001 /*
002 * Licensed to the Apache Software Foundation (ASF) under one or more
003 * contributor license agreements. See the NOTICE file distributed with
004 * this work for additional information regarding copyright ownership.
005 * The ASF licenses this file to You under the Apache License, Version 2.0
006 * (the "License"); you may not use this file except in compliance with
007 * the License. You may obtain a copy of the License at
008 *
009 * http://www.apache.org/licenses/LICENSE-2.0
010 *
011 * Unless required by applicable law or agreed to in writing, software
012 * distributed under the License is distributed on an "AS IS" BASIS,
013 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
014 * See the License for the specific language governing permissions and
015 * limitations under the License.
016 */
017
018 package org.apache.commons.math.ode.nonstiff;
019
020 import org.apache.commons.math.analysis.solvers.UnivariateRealSolver;
021 import org.apache.commons.math.exception.MathIllegalArgumentException;
022 import org.apache.commons.math.exception.MathIllegalStateException;
023 import org.apache.commons.math.ode.ExpandableStatefulODE;
024 import org.apache.commons.math.ode.events.EventHandler;
025 import org.apache.commons.math.ode.sampling.AbstractStepInterpolator;
026 import org.apache.commons.math.ode.sampling.StepHandler;
027 import org.apache.commons.math.util.FastMath;
028
029 /**
030 * This class implements a Gragg-Bulirsch-Stoer integrator for
031 * Ordinary Differential Equations.
032 *
033 * <p>The Gragg-Bulirsch-Stoer algorithm is one of the most efficient
034 * ones currently available for smooth problems. It uses Richardson
035 * extrapolation to estimate what would be the solution if the step
036 * size could be decreased down to zero.</p>
037 *
038 * <p>
039 * This method changes both the step size and the order during
040 * integration, in order to minimize computation cost. It is
041 * particularly well suited when a very high precision is needed. The
042 * limit where this method becomes more efficient than high-order
043 * embedded Runge-Kutta methods like {@link DormandPrince853Integrator
044 * Dormand-Prince 8(5,3)} depends on the problem. Results given in the
045 * Hairer, Norsett and Wanner book show for example that this limit
046 * occurs for accuracy around 1e-6 when integrating Saltzam-Lorenz
047 * equations (the authors note this problem is <i>extremely sensitive
048 * to the errors in the first integration steps</i>), and around 1e-11
049 * for a two dimensional celestial mechanics problems with seven
050 * bodies (pleiades problem, involving quasi-collisions for which
051 * <i>automatic step size control is essential</i>).
052 * </p>
053 *
054 * <p>
055 * This implementation is basically a reimplementation in Java of the
056 * <a
057 * href="http://www.unige.ch/math/folks/hairer/prog/nonstiff/odex.f">odex</a>
058 * fortran code by E. Hairer and G. Wanner. The redistribution policy
059 * for this code is available <a
060 * href="http://www.unige.ch/~hairer/prog/licence.txt">here</a>, for
061 * convenience, it is reproduced below.</p>
062 * </p>
063 *
064 * <table border="0" width="80%" cellpadding="10" align="center" bgcolor="#E0E0E0">
065 * <tr><td>Copyright (c) 2004, Ernst Hairer</td></tr>
066 *
067 * <tr><td>Redistribution and use in source and binary forms, with or
068 * without modification, are permitted provided that the following
069 * conditions are met:
070 * <ul>
071 * <li>Redistributions of source code must retain the above copyright
072 * notice, this list of conditions and the following disclaimer.</li>
073 * <li>Redistributions in binary form must reproduce the above copyright
074 * notice, this list of conditions and the following disclaimer in the
075 * documentation and/or other materials provided with the distribution.</li>
076 * </ul></td></tr>
077 *
078 * <tr><td><strong>THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND
079 * CONTRIBUTORS "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING,
080 * BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS
081 * FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR
082 * CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
083 * EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
