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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.ode.EquationsMapper;
26  import org.apache.commons.math3.ode.ExpandableStatefulODE;
27  import org.apache.commons.math3.ode.sampling.NordsieckStepInterpolator;
28  import org.apache.commons.math3.util.FastMath;
29  
30  
31  /**
32   * This class implements explicit Adams-Bashforth integrators for Ordinary
33   * Differential Equations.
34   *
35   * <p>Adams-Bashforth methods (in fact due to Adams alone) are explicit
36   * multistep ODE solvers. This implementation is a variation of the classical
37   * one: it uses adaptive stepsize to implement error control, whereas
38   * classical implementations are fixed step size. The value of state vector
39   * at step n+1 is a simple combination of the value at step n and of the
40   * derivatives at steps n, n-1, n-2 ... Depending on the number k of previous
41   * steps one wants to use for computing the next value, different formulas
42   * are available:</p>
43   * <ul>
44   *   <li>k = 1: y<sub>n+1</sub> = y<sub>n</sub> + h y'<sub>n</sub></li>
45   *   <li>k = 2: y<sub>n+1</sub> = y<sub>n</sub> + h (3y'<sub>n</sub>-y'<sub>n-1</sub>)/2</li>
46   *   <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>
47   *   <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>
48   *   <li>...</li>
49   * </ul>
50   *
51   * <p>A k-steps Adams-Bashforth method is of order k.</p>
52   *
53   * <h3>Implementation details</h3>
54   *
55   * <p>We define scaled derivatives s<sub>i</sub>(n) at step n as:
56   * <pre>
57   * s<sub>1</sub>(n) = h y'<sub>n</sub> for first derivative
58   * s<sub>2</sub>(n) = h<sup>2</sup>/2 y''<sub>n</sub> for second derivative
59   * s<sub>3</sub>(n) = h<sup>3</sup>/6 y'''<sub>n</sub> for third derivative
60   * ...
61   * s<sub>k</sub>(n) = h<sup>k</sup>/k! y<sup>(k)</sup><sub>n</sub> for k<sup>th</sup> derivative
62   * </pre></p>
63   *
64   * <p>The definitions above use the classical representation with several previous first
65   * derivatives. Lets define
66   * <pre>
67   *   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>
68   * </pre>
69   * (we omit the k index in the notation for clarity). With these definitions,
70   * Adams-Bashforth methods can be written:
71   * <ul>
72   *   <li>k = 1: y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n)</li>
73   *   <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>
74   *   <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>
75   *   <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>
76   *   <li>...</li>
77   * </ul></p>
78   *
79   * <p>Instead of using the classical representation with first derivatives only (y<sub>n</sub>,
80   * s<sub>1</sub>(n) and q<sub>n</sub>), our implementation uses the Nordsieck vector with
81   * higher degrees scaled derivatives all taken at the same step (y<sub>n</sub>, s<sub>1</sub>(n)
82   * and r<sub>n</sub>) where r<sub>n</sub> is defined as:
83   * <pre>
84   * r<sub>n</sub> = [ s<sub>2</sub>(n), s<sub>3</sub>(n) ... s<sub>k</sub>(n) ]<sup>T</sup>
85   * </pre>
86   * (here again we omit the k index in the notation for clarity)
87   * </p>
88   *
89   * <p>Taylor series formulas show that for any index offset i, s<sub>1</sub>(n-i) can be
90   * computed from s<sub>1</sub>(n), s<sub>2</sub>(n) ... s<sub>k</sub>(n), the formula being exact
91   * for degree k polynomials.
