    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 java.util.Arrays;
    021
    022    import org.apache.commons.math.linear.Array2DRowRealMatrix;
    023    import org.apache.commons.math.linear.RealMatrixPreservingVisitor;
    024    import org.apache.commons.math.ode.DerivativeException;
    025    import org.apache.commons.math.ode.FirstOrderDifferentialEquations;
    026    import org.apache.commons.math.ode.IntegratorException;
    027    import org.apache.commons.math.ode.sampling.NordsieckStepInterpolator;
    028    import org.apache.commons.math.ode.sampling.StepHandler;
    029    import org.apache.commons.math.util.FastMath;
    030
    031
    032    /**
    033     * This class implements implicit Adams-Moulton integrators for Ordinary
    034     * Differential Equations.
    035     *
    036     * <p>Adams-Moulton methods (in fact due to Adams alone) are implicit
    037     * multistep ODE solvers. This implementation is a variation of the classical
    038     * one: it uses adaptive stepsize to implement error control, whereas
    039     * classical implementations are fixed step size. The value of state vector
    040     * at step n+1 is a simple combination of the value at step n and of the
    041     * derivatives at steps n+1, n, n-1 ... Since y'<sub>n+1</sub> is needed to
    042     * compute y<sub>n+1</sub>,another method must be used to compute a first
    043     * estimate of y<sub>n+1</sub>, then compute y'<sub>n+1</sub>, then compute
    044     * a final estimate of y<sub>n+1</sub> using the following formulas. Depending
    045     * on the number k of previous steps one wants to use for computing the next
    046     * value, different formulas are available for the final estimate:</p>
    047     * <ul>
    048     *   <li>k = 1: y<sub>n+1</sub> = y<sub>n</sub> + h y'<sub>n+1</sub></li>
    049     *   <li>k = 2: y<sub>n+1</sub> = y<sub>n</sub> + h (y'<sub>n+1</sub>+y'<sub>n</sub>)/2</li>
    050     *   <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>
    051     *   <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>
    052     *   <li>...</li>
    053     * </ul>
    054     *
    055     * <p>A k-steps Adams-Moulton method is of order k+1.</p>
    056     *
    057     * <h3>Implementation details</h3>
    058     *
    059     * <p>We define scaled derivatives s<sub>i</sub>(n) at step n as:
    060     * <pre>
    061     * s<sub>1</sub>(n) = h y'<sub>n</sub> for first derivative
    062     * s<sub>2</sub>(n) = h<sup>2</sup>/2 y''<sub>n</sub> for second derivative
    063     * s<sub>3</sub>(n) = h<sup>3</sup>/6 y'''<sub>n</sub> for third derivative
    064     * ...
    065     * s<sub>k</sub>(n) = h<sup>k</sup>/k! y(k)<sub>n</sub> for k<sup>th</sup> derivative
    066     * </pre></p>
    067     *
    068     * <p>The definitions above use the classical representation with several previous first
    069     * derivatives. Lets define
    070     * <pre>
    071     *   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>
    072     * </pre>
    073     * (we omit the k index in the notation for clarity). With these definitions,
    074     * Adams-Moulton methods can be written:
    075     * <ul>
    076     *   <li>k = 1: y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n+1)</li>
    077     *   <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>
    078     *   <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>
    079     *   <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>
    080     *   <li>...</li>
    081     * </ul></p>
    082     *
    083     * <p>Instead of using the classical representation with first derivatives only (y<sub>n</sub>,
    084     * s<sub>1</sub>(n+1) and q<sub>n+1</sub>), our implementation uses the Nordsieck vector with
    085     * higher degrees scaled derivatives all taken at the same step (y<sub>n</sub>, s<sub>1</sub>(n)
    086     * and r<sub>n</sub>) where r<sub>n</sub> is defined as:
    087     * <pre>
    088     * r<sub>n</sub> = [ s<sub>2</sub>(n), s<sub>3</sub>(n) ... s<sub>k</sub>(n) ]<sup>T</sup>
    089     * </pre>
    090     * (here again we omit the k index in the notation for clarity)
    091     * </p>
    092     *
    093     * <p>Taylor series formulas show that for any index offset i, s<sub>1</sub>(n-i) can be
    094     * computed from s<sub>1</sub>(n), s<sub>2</sub>(n) ... s<sub>k</sub>(n), the formula being exact
    095     * for degree k polynomials.
