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.linear.Array2DRowRealMatrix;
021 import org.apache.commons.math.ode.DerivativeException;
022 import org.apache.commons.math.ode.FirstOrderDifferentialEquations;
023 import org.apache.commons.math.ode.IntegratorException;
024 import org.apache.commons.math.ode.events.CombinedEventsManager;
025 import org.apache.commons.math.ode.sampling.NordsieckStepInterpolator;
026 import org.apache.commons.math.ode.sampling.StepHandler;
027
028
029 /**
030 * This class implements explicit Adams-Bashforth integrators for Ordinary
031 * Differential Equations.
032 *
033 * <p>Adams-Bashforth methods (in fact due to Adams alone) are explicit
034 * multistep ODE solvers. This implementation is a variation of the classical
035 * one: it uses adaptive stepsize to implement error control, whereas
036 * classical implementations are fixed step size. The value of state vector
037 * at step n+1 is a simple combination of the value at step n and of the
038 * derivatives at steps n, n-1, n-2 ... Depending on the number k of previous
039 * steps one wants to use for computing the next value, different formulas
040 * are available:</p>
041 * <ul>
042 * <li>k = 1: y<sub>n+1</sub> = y<sub>n</sub> + h y'<sub>n</sub></li>
043 * <li>k = 2: y<sub>n+1</sub> = y<sub>n</sub> + h (3y'<sub>n</sub>-y'<sub>n-1</sub>)/2</li>
044 * <li>k = 3: y<sub>n+1</sub> = y<sub>n</sub> + h (23y'<sub>n</sub>-16y'<sub>n-1</sub>+5y'<sub>n-2</sub>)/12</li>
045 * <li>k = 4: y<sub>n+1</sub> = y<sub>n</sub> + h (55y'<sub>n</sub>-59y'<sub>n-1</sub>+37y'<sub>n-2</sub>-9y'<sub>n-3</sub>)/24</li>
046 * <li>...</li>
047 * </ul>
048 *
049 * <p>A k-steps Adams-Bashforth method is of order k.</p>
050 *
051 * <h3>Implementation details</h3>
052 *
053 * <p>We define scaled derivatives s<sub>i</sub>(n) at step n as:
054 * <pre>
055 * s<sub>1</sub>(n) = h y'<sub>n</sub> for first derivative
056 * s<sub>2</sub>(n) = h<sup>2</sup>/2 y''<sub>n</sub> for second derivative
057 * s<sub>3</sub>(n) = h<sup>3</sup>/6 y'''<sub>n</sub> for third derivative
058 * ...
059 * s<sub>k</sub>(n) = h<sup>k</sup>/k! y(k)<sub>n</sub> for k<sup>th</sup> derivative
060 * </pre></p>
061 *
062 * <p>The definitions above use the classical representation with several previous first
063 * derivatives. Lets define
064 * <pre>
065 * 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>
066 * </pre>
067 * (we omit the k index in the notation for clarity). With these definitions,
068 * Adams-Bashforth methods can be written:
069 * <ul>
070 * <li>k = 1: y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n)</li>
071 * <li>k = 2: y<sub>n+1</sub> = y<sub>n</sub> + 3/2 s<sub>1</sub>(n) + [ -1/2 ] q<sub>n</sub></li>
072 * <li>k = 3: y<sub>n+1</sub> = y<sub>n</sub> + 23/12 s<sub>1</sub>(n) + [ -16/12 5/12 ] q<sub>n</sub></li>
073 * <li>k = 4: y<sub>n+1</sub> = y<sub>n</sub> + 55/24 s<sub>1</sub>(n) + [ -59/24 37/24 -9/24 ] q<sub>n</sub></li>
074 * <li>...</li>
075 * </ul></p>
076 *
077 * <p>Instead of using the classical representation with first derivatives only (y<sub>n</sub>,
078 * s<sub>1</sub>(n) and q<sub>n</sub>), our implementation uses the Nordsieck vector with
079 * higher degrees scaled derivatives all taken at the same step (y<sub>n</sub>, s<sub>1</sub>(n)
080 * and r<sub>n</sub>) where r<sub>n</sub> is defined as:
081 * <pre>
082 * r<sub>n</sub> = [ s<sub>2</sub>(n), s<sub>3</sub>(n) ... s<sub>k</sub>(n) ]<sup>T</sup>
083 * </pre>
084 * (here again we omit the k index in the notation for clarity)
085 * </p>
086 *
087 * <p>Taylor series formulas show that for any index offset i, s<sub>1</sub>(n-i) can be
088 * computed from s<sub>1</sub>(n), s<sub>2</sub>(n) ... s<sub>k</sub>(n), the formula being exact
089 * for degree k polynomials.
