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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 java.io.IOException;
21  import java.io.ObjectInput;
22  import java.io.ObjectOutput;
23  
24  import org.apache.commons.math3.exception.MaxCountExceededException;
25  import org.apache.commons.math3.ode.AbstractIntegrator;
26  import org.apache.commons.math3.ode.EquationsMapper;
27  import org.apache.commons.math3.ode.sampling.StepInterpolator;
28  
29  /**
30   * This class represents an interpolator over the last step during an
31   * ODE integration for the 8(5,3) Dormand-Prince integrator.
32   *
33   * @see DormandPrince853Integrator
34   *
35   * @since 1.2
36   */
37  
38  class DormandPrince853StepInterpolator
39    extends RungeKuttaStepInterpolator {
40  
41      /** Serializable version identifier. */
42      private static final long serialVersionUID = 20111120L;
43  
44      /** Propagation weights, element 1. */
45      private static final double B_01 =         104257.0 / 1920240.0;
46  
47      // elements 2 to 5 are zero, so they are neither stored nor used
48  
49      /** Propagation weights, element 6. */
50      private static final double B_06 =        3399327.0 / 763840.0;
51  
52      /** Propagation weights, element 7. */
53      private static final double B_07 =       66578432.0 / 35198415.0;
54  
55      /** Propagation weights, element 8. */
56      private static final double B_08 =    -1674902723.0 / 288716400.0;
57  
58      /** Propagation weights, element 9. */
59      private static final double B_09 = 54980371265625.0 / 176692375811392.0;
60  
61      /** Propagation weights, element 10. */
62      private static final double B_10 =        -734375.0 / 4826304.0;
63  
64      /** Propagation weights, element 11. */
65      private static final double B_11 =      171414593.0 / 851261400.0;
66  
67      /** Propagation weights, element 12. */
68      private static final double B_12 =         137909.0 / 3084480.0;
69  
70      /** Time step for stage 14 (interpolation only). */
71      private static final double C14    = 1.0 / 10.0;
72  
73      /** Internal weights for stage 14, element 1. */
74      private static final double K14_01 =       13481885573.0 / 240030000000.0      - B_01;
75  
76      // elements 2 to 5 are zero, so they are neither stored nor used
77  
78      /** Internal weights for stage 14, element 6. */
79      private static final double K14_06 =                 0.0                       - B_06;
80  
81      /** Internal weights for stage 14, element 7. */
82      private static final double K14_07 =      139418837528.0 / 549975234375.0      - B_07;
83  
84      /** Internal weights for stage 14, element 8. */
85      private static final double K14_08 =   -11108320068443.0 / 45111937500000.0    - B_08;
86  
87      /** Internal weights for stage 14, element 9. */
88      private static final double K14_09 = -1769651421925959.0 / 14249385146080000.0 - B_09;
89  
90      /** Internal weights for stage 14, element 10. */
91      private static final double K14_10 =          57799439.0 / 377055000.0         - B_10;
92  
93      /** Internal weights for stage 14, element 11. */
94      private static final double K14_11 =      793322643029.0 / 96734250000000.0    - B_11;
95  
96      /** Internal weights for stage 14, element 12. */
97      private static final double K14_12 =        1458939311.0 / 192780000000.0      - B_12;
98  
99      /** Internal weights for stage 14, element 13. */
100     private static final double K14_13 =             -4149.0 / 500000.0;
101 
102     /** Time step for stage 15 (interpolation only). */
103     private static final double C15    = 1.0 / 5.0;
104 
105 
106     /** Internal weights for stage 15, element 1. */
107     private static final double K15_01 =     1595561272731.0 / 50120273500000.0    - B_01;
108 
109     // elements 2 to 5 are zero, so they are neither stored nor used
110 
111     /** Internal weights for stage 15, element 6. */
112     private static final double K15_06 =      975183916491.0 / 34457688031250.0    - B_06;
113 
114     /** Internal weights for stage 15, element 7. */
115     private static final double K15_07 =    38492013932672.0 / 718912673015625.0   - B_07;
116 
117     /** Internal weights for stage 15, element 8. */
118     private static final double K15_08 = -1114881286517557.0 / 20298710767500000.0 - B_08;
119 
120     /** Internal weights for stage 15, element 9. */
121     private static final double K15_09 =                 0.0                       - B_09;
122 
123     /** Internal weights for stage 15, element 10. */
124     private static final double K15_10 =                 0.0                       - B_10;
125 
126     /** Internal weights for stage 15, element 11. */
127     private static final double K15_11 =    -2538710946863.0 / 23431227861250000.0 - B_11;
128 
129     /** Internal weights for stage 15, element 12. */
