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.math3.ode;
019
020 import org.apache.commons.math3.exception.DimensionMismatchException;
021 import org.apache.commons.math3.exception.MaxCountExceededException;
022 import org.apache.commons.math3.exception.NoBracketingException;
023 import org.apache.commons.math3.exception.NumberIsTooSmallException;
024 import org.apache.commons.math3.exception.util.LocalizedFormats;
025 import org.apache.commons.math3.linear.Array2DRowRealMatrix;
026 import org.apache.commons.math3.ode.nonstiff.AdaptiveStepsizeIntegrator;
027 import org.apache.commons.math3.ode.nonstiff.DormandPrince853Integrator;
028 import org.apache.commons.math3.ode.sampling.StepHandler;
029 import org.apache.commons.math3.ode.sampling.StepInterpolator;
030 import org.apache.commons.math3.util.FastMath;
031
032 /**
033 * This class is the base class for multistep integrators for Ordinary
034 * Differential Equations.
035 * <p>We define scaled derivatives s<sub>i</sub>(n) at step n as:
036 * <pre>
037 * s<sub>1</sub>(n) = h y'<sub>n</sub> for first derivative
038 * s<sub>2</sub>(n) = h<sup>2</sup>/2 y''<sub>n</sub> for second derivative
039 * s<sub>3</sub>(n) = h<sup>3</sup>/6 y'''<sub>n</sub> for third derivative
040 * ...
041 * s<sub>k</sub>(n) = h<sup>k</sup>/k! y<sup>(k)</sup><sub>n</sub> for k<sup>th</sup> derivative
042 * </pre></p>
043 * <p>Rather than storing several previous steps separately, this implementation uses
044 * the Nordsieck vector with higher degrees scaled derivatives all taken at the same
045 * step (y<sub>n</sub>, s<sub>1</sub>(n) and r<sub>n</sub>) where r<sub>n</sub> is defined as:
046 * <pre>
047 * r<sub>n</sub> = [ s<sub>2</sub>(n), s<sub>3</sub>(n) ... s<sub>k</sub>(n) ]<sup>T</sup>
048 * </pre>
049 * (we omit the k index in the notation for clarity)</p>
050 * <p>
051 * Multistep integrators with Nordsieck representation are highly sensitive to
052 * large step changes because when the step is multiplied by factor a, the
053 * k<sup>th</sup> component of the Nordsieck vector is multiplied by a<sup>k</sup>
054 * and the last components are the least accurate ones. The default max growth
055 * factor is therefore set to a quite low value: 2<sup>1/order</sup>.
056 * </p>
057 *
058 * @see org.apache.commons.math3.ode.nonstiff.AdamsBashforthIntegrator
059 * @see org.apache.commons.math3.ode.nonstiff.AdamsMoultonIntegrator
060 * @version $Id: MultistepIntegrator.java 1463684 2013-04-02 19:04:13Z luc $
061 * @since 2.0
062 */
063 public abstract class MultistepIntegrator extends AdaptiveStepsizeIntegrator {
064
065 /** First scaled derivative (h y'). */
066 protected double[] scaled;
067
068 /** Nordsieck matrix of the higher scaled derivatives.
069 * <p>(h<sup>2</sup>/2 y'', h<sup>3</sup>/6 y''' ..., h<sup>k</sup>/k! y<sup>(k)</sup>)</p>
070 */
071 protected Array2DRowRealMatrix nordsieck;
072
073 /** Starter integrator. */
074 private FirstOrderIntegrator starter;
075
076 /** Number of steps of the multistep method (excluding the one being computed). */
077 private final int nSteps;
078
079 /** Stepsize control exponent. */
080 private double exp;
081
082 /** Safety factor for stepsize control. */
083 private double safety;
084
085 /** Minimal reduction factor for stepsize control. */
086 private double minReduction;
087
088 /** Maximal growth factor for stepsize control. */
089 private double maxGrowth;
090
091 /**
092 * Build a multistep integrator with the given stepsize bounds.
093 * <p>The default starter integrator is set to the {@link
094 * DormandPrince853Integrator Dormand-Prince 8(5,3)} integrator with
095 * some defaults settings.</p>
096 * <p>
097 * The default max growth factor is set to a quite low value: 2<sup>1/order</sup>.
