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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    *      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.math3.ode.nonstiff;
19  
20  import org.apache.commons.math3.exception.DimensionMismatchException;
21  import org.apache.commons.math3.exception.MaxCountExceededException;
22  import org.apache.commons.math3.exception.NoBracketingException;
23  import org.apache.commons.math3.exception.NumberIsTooSmallException;
24  import org.apache.commons.math3.linear.Array2DRowRealMatrix;
25  import org.apache.commons.math3.linear.RealMatrix;
26  import org.apache.commons.math3.ode.EquationsMapper;
27  import org.apache.commons.math3.ode.ExpandableStatefulODE;
28  import org.apache.commons.math3.ode.sampling.NordsieckStepInterpolator;
29  import org.apache.commons.math3.util.FastMath;
30  
31  
32  /**
33   * This class implements explicit Adams-Bashforth integrators for Ordinary
34   * Differential Equations.
35   *
36   * <p>Adams-Bashforth methods (in fact due to Adams alone) are explicit
37   * multistep ODE solvers. This implementation is a variation of the classical
38   * one: it uses adaptive stepsize to implement error control, whereas
39   * classical implementations are fixed step size. The value of state vector
40   * at step n+1 is a simple combination of the value at step n and of the
41   * derivatives at steps n, n-1, n-2 ... Depending on the number k of previous
42   * steps one wants to use for computing the next value, different formulas
43   * are available:</p>
44   * <ul>
45   *   <li>k = 1: y<sub>n+1</sub> = y<sub>n</sub> + h y'<sub>n</sub></li>
46   *   <li>k = 2: y<sub>n+1</sub> = y<sub>n</sub> + h (3y'<sub>n</sub>-y'<sub>n-1</sub>)/2</li>
47   *   <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>
48   *   <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>
49   *   <li>...</li>
50   * </ul>
51   *
52   * <p>A k-steps Adams-Bashforth method is of order k.</p>
53   *
54   * <h3>Implementation details</h3>
55   *
56   * <p>We define scaled derivatives s<sub>i</sub>(n) at step n as:
57   * <pre>
58   * s<sub>1</sub>(n) = h y'<sub>n</sub> for first derivative
59   * s<sub>2</sub>(n) = h<sup>2</sup>/2 y''<sub>n</sub> for second derivative
60   * s<sub>3</sub>(n) = h<sup>3</sup>/6 y'''<sub>n</sub> for third derivative
61   * ...
62   * s<sub>k</sub>(n) = h<sup>k</sup>/k! y<sup>(k)</sup><sub>n</sub> for k<sup>th</sup> derivative
63   * </pre></p>
64   *
65   * <p>The definitions above use the classical representation with several previous first
66   * derivatives. Lets define
67   * <pre>
68   *   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>
69   * </pre>
70   * (we omit the k index in the notation for clarity). With these definitions,
71   * Adams-Bashforth methods can be written:
72   * <ul>
73   *   <li>k = 1: y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n)</li>
74   *   <li>k = 2: y<sub>n+1</sub> = y<sub>n</sub> + 3/2 s<sub>1</sub>(n) + [ -1/2 ] q<sub>n</sub></li>
75   *   <li>k = 3: y<sub>n+1</sub> = y<sub>n</sub> + 23/12 s<sub>1</sub>(n) + [ -16/12 5/12 ] q<sub>n</sub></li>
76   *   <li>k = 4: y<sub>n+1</sub> = y<sub>n</sub> + 55/24 s<sub>1</sub>(n) + [ -59/24 37/24 -9/24 ] q<sub>n</sub></li>
77   *   <li>...</li>
78   * </ul></p>
79   *
80   * <p>Instead of using the classical representation with first derivatives only (y<sub>n</sub>,
81   * s<sub>1</sub>(n) and q<sub>n</sub>), our implementation uses the Nordsieck vector with
82   * higher degrees scaled derivatives all taken at the same step (y<sub>n</sub>, s<sub>1</sub>(n)
83   * and r<sub>n</sub>) where r<sub>n</sub> is defined as:
84   * <pre>
85   * r<sub>n</sub> = [ s<sub>2</sub>(n), s<sub>3</sub>(n) ... s<sub>k</sub>(n) ]<sup>T</sup>
86   * </pre>
87   * (here again we omit the k index in the notation for clarity)
88   * </p>
89   *
90   * <p>Taylor series formulas show that for any index offset i, s<sub>1</sub>(n-i) can be
91   * computed from s<sub>1</sub>(n), s<sub>2</sub>(n) ... s<sub>k</sub>(n), the formula being exact
92   * for degree k polynomials.