084 * PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
085 * PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
086 * LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
087 * NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
088 * SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.</strong></td></tr>
089 * </table>
090 *
091 * @version $Id: GraggBulirschStoerIntegrator.java 1176734 2011-09-28 05:56:42Z luc $
092 * @since 1.2
093 */
094
095 public class GraggBulirschStoerIntegrator extends AdaptiveStepsizeIntegrator {
096
097 /** Integrator method name. */
098 private static final String METHOD_NAME = "Gragg-Bulirsch-Stoer";
099
100 /** maximal order. */
101 private int maxOrder;
102
103 /** step size sequence. */
104 private int[] sequence;
105
106 /** overall cost of applying step reduction up to iteration k+1, in number of calls. */
107 private int[] costPerStep;
108
109 /** cost per unit step. */
110 private double[] costPerTimeUnit;
111
112 /** optimal steps for each order. */
113 private double[] optimalStep;
114
115 /** extrapolation coefficients. */
116 private double[][] coeff;
117
118 /** stability check enabling parameter. */
119 private boolean performTest;
120
121 /** maximal number of checks for each iteration. */
122 private int maxChecks;
123
124 /** maximal number of iterations for which checks are performed. */
125 private int maxIter;
126
127 /** stepsize reduction factor in case of stability check failure. */
128 private double stabilityReduction;
129
130 /** first stepsize control factor. */
131 private double stepControl1;
132
133 /** second stepsize control factor. */
134 private double stepControl2;
135
136 /** third stepsize control factor. */
137 private double stepControl3;
138
139 /** fourth stepsize control factor. */
140 private double stepControl4;
141
142 /** first order control factor. */
143 private double orderControl1;
144
145 /** second order control factor. */
146 private double orderControl2;
147
148 /** use interpolation error in stepsize control. */
149 private boolean useInterpolationError;
150
151 /** interpolation order control parameter. */
152 private int mudif;
153
154 /** Simple constructor.
155 * Build a Gragg-Bulirsch-Stoer integrator with the given step
156 * bounds. All tuning parameters are set to their default
157 * values. The default step handler does nothing.
158 * @param minStep minimal step (sign is irrelevant, regardless of
159 * integration direction, forward or backward), the last step can
160 * be smaller than this
161 * @param maxStep maximal step (sign is irrelevant, regardless of
162 * integration direction, forward or backward), the last step can
163 * be smaller than this
164 * @param scalAbsoluteTolerance allowed absolute error
165 * @param scalRelativeTolerance allowed relative error
166 */
167 public GraggBulirschStoerIntegrator(final double minStep, final double maxStep,
168 final double scalAbsoluteTolerance,
169 final double scalRelativeTolerance) {
170 super(METHOD_NAME, minStep, maxStep,
171 scalAbsoluteTolerance, scalRelativeTolerance);
172 setStabilityCheck(true, -1, -1, -1);
173 setStepsizeControl(-1, -1, -1, -1);
174 setOrderControl(-1, -1, -1);
175 setInterpolationControl(true, -1);
176 }
177
178 /** Simple constructor.
179 * Build a Gragg-Bulirsch-Stoer integrator with the given step
180 * bounds. All tuning parameters are set to their default
181 * values. The default step handler does nothing.
182 * @param minStep minimal step (must be positive even for backward
183 * integration), the last step can be smaller than this
184 * @param maxStep maximal step (must be positive even for backward
185 * integration)
186 * @param vecAbsoluteTolerance allowed absolute error
187 * @param vecRelativeTolerance allowed relative error
188 */
189 public GraggBulirschStoerIntegrator(final double minStep, final double maxStep,
190 final double[] vecAbsoluteTolerance,
191 final double[] vecRelativeTolerance) {
192 super(METHOD_NAME, minStep, maxStep,
193 vecAbsoluteTolerance, vecRelativeTolerance);
194 setStabilityCheck(true, -1, -1, -1);
195 setStepsizeControl(-1, -1, -1, -1);
196 setOrderControl(-1, -1, -1);
197 setInterpolationControl(true, -1);
198 }
199
200 /** Set the stability check controls.
