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.nonstiff;
019
020 import java.util.Arrays;
021
022 import org.apache.commons.math3.exception.DimensionMismatchException;
023 import org.apache.commons.math3.exception.MaxCountExceededException;
024 import org.apache.commons.math3.exception.NoBracketingException;
025 import org.apache.commons.math3.exception.NumberIsTooSmallException;
026 import org.apache.commons.math3.linear.Array2DRowRealMatrix;
027 import org.apache.commons.math3.linear.RealMatrixPreservingVisitor;
028 import org.apache.commons.math3.ode.ExpandableStatefulODE;
029 import org.apache.commons.math3.ode.sampling.NordsieckStepInterpolator;
030 import org.apache.commons.math3.util.FastMath;
031
032
033 /**
034 * This class implements implicit Adams-Moulton integrators for Ordinary
035 * Differential Equations.
036 *
037 * <p>Adams-Moulton methods (in fact due to Adams alone) are implicit
038 * multistep ODE solvers. This implementation is a variation of the classical
039 * one: it uses adaptive stepsize to implement error control, whereas
040 * classical implementations are fixed step size. The value of state vector
041 * at step n+1 is a simple combination of the value at step n and of the
042 * derivatives at steps n+1, n, n-1 ... Since y'<sub>n+1</sub> is needed to
043 * compute y<sub>n+1</sub>, another method must be used to compute a first
044 * estimate of y<sub>n+1</sub>, then compute y'<sub>n+1</sub>, then compute
045 * a final estimate of y<sub>n+1</sub> using the following formulas. Depending
046 * on the number k of previous steps one wants to use for computing the next
047 * value, different formulas are available for the final estimate:</p>
048 * <ul>
049 * <li>k = 1: y<sub>n+1</sub> = y<sub>n</sub> + h y'<sub>n+1</sub></li>
050 * <li>k = 2: y<sub>n+1</sub> = y<sub>n</sub> + h (y'<sub>n+1</sub>+y'<sub>n</sub>)/2</li>
051 * <li>k = 3: y<sub>n+1</sub> = y<sub>n</sub> + h (5y'<sub>n+1</sub>+8y'<sub>n</sub>-y'<sub>n-1</sub>)/12</li>
052 * <li>k = 4: y<sub>n+1</sub> = y<sub>n</sub> + h (9y'<sub>n+1</sub>+19y'<sub>n</sub>-5y'<sub>n-1</sub>+y'<sub>n-2</sub>)/24</li>
053 * <li>...</li>
054 * </ul>
055 *
056 * <p>A k-steps Adams-Moulton method is of order k+1.</p>
057 *
058 * <h3>Implementation details</h3>
059 *
060 * <p>We define scaled derivatives s<sub>i</sub>(n) at step n as:
061 * <pre>
062 * s<sub>1</sub>(n) = h y'<sub>n</sub> for first derivative
063 * s<sub>2</sub>(n) = h<sup>2</sup>/2 y''<sub>n</sub> for second derivative
064 * s<sub>3</sub>(n) = h<sup>3</sup>/6 y'''<sub>n</sub> for third derivative
065 * ...
066 * s<sub>k</sub>(n) = h<sup>k</sup>/k! y<sup>(k)</sup><sub>n</sub> for k<sup>th</sup> derivative
067 * </pre></p>
068 *
069 * <p>The definitions above use the classical representation with several previous first
070 * derivatives. Lets define
071 * <pre>
072 * 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>
073 * </pre>
074 * (we omit the k index in the notation for clarity). With these definitions,
075 * Adams-Moulton methods can be written:
076 * <ul>
077 * <li>k = 1: y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n+1)</li>
078 * <li>k = 2: y<sub>n+1</sub> = y<sub>n</sub> + 1/2 s<sub>1</sub>(n+1) + [ 1/2 ] q<sub>n+1</sub></li>
079 * <li>k = 3: y<sub>n+1</sub> = y<sub>n</sub> + 5/12 s<sub>1</sub>(n+1) + [ 8/12 -1/12 ] q<sub>n+1</sub></li>
080 * <li>k = 4: y<sub>n+1</sub> = y<sub>n</sub> + 9/24 s<sub>1</sub>(n+1) + [ 19/24 -5/24 1/24 ] q<sub>n+1</sub></li>
081 * <li>...</li>
082 * </ul></p>
083 *
084 * <p>Instead of using the classical representation with first derivatives only (y<sub>n</sub>,
085 * s<sub>1</sub>(n+1) and q<sub>n+1</sub>), our implementation uses the Nordsieck vector with
086 * higher degrees scaled derivatives all taken at the same step (y<sub>n</sub>, s<sub>1</sub>(n)
087 * and r<sub>n</sub>) where r<sub>n</sub> is defined as:
088 * <pre>
089 * r<sub>n</sub> = [ s<sub>2</sub>(n), s<sub>3</sub>(n) ... s<sub>k</sub>(n) ]<sup>T</sup>
090 * </pre>
091 * (here again we omit the k index in the notation for clarity)
092 * </p>
093 *
094 * <p>Taylor series formulas show that for any index offset i, s<sub>1</sub>(n-i) can be
095 * computed from s<sub>1</sub>(n), s<sub>2</sub>(n) ... s<sub>k</sub>(n), the formula being exact
096 * for degree k polynomials.