93   * <pre>
94   * s<sub>1</sub>(n-i) = s<sub>1</sub>(n) + &sum;<sub>j&gt;0</sub> (j+1) (-i)<sup>j</sup> s<sub>j+1</sub>(n)
95   * </pre>
96   * The previous formula can be used with several values for i to compute the transform between
97   * classical representation and Nordsieck vector. The transform between r<sub>n</sub>
98   * and q<sub>n</sub> resulting from the Taylor series formulas above is:
99   * <pre>
100  * q<sub>n</sub> = s<sub>1</sub>(n) u + P r<sub>n</sub>
101  * </pre>
102  * where u is the [ 1 1 ... 1 ]<sup>T</sup> vector and P is the (k-1)&times;(k-1) matrix built
103  * with the (j+1) (-i)<sup>j</sup> terms with i being the row number starting from 1 and j being
104  * the column number starting from 1:
105  * <pre>
106  *        [  -2   3   -4    5  ... ]
107  *        [  -4  12  -32   80  ... ]
108  *   P =  [  -6  27 -108  405  ... ]
109  *        [  -8  48 -256 1280  ... ]
110  *        [          ...           ]
111  * </pre></p>
112  *
113  * <p>Using the Nordsieck vector has several advantages:
114  * <ul>
115  *   <li>it greatly simplifies step interpolation as the interpolator mainly applies
116  *   Taylor series formulas,</li>
117  *   <li>it simplifies step changes that occur when discrete events that truncate
118  *   the step are triggered,</li>
119  *   <li>it allows to extend the methods in order to support adaptive stepsize.</li>
120  * </ul></p>
121  *
122  * <p>The Nordsieck vector at step n+1 is computed from the Nordsieck vector at step n as follows:
123  * <ul>
124  *   <li>y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n) + u<sup>T</sup> r<sub>n</sub></li>
125  *   <li>s<sub>1</sub>(n+1) = h f(t<sub>n+1</sub>, y<sub>n+1</sub>)</li>
126  *   <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>
127  * </ul>
128  * where A is a rows shifting matrix (the lower left part is an identity matrix):
129  * <pre>
130  *        [ 0 0   ...  0 0 | 0 ]
131  *        [ ---------------+---]
132  *        [ 1 0   ...  0 0 | 0 ]
133  *    A = [ 0 1   ...  0 0 | 0 ]
134  *        [       ...      | 0 ]
135  *        [ 0 0   ...  1 0 | 0 ]
136  *        [ 0 0   ...  0 1 | 0 ]
137  * </pre></p>
138  *
139  * <p>The P<sup>-1</sup>u vector and the P<sup>-1</sup> A P matrix do not depend on the state,
140  * they only depend on k and therefore are precomputed once for all.</p>
141  *
142  * @since 2.0
143  */
144 public class AdamsBashforthIntegrator extends AdamsIntegrator {
145 
146     /** Integrator method name. */
147     private static final String METHOD_NAME = "Adams-Bashforth";
148 
149     /**
150      * Build an Adams-Bashforth integrator with the given order and step control parameters.
151      * @param nSteps number of steps of the method excluding the one being computed
152      * @param minStep minimal step (sign is irrelevant, regardless of
153      * integration direction, forward or backward), the last step can
154      * be smaller than this
155      * @param maxStep maximal step (sign is irrelevant, regardless of
156      * integration direction, forward or backward), the last step can
157      * be smaller than this
158      * @param scalAbsoluteTolerance allowed absolute error
159      * @param scalRelativeTolerance allowed relative error
160      * @exception NumberIsTooSmallException if order is 1 or less
161      */
162     public AdamsBashforthIntegrator(final int nSteps,
163                                     final double minStep, final double maxStep,
164                                     final double scalAbsoluteTolerance,
165                                     final double scalRelativeTolerance)
166         throws NumberIsTooSmallException {
167         super(METHOD_NAME, nSteps, nSteps, minStep, maxStep,
168               scalAbsoluteTolerance, scalRelativeTolerance);
169     }
170 
171     /**
172      * Build an Adams-Bashforth integrator with the given order and step control parameters.