201 * <p>The stability check is performed on the first few iterations of
202 * the extrapolation scheme. If this test fails, the step is rejected
203 * and the stepsize is reduced.</p>
204 * <p>By default, the test is performed, at most during two
205 * iterations at each step, and at most once for each of these
206 * iterations. The default stepsize reduction factor is 0.5.</p>
207 * @param performStabilityCheck if true, stability check will be performed,
208 if false, the check will be skipped
209 * @param maxNumIter maximal number of iterations for which checks are
210 * performed (the number of iterations is reset to default if negative
211 * or null)
212 * @param maxNumChecks maximal number of checks for each iteration
213 * (the number of checks is reset to default if negative or null)
214 * @param stepsizeReductionFactor stepsize reduction factor in case of
215 * failure (the factor is reset to default if lower than 0.0001 or
216 * greater than 0.9999)
217 */
218 public void setStabilityCheck(final boolean performStabilityCheck,
219 final int maxNumIter, final int maxNumChecks,
220 final double stepsizeReductionFactor) {
221
222 this.performTest = performStabilityCheck;
223 this.maxIter = (maxNumIter <= 0) ? 2 : maxNumIter;
224 this.maxChecks = (maxNumChecks <= 0) ? 1 : maxNumChecks;
225
226 if ((stepsizeReductionFactor < 0.0001) || (stepsizeReductionFactor > 0.9999)) {
227 this.stabilityReduction = 0.5;
228 } else {
229 this.stabilityReduction = stepsizeReductionFactor;
230 }
231
232 }
233
234 /** Set the step size control factors.
235
236 * <p>The new step size hNew is computed from the old one h by:
237 * <pre>
238 * hNew = h * stepControl2 / (err/stepControl1)^(1/(2k+1))
239 * </pre>
240 * where err is the scaled error and k the iteration number of the
241 * extrapolation scheme (counting from 0). The default values are
242 * 0.65 for stepControl1 and 0.94 for stepControl2.</p>
243 * <p>The step size is subject to the restriction:
244 * <pre>
245 * stepControl3^(1/(2k+1))/stepControl4 <= hNew/h <= 1/stepControl3^(1/(2k+1))
246 * </pre>
247 * The default values are 0.02 for stepControl3 and 4.0 for
248 * stepControl4.</p>
249 * @param control1 first stepsize control factor (the factor is
250 * reset to default if lower than 0.0001 or greater than 0.9999)
251 * @param control2 second stepsize control factor (the factor
252 * is reset to default if lower than 0.0001 or greater than 0.9999)
253 * @param control3 third stepsize control factor (the factor is
254 * reset to default if lower than 0.0001 or greater than 0.9999)
255 * @param control4 fourth stepsize control factor (the factor
256 * is reset to default if lower than 1.0001 or greater than 999.9)
257 */
258 public void setStepsizeControl(final double control1, final double control2,
259 final double control3, final double control4) {
260
261 if ((control1 < 0.0001) || (control1 > 0.9999)) {
262 this.stepControl1 = 0.65;
263 } else {
264 this.stepControl1 = control1;
265 }
266
267 if ((control2 < 0.0001) || (control2 > 0.9999)) {
268 this.stepControl2 = 0.94;
269 } else {
270 this.stepControl2 = control2;
271 }
272
273 if ((control3 < 0.0001) || (control3 > 0.9999)) {
274 this.stepControl3 = 0.02;
275 } else {
276 this.stepControl3 = control3;
277 }
278
279 if ((control4 < 1.0001) || (control4 > 999.9)) {
280 this.stepControl4 = 4.0;
281 } else {
282 this.stepControl4 = control4;
283 }
284
285 }
286
287 /** Set the order control parameters.
288 * <p>The Gragg-Bulirsch-Stoer method changes both the step size and
289 * the order during integration, in order to minimize computation
290 * cost. Each extrapolation step increases the order by 2, so the
291 * maximal order that will be used is always even, it is twice the
292 * maximal number of columns in the extrapolation table.</p>
293 * <pre>
294 * order is decreased if w(k-1) <= w(k) * orderControl1
295 * order is increased if w(k) <= w(k-1) * orderControl2
296 * </pre>
297 * <p>where w is the table of work per unit step for each order
298 * (number of function calls divided by the step length), and k is
299 * the current order.</p>
300 * <p>The default maximal order after construction is 18 (i.e. the
301 * maximal number of columns is 9). The default values are 0.8 for
302 * orderControl1 and 0.9 for orderControl2.</p>
303 * @param maximalOrder maximal order in the extrapolation table (the