090 * <pre>
091 * 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)
092 * </pre>
093 * The previous formula can be used with several values for i to compute the transform between
094 * classical representation and Nordsieck vector. The transform between r<sub>n</sub>
095 * and q<sub>n</sub> resulting from the Taylor series formulas above is:
096 * <pre>
097 * q<sub>n</sub> = s<sub>1</sub>(n) u + P r<sub>n</sub>
098 * </pre>
099 * where u is the [ 1 1 ... 1 ]<sup>T</sup> vector and P is the (k-1)×(k-1) matrix built
100 * with the j (-i)<sup>j-1</sup> terms:
101 * <pre>
102 * [ -2 3 -4 5 ... ]
103 * [ -4 12 -32 80 ... ]
104 * P = [ -6 27 -108 405 ... ]
105 * [ -8 48 -256 1280 ... ]
106 * [ ... ]
107 * </pre></p>
108 *
109 * <p>Using the Nordsieck vector has several advantages:
110 * <ul>
111 * <li>it greatly simplifies step interpolation as the interpolator mainly applies
112 * Taylor series formulas,</li>
113 * <li>it simplifies step changes that occur when discrete events that truncate
114 * the step are triggered,</li>
115 * <li>it allows to extend the methods in order to support adaptive stepsize.</li>
116 * </ul></p>
117 *
118 * <p>The Nordsieck vector at step n+1 is computed from the Nordsieck vector at step n as follows:
119 * <ul>
120 * <li>y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n) + u<sup>T</sup> r<sub>n</sub></li>
121 * <li>s<sub>1</sub>(n+1) = h f(t<sub>n+1</sub>, y<sub>n+1</sub>)</li>
122 * <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>
123 * </ul>
124 * where A is a rows shifting matrix (the lower left part is an identity matrix):
125 * <pre>
126 * [ 0 0 ... 0 0 | 0 ]
127 * [ ---------------+---]
128 * [ 1 0 ... 0 0 | 0 ]
129 * A = [ 0 1 ... 0 0 | 0 ]
130 * [ ... | 0 ]
131 * [ 0 0 ... 1 0 | 0 ]
132 * [ 0 0 ... 0 1 | 0 ]
133 * </pre></p>
134 *
135 * <p>The P<sup>-1</sup>u vector and the P<sup>-1</sup> A P matrix do not depend on the state,
136 * they only depend on k and therefore are precomputed once for all.</p>
137 *
138 * @version $Revision: 927202 $ $Date: 2010-03-24 18:11:51 -0400 (Wed, 24 Mar 2010) $
139 * @since 2.0
140 */
141 public class AdamsBashforthIntegrator extends AdamsIntegrator {
142
143 /**
144 * Build an Adams-Bashforth integrator with the given order and step control parameters.
145 * @param nSteps number of steps of the method excluding the one being computed
146 * @param minStep minimal step (must be positive even for backward
147 * integration), the last step can be smaller than this
148 * @param maxStep maximal step (must be positive even for backward
149 * integration)
150 * @param scalAbsoluteTolerance allowed absolute error
151 * @param scalRelativeTolerance allowed relative error
152 * @exception IllegalArgumentException if order is 1 or less
153 */
154 public AdamsBashforthIntegrator(final int nSteps,
155 final double minStep, final double maxStep,
156 final double scalAbsoluteTolerance,
157 final double scalRelativeTolerance)
158 throws IllegalArgumentException {
159 super("Adams-Bashforth", nSteps, nSteps, minStep, maxStep,
160 scalAbsoluteTolerance, scalRelativeTolerance);
161 }
162
163 /**
164 * Build an Adams-Bashforth integrator with the given order and step control parameters.
