup>(k)</sup><sub>n</sub> for k<sup>th</sup> derivative
68 * </code></div>
69 *
70 * <p>The definitions above use the classical representation with several previous first
71 * derivatives. Lets define
72 * <div style="white-space: pre"><code>
73 * q<sub>n</sub> = [ s<sub>1</sub>(n-1) s<sub>1</sub>(n-2) ... s<sub>1</sub>(n-(k-1)) ]<sup>T</sup>
74 * </code></div>
75 * (we omit the k index in the notation for clarity). With these definitions,
76 * Adams-Moulton methods can be written:
77 * <ul>
78 * <li>k = 1: y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n+1)</li>
79 * <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>
80 * <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>
81 * <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>
82 * <li>...</li>
83 * </ul>
84 *
85 * <p>Instead of using the classical representation with first derivatives only (y<sub>n</sub>,
86 * s<sub>1</sub>(n+1) and q<sub>n+1</sub>), our implementation uses the Nordsieck vector with
87 * higher degrees scaled derivatives all taken at the same step (y<sub>n</sub>, s<sub>1</sub>(n)
88 * and r<sub>n</sub>) where r<sub>n</sub> is defined as:
89 * <div style="white-space: pre"><code>
90 * r<sub>n</sub> = [ s<sub>2</sub>(n), s<sub>3</sub>(n) ... s<sub>k</sub>(n) ]<sup>T</sup>
91 * </code></div>
92 * (here again we omit the k index in the notation for clarity)
93 *
94 * <p>Taylor series formulas show that for any index offset i, s<sub>1</sub>(n-i) can be
95 * computed from s<sub>1</sub>(n), s<sub>2</sub>(n) ... s<sub>k</sub>(n), the formula being exact
96 * for degree k polynomials.
97 * <div style="white-space: pre"><code>
98 * s<sub>1</sub>(n-i) = s<sub>1</sub>(n) + ∑<sub>j>0</sub> (j+1) (-i)<sup>j</sup> s<sub>j+1</sub>(n)
99 * </code></div>
100 * The previous formula can be used with several values for i to compute the transform between
101 * classical representation and Nordsieck vector. The transform between r<sub>n</sub>
102 * and q<sub>n</sub> resulting from the Taylor series formulas above is:
103 * <div style="white-space: pre"><code>
104 * q<sub>n</sub> = s<sub>1</sub>(n) u + P r<sub>n</sub>
105 * </code></div>
106 * where u is the [ 1 1 ... 1 ]<sup>T</sup> vector and P is the (k-1)×(k-1) matrix built
107 * with the (j+1) (-i)<sup>j</sup> terms with i being the row number starting from 1 and j being
108 * the column number starting from 1:
109 * <pre>
110 * [ -2 3 -4 5 ... ]
111 * [ -4 12 -32 80 ... ]
112 * P = [ -6 27 -108 405 ... ]
113 * [ -8 48 -256 1280 ... ]
114 * [ ... ]
115 * </pre>
116 *
117 * <p>Using the Nordsieck vector has several advantages:
118 * <ul>
119 * <li>it greatly simplifies step interpolation as the interpolator mainly applies
120 * Taylor series formulas,</li>
121 * <li>it simplifies step changes that occur when discrete events that truncate
122 * the step are triggered,</li>
123 * <li>it allows to extend the methods in order to support adaptive stepsize.</li>
124 * </ul>
125 *
126 * <p>The predicted Nordsieck vector at step n+1 is computed from the Nordsieck vector at step
127 * n as follows:
128 * <ul>
129 * <li>Y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n) + u<sup>T</sup> r<sub>n</sub></li>
130 * <li>S<sub>1</sub>(n+1) = h f(t<sub>n+1</sub>, Y<sub>n+1</sub>)</li>
131 * <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>
132 * </ul>
133 * where A is a rows shifting matrix (the lower left part is an identity matrix):
134 * <pre>
135 * [ 0 0 ... 0 0 | 0 ]
136 * [ ---------------+---]
137 * [ 1 0 ... 0 0 | 0 ]
138 * A = [ 0 1 ... 0 0 | 0 ]
139 * [ ... | 0 ]
140 * [ 0 0 ... 1 0 | 0 ]
141 * [ 0 0 ... 0 1 | 0 ]
142 * </pre>
143 * From this predicted vector, the corrected vector is computed as follows:
144 * <ul>
145 * <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>
146 * <li>s<sub>1</sub>(n+1) = h f(t<sub>n+1</sub>, y<sub>n+1</sub>)</li>
147 * <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>
148 * </ul>
149 * where the upper case Y<sub>n+1</sub>, S<sub>1</sub>(n+1) and R<sub>n+1</sub> represent the
150 * predicted states whereas the lower case y<sub>n+1</sub>, s<sub>n+1</sub> and r<sub>n+1</sub>
151 * represent the corrected states.
