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.ExpandableStatefulODE;
026 import org.apache.commons.math3.ode.sampling.NordsieckStepInterpolator;
027 import org.apache.commons.math3.util.FastMath;
028
029
030 /**
031 * This class implements explicit Adams-Bashforth integrators for Ordinary
032 * Differential Equations.
033 *
034 * <p>Adams-Bashforth methods (in fact due to Adams alone) are explicit
035 * multistep ODE solvers. This implementation is a variation of the classical
036 * one: it uses adaptive stepsize to implement error control, whereas
037 * classical implementations are fixed step size. The value of state vector
038 * at step n+1 is a simple combination of the value at step n and of the
039 * derivatives at steps n, n-1, n-2 ... Depending on the number k of previous
040 * steps one wants to use for computing the next value, different formulas
041 * are available:</p>
042 * <ul>
043 * <li>k = 1: y<sub>n+1</sub> = y<sub>n</sub> + h y'<sub>n</sub></li>
044 * <li>k = 2: y<sub>n+1</sub> = y<sub>n</sub> + h (3y'<sub>n</sub>-y'<sub>n-1</sub>)/2</li>
045 * <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>
046 * <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>
047 * <li>...</li>
048 * </ul>
049 *
050 * <p>A k-steps Adams-Bashforth method is of order k.</p>
051 *
052 * <h3>Implementation details</h3>
053 *
054 * <p>We define scaled derivatives s<sub>i</sub>(n) at step n as:
055 * <pre>
056 * s<sub>1</sub>(n) = h y'<sub>n</sub> for first derivative
057 * s<sub>2</sub>(n) = h<sup>2</sup>/2 y''<sub>n</sub> for second derivative
058 * s<sub>3</sub>(n) = h<sup>3</sup>/6 y'''<sub>n</sub> for third derivative
059 * ...
060 * s<sub>k</sub>(n) = h<sup>k</sup>/k! y<sup>(k)</sup><sub>n</sub> for k<sup>th</sup> derivative
061 * </pre></p>
062 *
063 * <p>The definitions above use the classical representation with several previous first
064 * derivatives. Lets define
065 * <pre>
066 * 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>
067 * </pre>
068 * (we omit the k index in the notation for clarity). With these definitions,
069 * Adams-Bashforth methods can be written:
070 * <ul>
071 * <li>k = 1: y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n)</li>
072 * <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>
073 * <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>
074 * <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>
075 * <li>...</li>
076 * </ul></p>
077 *
078 * <p>Instead of using the classical representation with first derivatives only (y<sub>n</sub>,
079 * s<sub>1</sub>(n) and q<sub>n</sub>), our implementation uses the Nordsieck vector with
080 * higher degrees scaled derivatives all taken at the same step (y<sub>n</sub>, s<sub>1</sub>(n)
081 * and r<sub>n</sub>) where r<sub>n</sub> is defined as:
082 * <pre>
083 * r<sub>n</sub> = [ s<sub>2</sub>(n), s<sub>3</sub>(n) ... s<sub>k</sub>(n) ]<sup>T</sup>
084 * </pre>
085 * (here again we omit the k index in the notation for clarity)
086 * </p>
087 *
088 * <p>Taylor series formulas show that for any index offset i, s<sub>1</sub>(n-i) can be
089 * computed from s<sub>1</sub>(n), s<sub>2</sub>(n) ... s<sub>k</sub>(n), the formula being exact
090 * for degree k polynomials.
