    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) + &sum;<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)&times;(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    }