152 *
153 * <p>The P<sup>-1</sup>u vector and the P<sup>-1</sup> A P matrix do not depend on the state,
154 * they only depend on k and therefore are precomputed once for all.</p>
155 *
156 * @since 2.0
157 */
158 public class AdamsMoultonIntegrator extends AdamsIntegrator {
159
160 /** Integrator method name. */
161 private static final String METHOD_NAME = "Adams-Moulton";
162
163 /**
164 * Build an Adams-Moulton integrator with the given order and error control parameters.
165 * @param nSteps number of steps of the method excluding the one being computed
166 * @param minStep minimal step (sign is irrelevant, regardless of
167 * integration direction, forward or backward), the last step can
168 * be smaller than this
169 * @param maxStep maximal step (sign is irrelevant, regardless of
170 * integration direction, forward or backward), the last step can
171 * be smaller than this
172 * @param scalAbsoluteTolerance allowed absolute error
173 * @param scalRelativeTolerance allowed relative error
174 * @exception NumberIsTooSmallException if order is 1 or less
175 */
176 public AdamsMoultonIntegrator(final int nSteps,
177 final double minStep, final double maxStep,
178 final double scalAbsoluteTolerance,
179 final double scalRelativeTolerance)
180 throws NumberIsTooSmallException {
181 super(METHOD_NAME, nSteps, nSteps + 1, minStep, maxStep,
182 scalAbsoluteTolerance, scalRelativeTolerance);
183 }
184
185 /**
186 * Build an Adams-Moulton integrator with the given order and error control parameters.
187 * @param nSteps number of steps of the method excluding the one being computed
188 * @param minStep minimal step (sign is irrelevant, regardless of
189 * integration direction, forward or backward), the last step can
190 * be smaller than this
191 * @param maxStep maximal step (sign is irrelevant, regardless of
192 * integration direction, forward or backward), the last step can
193 * be smaller than this
194 * @param vecAbsoluteTolerance allowed absolute error
195 * @param vecRelativeTolerance allowed relative error
196 * @exception IllegalArgumentException if order is 1 or less
197 */
198 public AdamsMoultonIntegrator(final int nSteps,
199 final double minStep, final double maxStep,
200 final double[] vecAbsoluteTolerance,
201 final double[] vecRelativeTolerance)
202 throws IllegalArgumentException {
203 super(METHOD_NAME, nSteps, nSteps + 1, minStep, maxStep,
204 vecAbsoluteTolerance, vecRelativeTolerance);
205 }
206
207 /** {@inheritDoc} */
208 @Override
209 public void integrate(final ExpandableStatefulODE equations,final double t)
210 throws NumberIsTooSmallException, DimensionMismatchException,
211 MaxCountExceededException, NoBracketingException {
212
213 sanityChecks(equations, t);
214 setEquations(equations);
215 final boolean forward = t > equations.getTime();
216
217 // initialize working arrays
218 final double[] y0 = equations.getCompleteState();
219 final double[] y = y0.clone();
220 final double[] yDot = new double[y.length];
221 final double[] yTmp = new double[y.length];
222 final double[] predictedScaled = new double[y.length];
223 Array2DRowRealMatrix nordsieckTmp = null;
224
225 // set up two interpolators sharing the integrator arrays
226 final NordsieckStepInterpolator interpolator = new NordsieckStepInterpolator();
227 interpolator.reinitialize(y, forward,
228 equations.getPrimaryMapper(), equations.getSecondaryMappers());
229
230 // set up integration control objects
231 initIntegration(equations.getTime(), y0, t);
232
233 // compute the initial Nordsieck vector using the configured starter integrator
234 start(equations.getTime(), y, t);
235 interpolator.reinitialize(stepStart, stepSize, scaled, nordsieck);
236 interpolator.storeTime(stepStart);
237
238 double hNew = stepSize;
239 interpolator.rescale(hNew);
240
241 isLastStep = false;
242 do {
243
244 double error = 10;
245 while (error >= 1.0) {
246
247 stepSize = hNew;
248
249 // predict a first estimate of the state at step end (P in the PECE sequence)
250 final double stepEnd = stepStart + stepSize;
251 interpolator.setInterpolatedTime(stepEnd);
252 final ExpandableStatefulODE expandable = getExpandable();
253 final EquationsMapper primary = expandable.getPrimaryMapper();
254 primary.insertEquationData(interpolator.getInterpolatedState(), yTmp);
255 int index = 0;
256 for (final EquationsMapper secondary : expandable.getSecondaryMappers()) {
257 secondary.insertEquationData(interpolator.getInterpolatedSecondaryState(index), yTmp);
258 ++index;
259 }
260
261 // evaluate a first estimate of the derivative (first E in the PECE sequence)
262 computeDerivatives(stepEnd, yTmp, yDot);
263
264 // update Nordsieck vector
265 for (int j = 0; j < y0.length; ++j) {
266 predictedScaled[j] = stepSize * yDot[j];
267 }
268 nordsieckTmp = updateHighOrderDerivativesPhase1(nordsieck);
269 updateHighOrderDerivativesPhase2(scaled, predictedScaled, nordsieckTmp);