130     private static final double K15_12 =        8824659001.0 / 23066716781250.0    - B_12;
131 
132     /** Internal weights for stage 15, element 13. */
133     private static final double K15_13 =      -11518334563.0 / 33831184612500.0;
134 
135     /** Internal weights for stage 15, element 14. */
136     private static final double K15_14 =        1912306948.0 / 13532473845.0;
137 
138     /** Time step for stage 16 (interpolation only). */
139     private static final double C16    = 7.0 / 9.0;
140 
141 
142     /** Internal weights for stage 16, element 1. */
143     private static final double K16_01 =      -13613986967.0 / 31741908048.0       - B_01;
144 
145     // elements 2 to 5 are zero, so they are neither stored nor used
146 
147     /** Internal weights for stage 16, element 6. */
148     private static final double K16_06 =       -4755612631.0 / 1012344804.0        - B_06;
149 
150     /** Internal weights for stage 16, element 7. */
151     private static final double K16_07 =    42939257944576.0 / 5588559685701.0     - B_07;
152 
153     /** Internal weights for stage 16, element 8. */
154     private static final double K16_08 =    77881972900277.0 / 19140370552944.0    - B_08;
155 
156     /** Internal weights for stage 16, element 9. */
157     private static final double K16_09 =    22719829234375.0 / 63689648654052.0    - B_09;
158 
159     /** Internal weights for stage 16, element 10. */
160     private static final double K16_10 =                 0.0                       - B_10;
161 
162     /** Internal weights for stage 16, element 11. */
163     private static final double K16_11 =                 0.0                       - B_11;
164 
165     /** Internal weights for stage 16, element 12. */
166     private static final double K16_12 =                 0.0                       - B_12;
167 
168     /** Internal weights for stage 16, element 13. */
169     private static final double K16_13 =       -1199007803.0 / 857031517296.0;
170 
171     /** Internal weights for stage 16, element 14. */
172     private static final double K16_14 =      157882067000.0 / 53564469831.0;
173 
174     /** Internal weights for stage 16, element 15. */
175     private static final double K16_15 =     -290468882375.0 / 31741908048.0;
176 
177     /** Interpolation weights.
178      * (beware that only the non-null values are in the table)
179      */
180     private static final double[][] D = {
181 
182       {        -17751989329.0 / 2106076560.0,               4272954039.0 / 7539864640.0,
183               -118476319744.0 / 38604839385.0,            755123450731.0 / 316657731600.0,
184         3692384461234828125.0 / 1744130441634250432.0,     -4612609375.0 / 5293382976.0,
185               2091772278379.0 / 933644586600.0,             2136624137.0 / 3382989120.0,
186                     -126493.0 / 1421424.0,                    98350000.0 / 5419179.0,
187                   -18878125.0 / 2053168.0,                 -1944542619.0 / 438351368.0},
188 
189       {         32941697297.0 / 3159114840.0,             456696183123.0 / 1884966160.0,
190              19132610714624.0 / 115814518155.0,       -177904688592943.0 / 474986597400.0,
191        -4821139941836765625.0 / 218016305204281304.0,      30702015625.0 / 3970037232.0,
192             -85916079474274.0 / 2800933759800.0,           -5919468007.0 / 634310460.0,
193                     2479159.0 / 157936.0,                    -18750000.0 / 602131.0,
194                   -19203125.0 / 2053168.0,                 15700361463.0 / 438351368.0},
195 
196       {         12627015655.0 / 631822968.0,              -72955222965.0 / 188496616.0,
197             -13145744952320.0 / 69488710893.0,          30084216194513.0 / 56998391688.0,
198         -296858761006640625.0 / 25648977082856624.0,         569140625.0 / 82709109.0,
199                -18684190637.0 / 18672891732.0,                69644045.0 / 89549712.0,
200                   -11847025.0 / 4264272.0,                  -978650000.0 / 16257537.0,
201                   519371875.0 / 6159504.0,                  5256837225.0 / 438351368.0},
202 
203       {          -450944925.0 / 17550638.0,               -14532122925.0 / 94248308.0,
204               -595876966400.0 / 2573655959.0,             188748653015.0 / 527762886.0,
205         2545485458115234375.0 / 27252038150535163.0,       -1376953125.0 / 36759604.0,
206                 53995596795.0 / 518691437.0,                 210311225.0 / 7047894.0,
207                    -1718875.0 / 39484.0,                      58000000.0 / 602131.0,
208                    -1546875.0 / 39484.0,                   -1262172375.0 / 8429834.0}
209 
210     };
211 
212     /** Last evaluations. */
213     private double[][] yDotKLast;
214 
215     /** Vectors for interpolation. */
216     private double[][] v;
217 
218     /** Initialization indicator for the interpolation vectors. */
219     private boolean vectorsInitialized;
220 
221   /** Simple constructor.