098 * </p>
099 * @param name name of the method
100 * @param nSteps number of steps of the multistep method
101 * (excluding the one being computed)
102 * @param order order of the method
103 * @param minStep minimal step (must be positive even for backward
104 * integration), the last step can be smaller than this
105 * @param maxStep maximal step (must be positive even for backward
106 * integration)
107 * @param scalAbsoluteTolerance allowed absolute error
108 * @param scalRelativeTolerance allowed relative error
109 * @exception NumberIsTooSmallException if number of steps is smaller than 2
110 */
111 protected MultistepIntegrator(final String name, final int nSteps,
112 final int order,
113 final double minStep, final double maxStep,
114 final double scalAbsoluteTolerance,
115 final double scalRelativeTolerance)
116 throws NumberIsTooSmallException {
117
118 super(name, minStep, maxStep, scalAbsoluteTolerance, scalRelativeTolerance);
119
120 if (nSteps < 2) {
121 throw new NumberIsTooSmallException(
122 LocalizedFormats.INTEGRATION_METHOD_NEEDS_AT_LEAST_TWO_PREVIOUS_POINTS,
123 nSteps, 2, true);
124 }
125
126 starter = new DormandPrince853Integrator(minStep, maxStep,
127 scalAbsoluteTolerance,
128 scalRelativeTolerance);
129 this.nSteps = nSteps;
130
131 exp = -1.0 / order;
132
133 // set the default values of the algorithm control parameters
134 setSafety(0.9);
135 setMinReduction(0.2);
136 setMaxGrowth(FastMath.pow(2.0, -exp));
137
138 }
139
140 /**
141 * Build a multistep integrator with the given stepsize bounds.
142 * <p>The default starter integrator is set to the {@link
143 * DormandPrince853Integrator Dormand-Prince 8(5,3)} integrator with
144 * some defaults settings.</p>
145 * <p>
146 * The default max growth factor is set to a quite low value: 2<sup>1/order</sup>.
147 * </p>
148 * @param name name of the method
149 * @param nSteps number of steps of the multistep method
150 * (excluding the one being computed)
151 * @param order order of the method
152 * @param minStep minimal step (must be positive even for backward
153 * integration), the last step can be smaller than this
154 * @param maxStep maximal step (must be positive even for backward
155 * integration)
156 * @param vecAbsoluteTolerance allowed absolute error
157 * @param vecRelativeTolerance allowed relative error
158 */
159 protected MultistepIntegrator(final String name, final int nSteps,
160 final int order,
161 final double minStep, final double maxStep,
162 final double[] vecAbsoluteTolerance,
163 final double[] vecRelativeTolerance) {
164 super(name, minStep, maxStep, vecAbsoluteTolerance, vecRelativeTolerance);
165 starter = new DormandPrince853Integrator(minStep, maxStep,
166 vecAbsoluteTolerance,
167 vecRelativeTolerance);
168 this.nSteps = nSteps;
169
170 exp = -1.0 / order;
171
172 // set the default values of the algorithm control parameters
173 setSafety(0.9);
174 setMinReduction(0.2);
175 setMaxGrowth(FastMath.pow(2.0, -exp));
176
177 }
178
179 /**
180 * Get the starter integrator.
181 * @return starter integrator
182 */
183 public ODEIntegrator getStarterIntegrator() {
184 return starter;
185 }
186
187 /**
188 * Set the starter integrator.
189 * <p>The various step and event handlers for this starter integrator
190 * will be managed automatically by the multi-step integrator. Any
191 * user configuration for these elements will be cleared before use.</p>
192 * @param starterIntegrator starter integrator
193 */
194 public void setStarterIntegrator(FirstOrderIntegrator starterIntegrator) {
195 this.starter = starterIntegrator;
196 }
197
198 /** Start the integration.
