    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 java.util.Arrays;
    021    import java.util.HashMap;
    022    import java.util.Map;
    023
    024    import org.apache.commons.math3.fraction.BigFraction;
    025    import org.apache.commons.math3.linear.Array2DRowFieldMatrix;
    026    import org.apache.commons.math3.linear.Array2DRowRealMatrix;
    027    import org.apache.commons.math3.linear.ArrayFieldVector;
    028    import org.apache.commons.math3.linear.FieldDecompositionSolver;
    029    import org.apache.commons.math3.linear.FieldLUDecomposition;
    030    import org.apache.commons.math3.linear.FieldMatrix;
    031    import org.apache.commons.math3.linear.MatrixUtils;
    032    import org.apache.commons.math3.linear.QRDecomposition;
    033    import org.apache.commons.math3.linear.RealMatrix;
    034
    035    /** Transformer to Nordsieck vectors for Adams integrators.
    036     * <p>This class is used by {@link AdamsBashforthIntegrator Adams-Bashforth} and
    037     * {@link AdamsMoultonIntegrator Adams-Moulton} integrators to convert between
    038     * classical representation with several previous first derivatives and Nordsieck
    039     * representation with higher order scaled derivatives.</p>
    040     *
    041     * <p>We define scaled derivatives s<sub>i</sub>(n) at step n as:
    042     * <pre>
    043     * s<sub>1</sub>(n) = h y'<sub>n</sub> for first derivative
    044     * s<sub>2</sub>(n) = h<sup>2</sup>/2 y''<sub>n</sub> for second derivative
    045     * s<sub>3</sub>(n) = h<sup>3</sup>/6 y'''<sub>n</sub> for third derivative
    046     * ...
    047     * s<sub>k</sub>(n) = h<sup>k</sup>/k! y<sup>(k)</sup><sub>n</sub> for k<sup>th</sup> derivative
    048     * </pre></p>
    049     *
    050     * <p>With the previous definition, the classical representation of multistep methods
    051     * uses first derivatives only, i.e. it handles y<sub>n</sub>, s<sub>1</sub>(n) and
    052     * q<sub>n</sub> where q<sub>n</sub> is defined as:
    053     * <pre>
    054     *   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>
    055     * </pre>
    056     * (we omit the k index in the notation for clarity).</p>
    057     *
    058     * <p>Another possible representation uses the Nordsieck vector with
    059     * higher degrees scaled derivatives all taken at the same step, i.e it handles y<sub>n</sub>,
    060     * s<sub>1</sub>(n) and r<sub>n</sub>) where r<sub>n</sub> is defined as:
    061     * <pre>
    062     * r<sub>n</sub> = [ s<sub>2</sub>(n), s<sub>3</sub>(n) ... s<sub>k</sub>(n) ]<sup>T</sup>
    063     * </pre>
    064     * (here again we omit the k index in the notation for clarity)
    065     * </p>
    066     *
    067     * <p>Taylor series formulas show that for any index offset i, s<sub>1</sub>(n-i) can be
    068     * computed from s<sub>1</sub>(n), s<sub>2</sub>(n) ... s<sub>k</sub>(n), the formula being exact
    069     * for degree k polynomials.