222    * This constructor builds an instance that is not usable yet, the
223    * {@link #reinitialize} method should be called before using the
224    * instance in order to initialize the internal arrays. This
225    * constructor is used only in order to delay the initialization in
226    * some cases. The {@link EmbeddedRungeKuttaIntegrator} uses the
227    * prototyping design pattern to create the step interpolators by
228    * cloning an uninitialized model and latter initializing the copy.
229    */
230   public DormandPrince853StepInterpolator() {
231     super();
232     yDotKLast = null;
233     v         = null;
234     vectorsInitialized = false;
235   }
236 
237   /** Copy constructor.
238    * @param interpolator interpolator to copy from. The copy is a deep
239    * copy: its arrays are separated from the original arrays of the
240    * instance
241    */
242   public DormandPrince853StepInterpolator(final DormandPrince853StepInterpolator interpolator) {
243 
244     super(interpolator);
245 
246     if (interpolator.currentState == null) {
247 
248       yDotKLast = null;
249       v         = null;
250       vectorsInitialized = false;
251 
252     } else {
253 
254       final int dimension = interpolator.currentState.length;
255 
256       yDotKLast    = new double[3][];
257       for (int k = 0; k < yDotKLast.length; ++k) {
258         yDotKLast[k] = new double[dimension];
259         System.arraycopy(interpolator.yDotKLast[k], 0, yDotKLast[k], 0,
260                          dimension);
261       }
262 
263       v = new double[7][];
264       for (int k = 0; k < v.length; ++k) {
265         v[k] = new double[dimension];
266         System.arraycopy(interpolator.v[k], 0, v[k], 0, dimension);
267       }
268 
269       vectorsInitialized = interpolator.vectorsInitialized;
270 
271     }
272 
273   }
274 
275   /** {@inheritDoc} */
276   @Override
277   protected StepInterpolator doCopy() {
278     return new DormandPrince853StepInterpolator(this);
279   }
280 
281   /** {@inheritDoc} */
282   @Override
283   public void reinitialize(final AbstractIntegrator integrator,
284                            final double[] y, final double[][] yDotK, final boolean forward,
285                            final EquationsMapper primaryMapper,
286                            final EquationsMapper[] secondaryMappers) {
287 
288     super.reinitialize(integrator, y, yDotK, forward, primaryMapper, secondaryMappers);
289 
290     final int dimension = currentState.length;
291 
292     yDotKLast = new double[3][];
293     for (int k = 0; k < yDotKLast.length; ++k) {
294       yDotKLast[k] = new double[dimension];
295     }
296 
297     v = new double[7][];
298     for (int k = 0; k < v.length; ++k) {
299       v[k]  = new double[dimension];
300     }
301 
302     vectorsInitialized = false;
303 
304   }
305 
306   /** {@inheritDoc} */
307   @Override
308   public void storeTime(final double t) {
309     super.storeTime(t);
310     vectorsInitialized = false;
311   }
312 
313   /** {@inheritDoc} */
314   @Override
315   protected void computeInterpolatedStateAndDerivatives(final double theta,
316                                           final double oneMinusThetaH)
317       throws MaxCountExceededException {
318 
319     if (! vectorsInitialized) {
320 
321       if (v == null) {
322         v = new double[7][];
323         for (int k = 0; k < 7; ++k) {
324           v[k] = new double[interpolatedState.length];
325         }
326       }
327 
328       // perform the last evaluations if they have not been done yet
329       finalizeStep();
330 
331       // compute the interpolation vectors for this time step