92   * <pre>
93   * s<sub>1</sub>(n-i) = s<sub>1</sub>(n) + &sum;<sub>j</sub> j (-i)<sup>j-1</sup> s<sub>j</sub>(n)
94   * </pre>
95   * The previous formula can be used with several values for i to compute the transform between
96   * classical representation and Nordsieck vector. The transform between r<sub>n</sub>
97   * and q<sub>n</sub> resulting from the Taylor series formulas above is:
98   * <pre>
99   * q<sub>n</sub> = s<sub>1</sub>(n) u + P r<sub>n</sub>
100  * </pre>
101  * where u is the [ 1 1 ... 1 ]<sup>T</sup> vector and P is the (k-1)&times;(k-1) matrix built
102  * with the j (-i)<sup>j-1</sup> terms:
103  * <pre>
104  *        [  -2   3   -4    5  ... ]
105  *        [  -4  12  -32   80  ... ]
106  *   P =  [  -6  27 -108  405  ... ]
107  *        [  -8  48 -256 1280  ... ]
108  *        [          ...           ]
109  * </pre></p>
110  *
111  * <p>Using the Nordsieck vector has several advantages:
112  * <ul>
113  *   <li>it greatly simplifies step interpolation as the interpolator mainly applies
114  *   Taylor series formulas,</li>
115  *   <li>it simplifies step changes that occur when discrete events that truncate
116  *   the step are triggered,</li>
117  *   <li>it allows to extend the methods in order to support adaptive stepsize.</li>
118  * </ul></p>
119  *
120  * <p>The Nordsieck vector at step n+1 is computed from the Nordsieck vector at step n as follows:
121  * <ul>
122  *   <li>y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n) + u<sup>T</sup> r<sub>n</sub></li>
123  *   <li>s<sub>1</sub>(n+1) = h f(t<sub>n+1</sub>, y<sub>n+1</sub>)</li>
124  *   <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>
125  * </ul>
126  * where A is a rows shifting matrix (the lower left part is an identity matrix):
127  * <pre>
128  *        [ 0 0   ...  0 0 | 0 ]
129  *        [ ---------------+---]
130  *        [ 1 0   ...  0 0 | 0 ]
131  *    A = [ 0 1   ...  0 0 | 0 ]
132  *        [       ...      | 0 ]
133  *        [ 0 0   ...  1 0 | 0 ]
134  *        [ 0 0   ...  0 1 | 0 ]
135  * </pre></p>
136  *
137  * <p>The P<sup>-1</sup>u vector and the P<sup>-1</sup> A P matrix do not depend on the state,
138  * they only depend on k and therefore are precomputed once for all.</p>
139  *
140  * @since 2.0
141  */
142 public class AdamsBashforthIntegrator extends AdamsIntegrator {
143 
144     /** Integrator method name. */
145     private static final String METHOD_NAME = "Adams-Bashforth";
146 
147     /**
148      * Build an Adams-Bashforth integrator with the given order and step control parameters.
149      * @param nSteps number of steps of the method excluding the one being computed
150      * @param minStep minimal step (sign is irrelevant, regardless of
151      * integration direction, forward or backward), the last step can
152      * be smaller than this
153      * @param maxStep maximal step (sign is irrelevant, regardless of
154      * integration direction, forward or backward), the last step can
155      * be smaller than this
156      * @param scalAbsoluteTolerance allowed absolute error
157      * @param scalRelativeTolerance allowed relative error
158      * @exception NumberIsTooSmallException if order is 1 or less
159      */
160     public AdamsBashforthIntegrator(final int nSteps,
161                                     final double minStep, final double maxStep,
162                                     final double scalAbsoluteTolerance,
163                                     final double scalRelativeTolerance)
164         throws NumberIsTooSmallException {
165         super(METHOD_NAME, nSteps, nSteps, minStep, maxStep,
166               scalAbsoluteTolerance, scalRelativeTolerance);
167     }
168 
169     /**
170      * Build an Adams-Bashforth integrator with the given order and step control parameters.
171      * @param nSteps number of steps of the method excluding the one being computed
172      * @param minStep minimal step (sign is irrelevant, regardless of
173      * integration direction, forward or backward), the last step can
174      * be smaller than this
175      * @param maxStep maximal step (sign is irrelevant, regardless of
176      * integration direction, forward or backward), the last step can
177      * be smaller than this
178      * @param vecAbsoluteTolerance allowed absolute error
179      * @param vecRelativeTolerance allowed relative error
180      * @exception IllegalArgumentException if order is 1 or less
181      */
182     public AdamsBashforthIntegrator(final int nSteps,
183                                     final double minStep, final double maxStep,
184                                     final double[] vecAbsoluteTolerance,
185                                     final double[] vecRelativeTolerance)
186         throws IllegalArgumentException {
187         super(METHOD_NAME, nSteps, nSteps, minStep, maxStep,
188               vecAbsoluteTolerance, vecRelativeTolerance);
189     }
190 
191     /** {@inheritDoc} */
192     @Override
193     public void integrate(final ExpandableStatefulODE equations, final double t)
194         throws NumberIsTooSmallException, DimensionMismatchException,
195                MaxCountExceededException, NoBracketingException {
196 
197         sanityChecks(equations, t);
198         setEquations(equations);
199         final boolean forward = t > equations.getTime();
200 
201         // initialize working arrays
202         final double[] y0   = equations.getCompleteState();
203         final double[] y    = y0.clone();
204         final double[] yDot = new double[y.length];
205 
206         // set up an interpolator sharing the integrator arrays
207         final NordsieckStepInterpolator interpolator = new NordsieckStepInterpolator();
208         interpolator.reinitialize(y, forward,
209                                   equations.getPrimaryMapper(), equations.getSecondaryMappers());
210 
211         // set up integration control objects
212         initIntegration(equations.getTime(), y0, t);
213 
214         // compute the initial Nordsieck vector using the configured starter integrator
215         start(equations.getTime(), y, t);
216         interpolator.reinitialize(stepStart, stepSize, scaled, nordsieck);
217         interpolator.storeTime(stepStart);
218         final int lastRow = nordsieck.getRowDimension() - 1;
219 
220         // reuse the step that was chosen by the starter integrator
221         double hNew = stepSize;
222         interpolator.rescale(hNew);
223 
224         // main integration loop
225         isLastStep = false;
226         do {
227 
228             double error = 10;
229             while (error >= 1.0) {
230 
231                 stepSize = hNew;
232 
233                 // evaluate error using the last term of the Taylor expansion
234                 error = 0;
235                 for (int i = 0; i < mainSetDimension; ++i) {
236                     final double yScale = FastMath.abs(y[i]);
237                     final double tol = (vecAbsoluteTolerance == null) ?