091 * <pre>
092 * 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)
093 * </pre>
094 * The previous formula can be used with several values for i to compute the transform between
095 * classical representation and Nordsieck vector. The transform between r<sub>n</sub>
096 * and q<sub>n</sub> resulting from the Taylor series formulas above is:
097 * <pre>
098 * q<sub>n</sub> = s<sub>1</sub>(n) u + P r<sub>n</sub>
099 * </pre>
100 * where u is the [ 1 1 ... 1 ]<sup>T</sup> vector and P is the (k-1)×(k-1) matrix built
101 * with the j (-i)<sup>j-1</sup> terms:
102 * <pre>
103 * [ -2 3 -4 5 ... ]
104 * [ -4 12 -32 80 ... ]
105 * P = [ -6 27 -108 405 ... ]
106 * [ -8 48 -256 1280 ... ]
107 * [ ... ]
108 * </pre></p>
109 *
110 * <p>Using the Nordsieck vector has several advantages:
111 * <ul>
112 * <li>it greatly simplifies step interpolation as the interpolator mainly applies
113 * Taylor series formulas,</li>
114 * <li>it simplifies step changes that occur when discrete events that truncate
115 * the step are triggered,</li>
116 * <li>it allows to extend the methods in order to support adaptive stepsize.</li>
117 * </ul></p>
118 *
119 * <p>The Nordsieck vector at step n+1 is computed from the Nordsieck vector at step n as follows:
120 * <ul>
121 * <li>y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n) + u<sup>T</sup> r<sub>n</sub></li>
122 * <li>s<sub>1</sub>(n+1) = h f(t<sub>n+1</sub>, y<sub>n+1</sub>)</li>
123 * <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>
124 * </ul>
125 * where A is a rows shifting matrix (the lower left part is an identity matrix):
126 * <pre>
127 * [ 0 0 ... 0 0 | 0 ]
128 * [ ---------------+---]
129 * [ 1 0 ... 0 0 | 0 ]
130 * A = [ 0 1 ... 0 0 | 0 ]
131 * [ ... | 0 ]
132 * [ 0 0 ... 1 0 | 0 ]
133 * [ 0 0 ... 0 1 | 0 ]
134 * </pre></p>
135 *
136 * <p>The P<sup>-1</sup>u vector and the P<sup>-1</sup> A P matrix do not depend on the state,
137 * they only depend on k and therefore are precomputed once for all.</p>
138 *
139 * @version $Id: AdamsBashforthIntegrator.java 1416643 2012-12-03 19:37:14Z tn $
140 * @since 2.0
141 */
142 public class AdamsBashforthIntegrator extends AdamsIntegrator {
143
144 /** Integrator method name. */
145 private static final String METHOD_NAME = "Adams-Bashforth";
146
147 /**
148 * Build an Adams-Bashforth integrator with the given order and step control parameters.
149 * @param nSteps number of steps of the method excluding the one being computed
150 * @param minStep minimal step (sign is irrelevant, regardless of
151 * integration direction, forward or backward), the last step can
152 * be smaller than this
153 * @param maxStep maximal step (sign is irrelevant, regardless of
154 * integration direction, forward or backward), the last step can
155 * be smaller than this
156 * @param scalAbsoluteTolerance allowed absolute error
157 * @param scalRelativeTolerance allowed relative error
158 * @exception NumberIsTooSmallException if order is 1 or less
159 */
160 public AdamsBashforthIntegrator(final int nSteps,
161 final double minStep, final double maxStep,
162 final double scalAbsoluteTolerance,
163 final double scalRelativeTolerance)
164 throws NumberIsTooSmallException {
165 super(METHOD_NAME, nSteps, nSteps, minStep, maxStep,
166 scalAbsoluteTolerance, scalRelativeTolerance);
167 }
168
169 /**
170 * Build an Adams-Bashforth integrator with the given order and step control parameters.
171 * @param nSteps number of steps of the method excluding the one being computed
172 * @param minStep minimal step (sign is irrelevant, regardless of
173 * integration direction, forward or backward), the last step can
174 * be smaller than this
175 * @param maxStep maximal step (sign is irrelevant, regardless of
176 * integration direction, forward or backward), the last step can
177 * be smaller than this
178 * @param vecAbsoluteTolerance allowed absolute error
179 * @param vecRelativeTolerance allowed relative error
180 * @exception IllegalArgumentException if order is 1 or less
181 */
182 public AdamsBashforthIntegrator(final int nSteps,
183 final double minStep, final double maxStep,
184 final double[] vecAbsoluteTolerance,
185 final double[] vecRelativeTolerance)
186 throws IllegalArgumentException {
187 super(METHOD_NAME, nSteps, nSteps, minStep, maxStep,
188 vecAbsoluteTolerance, vecRelativeTolerance);
189 }
190
191 /** {@inheritDoc} */
192 @Override
193 public void integrate(final ExpandableStatefulODE equations, final double t)
194 throws NumberIsTooSmallException, DimensionMismatchException,
195 MaxCountExceededException, NoBracketingException {
196
197 sanityChecks(equations, t);
198 setEquations(equations);
199 final boolean forward = t > equations.getTime();
200
201 // initialize working arrays
202 final double[] y0 = equations.getCompleteState();
203 final double[] y = y0.clone();
204 final double[] yDot = new double[y.length];
205
206 // set up an interpolator sharing the integrator arrays
207 final NordsieckStepInterpolator interpolator = new NordsieckStepInterpolator();
208 interpolator.reinitialize(y, forward,
209 equations.getPrimaryMapper(), equations.getSecondaryMappers());
210
211 // set up integration control objects
212 initIntegration(equations.getTime(), y0, t);
213
214 // compute the initial Nordsieck vector using the configured starter integrator
215 start(equations.getTime(), y, t);
216 interpolator.reinitialize(stepStart, stepSize, scaled, nordsieck);
217 interpolator.storeTime(stepStart);
218 final int lastRow = nordsieck.getRowDimension() - 1;
219
220 // reuse the step that was chosen by the starter integrator
221 double hNew = stepSize;
222 interpolator.rescale(hNew);
223
224 // main integration loop
225 isLastStep = false;
226 do {
227
228 double error = 10;
229 while (error >= 1.0) {
230
231 stepSize = hNew;
232
233 // evaluate error using the last term of the Taylor expansion
234 error = 0;
235 for (int i = 0; i < mainSetDimension; ++i) {
236 final double yScale = FastMath.abs(y[i]);
237 final double tol = (vecAbsoluteTolerance == null) ?