199 * <p>This method computes one step using the underlying starter integrator,
200 * and initializes the Nordsieck vector at step start. The starter integrator
201 * purpose is only to establish initial conditions, it does not really change
202 * time by itself. The top level multistep integrator remains in charge of
203 * handling time propagation and events handling as it will starts its own
204 * computation right from the beginning. In a sense, the starter integrator
205 * can be seen as a dummy one and so it will never trigger any user event nor
206 * call any user step handler.</p>
207 * @param t0 initial time
208 * @param y0 initial value of the state vector at t0
209 * @param t target time for the integration
210 * (can be set to a value smaller than <code>t0</code> for backward integration)
211 * @exception DimensionMismatchException if arrays dimension do not match equations settings
212 * @exception NumberIsTooSmallException if integration step is too small
213 * @exception MaxCountExceededException if the number of functions evaluations is exceeded
214 * @exception NoBracketingException if the location of an event cannot be bracketed
215 */
216 protected void start(final double t0, final double[] y0, final double t)
217 throws DimensionMismatchException, NumberIsTooSmallException,
218 MaxCountExceededException, NoBracketingException {
219
220 // make sure NO user event nor user step handler is triggered,
221 // this is the task of the top level integrator, not the task
222 // of the starter integrator
223 starter.clearEventHandlers();
224 starter.clearStepHandlers();
225
226 // set up one specific step handler to extract initial Nordsieck vector
227 starter.addStepHandler(new NordsieckInitializer(nSteps, y0.length));
228
229 // start integration, expecting a InitializationCompletedMarkerException
230 try {
231
232 if (starter instanceof AbstractIntegrator) {
233 ((AbstractIntegrator) starter).integrate(getExpandable(), t);
234 } else {
235 starter.integrate(new FirstOrderDifferentialEquations() {
236
237 /** {@inheritDoc} */
238 public int getDimension() {
239 return getExpandable().getTotalDimension();
240 }
241
242 /** {@inheritDoc} */
243 public void computeDerivatives(double t, double[] y, double[] yDot) {
244 getExpandable().computeDerivatives(t, y, yDot);
245 }
246
247 }, t0, y0, t, new double[y0.length]);
248 }
249
250 } catch (InitializationCompletedMarkerException icme) { // NOPMD
251 // this is the expected nominal interruption of the start integrator
252
253 // count the evaluations used by the starter
254 getEvaluationsCounter().incrementCount(starter.getEvaluations());
255
256 }
257
258 // remove the specific step handler
259 starter.clearStepHandlers();
260
261 }
262
263 /** Initialize the high order scaled derivatives at step start.
264 * @param h step size to use for scaling
265 * @param t first steps times
266 * @param y first steps states
267 * @param yDot first steps derivatives
268 * @return Nordieck vector at first step (h<sup>2</sup>/2 y''<sub>n</sub>,
269 * h<sup>3</sup>/6 y'''<sub>n</sub> ... h<sup>k</sup>/k! y<sup>(k)</sup><sub>n</sub>)
270 */
271 protected abstract Array2DRowRealMatrix initializeHighOrderDerivatives(final double h, final double[] t,
272 final double[][] y,
273 final double[][] yDot);
274
275 /** Get the minimal reduction factor for stepsize control.
276 * @return minimal reduction factor
277 */
278 public double getMinReduction() {
279 return minReduction;
280 }
281
282 /** Set the minimal reduction factor for stepsize control.
283 * @param minReduction minimal reduction factor
284 */
285 public void setMinReduction(final double minReduction) {
286 this.minReduction = minReduction;
287 }
288
289 /** Get the maximal growth factor for stepsize control.
290 * @return maximal growth factor
291 */
292 public double getMaxGrowth() {
293 return maxGrowth;
294 }
295
296 /** Set the maximal growth factor for stepsize control.
297 * @param maxGrowth maximal growth factor
298 */
299 public void setMaxGrowth(final double maxGrowth) {
300 this.maxGrowth = maxGrowth;
301 }
302
303 /** Get the safety factor for stepsize control.
304 * @return safety factor
305 */
306 public double getSafety() {
307 return safety;
308 }
309
310 /** Set the safety factor for stepsize control.
311 * @param safety safety factor
312 */
313 public void setSafety(final double safety) {
314 this.safety = safety;
315 }
316
317 /** Compute step grow/shrink factor according to normalized error.
318 * @param error normalized error of the current step
319 * @return grow/shrink factor for next step
320 */
321 protected double computeStepGrowShrinkFactor(final double error) {
322 return FastMath.min(maxGrowth, FastMath.max(minReduction, safety * FastMath.pow(error, exp)));
323 }
324
325 /** Transformer used to convert the first step to Nordsieck representation. */
326 public interface NordsieckTransformer {
327 /** Initialize the high order scaled derivatives at step start.