332       for (int i = 0; i < interpolatedState.length; ++i) {
333           final double yDot1  = yDotK[0][i];
334           final double yDot6  = yDotK[5][i];
335           final double yDot7  = yDotK[6][i];
336           final double yDot8  = yDotK[7][i];
337           final double yDot9  = yDotK[8][i];
338           final double yDot10 = yDotK[9][i];
339           final double yDot11 = yDotK[10][i];
340           final double yDot12 = yDotK[11][i];
341           final double yDot13 = yDotK[12][i];
342           final double yDot14 = yDotKLast[0][i];
343           final double yDot15 = yDotKLast[1][i];
344           final double yDot16 = yDotKLast[2][i];
345           v[0][i] = B_01 * yDot1  + B_06 * yDot6 + B_07 * yDot7 +
346                     B_08 * yDot8  + B_09 * yDot9 + B_10 * yDot10 +
347                     B_11 * yDot11 + B_12 * yDot12;
348           v[1][i] = yDot1 - v[0][i];
349           v[2][i] = v[0][i] - v[1][i] - yDotK[12][i];
350           for (int k = 0; k < D.length; ++k) {
351               v[k+3][i] = D[k][0] * yDot1  + D[k][1]  * yDot6  + D[k][2]  * yDot7  +
352                           D[k][3] * yDot8  + D[k][4]  * yDot9  + D[k][5]  * yDot10 +
353                           D[k][6] * yDot11 + D[k][7]  * yDot12 + D[k][8]  * yDot13 +
354                           D[k][9] * yDot14 + D[k][10] * yDot15 + D[k][11] * yDot16;
355           }
356       }
357 
358       vectorsInitialized = true;
359 
360     }
361 
362     final double eta      = 1 - theta;
363     final double twoTheta = 2 * theta;
364     final double theta2   = theta * theta;
365     final double dot1 = 1 - twoTheta;
366     final double dot2 = theta * (2 - 3 * theta);
367     final double dot3 = twoTheta * (1 + theta * (twoTheta -3));
368     final double dot4 = theta2 * (3 + theta * (5 * theta - 8));
369     final double dot5 = theta2 * (3 + theta * (-12 + theta * (15 - 6 * theta)));
370     final double dot6 = theta2 * theta * (4 + theta * (-15 + theta * (18 - 7 * theta)));
371 
372     if ((previousState != null) && (theta <= 0.5)) {
373         for (int i = 0; i < interpolatedState.length; ++i) {
374             interpolatedState[i] = previousState[i] +
375                     theta * h * (v[0][i] +
376                             eta * (v[1][i] +
377                                     theta * (v[2][i] +
378                                             eta * (v[3][i] +
379                                                     theta * (v[4][i] +
380                                                             eta * (v[5][i] +
381                                                                     theta * (v[6][i])))))));
382             interpolatedDerivatives[i] =  v[0][i] + dot1 * v[1][i] + dot2 * v[2][i] +
383                     dot3 * v[3][i] + dot4 * v[4][i] +
384                     dot5 * v[5][i] + dot6 * v[6][i];
385         }
386     } else {
387         for (int i = 0; i < interpolatedState.length; ++i) {
388             interpolatedState[i] = currentState[i] -
389                     oneMinusThetaH * (v[0][i] -
390                             theta * (v[1][i] +
391                                     theta * (v[2][i] +
392                                             eta * (v[3][i] +
393                                                     theta * (v[4][i] +
394                                                             eta * (v[5][i] +
395                                                                     theta * (v[6][i])))))));
396             interpolatedDerivatives[i] =  v[0][i] + dot1 * v[1][i] + dot2 * v[2][i] +
397                     dot3 * v[3][i] + dot4 * v[4][i] +
