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 1421448 2012-12-13 19:45:57Z tn $
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 starter.integrate(new CountingDifferentialEquations(y0.length),
232 t0, y0, t, new double[y0.length]);
233 } catch (InitializationCompletedMarkerException icme) { // NOPMD
234 // this is the expected nominal interruption of the start integrator
235 }
236
237 // remove the specific step handler
238 starter.clearStepHandlers();
239
240 }
241
242 /** Initialize the high order scaled derivatives at step start.
243 * @param h step size to use for scaling
244 * @param t first steps times
245 * @param y first steps states
246 * @param yDot first steps derivatives
247 * @return Nordieck vector at first step (h<sup>2</sup>/2 y''<sub>n</sub>,
248 * h<sup>3</sup>/6 y'''<sub>n</sub> ... h<sup>k</sup>/k! y<sup>(k)</sup><sub>n</sub>)
249 */
250 protected abstract Array2DRowRealMatrix initializeHighOrderDerivatives(final double h, final double[] t,
251 final double[][] y,
252 final double[][] yDot);
253
254 /** Get the minimal reduction factor for stepsize control.
255 * @return minimal reduction factor
256 */
257 public double getMinReduction() {
258 return minReduction;
259 }
260
261 /** Set the minimal reduction factor for stepsize control.
262 * @param minReduction minimal reduction factor
263 */
264 public void setMinReduction(final double minReduction) {
265 this.minReduction = minReduction;
266 }
267
268 /** Get the maximal growth factor for stepsize control.
269 * @return maximal growth factor
270 */
271 public double getMaxGrowth() {
272 return maxGrowth;
273 }
274
275 /** Set the maximal growth factor for stepsize control.
276 * @param maxGrowth maximal growth factor
277 */
278 public void setMaxGrowth(final double maxGrowth) {
279 this.maxGrowth = maxGrowth;
280 }
281
282 /** Get the safety factor for stepsize control.
283 * @return safety factor
284 */
285 public double getSafety() {
286 return safety;
287 }
288
289 /** Set the safety factor for stepsize control.
290 * @param safety safety factor
291 */
292 public void setSafety(final double safety) {
293 this.safety = safety;
294 }
295
296 /** Compute step grow/shrink factor according to normalized error.
297 * @param error normalized error of the current step
298 * @return grow/shrink factor for next step
299 */
300 protected double computeStepGrowShrinkFactor(final double error) {
301 return FastMath.min(maxGrowth, FastMath.max(minReduction, safety * FastMath.pow(error, exp)));
302 }
303
304 /** Transformer used to convert the first step to Nordsieck representation. */
305 public interface NordsieckTransformer {
306 /** Initialize the high order scaled derivatives at step start.
307 * @param h step size to use for scaling
308 * @param t first steps times
309 * @param y first steps states
310 * @param yDot first steps derivatives
311 * @return Nordieck vector at first step (h<sup>2</sup>/2 y''<sub>n</sub>,
312 * h<sup>3</sup>/6 y'''<sub>n</sub> ... h<sup>k</sup>/k! y<sup>(k)</sup><sub>n</sub>)
313 */
314 Array2DRowRealMatrix initializeHighOrderDerivatives(final double h, final double[] t,
315 final double[][] y,
316 final double[][] yDot);
317 }
318
319 /** Specialized step handler storing the first step. */
320 private class NordsieckInitializer implements StepHandler {
321
322 /** Steps counter. */
323 private int count;
324
325 /** First steps times. */
326 private final double[] t;
327
328 /** First steps states. */
329 private final double[][] y;
330
331 /** First steps derivatives. */
332 private final double[][] yDot;
333
334 /** Simple constructor.
335 * @param nSteps number of steps of the multistep method (excluding the one being computed)
336 * @param n problem dimension
337 */
338 public NordsieckInitializer(final int nSteps, final int n) {
339 this.count = 0;
340 this.t = new double[nSteps];
341 this.y = new double[nSteps][n];
342 this.yDot = new double[nSteps][n];
343 }
344
345 /** {@inheritDoc} */
346 public void handleStep(StepInterpolator interpolator, boolean isLast)
347 throws MaxCountExceededException {
348
349 final double prev = interpolator.getPreviousTime();
350 final double curr = interpolator.getCurrentTime();
351
352 if (count == 0) {
353 // first step, we need to store also the beginning of the step
354 interpolator.setInterpolatedTime(prev);
355 t[0] = prev;
356 System.arraycopy(interpolator.getInterpolatedState(), 0,
357 y[0], 0, y[0].length);
358 System.arraycopy(interpolator.getInterpolatedDerivatives(), 0,
359 yDot[0], 0, yDot[0].length);
360 }
361
362 // store the end of the step
363 ++count;
364 interpolator.setInterpolatedTime(curr);
365 t[count] = curr;
366 System.arraycopy(interpolator.getInterpolatedState(), 0,
367 y[count], 0, y[count].length);
368 System.arraycopy(interpolator.getInterpolatedDerivatives(), 0,
369 yDot[count], 0, yDot[count].length);
370
371 if (count == t.length - 1) {
372
373 // this was the last step we needed, we can compute the derivatives
374 stepStart = t[0];
375 stepSize = (t[t.length - 1] - t[0]) / (t.length - 1);
376
377 // first scaled derivative
378 scaled = yDot[0].clone();
379 for (int j = 0; j < scaled.length; ++j) {
380 scaled[j] *= stepSize;
381 }
382
383 // higher order derivatives
384 nordsieck = initializeHighOrderDerivatives(stepSize, t, y, yDot);
385
386 // stop the integrator now that all needed steps have been handled
387 throw new InitializationCompletedMarkerException();
388
389 }
390
391 }
392
393 /** {@inheritDoc} */
394 public void init(double t0, double[] y0, double time) {
395 // nothing to do
396 }
397
398 }
399
400 /** Marker exception used ONLY to stop the starter integrator after first step. */
401 private static class InitializationCompletedMarkerException
402 extends RuntimeException {
403
404 /** Serializable version identifier. */
405 private static final long serialVersionUID = -1914085471038046418L;
406
407 /** Simple constructor. */
408 public InitializationCompletedMarkerException() {
409 super((Throwable) null);
410 }
411
412 }
413
414 /** Wrapper for differential equations, ensuring start evaluations are counted. */
415 private class CountingDifferentialEquations implements FirstOrderDifferentialEquations {
416
417 /** Dimension of the problem. */
418 private final int dimension;
419
420 /** Simple constructor.
421 * @param dimension dimension of the problem
422 */
423 public CountingDifferentialEquations(final int dimension) {
424 this.dimension = dimension;
425 }
426
427 /** {@inheritDoc} */
428 public void computeDerivatives(double t, double[] y, double[] dot)
429 throws MaxCountExceededException, DimensionMismatchException {
430 MultistepIntegrator.this.computeDerivatives(t, y, dot);
431 }
432
433 /** {@inheritDoc} */
434 public int getDimension() {
435 return dimension;
436 }
437
438 }
439
440 }