328 * @param h step size to use for scaling
329 * @param t first steps times
330 * @param y first steps states
331 * @param yDot first steps derivatives
332 * @return Nordieck vector at first step (h<sup>2</sup>/2 y''<sub>n</sub>,
333 * h<sup>3</sup>/6 y'''<sub>n</sub> ... h<sup>k</sup>/k! y<sup>(k)</sup><sub>n</sub>)
334 */
335 Array2DRowRealMatrix initializeHighOrderDerivatives(final double h, final double[] t,
336 final double[][] y,
337 final double[][] yDot);
338 }
339
340 /** Specialized step handler storing the first step. */
341 private class NordsieckInitializer implements StepHandler {
342
343 /** Steps counter. */
344 private int count;
345
346 /** First steps times. */
347 private final double[] t;
348
349 /** First steps states. */
350 private final double[][] y;
351
352 /** First steps derivatives. */
353 private final double[][] yDot;
354
355 /** Simple constructor.
356 * @param nSteps number of steps of the multistep method (excluding the one being computed)
357 * @param n problem dimension
358 */
359 public NordsieckInitializer(final int nSteps, final int n) {
360 this.count = 0;
361 this.t = new double[nSteps];
362 this.y = new double[nSteps][n];
363 this.yDot = new double[nSteps][n];
364 }
365
366 /** {@inheritDoc} */
367 public void handleStep(StepInterpolator interpolator, boolean isLast)
368 throws MaxCountExceededException {
369
370 final double prev = interpolator.getPreviousTime();
371 final double curr = interpolator.getCurrentTime();
372
373 if (count == 0) {
374 // first step, we need to store also the beginning of the step
375 interpolator.setInterpolatedTime(prev);
376 t[0] = prev;
377 final ExpandableStatefulODE expandable = getExpandable();
378 final EquationsMapper primary = expandable.getPrimaryMapper();
379 primary.insertEquationData(interpolator.getInterpolatedState(), y[count]);
380 primary.insertEquationData(interpolator.getInterpolatedDerivatives(), yDot[count]);
381 int index = 0;
382 for (final EquationsMapper secondary : expandable.getSecondaryMappers()) {
383 secondary.insertEquationData(interpolator.getInterpolatedSecondaryState(index), y[count]);
384 secondary.insertEquationData(interpolator.getInterpolatedSecondaryDerivatives(index), yDot[count]);
385 ++index;
386 }
387 }
388
389 // store the end of the step
390 ++count;
391 interpolator.setInterpolatedTime(curr);
392 t[count] = curr;
393
394 final ExpandableStatefulODE expandable = getExpandable();
395 final EquationsMapper primary = expandable.getPrimaryMapper();
396 primary.insertEquationData(interpolator.getInterpolatedState(), y[count]);
397 primary.insertEquationData(interpolator.getInterpolatedDerivatives(), yDot[count]);
398 int index = 0;
399 for (final EquationsMapper secondary : expandable.getSecondaryMappers()) {
400 secondary.insertEquationData(interpolator.getInterpolatedSecondaryState(index), y[count]);
401 secondary.insertEquationData(interpolator.getInterpolatedSecondaryDerivatives(index), yDot[count]);
402 ++index;
403 }
404
405 if (count == t.length - 1) {
406
407 // this was the last step we needed, we can compute the derivatives
408 stepStart = t[0];
409 stepSize = (t[t.length - 1] - t[0]) / (t.length - 1);
410
411 // first scaled derivative
412 scaled = yDot[0].clone();
413 for (int j = 0; j < scaled.length; ++j) {
414 scaled[j] *= stepSize;
415 }
416
417 // higher order derivatives
418 nordsieck = initializeHighOrderDerivatives(stepSize, t, y, yDot);
419
420 // stop the integrator now that all needed steps have been handled
421 throw new InitializationCompletedMarkerException();
422
423 }
424
425 }
426
427 /** {@inheritDoc} */
428 public void init(double t0, double[] y0, double time) {
429 // nothing to do
430 }
431
432 }
433
434 /** Marker exception used ONLY to stop the starter integrator after first step. */
435 private static class InitializationCompletedMarkerException
436 extends RuntimeException {
437
438 /** Serializable version identifier. */
439 private static final long serialVersionUID = -1914085471038046418L;
440
441 /** Simple constructor. */
442 public InitializationCompletedMarkerException() {
443 super((Throwable) null);
444 }
445
446 }
447
448 }