398                     dot5 * v[5][i] + dot6 * v[6][i];
399         }
400     }
401 
402   }
403 
404   /** {@inheritDoc} */
405   @Override
406   protected void doFinalize() throws MaxCountExceededException {
407 
408       if (currentState == null) {
409           // we are finalizing an uninitialized instance
410           return;
411       }
412 
413       double s;
414       final double[] yTmp = new double[currentState.length];
415       final double pT = getGlobalPreviousTime();
416 
417       // k14
418       for (int j = 0; j < currentState.length; ++j) {
419           s = K14_01 * yDotK[0][j]  + K14_06 * yDotK[5][j]  + K14_07 * yDotK[6][j] +
420                   K14_08 * yDotK[7][j]  + K14_09 * yDotK[8][j]  + K14_10 * yDotK[9][j] +
421                   K14_11 * yDotK[10][j] + K14_12 * yDotK[11][j] + K14_13 * yDotK[12][j];
422           yTmp[j] = currentState[j] + h * s;
423       }
424       integrator.computeDerivatives(pT + C14 * h, yTmp, yDotKLast[0]);
425 
426       // k15
427       for (int j = 0; j < currentState.length; ++j) {
428           s = K15_01 * yDotK[0][j]  + K15_06 * yDotK[5][j]  + K15_07 * yDotK[6][j] +
429                   K15_08 * yDotK[7][j]  + K15_09 * yDotK[8][j]  + K15_10 * yDotK[9][j] +
430                   K15_11 * yDotK[10][j] + K15_12 * yDotK[11][j] + K15_13 * yDotK[12][j] +
431                   K15_14 * yDotKLast[0][j];
432           yTmp[j] = currentState[j] + h * s;
433       }
434       integrator.computeDerivatives(pT + C15 * h, yTmp, yDotKLast[1]);
435 
436       // k16
437       for (int j = 0; j < currentState.length; ++j) {
438           s = K16_01 * yDotK[0][j]  + K16_06 * yDotK[5][j]  + K16_07 * yDotK[6][j] +
439                   K16_08 * yDotK[7][j]  + K16_09 * yDotK[8][j]  + K16_10 * yDotK[9][j] +
440                   K16_11 * yDotK[10][j] + K16_12 * yDotK[11][j] + K16_13 * yDotK[12][j] +
441                   K16_14 * yDotKLast[0][j] +  K16_15 * yDotKLast[1][j];
442           yTmp[j] = currentState[j] + h * s;
443       }
444       integrator.computeDerivatives(pT + C16 * h, yTmp, yDotKLast[2]);
445 
446   }
447 
448   /** {@inheritDoc} */
449   @Override
450   public void writeExternal(final ObjectOutput out)
451     throws IOException {
452 
453     try {
454         // save the local attributes
455         finalizeStep();
456     } catch (MaxCountExceededException mcee) {
457         final IOException ioe = new IOException(mcee.getLocalizedMessage());
458         ioe.initCause(mcee);
459         throw ioe;
460     }
461 
462     final int dimension = (currentState == null) ? -1 : currentState.length;
463     out.writeInt(dimension);
464     for (int i = 0; i < dimension; ++i) {
465       out.writeDouble(yDotKLast[0][i]);
466       out.writeDouble(yDotKLast[1][i]);
467       out.writeDouble(yDotKLast[2][i]);
468     }
469 
470     // save the state of the base class
471     super.writeExternal(out);
472 
473   }
474 
475   /** {@inheritDoc} */
476   @Override
477   public void readExternal(final ObjectInput in)
478     throws IOException, ClassNotFoundException {
479 
480     // read the local attributes
481     yDotKLast = new double[3][];
482     final int dimension = in.readInt();
483     yDotKLast[0] = (dimension < 0) ? null : new double[dimension];
484     yDotKLast[1] = (dimension < 0) ? null : new double[dimension];
485     yDotKLast[2] = (dimension < 0) ? null : new double[dimension];
486 
487     for (int i = 0; i < dimension; ++i) {
488       yDotKLast[0][i] = in.readDouble();
489       yDotKLast[1][i] = in.readDouble();
490       yDotKLast[2][i] = in.readDouble();
491     }
492 
493     // read the base state
494     super.readExternal(in);
495 
496   }
497 
498 }