    096     * <pre>
    097     * 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)
    098     * </pre>
    099     * The previous formula can be used with several values for i to compute the transform between
    100     * classical representation and Nordsieck vector. The transform between r<sub>n</sub>
    101     * and q<sub>n</sub> resulting from the Taylor series formulas above is:
    102     * <pre>
    103     * q<sub>n</sub> = s<sub>1</sub>(n) u + P r<sub>n</sub>
    104     * </pre>
    105     * where u is the [ 1 1 ... 1 ]<sup>T</sup> vector and P is the (k-1)&times;(k-1) matrix built
    106     * with the j (-i)<sup>j-1</sup> terms:
    107     * <pre>
    108     *        [  -2   3   -4    5  ... ]
    109     *        [  -4  12  -32   80  ... ]
    110     *   P =  [  -6  27 -108  405  ... ]
    111     *        [  -8  48 -256 1280  ... ]
    112     *        [          ...           ]
    113     * </pre></p>
    114     *
    115     * <p>Using the Nordsieck vector has several advantages:
    116     * <ul>
    117     *   <li>it greatly simplifies step interpolation as the interpolator mainly applies
    118     *   Taylor series formulas,</li>
    119     *   <li>it simplifies step changes that occur when discrete events that truncate
    120     *   the step are triggered,</li>
    121     *   <li>it allows to extend the methods in order to support adaptive stepsize.</li>
    122     * </ul></p>
    123     *
    124     * <p>The predicted Nordsieck vector at step n+1 is computed from the Nordsieck vector at step
    125     * n as follows:
    126     * <ul>
    127     *   <li>Y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n) + u<sup>T</sup> r<sub>n</sub></li>
    128     *   <li>S<sub>1</sub>(n+1) = h f(t<sub>n+1</sub>, Y<sub>n+1</sub>)</li>
    129     *   <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>
    130     * </ul>
    131     * where A is a rows shifting matrix (the lower left part is an identity matrix):
    132     * <pre>
    133     *        [ 0 0   ...  0 0 | 0 ]
    134     *        [ ---------------+---]
    135     *        [ 1 0   ...  0 0 | 0 ]
    136     *    A = [ 0 1   ...  0 0 | 0 ]
    137     *        [       ...      | 0 ]
    138     *        [ 0 0   ...  1 0 | 0 ]
    139     *        [ 0 0   ...  0 1 | 0 ]
    140     * </pre>
    141     * From this predicted vector, the corrected vector is computed as follows:
    142     * <ul>
    143     *   <li>y<sub>n+1</sub> = y<sub>n</sub> + S<sub>1</sub>(n+1) + [ -1 +1 -1 +1 ... &plusmn;1 ] r<sub>n+1</sub></li>
    144     *   <li>s<sub>1</sub>(n+1) = h f(t<sub>n+1</sub>, y<sub>n+1</sub>)</li>
    145     *   <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>
    146     * </ul>
    147     * where the upper case Y<sub>n+1</sub>, S<sub>1</sub>(n+1) and R<sub>n+1</sub> represent the
    148     * predicted states whereas the lower case y<sub>n+1</sub>, s<sub>n+1</sub> and r<sub>n+1</sub>
    149     * represent the corrected states.</p>
    150     *
    151     * <p>The P<sup>-1</sup>u vector and the P<sup>-1</sup> A P matrix do not depend on the state,
    152     * they only depend on k and therefore are precomputed once for all.</p>
    153     *
    154     * @version $Revision: 1073158 $ $Date: 2011-02-21 22:46:52 +0100 (lun. 21 f??vr. 2011) $
    155     * @since 2.0
    156     */
    157    public class AdamsMoultonIntegrator extends AdamsIntegrator {
    158
    159        /** Integrator method name. */
    160        private static final String METHOD_NAME = "Adams-Moulton";
    161
    162        /**
    163         * Build an Adams-Moulton integrator with the given order and error control parameters.