173      * @param nSteps number of steps of the method excluding the one being computed
174      * @param minStep minimal step (sign is irrelevant, regardless of
175      * integration direction, forward or backward), the last step can
176      * be smaller than this
177      * @param maxStep maximal step (sign is irrelevant, regardless of
178      * integration direction, forward or backward), the last step can
179      * be smaller than this
180      * @param vecAbsoluteTolerance allowed absolute error
181      * @param vecRelativeTolerance allowed relative error
182      * @exception IllegalArgumentException if order is 1 or less
183      */
184     public AdamsBashforthIntegrator(final int nSteps,
185                                     final double minStep, final double maxStep,
186                                     final double[] vecAbsoluteTolerance,
187                                     final double[] vecRelativeTolerance)
188         throws IllegalArgumentException {
189         super(METHOD_NAME, nSteps, nSteps, minStep, maxStep,
190               vecAbsoluteTolerance, vecRelativeTolerance);
191     }
192 
193     /** Estimate error.
194      * <p>
195      * Error is estimated by interpolating back to previous state using
196      * the state Taylor expansion and comparing to real previous state.
197      * </p>
198      * @param previousState state vector at step start
199      * @param predictedState predicted state vector at step end
200      * @param predictedScaled predicted value of the scaled derivatives at step end
201      * @param predictedNordsieck predicted value of the Nordsieck vector at step end
202      * @return estimated normalized local discretization error
203      */
204     private double errorEstimation(final double[] previousState,
205                                    final double[] predictedState,
206                                    final double[] predictedScaled,
207                                    final RealMatrix predictedNordsieck) {
208 
209         double error = 0;
210         for (int i = 0; i < mainSetDimension; ++i) {
211             final double yScale = FastMath.abs(predictedState[i]);
212             final double tol = (vecAbsoluteTolerance == null) ?
213                                (scalAbsoluteTolerance + scalRelativeTolerance * yScale) :
214                                (vecAbsoluteTolerance[i] + vecRelativeTolerance[i] * yScale);
215 
216             // apply Taylor formula from high order to low order,
217             // for the sake of numerical accuracy
218             double variation = 0;
219             int sign = predictedNordsieck.getRowDimension() % 2 == 0 ? -1 : 1;
220             for (int k = predictedNordsieck.getRowDimension() - 1; k >= 0; --k) {
221                 variation += sign * predictedNordsieck.getEntry(k, i);
222                 sign       = -sign;
223             }
224             variation -= predictedScaled[i];
225 
226             final double ratio  = (predictedState[i] - previousState[i] + variation) / tol;
227             error              += ratio * ratio;
228 
229         }
230 
231         return FastMath.sqrt(error / mainSetDimension);
232 
233     }
234 
235     /** {@inheritDoc} */
236     @Override
237     public void integrate(final ExpandableStatefulODE equations, final double t)
238         throws NumberIsTooSmallException, DimensionMismatchException,
239                MaxCountExceededException, NoBracketingException {
240 
241         sanityChecks(equations, t);
242         setEquations(equations);
243         final boolean forward = t > equations.getTime();
244 
245         // initialize working arrays
246         final double[] y    = equations.getCompleteState();
247         final double[] yDot = new double[y.length];
248 
249         // set up an interpolator sharing the integrator arrays
250         final NordsieckStepInterpolator interpolator = new NordsieckStepInterpolator();
251         interpolator.reinitialize(y, forward,
252                                   equations.getPrimaryMapper(), equations.getSecondaryMappers());
253 
254         // set up integration control objects
255         initIntegration(equations.getTime(), y, t);
256 
257         // compute the initial Nordsieck vector using the configured starter integrator
258         start(equations.getTime(), y, t);
259         interpolator.reinitialize(stepStart, stepSize, scaled, nordsieck);
260         interpolator.storeTime(stepStart);
261 