304 * maximal order is reset to default if order <= 6 or odd)
305 * @param control1 first order control factor (the factor is
306 * reset to default if lower than 0.0001 or greater than 0.9999)
307 * @param control2 second order control factor (the factor
308 * is reset to default if lower than 0.0001 or greater than 0.9999)
309 */
310 public void setOrderControl(final int maximalOrder,
311 final double control1, final double control2) {
312
313 if ((maximalOrder <= 6) || (maximalOrder % 2 != 0)) {
314 this.maxOrder = 18;
315 }
316
317 if ((control1 < 0.0001) || (control1 > 0.9999)) {
318 this.orderControl1 = 0.8;
319 } else {
320 this.orderControl1 = control1;
321 }
322
323 if ((control2 < 0.0001) || (control2 > 0.9999)) {
324 this.orderControl2 = 0.9;
325 } else {
326 this.orderControl2 = control2;
327 }
328
329 // reinitialize the arrays
330 initializeArrays();
331
332 }
333
334 /** {@inheritDoc} */
335 @Override
336 public void addStepHandler (final StepHandler handler) {
337
338 super.addStepHandler(handler);
339
340 // reinitialize the arrays
341 initializeArrays();
342
343 }
344
345 /** {@inheritDoc} */
346 @Override
347 public void addEventHandler(final EventHandler function,
348 final double maxCheckInterval,
349 final double convergence,
350 final int maxIterationCount,
351 final UnivariateRealSolver solver) {
352 super.addEventHandler(function, maxCheckInterval, convergence,
353 maxIterationCount, solver);
354
355 // reinitialize the arrays
356 initializeArrays();
357
358 }
359
360 /** Initialize the integrator internal arrays. */
361 private void initializeArrays() {
362
363 final int size = maxOrder / 2;
364
365 if ((sequence == null) || (sequence.length != size)) {
366 // all arrays should be reallocated with the right size
367 sequence = new int[size];
368 costPerStep = new int[size];
369 coeff = new double[size][];
370 costPerTimeUnit = new double[size];
371 optimalStep = new double[size];
372 }
373
374 // step size sequence: 2, 6, 10, 14, ...
375 for (int k = 0; k < size; ++k) {
376 sequence[k] = 4 * k + 2;
377 }
378
379 // initialize the order selection cost array
380 // (number of function calls for each column of the extrapolation table)
381 costPerStep[0] = sequence[0] + 1;
382 for (int k = 1; k < size; ++k) {
383 costPerStep[k] = costPerStep[k-1] + sequence[k];
384 }
385
386 // initialize the extrapolation tables
387 for (int k = 0; k < size; ++k) {
388 coeff[k] = (k > 0) ? new double[k] : null;
389 for (int l = 0; l < k; ++l) {
390 final double ratio = ((double) sequence[k]) / sequence[k-l-1];
391 coeff[k][l] = 1.0 / (ratio * ratio - 1.0);
392 }
393 }
394
395 }
396
397 /** Set the interpolation order control parameter.
398 * The interpolation order for dense output is 2k - mudif + 1. The
399 * default value for mudif is 4 and the interpolation error is used
400 * in stepsize control by default.
401
402 * @param useInterpolationErrorForControl if true, interpolation error is used
403 * for stepsize control
404 * @param mudifControlParameter interpolation order control parameter (the parameter
405 * is reset to default if <= 0 or >= 7)
406 */
407 public void setInterpolationControl(final boolean useInterpolationErrorForControl,
408 final int mudifControlParameter) {
409
410 this.useInterpolationError = useInterpolationErrorForControl;
411
412 if ((mudifControlParameter <= 0) || (mudifControlParameter >= 7)) {
413 this.mudif = 4;
414 } else {
415 this.mudif = mudifControlParameter;
416 }
417
418 }
419
420 /** Update scaling array.
421 * @param y1 first state vector to use for scaling
422 * @param y2 second state vector to use for scaling
423 * @param scale scaling array to update (can be shorter than state)
424 */
425 private void rescale(final double[] y1, final double[] y2, final double[] scale) {
426 if (vecAbsoluteTolerance == null) {
427 for (int i = 0; i < scale.length; ++i) {
428 final double yi = FastMath.max(FastMath.abs(y1[i]), FastMath.abs(y2[i]));
429 scale[i] = scalAbsoluteTolerance + scalRelativeTolerance * yi;
430 }
431 } else {
432 for (int i = 0; i < scale.length; ++i) {
433 final double yi = FastMath.max(FastMath.abs(y1[i]), FastMath.abs(y2[i]));
434 scale[i] = vecAbsoluteTolerance[i] + vecRelativeTolerance[i] * yi;
435 }
436 }
437 }
438
439 /** Perform integration over one step using substeps of a modified
440 * midpoint method.
