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 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.linear.Array2DRowRealMatrix;
025 import org.apache.commons.math3.ode.EquationsMapper;
026 import org.apache.commons.math3.ode.ExpandableStatefulODE;
027 import org.apache.commons.math3.ode.sampling.NordsieckStepInterpolator;
028 import org.apache.commons.math3.util.FastMath;
029
030
031 /**
032 * This class implements explicit Adams-Bashforth integrators for Ordinary
033 * Differential Equations.
034 *
035 * <p>Adams-Bashforth methods (in fact due to Adams alone) are explicit
036 * multistep ODE solvers. This implementation is a variation of the classical
037 * one: it uses adaptive stepsize to implement error control, whereas
038 * classical implementations are fixed step size. The value of state vector
039 * at step n+1 is a simple combination of the value at step n and of the
040 * derivatives at steps n, n-1, n-2 ... Depending on the number k of previous
041 * steps one wants to use for computing the next value, different formulas
042 * are available:</p>
043 * <ul>
044 * <li>k = 1: y<sub>n+1</sub> = y<sub>n</sub> + h y'<sub>n</sub></li>
045 * <li>k = 2: y<sub>n+1</sub> = y<sub>n</sub> + h (3y'<sub>n</sub>-y'<sub>n-1</sub>)/2</li>
046 * <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>
047 * <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>
048 * <li>...</li>
049 * </ul>
050 *
051 * <p>A k-steps Adams-Bashforth method is of order k.</p>
052 *
053 * <h3>Implementation details</h3>
054 *
055 * <p>We define scaled derivatives s<sub>i</sub>(n) at step n as:
056 * <pre>
057 * s<sub>1</sub>(n) = h y'<sub>n</sub> for first derivative
058 * s<sub>2</sub>(n) = h<sup>2</sup>/2 y''<sub>n</sub> for second derivative
059 * s<sub>3</sub>(n) = h<sup>3</sup>/6 y'''<sub>n</sub> for third derivative
060 * ...
061 * s<sub>k</sub>(n) = h<sup>k</sup>/k! y<sup>(k)</sup><sub>n</sub> for k<sup>th</sup> derivative
062 * </pre></p>
063 *
064 * <p>The definitions above use the classical representation with several previous first
065 * derivatives. Lets define
066 * <pre>
067 * 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>
068 * </pre>
069 * (we omit the k index in the notation for clarity). With these definitions,
070 * Adams-Bashforth methods can be written:
071 * <ul>
072 * <li>k = 1: y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n)</li>
073 * <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>
074 * <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>
075 * <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>
076 * <li>...</li>
077 * </ul></p>
078 *
079 * <p>Instead of using the classical representation with first derivatives only (y<sub>n</sub>,
080 * s<sub>1</sub>(n) and q<sub>n</sub>), our implementation uses the Nordsieck vector with
081 * higher degrees scaled derivatives all taken at the same step (y<sub>n</sub>, s<sub>1</sub>(n)
082 * and r<sub>n</sub>) where r<sub>n</sub> is defined as:
083 * <pre>
084 * r<sub>n</sub> = [ s<sub>2</sub>(n), s<sub>3</sub>(n) ... s<sub>k</sub>(n) ]<sup>T</sup>
085 * </pre>
086 * (here again we omit the k index in the notation for clarity)
087 * </p>
088 *
089 * <p>Taylor series formulas show that for any index offset i, s<sub>1</sub>(n-i) can be
090 * computed from s<sub>1</sub>(n), s<sub>2</sub>(n) ... s<sub>k</sub>(n), the formula being exact
091 * for degree k polynomials.