165 * @param nSteps number of steps of the method excluding the one being computed
166 * @param minStep minimal step (must be positive even for backward
167 * integration), the last step can be smaller than this
168 * @param maxStep maximal step (must be positive even for backward
169 * integration)
170 * @param vecAbsoluteTolerance allowed absolute error
171 * @param vecRelativeTolerance allowed relative error
172 * @exception IllegalArgumentException if order is 1 or less
173 */
174 public AdamsBashforthIntegrator(final int nSteps,
175 final double minStep, final double maxStep,
176 final double[] vecAbsoluteTolerance,
177 final double[] vecRelativeTolerance)
178 throws IllegalArgumentException {
179 super("Adams-Bashforth", nSteps, nSteps, minStep, maxStep,
180 vecAbsoluteTolerance, vecRelativeTolerance);
181 }
182
183 /** {@inheritDoc} */
184 @Override
185 public double integrate(final FirstOrderDifferentialEquations equations,
186 final double t0, final double[] y0,
187 final double t, final double[] y)
188 throws DerivativeException, IntegratorException {
189
190 final int n = y0.length;
191 sanityChecks(equations, t0, y0, t, y);
192 setEquations(equations);
193 resetEvaluations();
194 final boolean forward = t > t0;
195
196 // initialize working arrays
197 if (y != y0) {
198 System.arraycopy(y0, 0, y, 0, n);
199 }
200 final double[] yDot = new double[n];
201 final double[] yTmp = new double[y0.length];
202
203 // set up an interpolator sharing the integrator arrays
204 final NordsieckStepInterpolator interpolator = new NordsieckStepInterpolator();
205 interpolator.reinitialize(y, forward);
206 final NordsieckStepInterpolator interpolatorTmp = new NordsieckStepInterpolator();
207 interpolatorTmp.reinitialize(yTmp, forward);
208
209 // set up integration control objects
210 for (StepHandler handler : stepHandlers) {
211 handler.reset();
212 }
213 CombinedEventsManager manager = addEndTimeChecker(t0, t, eventsHandlersManager);
214
215 // compute the initial Nordsieck vector using the configured starter integrator
216 start(t0, y, t);
217 interpolator.reinitialize(stepStart, stepSize, scaled, nordsieck);
218 interpolator.storeTime(stepStart);
219 final int lastRow = nordsieck.getRowDimension() - 1;
220
221 // reuse the step that was chosen by the starter integrator
222 double hNew = stepSize;
223 interpolator.rescale(hNew);
224
225 boolean lastStep = false;
226 while (!lastStep) {
227
228 // shift all data
229 interpolator.shift();
230
231 double error = 0;
232 for (boolean loop = true; loop;) {
233
234 stepSize = hNew;
235
236 // evaluate error using the last term of the Taylor expansion
237 error = 0;
238 for (int i = 0; i < y0.length; ++i) {
239 final double yScale = Math.abs(y[i]);
240 final double tol = (vecAbsoluteTolerance == null) ?