238                                        (scalAbsoluteTolerance + scalRelativeTolerance * yScale) :
239                                        (vecAbsoluteTolerance[i] + vecRelativeTolerance[i] * yScale);
240                     final double ratio  = nordsieck.getEntry(lastRow, i) / tol;
241                     error += ratio * ratio;
242                 }
243                 error = FastMath.sqrt(error / mainSetDimension);
244 
245                 if (error >= 1.0) {
246                     // reject the step and attempt to reduce error by stepsize control
247                     final double factor = computeStepGrowShrinkFactor(error);
248                     hNew = filterStep(stepSize * factor, forward, false);
249                     interpolator.rescale(hNew);
250 
251                 }
252             }
253 
254             // predict a first estimate of the state at step end
255             final double stepEnd = stepStart + stepSize;
256             interpolator.shift();
257             interpolator.setInterpolatedTime(stepEnd);
258             final ExpandableStatefulODE expandable = getExpandable();
259             final EquationsMapper primary = expandable.getPrimaryMapper();
260             primary.insertEquationData(interpolator.getInterpolatedState(), y);
261             int index = 0;
262             for (final EquationsMapper secondary : expandable.getSecondaryMappers()) {
263                 secondary.insertEquationData(interpolator.getInterpolatedSecondaryState(index), y);
264                 ++index;
265             }
266 
267             // evaluate the derivative
268             computeDerivatives(stepEnd, y, yDot);
269 
270             // update Nordsieck vector
271             final double[] predictedScaled = new double[y0.length];
272             for (int j = 0; j < y0.length; ++j) {
273                 predictedScaled[j] = stepSize * yDot[j];
274             }
275             final Array2DRowRealMatrix nordsieckTmp = updateHighOrderDerivativesPhase1(nordsieck);
276             updateHighOrderDerivativesPhase2(scaled, predictedScaled, nordsieckTmp);
277             interpolator.reinitialize(stepEnd, stepSize, predictedScaled, nordsieckTmp);
278 
279             // discrete events handling
280             interpolator.storeTime(stepEnd);
281             stepStart = acceptStep(interpolator, y, yDot, t);
282             scaled    = predictedScaled;
283             nordsieck = nordsieckTmp;
284             interpolator.reinitialize(stepEnd, stepSize, scaled, nordsieck);
285 
286             if (!isLastStep) {
287 
288                 // prepare next step
289                 interpolator.storeTime(stepStart);
290 
291                 if (resetOccurred) {
292                     // some events handler has triggered changes that
293                     // invalidate the derivatives, we need to restart from scratch
294                     start(stepStart, y, t);
295                     interpolator.reinitialize(stepStart, stepSize, scaled, nordsieck);
296                 }
297 
298                 // stepsize control for next step
299                 final double  factor     = computeStepGrowShrinkFactor(error);
300                 final double  scaledH    = stepSize * factor;
301                 final double  nextT      = stepStart + scaledH;
302                 final boolean nextIsLast = forward ? (nextT >= t) : (nextT <= t);
303                 hNew = filterStep(scaledH, forward, nextIsLast);
304 
305                 final double  filteredNextT      = stepStart + hNew;
306                 final boolean filteredNextIsLast = forward ? (filteredNextT >= t) : (filteredNextT <= t);
307                 if (filteredNextIsLast) {
308                     hNew = t - stepStart;
309                 }
310 
311                 interpolator.rescale(hNew);
312 
313             }
314 
315         } while (!isLastStep);
316 
317         // dispatch results
318         equations.setTime(stepStart);
319         equations.setCompleteState(y);
320 
321         resetInternalState();
322 
323     }
324 
325 }