    070     * <pre>
    071     * s<sub>1</sub>(n-i) = s<sub>1</sub>(n) + &sum;<sub>j&gt;1</sub> j (-i)<sup>j-1</sup> s<sub>j</sub>(n)
    072     * </pre>
    073     * The previous formula can be used with several values for i to compute the transform between
    074     * classical representation and Nordsieck vector at step end. The transform between r<sub>n</sub>
    075     * and q<sub>n</sub> resulting from the Taylor series formulas above is:
    076     * <pre>
    077     * q<sub>n</sub> = s<sub>1</sub>(n) u + P r<sub>n</sub>
    078     * </pre>
    079     * where u is the [ 1 1 ... 1 ]<sup>T</sup> vector and P is the (k-1)&times;(k-1) matrix built
    080     * with the j (-i)<sup>j-1</sup> terms:
    081     * <pre>
    082     *        [  -2   3   -4    5  ... ]
    083     *        [  -4  12  -32   80  ... ]
    084     *   P =  [  -6  27 -108  405  ... ]
    085     *        [  -8  48 -256 1280  ... ]
    086     *        [          ...           ]
    087     * </pre></p>
    088     *
    089     * <p>Changing -i into +i in the formula above can be used to compute a similar transform between
    090     * classical representation and Nordsieck vector at step start. The resulting matrix is simply
    091     * the absolute value of matrix P.</p>
    092     *
    093     * <p>For {@link AdamsBashforthIntegrator Adams-Bashforth} method, the Nordsieck vector
    094     * at step n+1 is computed from the Nordsieck vector at step n as follows:
    095     * <ul>
    096     *   <li>y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n) + u<sup>T</sup> r<sub>n</sub></li>
    097     *   <li>s<sub>1</sub>(n+1) = h f(t<sub>n+1</sub>, y<sub>n+1</sub>)</li>
    098     *   <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>
    099     * </ul>
    100     * where A is a rows shifting matrix (the lower left part is an identity matrix):
    101     * <pre>
    102     *        [ 0 0   ...  0 0 | 0 ]
    103     *        [ ---------------+---]
    104     *        [ 1 0   ...  0 0 | 0 ]
    105     *    A = [ 0 1   ...  0 0 | 0 ]
    106     *        [       ...      | 0 ]
    107     *        [ 0 0   ...  1 0 | 0 ]
    108     *        [ 0 0   ...  0 1 | 0 ]
    109     * </pre></p>
    110     *
    111     * <p>For {@link AdamsMoultonIntegrator Adams-Moulton} method, the predicted Nordsieck vector
    112     * at step n+1 is computed from the Nordsieck vector at step n as follows:
    113     * <ul>
    114     *   <li>Y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n) + u<sup>T</sup> r<sub>n</sub></li>
    115     *   <li>S<sub>1</sub>(n+1) = h f(t<sub>n+1</sub>, Y<sub>n+1</sub>)</li>
    116     *   <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>
    117     * </ul>
    118     * From this predicted vector, the corrected vector is computed as follows:
    119     * <ul>
    120     *   <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>
    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> = R<sub>n+1</sub> + (s<sub>1</sub>(n+1) - S<sub>1</sub>(n+1)) P<sup>-1</sup> u</li>
    123     * </ul>
    124     * where the upper case Y<sub>n+1</sub>, S<sub>1</sub>(n+1) and R<sub>n+1</sub> represent the
    125     * predicted states whereas the lower case y<sub>n+1</sub>, s<sub>n+1</sub> and r<sub>n+1</sub>
    126     * represent the corrected states.</p>
    127     *
    128     * <p>We observe that both methods use similar update formulas. In both cases a P<sup>-1</sup>u
    129     * vector and a P<sup>-1</sup> A P matrix are used that do not depend on the state,
    130     * they only depend on k. This class handles these transformations.</p>
    131     *
    132     * @version $Id: AdamsNordsieckTransformer.java 1416643 2012-12-03 19:37:14Z tn $
    133     * @since 2.0
    134     */
    135    public class AdamsNordsieckTransformer {
    136
    137        /** Cache for already computed coefficients. */
    138        private static final Map<Integer, AdamsNordsieckTransformer> CACHE =
    139            new HashMap<Integer, AdamsNordsieckTransformer>();
    140
    141        /** Update matrix for the higher order derivatives h<sup>2</sup>/2y'', h<sup>3</sup>/6 y''' ... */
    142        private final Array2DRowRealMatrix update;
    143
    144        /** Update coefficients of the higher order derivatives wrt y'. */
    145        private final double[] c1;
    146
    147        /** Simple constructor.