441 * @param t0 initial time
442 * @param y0 initial value of the state vector at t0
443 * @param step global step
444 * @param k iteration number (from 0 to sequence.length - 1)
445 * @param scale scaling array (can be shorter than state)
446 * @param f placeholder where to put the state vector derivatives at each substep
447 * (element 0 already contains initial derivative)
448 * @param yMiddle placeholder where to put the state vector at the middle of the step
449 * @param yEnd placeholder where to put the state vector at the end
450 * @param yTmp placeholder for one state vector
451 * @return true if computation was done properly,
452 * false if stability check failed before end of computation
453 */
454 private boolean tryStep(final double t0, final double[] y0, final double step, final int k,
455 final double[] scale, final double[][] f,
456 final double[] yMiddle, final double[] yEnd,
457 final double[] yTmp) {
458
459 final int n = sequence[k];
460 final double subStep = step / n;
461 final double subStep2 = 2 * subStep;
462
463 // first substep
464 double t = t0 + subStep;
465 for (int i = 0; i < y0.length; ++i) {
466 yTmp[i] = y0[i];
467 yEnd[i] = y0[i] + subStep * f[0][i];
468 }
469 computeDerivatives(t, yEnd, f[1]);
470
471 // other substeps
472 for (int j = 1; j < n; ++j) {
473
474 if (2 * j == n) {
475 // save the point at the middle of the step
476 System.arraycopy(yEnd, 0, yMiddle, 0, y0.length);
477 }
478
479 t += subStep;
480 for (int i = 0; i < y0.length; ++i) {
481 final double middle = yEnd[i];
482 yEnd[i] = yTmp[i] + subStep2 * f[j][i];
483 yTmp[i] = middle;
484 }
485
486 computeDerivatives(t, yEnd, f[j+1]);
487
488 // stability check
489 if (performTest && (j <= maxChecks) && (k < maxIter)) {
490 double initialNorm = 0.0;
491 for (int l = 0; l < scale.length; ++l) {
492 final double ratio = f[0][l] / scale[l];
493 initialNorm += ratio * ratio;
494 }
495 double deltaNorm = 0.0;
496 for (int l = 0; l < scale.length; ++l) {
497 final double ratio = (f[j+1][l] - f[0][l]) / scale[l];
498 deltaNorm += ratio * ratio;
499 }
500 if (deltaNorm > 4 * FastMath.max(1.0e-15, initialNorm)) {
501 return false;
502 }
503 }
504
505 }
506
507 // correction of the last substep (at t0 + step)
508 for (int i = 0; i < y0.length; ++i) {
509 yEnd[i] = 0.5 * (yTmp[i] + yEnd[i] + subStep * f[n][i]);
510 }
511
512 return true;
513
514 }
515
516 /** Extrapolate a vector.
517 * @param offset offset to use in the coefficients table
518 * @param k index of the last updated point
519 * @param diag working diagonal of the Aitken-Neville's
520 * triangle, without the last element
521 * @param last last element
522 */
523 private void extrapolate(final int offset, final int k,
524 final double[][] diag, final double[] last) {
525
526 // update the diagonal
527 for (int j = 1; j < k; ++j) {
528 for (int i = 0; i < last.length; ++i) {
529 // Aitken-Neville's recursive formula
530 diag[k-j-1][i] = diag[k-j][i] +
531 coeff[k+offset][j-1] * (diag[k-j][i] - diag[k-j-1][i]);
532 }
533 }
534
535 // update the last element
536 for (int i = 0; i < last.length; ++i) {
537 // Aitken-Neville's recursive formula
538 last[i] = diag[0][i] + coeff[k+offset][k-1] * (diag[0][i] - last[i]);
539 }
540 }
541
542 /** {@inheritDoc} */
543 @Override
544 public void integrate(final ExpandableStatefulODE equations, final double t)
545 throws MathIllegalStateException, MathIllegalArgumentException {
546
547 sanityChecks(equations, t);
548 setEquations(equations);
549 resetEvaluations();
550 final boolean forward = t > equations.getTime();
551
552 // create some internal working arrays
553 final double[] y0 = equations.getCompleteState();
554 final double[] y = y0.clone();