238 (scalAbsoluteTolerance + scalRelativeTolerance * yScale) :
239 (vecAbsoluteTolerance[i] + vecRelativeTolerance[i] * yScale);
240 final double ratio = nordsieck.getEntry(lastRow, i) / tol;
241 error += ratio * ratio;
242 }
243 error = FastMath.sqrt(error / mainSetDimension);
244
245 if (error >= 1.0) {
246 // reject the step and attempt to reduce error by stepsize control
247 final double factor = computeStepGrowShrinkFactor(error);
248 hNew = filterStep(stepSize * factor, forward, false);
249 interpolator.rescale(hNew);
250
251 }
252 }
253
254 // predict a first estimate of the state at step end
255 final double stepEnd = stepStart + stepSize;
256 interpolator.shift();
257 interpolator.setInterpolatedTime(stepEnd);
258 System.arraycopy(interpolator.getInterpolatedState(), 0, y, 0, y0.length);
259
260 // evaluate the derivative
261 computeDerivatives(stepEnd, y, yDot);
262
263 // update Nordsieck vector
264 final double[] predictedScaled = new double[y0.length];
265 for (int j = 0; j < y0.length; ++j) {
266 predictedScaled[j] = stepSize * yDot[j];
267 }
268 final Array2DRowRealMatrix nordsieckTmp = updateHighOrderDerivativesPhase1(nordsieck);
269 updateHighOrderDerivativesPhase2(scaled, predictedScaled, nordsieckTmp);
270 interpolator.reinitialize(stepEnd, stepSize, predictedScaled, nordsieckTmp);
271
272 // discrete events handling
273 interpolator.storeTime(stepEnd);
274 stepStart = acceptStep(interpolator, y, yDot, t);
275 scaled = predictedScaled;
276 nordsieck = nordsieckTmp;
277 interpolator.reinitialize(stepEnd, stepSize, scaled, nordsieck);
278
279 if (!isLastStep) {
280
281 // prepare next step
282 interpolator.storeTime(stepStart);
283
284 if (resetOccurred) {
285 // some events handler has triggered changes that
286 // invalidate the derivatives, we need to restart from scratch
287 start(stepStart, y, t);
288 interpolator.reinitialize(stepStart, stepSize, scaled, nordsieck);
289 }
290
291 // stepsize control for next step
292 final double factor = computeStepGrowShrinkFactor(error);
293 final double scaledH = stepSize * factor;
294 final double nextT = stepStart + scaledH;
295 final boolean nextIsLast = forward ? (nextT >= t) : (nextT <= t);
296 hNew = filterStep(scaledH, forward, nextIsLast);
297
298 final double filteredNextT = stepStart + hNew;
299 final boolean filteredNextIsLast = forward ? (filteredNextT >= t) : (filteredNextT <= t);
300 if (filteredNextIsLast) {
301 hNew = t - stepStart;
302 }
303
304 interpolator.rescale(hNew);
305
306 }
307
308 } while (!isLastStep);
309
310 // dispatch results
311 equations.setTime(stepStart);
312 equations.setCompleteState(y);
313
314 resetInternalState();
315
316 }
317
318 }