092 * <pre>
093 * 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)
094 * </pre>
095 * The previous formula can be used with several values for i to compute the transform between
096 * classical representation and Nordsieck vector. The transform between r<sub>n</sub>
097 * and q<sub>n</sub> resulting from the Taylor series formulas above is:
098 * <pre>
099 * q<sub>n</sub> = s<sub>1</sub>(n) u + P r<sub>n</sub>
100 * </pre>
101 * where u is the [ 1 1 ... 1 ]<sup>T</sup> vector and P is the (k-1)×(k-1) matrix built
102 * with the j (-i)<sup>j-1</sup> terms:
103 * <pre>
104 * [ -2 3 -4 5 ... ]
105 * [ -4 12 -32 80 ... ]
106 * P = [ -6 27 -108 405 ... ]
107 * [ -8 48 -256 1280 ... ]
108 * [ ... ]
109 * </pre></p>
110 *
111 * <p>Using the Nordsieck vector has several advantages:
112 * <ul>
113 * <li>it greatly simplifies step interpolation as the interpolator mainly applies
114 * Taylor series formulas,</li>
115 * <li>it simplifies step changes that occur when discrete events that truncate
116 * the step are triggered,</li>
117 * <li>it allows to extend the methods in order to support adaptive stepsize.</li>
118 * </ul></p>
119 *
120 * <p>The Nordsieck vector at step n+1 is computed from the Nordsieck vector at step n as follows:
121 * <ul>
122 * <li>y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n) + u<sup>T</sup> r<sub>n</sub></li>
123 * <li>s<sub>1</sub>(n+1) = h f(t<sub>n+1</sub>, y<sub>n+1</sub>)</li>
124 * <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>
125 * </ul>
126 * where A is a rows shifting matrix (the lower left part is an identity matrix):
127 * <pre>
128 * [ 0 0 ... 0 0 | 0 ]
129 * [ ---------------+---]
130 * [ 1 0 ... 0 0 | 0 ]
131 * A = [ 0 1 ... 0 0 | 0 ]
132 * [ ... | 0 ]
133 * [ 0 0 ... 1 0 | 0 ]
134 * [ 0 0 ... 0 1 | 0 ]
135 * </pre></p>
136 *
137 * <p>The P<sup>-1</sup>u vector and the P<sup>-1</sup> A P matrix do not depend on the state,
138 * they only depend on k and therefore are precomputed once for all.</p>
139 *
140 * @version $Id: AdamsBashforthIntegrator.java 1463684 2013-04-02 19:04:13Z luc $
141 * @since 2.0
142 */
143 public class AdamsBashforthIntegrator extends AdamsIntegrator {
144
145 /** Integrator method name. */
146 private static final String METHOD_NAME = "Adams-Bashforth";
147
148 /**
149 * Build an Adams-Bashforth integrator with the given order and step control parameters.
150 * @param nSteps number of steps of the method excluding the one being computed
151 * @param minStep minimal step (sign is irrelevant, regardless of
152 * integration direction, forward or backward), the last step can
153 * be smaller than this
154 * @param maxStep maximal step (sign is irrelevant, regardless of
155 * integration direction, forward or backward), the last step can
156 * be smaller than this
157 * @param scalAbsoluteTolerance allowed absolute error
158 * @param scalRelativeTolerance allowed relative error
159 * @exception NumberIsTooSmallException if order is 1 or less
160 */
161 public AdamsBashforthIntegrator(final int nSteps,
162 final double minStep, final double maxStep,
163 final double scalAbsoluteTolerance,
164 final double scalRelativeTolerance)
165 throws NumberIsTooSmallException {
166 super(METHOD_NAME, nSteps, nSteps, minStep, maxStep,
167 scalAbsoluteTolerance, scalRelativeTolerance);
168 }
169
170 /**
171 * Build an Adams-Bashforth integrator with the given order and step control parameters.
172 * @param nSteps number of steps of the method excluding the one being computed
173 * @param minStep minimal step (sign is irrelevant, regardless of
174 * integration direction, forward or backward), the last step can
175 * be smaller than this
176 * @param maxStep maximal step (sign is irrelevant, regardless of
177 * integration direction, forward or backward), the last step can
178 * be smaller than this
179 * @param vecAbsoluteTolerance allowed absolute error
180 * @param vecRelativeTolerance allowed relative error
181 * @exception IllegalArgumentException if order is 1 or less
182 */
183 public AdamsBashforthIntegrator(final int nSteps,
184 final double minStep, final double maxStep,
185 final double[] vecAbsoluteTolerance,
186 final double[] vecRelativeTolerance)
187 throws IllegalArgumentException {
188 super(METHOD_NAME, nSteps, nSteps, minStep, maxStep,
189 vecAbsoluteTolerance, vecRelativeTolerance);
190 }
191
192 /** {@inheritDoc} */
193 @Override
194 public void integrate(final ExpandableStatefulODE equations, final double t)
195 throws NumberIsTooSmallException, DimensionMismatchException,
196 MaxCountExceededException, NoBracketingException {
197
198 sanityChecks(equations, t);
199 setEquations(equations);
200 final boolean forward = t > equations.getTime();
201
202 // initialize working arrays
203 final double[] y0 = equations.getCompleteState();
204 final double[] y = y0.clone();
205 final double[] yDot = new double[y.length];
206
207 // set up an interpolator sharing the integrator arrays
208 final NordsieckStepInterpolator interpolator = new NordsieckStepInterpolator();
209 interpolator.reinitialize(y, forward,
210 equations.getPrimaryMapper(), equations.getSecondaryMappers());
211
212 // set up integration control objects
213 initIntegration(equations.getTime(), y0, t);
214
215 // compute the initial Nordsieck vector using the configured starter integrator
216 start(equations.getTime(), 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 // main integration loop
226 isLastStep = false;
227 do {
228
229 double error = 10;
230 while (error >= 1.0) {
231
232 stepSize = hNew;
233
234 // evaluate error using the last term of the Taylor expansion
235 error = 0;
236 for (int i = 0; i < mainSetDimension; ++i) {
237 final double yScale = FastMath.abs(y[i]);
238 final double tol = (vecAbsoluteTolerance == null) ?