555 final double[] yDot0 = new double[y.length];
556 final double[] y1 = new double[y.length];
557 final double[] yTmp = new double[y.length];
558 final double[] yTmpDot = new double[y.length];
559
560 final double[][] diagonal = new double[sequence.length-1][];
561 final double[][] y1Diag = new double[sequence.length-1][];
562 for (int k = 0; k < sequence.length-1; ++k) {
563 diagonal[k] = new double[y.length];
564 y1Diag[k] = new double[y.length];
565 }
566
567 final double[][][] fk = new double[sequence.length][][];
568 for (int k = 0; k < sequence.length; ++k) {
569
570 fk[k] = new double[sequence[k] + 1][];
571
572 // all substeps start at the same point, so share the first array
573 fk[k][0] = yDot0;
574
575 for (int l = 0; l < sequence[k]; ++l) {
576 fk[k][l+1] = new double[y0.length];
577 }
578
579 }
580
581 if (y != y0) {
582 System.arraycopy(y0, 0, y, 0, y0.length);
583 }
584
585 final double[] yDot1 = new double[y0.length];
586 final double[][] yMidDots = new double[1 + 2 * sequence.length][y0.length];
587
588 // initial scaling
589 final double[] scale = new double[mainSetDimension];
590 rescale(y, y, scale);
591
592 // initial order selection
593 final double tol =
594 (vecRelativeTolerance == null) ? scalRelativeTolerance : vecRelativeTolerance[0];
595 final double log10R = FastMath.log10(FastMath.max(1.0e-10, tol));
596 int targetIter = FastMath.max(1,
597 FastMath.min(sequence.length - 2,
598 (int) FastMath.floor(0.5 - 0.6 * log10R)));
599
600 // set up an interpolator sharing the integrator arrays
601 final AbstractStepInterpolator interpolator =
602 new GraggBulirschStoerStepInterpolator(y, yDot0,
603 y1, yDot1,
604 yMidDots, forward,
605 equations.getPrimaryMapper(),
606 equations.getSecondaryMappers());
607 interpolator.storeTime(equations.getTime());
608
609 stepStart = equations.getTime();
610 double hNew = 0;
611 double maxError = Double.MAX_VALUE;
612 boolean previousRejected = false;
613 boolean firstTime = true;
614 boolean newStep = true;
615 boolean firstStepAlreadyComputed = false;
616 for (StepHandler handler : stepHandlers) {
617 handler.reset();
618 }
619 setStateInitialized(false);
620 costPerTimeUnit[0] = 0;
621 isLastStep = false;
622 do {
623
624 double error;
625 boolean reject = false;
626
627 if (newStep) {
628
629 interpolator.shift();
630
631 // first evaluation, at the beginning of the step
632 if (! firstStepAlreadyComputed) {
633 computeDerivatives(stepStart, y, yDot0);
634 }
635
636 if (firstTime) {
637 hNew = initializeStep(forward, 2 * targetIter + 1, scale,
638 stepStart, y, yDot0, yTmp, yTmpDot);
639 }
640
641 newStep = false;
642
643 }
644
645 stepSize = hNew;
646
647 // step adjustment near bounds
648 if ((forward && (stepStart + stepSize > t)) ||
649 ((! forward) && (stepStart + stepSize < t))) {
650 stepSize = t - stepStart;
651 }
652 final double nextT = stepStart + stepSize;
653 isLastStep = forward ? (nextT >= t) : (nextT <= t);
654
655 // iterate over several substep sizes
656 int k = -1;
657 for (boolean loop = true; loop; ) {
658
659 ++k;
660
661 // modified midpoint integration with the current substep
662 if ( ! tryStep(stepStart, y, stepSize, k, scale, fk[k],
663 (k == 0) ? yMidDots[0] : diagonal[k-1],
664 (k == 0) ? y1 : y1Diag[k-1],
665 yTmp)) {
666
667 // the stability check failed, we reduce the global step
668 hNew = FastMath.abs(filterStep(stepSize * stabilityReduction, forward, false));
669 reject = true;
670 loop = false;
671
672 } else {
673
674 // the substep was computed successfully
675 if (k > 0) {
676
677 // extrapolate the state at the end of the step
678 // using last iteration data
679 extrapolate(0, k, y1Diag, y1);
680 rescale(y, y1, scale);
681
682 // estimate the error at the end of the step.