    148         * @param nSteps number of steps of the multistep method
    149         * (excluding the one being computed)
    150         */
    151        private AdamsNordsieckTransformer(final int nSteps) {
    152
    153            // compute exact coefficients
    154            FieldMatrix<BigFraction> bigP = buildP(nSteps);
    155            FieldDecompositionSolver<BigFraction> pSolver =
    156                new FieldLUDecomposition<BigFraction>(bigP).getSolver();
    157
    158            BigFraction[] u = new BigFraction[nSteps];
    159            Arrays.fill(u, BigFraction.ONE);
    160            BigFraction[] bigC1 = pSolver
    161                .solve(new ArrayFieldVector<BigFraction>(u, false)).toArray();
    162
    163            // update coefficients are computed by combining transform from
    164            // Nordsieck to multistep, then shifting rows to represent step advance
    165            // then applying inverse transform
    166            BigFraction[][] shiftedP = bigP.getData();
    167            for (int i = shiftedP.length - 1; i > 0; --i) {
    168                // shift rows
    169                shiftedP[i] = shiftedP[i - 1];
    170            }
    171            shiftedP[0] = new BigFraction[nSteps];
    172            Arrays.fill(shiftedP[0], BigFraction.ZERO);
    173            FieldMatrix<BigFraction> bigMSupdate =
    174                pSolver.solve(new Array2DRowFieldMatrix<BigFraction>(shiftedP, false));
    175
    176            // convert coefficients to double
    177            update         = MatrixUtils.bigFractionMatrixToRealMatrix(bigMSupdate);
    178            c1             = new double[nSteps];
    179            for (int i = 0; i < nSteps; ++i) {
    180                c1[i] = bigC1[i].doubleValue();
    181            }
    182
    183        }
    184
    185        /** Get the Nordsieck transformer for a given number of steps.
    186         * @param nSteps number of steps of the multistep method
    187         * (excluding the one being computed)
    188         * @return Nordsieck transformer for the specified number of steps
    189         */
    190        public static AdamsNordsieckTransformer getInstance(final int nSteps) {
    191            synchronized(CACHE) {
    192                AdamsNordsieckTransformer t = CACHE.get(nSteps);
    193                if (t == null) {
    194                    t = new AdamsNordsieckTransformer(nSteps);
    195                    CACHE.put(nSteps, t);
    196                }
    197                return t;
    198            }
    199        }
    200
    201        /** Get the number of steps of the method
    202         * (excluding the one being computed).
    203         * @return number of steps of the method
    204         * (excluding the one being computed)
    205         */
    206        public int getNSteps() {
    207            return c1.length;
    208        }
    209
    210        /** Build the P matrix.
    211         * <p>The P matrix general terms are shifted j (-i)<sup>j-1</sup> terms:
    212         * <pre>
    213         *        [  -2   3   -4    5  ... ]
    214         *        [  -4  12  -32   80  ... ]
    215         *   P =  [  -6  27 -108  405  ... ]
    216         *        [  -8  48 -256 1280  ... ]
    217         *        [          ...           ]
    218         * </pre></p>
    219         * @param nSteps number of steps of the multistep method
    220         * (excluding the one being computed)
    221         * @return P matrix
    222         */
    223        private FieldMatrix<BigFraction> buildP(final int nSteps) {
    224
    225            final BigFraction[][] pData = new BigFraction[nSteps][nSteps];
    226
    227            for (int i = 0; i < pData.length; ++i) {
    228                // build the P matrix elements from Taylor series formulas
    229                final BigFraction[] pI = pData[i];
    230                final int factor = -(i + 1);
    231                int aj = factor;
    232                for (int j = 0; j < pI.length; ++j) {
    233                    pI[j] = new BigFraction(aj * (j + 2));
    234                    aj *= factor;
    235                }
    236            }
    237
    238            return new Array2DRowFieldMatrix<BigFraction>(pData, false);
    239
    240        }
    241
    242        /** Initialize the high order scaled derivatives at step start.