097 * <pre>
098 * s<sub>1</sub>(n-i) = s<sub>1</sub>(n) + ∑<sub>j</sub> j (-i)<sup>j-1</sup> s<sub>j</sub>(n)
099 * </pre>
100 * The previous formula can be used with several values for i to compute the transform between
101 * classical representation and Nordsieck vector. The transform between r<sub>n</sub>
102 * and q<sub>n</sub> resulting from the Taylor series formulas above is:
103 * <pre>
104 * q<sub>n</sub> = s<sub>1</sub>(n) u + P r<sub>n</sub>
105 * </pre>
106 * where u is the [ 1 1 ... 1 ]<sup>T</sup> vector and P is the (k-1)×(k-1) matrix built
107 * with the j (-i)<sup>j-1</sup> terms:
108 * <pre>
109 * [ -2 3 -4 5 ... ]
110 * [ -4 12 -32 80 ... ]
111 * P = [ -6 27 -108 405 ... ]
112 * [ -8 48 -256 1280 ... ]
113 * [ ... ]
114 * </pre></p>
115 *
116 * <p>Using the Nordsieck vector has several advantages:
117 * <ul>
118 * <li>it greatly simplifies step interpolation as the interpolator mainly applies
119 * Taylor series formulas,</li>
120 * <li>it simplifies step changes that occur when discrete events that truncate
121 * the step are triggered,</li>
122 * <li>it allows to extend the methods in order to support adaptive stepsize.</li>
123 * </ul></p>
124 *
125 * <p>The predicted Nordsieck vector at step n+1 is computed from the Nordsieck vector at step
126 * n as follows:
127 * <ul>
128 * <li>Y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n) + u<sup>T</sup> r<sub>n</sub></li>
129 * <li>S<sub>1</sub>(n+1) = h f(t<sub>n+1</sub>, Y<sub>n+1</sub>)</li>
130 * <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>
131 * </ul>
132 * where A is a rows shifting matrix (the lower left part is an identity matrix):
133 * <pre>
134 * [ 0 0 ... 0 0 | 0 ]
135 * [ ---------------+---]
136 * [ 1 0 ... 0 0 | 0 ]
137 * A = [ 0 1 ... 0 0 | 0 ]
138 * [ ... | 0 ]
139 * [ 0 0 ... 1 0 | 0 ]
140 * [ 0 0 ... 0 1 | 0 ]
141 * </pre>
142 * From this predicted vector, the corrected vector is computed as follows:
143 * <ul>
144 * <li>y<sub>n+1</sub> = y<sub>n</sub> + S<sub>1</sub>(n+1) + [ -1 +1 -1 +1 ... ±1 ] r<sub>n+1</sub></li>
145 * <li>s<sub>1</sub>(n+1) = h f(t<sub>n+1</sub>, y<sub>n+1</sub>)</li>
146 * <li>r<sub>n+1</sub> = R<sub>n+1</sub> + (s<sub>1</sub>(n+1) - S<sub>1</sub>(n+1)) P<sup>-1</sup> u</li>
147 * </ul>
148 * where the upper case Y<sub>n+1</sub>, S<sub>1</sub>(n+1) and R<sub>n+1</sub> represent the
149 * predicted states whereas the lower case y<sub>n+1</sub>, s<sub>n+1</sub> and r<sub>n+1</sub>
150 * represent the corrected states.</p>
151 *
152 * <p>The P<sup>-1</sup>u vector and the P<sup>-1</sup> A P matrix do not depend on the state,
153 * they only depend on k and therefore are precomputed once for all.</p>
154 *
155 * @version $Id: AdamsMoultonIntegrator.java 1416643 2012-12-03 19:37:14Z tn $
156 * @since 2.0
157 */
158 public class AdamsMoultonIntegrator extends AdamsIntegrator {
159
160 /** Integrator method name. */
161 private static final String METHOD_NAME = "Adams-Moulton";
162
163 /**
164 * Build an Adams-Moulton integrator with the given order and error control parameters.