    164         * @param nSteps number of steps of the method excluding the one being computed
    165         * @param minStep minimal step (must be positive even for backward
    166         * integration), the last step can be smaller than this
    167         * @param maxStep maximal step (must be positive even for backward
    168         * integration)
    169         * @param scalAbsoluteTolerance allowed absolute error
    170         * @param scalRelativeTolerance allowed relative error
    171         * @exception IllegalArgumentException if order is 1 or less
    172         */
    173        public AdamsMoultonIntegrator(final int nSteps,
    174                                      final double minStep, final double maxStep,
    175                                      final double scalAbsoluteTolerance,
    176                                      final double scalRelativeTolerance)
    177            throws IllegalArgumentException {
    178            super(METHOD_NAME, nSteps, nSteps + 1, minStep, maxStep,
    179                  scalAbsoluteTolerance, scalRelativeTolerance);
    180        }
    181
    182        /**
    183         * Build an Adams-Moulton integrator with the given order and error control parameters.
    184         * @param nSteps number of steps of the method excluding the one being computed
    185         * @param minStep minimal step (must be positive even for backward
    186         * integration), the last step can be smaller than this
    187         * @param maxStep maximal step (must be positive even for backward
    188         * integration)
    189         * @param vecAbsoluteTolerance allowed absolute error
    190         * @param vecRelativeTolerance allowed relative error
    191         * @exception IllegalArgumentException if order is 1 or less
    192         */
    193        public AdamsMoultonIntegrator(final int nSteps,
    194                                      final double minStep, final double maxStep,
    195                                      final double[] vecAbsoluteTolerance,
    196                                      final double[] vecRelativeTolerance)
    197            throws IllegalArgumentException {
    198            super(METHOD_NAME, nSteps, nSteps + 1, minStep, maxStep,
    199                  vecAbsoluteTolerance, vecRelativeTolerance);
    200        }
    201
    202
    203        /** {@inheritDoc} */
    204        @Override
    205        public double integrate(final FirstOrderDifferentialEquations equations,
    206                                final double t0, final double[] y0,
    207                                final double t, final double[] y)
    208            throws DerivativeException, IntegratorException {
    209
    210            final int n = y0.length;
    211            sanityChecks(equations, t0, y0, t, y);
    212            setEquations(equations);
    213            resetEvaluations();
    214            final boolean forward = t > t0;
    215
    216            // initialize working arrays
    217            if (y != y0) {
    218                System.arraycopy(y0, 0, y, 0, n);
    219            }
    220            final double[] yDot = new double[y0.length];
    221            final double[] yTmp = new double[y0.length];
    222            final double[] predictedScaled = new double[y0.length];
    223            Array2DRowRealMatrix nordsieckTmp = null;
    224
    225            // set up two interpolators sharing the integrator arrays
    226            final NordsieckStepInterpolator interpolator = new NordsieckStepInterpolator();
    227            interpolator.reinitialize(y, forward);
    228
    229            // set up integration control objects
    230            for (StepHandler handler : stepHandlers) {
    231                handler.reset();
    232            }
    233            setStateInitialized(false);
    234
    235            // compute the initial Nordsieck vector using the configured starter integrator
    236            start(t0, y, t);
    237            interpolator.reinitialize(stepStart, stepSize, scaled, nordsieck);
    238            interpolator.storeTime(stepStart);
    239
    240            double hNew = stepSize;
    241            interpolator.rescale(hNew);
    242
    243            isLastStep = false;
    244            do {
    245
    246                double error = 10;
    247                while (error >= 1.0) {
    248
    249                    stepSize = hNew;
    250