    243         * @param h step size to use for scaling
    244         * @param t first steps times
    245         * @param y first steps states
    246         * @param yDot first steps derivatives
    247         * @return Nordieck vector at first step (h<sup>2</sup>/2 y''<sub>n</sub>,
    248         * h<sup>3</sup>/6 y'''<sub>n</sub> ... h<sup>k</sup>/k! y<sup>(k)</sup><sub>n</sub>)
    249         */
    250        public Array2DRowRealMatrix initializeHighOrderDerivatives(final double h, final double[] t,
    251                                                                   final double[][] y,
    252                                                                   final double[][] yDot) {
    253
    254            // using Taylor series with di = ti - t0, we get:
    255            //  y(ti)  - y(t0)  - di y'(t0) =   di^2 / h^2 s2 + ... +   di^k     / h^k sk + O(h^(k+1))
    256            //  y'(ti) - y'(t0)             = 2 di   / h^2 s2 + ... + k di^(k-1) / h^k sk + O(h^k)
    257            // we write these relations for i = 1 to i= n-1 as a set of 2(n-1) linear
    258            // equations depending on the Nordsieck vector [s2 ... sk]
    259            final double[][] a     = new double[2 * (y.length - 1)][c1.length];
    260            final double[][] b     = new double[2 * (y.length - 1)][y[0].length];
    261            final double[]   y0    = y[0];
    262            final double[]   yDot0 = yDot[0];
    263            for (int i = 1; i < y.length; ++i) {
    264
    265                final double di    = t[i] - t[0];
    266                final double ratio = di / h;
    267                double dikM1Ohk    =  1 / h;
    268
    269                // linear coefficients of equations
    270                // y(ti) - y(t0) - di y'(t0) and y'(ti) - y'(t0)
    271                final double[] aI    = a[2 * i - 2];
    272                final double[] aDotI = a[2 * i - 1];
    273                for (int j = 0; j < aI.length; ++j) {
    274                    dikM1Ohk *= ratio;
    275                    aI[j]     = di      * dikM1Ohk;
    276                    aDotI[j]  = (j + 2) * dikM1Ohk;
    277                }
    278
    279                // expected value of the previous equations
    280                final double[] yI    = y[i];
    281                final double[] yDotI = yDot[i];
    282                final double[] bI    = b[2 * i - 2];
    283                final double[] bDotI = b[2 * i - 1];
    284                for (int j = 0; j < yI.length; ++j) {
    285                    bI[j]    = yI[j] - y0[j] - di * yDot0[j];
    286                    bDotI[j] = yDotI[j] - yDot0[j];
    287                }
    288
    289            }
    290
    291            // solve the rectangular system in the least square sense
    292            // to get the best estimate of the Nordsieck vector [s2 ... sk]
    293            QRDecomposition decomposition;
    294            decomposition = new QRDecomposition(new Array2DRowRealMatrix(a, false));
    295            RealMatrix x = decomposition.getSolver().solve(new Array2DRowRealMatrix(b, false));
    296            return new Array2DRowRealMatrix(x.getData(), false);
    297        }
    298
    299        /** Update the high order scaled derivatives for Adams integrators (phase 1).
    300         * <p>The complete update of high order derivatives has a form similar to:
    301         * <pre>
    302         * 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>
    303         * </pre>
    304         * this method computes the P<sup>-1</sup> A P r<sub>n</sub> part.</p>
    305         * @param highOrder high order scaled derivatives
    306         * (h<sup>2</sup>/2 y'', ... h<sup>k</sup>/k! y(k))
    307         * @return updated high order derivatives
    308         * @see #updateHighOrderDerivativesPhase2(double[], double[], Array2DRowRealMatrix)
    309         */
    310        public Array2DRowRealMatrix updateHighOrderDerivativesPhase1(final Array2DRowRealMatrix highOrder) {
    311            return update.multiply(highOrder);
    312        }
    313
    314        /** Update the high order scaled derivatives Adams integrators (phase 2).
    315         * <p>The complete update of high order derivatives has a form similar to:
    316         * <pre>
    317         * 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>
    318         * </pre>
    319         * this method computes the (s<sub>1</sub>(n) - s<sub>1</sub>(n+1)) P<sup>-1</sup> u part.</p>
    320         * <p>Phase 1 of the update must already have been performed.</p>
    321         * @param start first order scaled derivatives at step start
    322         * @param end first order scaled derivatives at step end
    323         * @param highOrder high order scaled derivatives, will be modified
    324         * (h<sup>2</sup>/2 y'', ... h<sup>k</sup>/k! y(k))
    325         * @see #updateHighOrderDerivativesPhase1(Array2DRowRealMatrix)
    326         */
    327        public void updateHighOrderDerivativesPhase2(final double[] start,
    328                                                     final double[] end,
    329                                                     final Array2DRowRealMatrix highOrder) {
    330            final double[][] data = highOrder.getDataRef();
    331            for (int i = 0; i < data.length; ++i) {
    332                final double[] dataI = data[i];
    333                final double c1I = c1[i];
    334                for (int j = 0; j < dataI.length; ++j) {
    335                    dataI[j] += c1I * (start[j] - end[j]);
    336                }
    337            }
    338        }
    339
    340    }