683 error = 0;
684 for (int j = 0; j < mainSetDimension; ++j) {
685 final double e = FastMath.abs(y1[j] - y1Diag[0][j]) / scale[j];
686 error += e * e;
687 }
688 error = FastMath.sqrt(error / mainSetDimension);
689
690 if ((error > 1.0e15) || ((k > 1) && (error > maxError))) {
691 // error is too big, we reduce the global step
692 hNew = FastMath.abs(filterStep(stepSize * stabilityReduction, forward, false));
693 reject = true;
694 loop = false;
695 } else {
696
697 maxError = FastMath.max(4 * error, 1.0);
698
699 // compute optimal stepsize for this order
700 final double exp = 1.0 / (2 * k + 1);
701 double fac = stepControl2 / FastMath.pow(error / stepControl1, exp);
702 final double pow = FastMath.pow(stepControl3, exp);
703 fac = FastMath.max(pow / stepControl4, FastMath.min(1 / pow, fac));
704 optimalStep[k] = FastMath.abs(filterStep(stepSize * fac, forward, true));
705 costPerTimeUnit[k] = costPerStep[k] / optimalStep[k];
706
707 // check convergence
708 switch (k - targetIter) {
709
710 case -1 :
711 if ((targetIter > 1) && ! previousRejected) {
712
713 // check if we can stop iterations now
714 if (error <= 1.0) {
715 // convergence have been reached just before targetIter
716 loop = false;
717 } else {
718 // estimate if there is a chance convergence will
719 // be reached on next iteration, using the
720 // asymptotic evolution of error
721 final double ratio = ((double) sequence [targetIter] * sequence[targetIter + 1]) /
722 (sequence[0] * sequence[0]);
723 if (error > ratio * ratio) {
724 // we don't expect to converge on next iteration
725 // we reject the step immediately and reduce order
726 reject = true;
727 loop = false;
728 targetIter = k;
729 if ((targetIter > 1) &&
730 (costPerTimeUnit[targetIter-1] <
731 orderControl1 * costPerTimeUnit[targetIter])) {
732 --targetIter;
733 }
734 hNew = optimalStep[targetIter];
735 }
736 }
737 }
738 break;
739
740 case 0:
741 if (error <= 1.0) {
742 // convergence has been reached exactly at targetIter
743 loop = false;
744 } else {
745 // estimate if there is a chance convergence will
746 // be reached on next iteration, using the
747 // asymptotic evolution of error
748 final double ratio = ((double) sequence[k+1]) / sequence[0];
749 if (error > ratio * ratio) {
750 // we don't expect to converge on next iteration
751 // we reject the step immediately
752 reject = true;
753 loop = false;
754 if ((targetIter > 1) &&
755 (costPerTimeUnit[targetIter-1] <
756 orderControl1 * costPerTimeUnit[targetIter])) {
757 --targetIter;
758 }
759 hNew = optimalStep[targetIter];
760 }
761 }
762 break;
763
764 case 1 :
765 if (error > 1.0) {
766 reject = true;
767 if ((targetIter > 1) &&
768 (costPerTimeUnit[targetIter-1] <
769 orderControl1 * costPerTimeUnit[targetIter])) {
770 --targetIter;
771 }
772 hNew = optimalStep[targetIter];
773 }
774 loop = false;
775 break;
776
777 default :
778 if ((firstTime || isLastStep) && (error <= 1.0)) {
779 loop = false;
780 }
781 break;
782
783 }
784
785 }
786 }
787 }
788 }
789
790 if (! reject) {
791 // derivatives at end of step
792 computeDerivatives(stepStart + stepSize, y1, yDot1);
793 }
794
795 // dense output handling
796 double hInt = getMaxStep();
797 if (! reject) {
798
799 // extrapolate state at middle point of the step
800 for (int j = 1; j <= k; ++j) {
801 extrapolate(0, j, diagonal, yMidDots[0]);
802 }
803
804 final int mu = 2 * k - mudif + 3;
805
806 for (int l = 0; l < mu; ++l) {
807
808 // derivative at middle point of the step
809 final int l2 = l / 2;
810 double factor = FastMath.pow(0.5 * sequence[l2], l);
811 int middleIndex = fk[l2].length / 2;