    251                    // predict a first estimate of the state at step end (P in the PECE sequence)
    252                    final double stepEnd = stepStart + stepSize;
    253                    interpolator.setInterpolatedTime(stepEnd);
    254                    System.arraycopy(interpolator.getInterpolatedState(), 0, yTmp, 0, y0.length);
    255
    256                    // evaluate a first estimate of the derivative (first E in the PECE sequence)
    257                    computeDerivatives(stepEnd, yTmp, yDot);
    258
    259                    // update Nordsieck vector
    260                    for (int j = 0; j < y0.length; ++j) {
    261                        predictedScaled[j] = stepSize * yDot[j];
    262                    }
    263                    nordsieckTmp = updateHighOrderDerivativesPhase1(nordsieck);
    264                    updateHighOrderDerivativesPhase2(scaled, predictedScaled, nordsieckTmp);
    265
    266                    // apply correction (C in the PECE sequence)
    267                    error = nordsieckTmp.walkInOptimizedOrder(new Corrector(y, predictedScaled, yTmp));
    268
    269                    if (error >= 1.0) {
    270                        // reject the step and attempt to reduce error by stepsize control
    271                        final double factor = computeStepGrowShrinkFactor(error);
    272                        hNew = filterStep(stepSize * factor, forward, false);
    273                        interpolator.rescale(hNew);
    274                    }
    275                }
    276
    277                // evaluate a final estimate of the derivative (second E in the PECE sequence)
    278                final double stepEnd = stepStart + stepSize;
    279                computeDerivatives(stepEnd, yTmp, yDot);
    280
    281                // update Nordsieck vector
    282                final double[] correctedScaled = new double[y0.length];
    283                for (int j = 0; j < y0.length; ++j) {
    284                    correctedScaled[j] = stepSize * yDot[j];
    285                }
    286                updateHighOrderDerivativesPhase2(predictedScaled, correctedScaled, nordsieckTmp);
    287
    288                // discrete events handling
    289                System.arraycopy(yTmp, 0, y, 0, n);
    290                interpolator.reinitialize(stepEnd, stepSize, correctedScaled, nordsieckTmp);
    291                interpolator.storeTime(stepStart);
    292                interpolator.shift();
    293                interpolator.storeTime(stepEnd);
    294                stepStart = acceptStep(interpolator, y, yDot, t);
    295                scaled    = correctedScaled;
    296                nordsieck = nordsieckTmp;
    297
    298                if (!isLastStep) {
    299
    300                    // prepare next step
    301                    interpolator.storeTime(stepStart);
    302
    303                    if (resetOccurred) {
    304                        // some events handler has triggered changes that
    305                        // invalidate the derivatives, we need to restart from scratch
    306                        start(stepStart, y, t);
    307                        interpolator.reinitialize(stepStart, stepSize, scaled, nordsieck);
    308
    309                    }
    310
    311                    // stepsize control for next step
    312                    final double  factor     = computeStepGrowShrinkFactor(error);
    313                    final double  scaledH    = stepSize * factor;
    314                    final double  nextT      = stepStart + scaledH;
    315                    final boolean nextIsLast = forward ? (nextT >= t) : (nextT <= t);
    316                    hNew = filterStep(scaledH, forward, nextIsLast);
    317
    318                    final double  filteredNextT      = stepStart + hNew;
    319                    final boolean filteredNextIsLast = forward ? (filteredNextT >= t) : (filteredNextT <= t);
    320                    if (filteredNextIsLast) {
    321                        hNew = t - stepStart;
    322                    }
    323
    324                    interpolator.rescale(hNew);
    325                }
    326
    327            } while (!isLastStep);
    328
    329            final double stopTime  = stepStart;
    330            stepStart = Double.NaN;
    331            stepSize  = Double.NaN;
    332            return stopTime;
    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 ... &plusmn;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    }