165 * @param nSteps number of steps of the method excluding the one being computed
166 * @param minStep minimal step (sign is irrelevant, regardless of
167 * integration direction, forward or backward), the last step can
168 * be smaller than this
169 * @param maxStep maximal step (sign is irrelevant, regardless of
170 * integration direction, forward or backward), the last step can
171 * be smaller than this
172 * @param scalAbsoluteTolerance allowed absolute error
173 * @param scalRelativeTolerance allowed relative error
174 * @exception NumberIsTooSmallException if order is 1 or less
175 */
176 public AdamsMoultonIntegrator(final int nSteps,
177 final double minStep, final double maxStep,
178 final double scalAbsoluteTolerance,
179 final double scalRelativeTolerance)
180 throws NumberIsTooSmallException {
181 super(METHOD_NAME, nSteps, nSteps + 1, minStep, maxStep,
182 scalAbsoluteTolerance, scalRelativeTolerance);
183 }
184
185 /**
186 * Build an Adams-Moulton integrator with the given order and error control parameters.
187 * @param nSteps number of steps of the method excluding the one being computed
188 * @param minStep minimal step (sign is irrelevant, regardless of
189 * integration direction, forward or backward), the last step can
190 * be smaller than this
191 * @param maxStep maximal step (sign is irrelevant, regardless of
192 * integration direction, forward or backward), the last step can
193 * be smaller than this
194 * @param vecAbsoluteTolerance allowed absolute error
195 * @param vecRelativeTolerance allowed relative error
196 * @exception IllegalArgumentException if order is 1 or less
197 */
198 public AdamsMoultonIntegrator(final int nSteps,
199 final double minStep, final double maxStep,
200 final double[] vecAbsoluteTolerance,
201 final double[] vecRelativeTolerance)
202 throws IllegalArgumentException {
203 super(METHOD_NAME, nSteps, nSteps + 1, minStep, maxStep,
204 vecAbsoluteTolerance, vecRelativeTolerance);
205 }
206
207
208 /** {@inheritDoc} */
209 @Override
210 public void integrate(final ExpandableStatefulODE equations,final double t)
211 throws NumberIsTooSmallException, DimensionMismatchException,
212 MaxCountExceededException, NoBracketingException {
213
214 sanityChecks(equations, t);
215 setEquations(equations);
216 final boolean forward = t > equations.getTime();
217
218 // initialize working arrays
219 final double[] y0 = equations.getCompleteState();
220 final double[] y = y0.clone();
221 final double[] yDot = new double[y.length];
222 final double[] yTmp = new double[y.length];
223 final double[] predictedScaled = new double[y.length];
224 Array2DRowRealMatrix nordsieckTmp = null;
225
226 // set up two interpolators sharing the integrator arrays
227 final NordsieckStepInterpolator interpolator = new NordsieckStepInterpolator();
228 interpolator.reinitialize(y, forward,
229 equations.getPrimaryMapper(), equations.getSecondaryMappers());
230
231 // set up integration control objects
232 initIntegration(equations.getTime(), y0, t);
233
234 // compute the initial Nordsieck vector using the configured starter integrator
235 start(equations.getTime(), y, t);
236 interpolator.reinitialize(stepStart, stepSize, scaled, nordsieck);
237 interpolator.storeTime(stepStart);
238
239 double hNew = stepSize;
240 interpolator.rescale(hNew);
241
242 isLastStep = false;
243 do {
244
245 double error = 10;
246 while (error >= 1.0) {
247
248 stepSize = hNew;
249
250 // predict a first estimate of the state at step end (P in the PECE sequence)
251 final double stepEnd = stepStart + stepSize;
252 interpolator.setInterpolatedTime(stepEnd);
253 System.arraycopy(interpolator.getInterpolatedState(), 0, yTmp, 0, y0.length);
254
255 // evaluate a first estimate of the derivative (first E in the PECE sequence)
256 computeDerivatives(stepEnd, yTmp, yDot);
257
258 // update Nordsieck vector
259 for (int j = 0; j < y0.length; ++j) {
260 predictedScaled[j] = stepSize * yDot[j];
261 }
262 nordsieckTmp = updateHighOrderDerivativesPhase1(nordsieck);
263 updateHighOrderDerivativesPhase2(scaled, predictedScaled, nordsieckTmp);
264
265 // apply correction (C in the PECE sequence)
266 error = nordsieckTmp.walkInOptimizedOrder(new Corrector(y, predictedScaled, yTmp));
267
268 if (error >= 1.0) {
269 // reject the step and attempt to reduce error by stepsize control
270 final double factor = computeStepGrowShrinkFactor(error);
271 hNew = filterStep(stepSize * factor, forward, false);