812 for (int i = 0; i < y0.length; ++i) {
813 yMidDots[l+1][i] = factor * fk[l2][middleIndex + l][i];
814 }
815 for (int j = 1; j <= k - l2; ++j) {
816 factor = FastMath.pow(0.5 * sequence[j + l2], l);
817 middleIndex = fk[l2+j].length / 2;
818 for (int i = 0; i < y0.length; ++i) {
819 diagonal[j-1][i] = factor * fk[l2+j][middleIndex+l][i];
820 }
821 extrapolate(l2, j, diagonal, yMidDots[l+1]);
822 }
823 for (int i = 0; i < y0.length; ++i) {
824 yMidDots[l+1][i] *= stepSize;
825 }
826
827 // compute centered differences to evaluate next derivatives
828 for (int j = (l + 1) / 2; j <= k; ++j) {
829 for (int m = fk[j].length - 1; m >= 2 * (l + 1); --m) {
830 for (int i = 0; i < y0.length; ++i) {
831 fk[j][m][i] -= fk[j][m-2][i];
832 }
833 }
834 }
835
836 }
837
838 if (mu >= 0) {
839
840 // estimate the dense output coefficients
841 final GraggBulirschStoerStepInterpolator gbsInterpolator
842 = (GraggBulirschStoerStepInterpolator) interpolator;
843 gbsInterpolator.computeCoefficients(mu, stepSize);
844
845 if (useInterpolationError) {
846 // use the interpolation error to limit stepsize
847 final double interpError = gbsInterpolator.estimateError(scale);
848 hInt = FastMath.abs(stepSize / FastMath.max(FastMath.pow(interpError, 1.0 / (mu+4)),
849 0.01));
850 if (interpError > 10.0) {
851 hNew = hInt;
852 reject = true;
853 }
854 }
855
856 }
857
858 }
859
860 if (! reject) {
861
862 // Discrete events handling
863 interpolator.storeTime(stepStart + stepSize);
864 stepStart = acceptStep(interpolator, y1, yDot1, t);
865
866 // prepare next step
867 interpolator.storeTime(stepStart);
868 System.arraycopy(y1, 0, y, 0, y0.length);
869 System.arraycopy(yDot1, 0, yDot0, 0, y0.length);
870 firstStepAlreadyComputed = true;
871
872 int optimalIter;
873 if (k == 1) {
874 optimalIter = 2;
875 if (previousRejected) {
876 optimalIter = 1;
877 }
878 } else if (k <= targetIter) {
879 optimalIter = k;
880 if (costPerTimeUnit[k-1] < orderControl1 * costPerTimeUnit[k]) {
881 optimalIter = k-1;
882 } else if (costPerTimeUnit[k] < orderControl2 * costPerTimeUnit[k-1]) {
883 optimalIter = FastMath.min(k+1, sequence.length - 2);
884 }
885 } else {
886 optimalIter = k - 1;
887 if ((k > 2) &&
888 (costPerTimeUnit[k-2] < orderControl1 * costPerTimeUnit[k-1])) {
889 optimalIter = k - 2;
890 }
891 if (costPerTimeUnit[k] < orderControl2 * costPerTimeUnit[optimalIter]) {
892 optimalIter = FastMath.min(k, sequence.length - 2);
893 }
894 }
895
896 if (previousRejected) {
897 // after a rejected step neither order nor stepsize
898 // should increase
899 targetIter = FastMath.min(optimalIter, k);
900 hNew = FastMath.min(FastMath.abs(stepSize), optimalStep[targetIter]);
901 } else {
902 // stepsize control
903 if (optimalIter <= k) {
904 hNew = optimalStep[optimalIter];
905 } else {
906 if ((k < targetIter) &&
907 (costPerTimeUnit[k] < orderControl2 * costPerTimeUnit[k-1])) {
908 hNew = filterStep(optimalStep[k] * costPerStep[optimalIter+1] / costPerStep[k],
909 forward, false);
910 } else {
911 hNew = filterStep(optimalStep[k] * costPerStep[optimalIter] / costPerStep[k],
912 forward, false);
913 }
914 }
915
916 targetIter = optimalIter;
917
918 }
919
920 newStep = true;
921
922 }
923
924 hNew = FastMath.min(hNew, hInt);
925 if (! forward) {
926 hNew = -hNew;
927 }
928
929 firstTime = false;
930
931 if (reject) {
932 isLastStep = false;
933 previousRejected = true;
934 } else {
935 previousRejected = false;
936 }
937
938 } while (!isLastStep);
939
940 // dispatch results
941 equations.setTime(stepStart);
942 equations.setCompleteState(y);
943
944 resetInternalState();
945
946 }
947
948 }