241 (scalAbsoluteTolerance + scalRelativeTolerance * yScale) :
242 (vecAbsoluteTolerance[i] + vecRelativeTolerance[i] * yScale);
243 final double ratio = nordsieck.getEntry(lastRow, i) / tol;
244 error += ratio * ratio;
245 }
246 error = Math.sqrt(error / y0.length);
247
248 if (error <= 1.0) {
249
250 // predict a first estimate of the state at step end
251 final double stepEnd = stepStart + stepSize;
252 interpolator.setInterpolatedTime(stepEnd);
253 System.arraycopy(interpolator.getInterpolatedState(), 0, yTmp, 0, y0.length);
254
255 // evaluate the derivative
256 computeDerivatives(stepEnd, yTmp, yDot);
257
258 // update Nordsieck vector
259 final double[] predictedScaled = new double[y0.length];
260 for (int j = 0; j < y0.length; ++j) {
261 predictedScaled[j] = stepSize * yDot[j];
262 }
263 final Array2DRowRealMatrix nordsieckTmp = updateHighOrderDerivativesPhase1(nordsieck);
264 updateHighOrderDerivativesPhase2(scaled, predictedScaled, nordsieckTmp);
265
266 // discrete events handling
267 interpolatorTmp.reinitialize(stepEnd, stepSize, predictedScaled, nordsieckTmp);
268 interpolatorTmp.storeTime(stepStart);
269 interpolatorTmp.shift();
270 interpolatorTmp.storeTime(stepEnd);
271 if (manager.evaluateStep(interpolatorTmp)) {
272 final double dt = manager.getEventTime() - stepStart;
273 if (Math.abs(dt) <= Math.ulp(stepStart)) {
274 // we cannot simply truncate the step, reject the current computation
275 // and let the loop compute another state with the truncated step.
276 // it is so small (much probably exactly 0 due to limited accuracy)
277 // that the code above would fail handling it.
278 // So we set up an artificial 0 size step by copying states
279 interpolator.storeTime(stepStart);
280 System.arraycopy(y, 0, yTmp, 0, y0.length);
281 hNew = 0;
282 stepSize = 0;
283 loop = false;
284 } else {
285 // reject the step to match exactly the next switch time
286 hNew = dt;
287 interpolator.rescale(hNew);
288 }
289 } else {
290 // accept the step
291 scaled = predictedScaled;
292 nordsieck = nordsieckTmp;
293 interpolator.reinitialize(stepEnd, stepSize, scaled, nordsieck);
294 loop = false;
295 }
296
297 } else {
298 // reject the step and attempt to reduce error by stepsize control
299 final double factor = computeStepGrowShrinkFactor(error);
300 hNew = filterStep(stepSize * factor, forward, false);
301 interpolator.rescale(hNew);
302 }
303
304 }
305
306 // the step has been accepted (may have been truncated)
307 final double nextStep = stepStart + stepSize;
308 System.arraycopy(yTmp, 0, y, 0, n);
309 interpolator.storeTime(nextStep);
310 manager.stepAccepted(nextStep, y);
311 lastStep = manager.stop();
312
313 // provide the step data to the step handler
314 for (StepHandler handler : stepHandlers) {
315 interpolator.setInterpolatedTime(nextStep);
316 handler.handleStep(interpolator, lastStep);
317 }
318 stepStart = nextStep;
319
320 if (!lastStep && manager.reset(stepStart, y)) {
321
322 // some events handler has triggered changes that
323 // invalidate the derivatives, we need to restart from scratch
324 start(stepStart, y, t);
325 interpolator.reinitialize(stepStart, stepSize, scaled, nordsieck);
326
327 }
328
329 if (! lastStep) {
330 // in some rare cases we may get here with stepSize = 0, for example
331 // when an event occurs at integration start, reducing the first step
332 // to zero; we have to reset the step to some safe non zero value
333 stepSize = filterStep(stepSize, forward, true);
334
335 // stepsize control for next step
336 final double factor = computeStepGrowShrinkFactor(error);
337 final double scaledH = stepSize * factor;
338 final double nextT = stepStart + scaledH;
339 final boolean nextIsLast = forward ? (nextT >= t) : (nextT <= t);
340 hNew = filterStep(scaledH, forward, nextIsLast);
341 interpolator.rescale(hNew);
342 }
343
344 }
345
346 final double stopTime = stepStart;
347 stepStart = Double.NaN;
348 stepSize = Double.NaN;
349 return stopTime;
350
351 }
352
353 }