272 interpolator.rescale(hNew);
273 }
274 }
275
276 // evaluate a final estimate of the derivative (second E in the PECE sequence)
277 final double stepEnd = stepStart + stepSize;
278 computeDerivatives(stepEnd, yTmp, yDot);
279
280 // update Nordsieck vector
281 final double[] correctedScaled = new double[y0.length];
282 for (int j = 0; j < y0.length; ++j) {
283 correctedScaled[j] = stepSize * yDot[j];
284 }
285 updateHighOrderDerivativesPhase2(predictedScaled, correctedScaled, nordsieckTmp);
286
287 // discrete events handling
288 System.arraycopy(yTmp, 0, y, 0, y.length);
289 interpolator.reinitialize(stepEnd, stepSize, correctedScaled, nordsieckTmp);
290 interpolator.storeTime(stepStart);
291 interpolator.shift();
292 interpolator.storeTime(stepEnd);
293 stepStart = acceptStep(interpolator, y, yDot, t);
294 scaled = correctedScaled;
295 nordsieck = nordsieckTmp;
296
297 if (!isLastStep) {
298
299 // prepare next step
300 interpolator.storeTime(stepStart);
301
302 if (resetOccurred) {
303 // some events handler has triggered changes that
304 // invalidate the derivatives, we need to restart from scratch
305 start(stepStart, y, t);
306 interpolator.reinitialize(stepStart, stepSize, scaled, nordsieck);
307
308 }
309
310 // stepsize control for next step
311 final double factor = computeStepGrowShrinkFactor(error);
312 final double scaledH = stepSize * factor;
313 final double nextT = stepStart + scaledH;
314 final boolean nextIsLast = forward ? (nextT >= t) : (nextT <= t);
315 hNew = filterStep(scaledH, forward, nextIsLast);
316
317 final double filteredNextT = stepStart + hNew;
318 final boolean filteredNextIsLast = forward ? (filteredNextT >= t) : (filteredNextT <= t);
319 if (filteredNextIsLast) {
320 hNew = t - stepStart;
321 }
322
323 interpolator.rescale(hNew);
324 }
325
326 } while (!isLastStep);
327
328 // dispatch results
329 equations.setTime(stepStart);
330 equations.setCompleteState(y);
331
332 resetInternalState();
333
334 }
335
336 /** Corrector for current state in Adams-Moulton method.
337 * <p>
338 * This visitor implements the Taylor series formula:
339 * <pre>
340 * Y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n+1) + [ -1 +1 -1 +1 ... ±1 ] r<sub>n+1</sub>
341 * </pre>
342 * </p>
343 */
344 private class Corrector implements RealMatrixPreservingVisitor {
345
346 /** Previous state. */
347 private final double[] previous;
348
349 /** Current scaled first derivative. */
350 private final double[] scaled;
351
352 /** Current state before correction. */
353 private final double[] before;
354
355 /** Current state after correction. */
356 private final double[] after;
357
358 /** Simple constructor.
359 * @param previous previous state
360 * @param scaled current scaled first derivative
361 * @param state state to correct (will be overwritten after visit)
362 */
363 public Corrector(final double[] previous, final double[] scaled, final double[] state) {
364 this.previous = previous;
365 this.scaled = scaled;
366 this.after = state;
367 this.before = state.clone();
368 }
369
370 /** {@inheritDoc} */
371 public void start(int rows, int columns,
372 int startRow, int endRow, int startColumn, int endColumn) {
373 Arrays.fill(after, 0.0);
374 }
375
376 /** {@inheritDoc} */
377 public void visit(int row, int column, double value) {
378 if ((row & 0x1) == 0) {
379 after[column] -= value;
380 } else {
381 after[column] += value;
382 }
383 }
384
385 /**
386 * End visiting the Nordsieck vector.
387 * <p>The correction is used to control stepsize. So its amplitude is
388 * considered to be an error, which must be normalized according to
389 * error control settings. If the normalized value is greater than 1,
390 * the correction was too large and the step must be rejected.</p>
391 * @return the normalized correction, if greater than 1, the step
392 * must be rejected
393 */
394 public double end() {
395
396 double error = 0;
397 for (int i = 0; i < after.length; ++i) {
398 after[i] += previous[i] + scaled[i];
399 if (i < mainSetDimension) {
400 final double yScale = FastMath.max(FastMath.abs(previous[i]), FastMath.abs(after[i]));
401 final double tol = (vecAbsoluteTolerance == null) ?
402 (scalAbsoluteTolerance + scalRelativeTolerance * yScale) :
403 (vecAbsoluteTolerance[i] + vecRelativeTolerance[i] * yScale);
404 final double ratio = (after[i] - before[i]) / tol;
405 error += ratio * ratio;
406 }
407 }
408
409 return FastMath.sqrt(error / mainSetDimension);
410
411 }
412 }
413
414 }