262         // reuse the step that was chosen by the starter integrator
263         double hNew = stepSize;
264         interpolator.rescale(hNew);
265 
266         // main integration loop
267         isLastStep = false;
268         do {
269 
270             interpolator.shift();
271             final double[] predictedY      = new double[y.length];
272             final double[] predictedScaled = new double[y.length];
273             Array2DRowRealMatrix predictedNordsieck = null;
274             double error = 10;
275             while (error >= 1.0) {
276 
277                 // predict a first estimate of the state at step end
278                 final double stepEnd = stepStart + hNew;
279                 interpolator.storeTime(stepEnd);
280                 final ExpandableStatefulODE expandable = getExpandable();
281                 final EquationsMapper primary = expandable.getPrimaryMapper();
282                 primary.insertEquationData(interpolator.getInterpolatedState(), predictedY);
283                 int index = 0;
284                 for (final EquationsMapper secondary : expandable.getSecondaryMappers()) {
285                     secondary.insertEquationData(interpolator.getInterpolatedSecondaryState(index), predictedY);
286                     ++index;
287                 }
288 
289                 // evaluate the derivative
290                 computeDerivatives(stepEnd, predictedY, yDot);
291 
292                 // predict Nordsieck vector at step end
293                 for (int j = 0; j < predictedScaled.length; ++j) {
294                     predictedScaled[j] = hNew * yDot[j];
295                 }
296                 predictedNordsieck = updateHighOrderDerivativesPhase1(nordsieck);
297                 updateHighOrderDerivativesPhase2(scaled, predictedScaled, predictedNordsieck);
298 
299                 // evaluate error
300                 error = errorEstimation(y, predictedY, predictedScaled, predictedNordsieck);
301 
302                 if (error >= 1.0) {
303                     // reject the step and attempt to reduce error by stepsize control
304                     final double factor = computeStepGrowShrinkFactor(error);
305                     hNew = filterStep(hNew * factor, forward, false);
306                     interpolator.rescale(hNew);
307 
308                 }
309             }
310 
311             stepSize = hNew;
312             final double stepEnd = stepStart + stepSize;
313             interpolator.reinitialize(stepEnd, stepSize, predictedScaled, predictedNordsieck);
314 
315             // discrete events handling
316             interpolator.storeTime(stepEnd);
317             System.arraycopy(predictedY, 0, y, 0, y.length);
318             stepStart = acceptStep(interpolator, y, yDot, t);
319             scaled    = predictedScaled;
320             nordsieck = predictedNordsieck;
321             interpolator.reinitialize(stepEnd, stepSize, scaled, nordsieck);
322 
323             if (!isLastStep) {
324 
325                 // prepare next step
326                 interpolator.storeTime(stepStart);
327 
328                 if (resetOccurred) {
329                     // some events handler has triggered changes that
330                     // invalidate the derivatives, we need to restart from scratch
331                     start(stepStart, y, t);
332                     interpolator.reinitialize(stepStart, stepSize, scaled, nordsieck);
333                 }
334 
335                 // stepsize control for next step
336                 final double  factor     = computeStepGrowShrinkFactor(error);
337                 final double  scaledH    = stepSize * factor;
338                 final double  nextT      = stepStart + scaledH;
339                 final boolean nextIsLast = forward ? (nextT >= t) : (nextT <= t);
340                 hNew = filterStep(scaledH, forward, nextIsLast);
341 
342                 final double  filteredNextT      = stepStart + hNew;
343                 final boolean filteredNextIsLast = forward ? (filteredNextT >= t) : (filteredNextT <= t);
344                 if (filteredNextIsLast) {
345                     hNew = t - stepStart;
346                 }
347 
348                 interpolator.rescale(hNew);
349 
350             }
351 
352         } while (!isLastStep);
353 
354         // dispatch results
355         equations.setTime(stepStart);
356         equations.setCompleteState(y);
357 
358         resetInternalState();
359 
360     }
361 
362 }