239 (scalAbsoluteTolerance + scalRelativeTolerance * yScale) :
240 (vecAbsoluteTolerance[i] + vecRelativeTolerance[i] * yScale);
241 final double ratio = nordsieck.getEntry(lastRow, i) / tol;
242 error += ratio * ratio;
243 }
244 error = FastMath.sqrt(error / mainSetDimension);
245
246 if (error >= 1.0) {
247 // reject the step and attempt to reduce error by stepsize control
248 final double factor = computeStepGrowShrinkFactor(error);
249 hNew = filterStep(stepSize * factor, forward, false);
250 interpolator.rescale(hNew);
251
252 }
253 }
254
255 // predict a first estimate of the state at step end
256 final double stepEnd = stepStart + stepSize;
257 interpolator.shift();
258 interpolator.setInterpolatedTime(stepEnd);
259 final ExpandableStatefulODE expandable = getExpandable();
260 final EquationsMapper primary = expandable.getPrimaryMapper();
261 primary.insertEquationData(interpolator.getInterpolatedState(), y);
262 int index = 0;
263 for (final EquationsMapper secondary : expandable.getSecondaryMappers()) {
264 secondary.insertEquationData(interpolator.getInterpolatedSecondaryState(index), y);
265 ++index;
266 }
267
268 // evaluate the derivative
269 computeDerivatives(stepEnd, y, yDot);
270
271 // update Nordsieck vector
272 final double[] predictedScaled = new double[y0.length];
273 for (int j = 0; j < y0.length; ++j) {
274 predictedScaled[j] = stepSize * yDot[j];
275 }
276 final Array2DRowRealMatrix nordsieckTmp = updateHighOrderDerivativesPhase1(nordsieck);
277 updateHighOrderDerivativesPhase2(scaled, predictedScaled, nordsieckTmp);
278 interpolator.reinitialize(stepEnd, stepSize, predictedScaled, nordsieckTmp);
279
280 // discrete events handling
281 interpolator.storeTime(stepEnd);
282 stepStart = acceptStep(interpolator, y, yDot, t);
283 scaled = predictedScaled;
284 nordsieck = nordsieckTmp;
285 interpolator.reinitialize(stepEnd, stepSize, scaled, nordsieck);
286
287 if (!isLastStep) {
288
289 // prepare next step
290 interpolator.storeTime(stepStart);
291
292 if (resetOccurred) {
293 // some events handler has triggered changes that
294 // invalidate the derivatives, we need to restart from scratch
295 start(stepStart, y, t);
296 interpolator.reinitialize(stepStart, stepSize, scaled, nordsieck);
297 }
298
299 // stepsize control for next step
300 final double factor = computeStepGrowShrinkFactor(error);
301 final double scaledH = stepSize * factor;
302 final double nextT = stepStart + scaledH;
303 final boolean nextIsLast = forward ? (nextT >= t) : (nextT <= t);
304 hNew = filterStep(scaledH, forward, nextIsLast);
305
306 final double filteredNextT = stepStart + hNew;
307 final boolean filteredNextIsLast = forward ? (filteredNextT >= t) : (filteredNextT <= t);
308 if (filteredNextIsLast) {
309 hNew = t - stepStart;
310 }
311
312 interpolator.rescale(hNew);
313
314 }
315
316 } while (!isLastStep);
317
318 // dispatch results
319 equations.setTime(stepStart);
320 equations.setCompleteState(y);
321
322 resetInternalState();
323
324 }
325
326 }