270
271 // apply correction (C in the PECE sequence)
272 error = nordsieckTmp.walkInOptimizedOrder(new Corrector(y, predictedScaled, yTmp));
273
274 if (error >= 1.0) {
275 // reject the step and attempt to reduce error by stepsize control
276 final double factor = computeStepGrowShrinkFactor(error);
277 hNew = filterStep(stepSize * factor, forward, false);
278 interpolator.rescale(hNew);
279 }
280 }
281
282 // evaluate a final estimate of the derivative (second E in the PECE sequence)
283 final double stepEnd = stepStart + stepSize;
284 computeDerivatives(stepEnd, yTmp, yDot);
285
286 // update Nordsieck vector
287 final double[] correctedScaled = new double[y0.length];
288 for (int j = 0; j < y0.length; ++j) {
289 correctedScaled[j] = stepSize * yDot[j];
290 }
291 updateHighOrderDerivativesPhase2(predictedScaled, correctedScaled, nordsieckTmp);
292
293 // discrete events handling
294 System.arraycopy(yTmp, 0, y, 0, y.length);
295 interpolator.reinitialize(stepEnd, stepSize, correctedScaled, nordsieckTmp);
296 interpolator.storeTime(stepStart);
297 interpolator.shift();
298 interpolator.storeTime(stepEnd);
299 stepStart = acceptStep(interpolator, y, yDot, t);
300 scaled = correctedScaled;
301 nordsieck = nordsieckTmp;
302
303 if (!isLastStep) {
304
305 // prepare next step
306 interpolator.storeTime(stepStart);
307
308 if (resetOccurred) {
309 // some events handler has triggered changes that
310 // invalidate the derivatives, we need to restart from scratch
311 start(stepStart, y, t);
312 interpolator.reinitialize(stepStart, stepSize, scaled, nordsieck);
313 }
314
315 // stepsize control for next step
316 final double factor = computeStepGrowShrinkFactor(error);
317 final double scaledH = stepSize * factor;
318 final double nextT = stepStart + scaledH;
319 final boolean nextIsLast = forward ? (nextT >= t) : (nextT <= t);
320 hNew = filterStep(scaledH, forward, nextIsLast);
321
322 final double filteredNextT = stepStart + hNew;
323 final boolean filteredNextIsLast = forward ? (filteredNextT >= t) : (filteredNextT <= t);
324 if (filteredNextIsLast) {
325 hNew = t - stepStart;
326 }
327
328 interpolator.rescale(hNew);
329 }
330 } while (!isLastStep);
331
332 // dispatch results
333 equations.setTime(stepStart);
334 equations.setCompleteState(y);
335
336 resetInternalState();
337 }
338
339 /** Corrector for current state in Adams-Moulton method.
340 * <p>
341 * This visitor implements the Taylor series formula:
342 * <pre>
343 * 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>
344 * </pre>
345 * </p>
346 */
347 private final class Corrector implements RealMatrixPreservingVisitor {
348
349 /** Previous state. */
350 private final double[] previous;
351
352 /** Current scaled first derivative. */
353 private final double[] scaled;
354
355 /** Current state before correction. */
356 private final double[] before;
357
358 /** Current state after correction. */
359 private final double[] after;
360
361 /** Simple constructor.
362 * @param previous previous state
363 * @param scaled current scaled first derivative
364 * @param state state to correct (will be overwritten after visit)
365 */
366 Corrector(final double[] previous, final double[] scaled, final double[] state) {
367 this.previous = previous;
368 this.scaled = scaled;
369 this.after = state;
370 this.before = state.clone();
371 }
372
373 /** {@inheritDoc} */
374 @Override
375 public void start(int rows, int columns,
376 int startRow, int endRow, int startColumn, int endColumn) {
377 Arrays.fill(after, 0.0);
378 }
379
380 /** {@inheritDoc} */
381 @Override
382 public void visit(int row, int column, double value) {
383 if ((row & 0x1) == 0) {
384 after[column] -= value;
385 } else {
386 after[column] += value;
387 }
388 }
389
390 /**
391 * End visiting the Nordsieck vector.
392 * <p>The correction is used to control stepsize. So its amplitude is
393 * considered to be an error, which must be normalized according to
394 * error control settings. If the normalized value is greater than 1,
395 * the correction was too large and the step must be rejected.</p>
396 * @return the normalized correction, if greater than 1, the step
397 * must be rejected
398 */
399 @Override
400 public double end() {
401
402 double error = 0;
403 for (int i = 0; i < after.length; ++i) {
404 after[i] += previous[i] + scaled[i];
405 if (i < mainSetDimension) {
406 final double yScale = JdkMath.max(JdkMath.abs(previous[i]), JdkMath.abs(after[i]));
407 final double tol = (vecAbsoluteTolerance == null) ?
408 (scalAbsoluteTolerance + scalRelativeTolerance * yScale) :
409 (vecAbsoluteTolerance[i] + vecRelativeTolerance[i] * yScale);
410 final double ratio = (after[i] - before[i]) / tol; // (corrected-predicted)/tol
411 error += ratio * ratio;
412 }
413 }
414
415 return JdkMath.sqrt(error / mainSetDimension);
416 }
417 }
418 }