AdamsNordsieckTransformer.java

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package org.hipparchus.ode.nonstiff;

import java.util.Arrays;
import java.util.HashMap;
import java.util.Map;

import org.hipparchus.fraction.BigFraction;
import org.hipparchus.linear.Array2DRowFieldMatrix;
import org.hipparchus.linear.Array2DRowRealMatrix;
import org.hipparchus.linear.ArrayFieldVector;
import org.hipparchus.linear.FieldDecompositionSolver;
import org.hipparchus.linear.FieldLUDecomposition;
import org.hipparchus.linear.FieldMatrix;
import org.hipparchus.linear.MatrixUtils;
import org.hipparchus.linear.QRDecomposition;
import org.hipparchus.linear.RealMatrix;

/** Transformer to Nordsieck vectors for Adams integrators.
 * <p>This class is used by {@link AdamsBashforthIntegrator Adams-Bashforth} and
 * {@link AdamsMoultonIntegrator Adams-Moulton} integrators to convert between
 * classical representation with several previous first derivatives and Nordsieck
 * representation with higher order scaled derivatives.</p>
 *
 * <p>We define scaled derivatives s<sub>i</sub>(n) at step n as:
 * <pre>
 * s<sub>1</sub>(n) = h y'<sub>n</sub> for first derivative
 * s<sub>2</sub>(n) = h<sup>2</sup>/2 y''<sub>n</sub> for second derivative
 * s<sub>3</sub>(n) = h<sup>3</sup>/6 y'''<sub>n</sub> for third derivative
 * ...
 * s<sub>k</sub>(n) = h<sup>k</sup>/k! y<sup>(k)</sup><sub>n</sub> for k<sup>th</sup> derivative
 * </pre></p>
 *
 * <p>With the previous definition, the classical representation of multistep methods
 * uses first derivatives only, i.e. it handles y<sub>n</sub>, s<sub>1</sub>(n) and
 * q<sub>n</sub> where q<sub>n</sub> is defined as:
 * <pre>
 *   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>
 * </pre>
 * (we omit the k index in the notation for clarity).</p>
 *
 * <p>Another possible representation uses the Nordsieck vector with
 * higher degrees scaled derivatives all taken at the same step, i.e it handles y<sub>n</sub>,
 * s<sub>1</sub>(n) and r<sub>n</sub>) where r<sub>n</sub> is defined as:
 * <pre>
 * r<sub>n</sub> = [ s<sub>2</sub>(n), s<sub>3</sub>(n) ... s<sub>k</sub>(n) ]<sup>T</sup>
 * </pre>
 * (here again we omit the k index in the notation for clarity)
 * </p>
 *
 * <p>Taylor series formulas show that for any index offset i, s<sub>1</sub>(n-i) can be
 * computed from s<sub>1</sub>(n), s<sub>2</sub>(n) ... s<sub>k</sub>(n), the formula being exact
 * for degree k polynomials.
 * <pre>
 * s<sub>1</sub>(n-i) = s<sub>1</sub>(n) + &sum;<sub>j&gt;0</sub> (j+1) (-i)<sup>j</sup> s<sub>j+1</sub>(n)
 * </pre>
 * The previous formula can be used with several values for i to compute the transform between
 * classical representation and Nordsieck vector at step end. The transform between r<sub>n</sub>
 * and q<sub>n</sub> resulting from the Taylor series formulas above is:
 * <pre>
 * q<sub>n</sub> = s<sub>1</sub>(n) u + P r<sub>n</sub>
 * </pre>
 * where u is the [ 1 1 ... 1 ]<sup>T</sup> vector and P is the (k-1)&times;(k-1) matrix built
 * with the (j+1) (-i)<sup>j</sup> terms with i being the row number starting from 1 and j being
 * the column number starting from 1:
 * <pre>
 *        [  -2   3   -4    5  ... ]
 *        [  -4  12  -32   80  ... ]
 *   P =  [  -6  27 -108  405  ... ]
 *        [  -8  48 -256 1280  ... ]
 *        [          ...           ]
 * </pre></p>
 *
 * <p>Changing -i into +i in the formula above can be used to compute a similar transform between
 * classical representation and Nordsieck vector at step start. The resulting matrix is simply
 * the absolute value of matrix P.</p>
 *
 * <p>For {@link AdamsBashforthIntegrator Adams-Bashforth} method, the Nordsieck vector
 * at step n+1 is computed from the Nordsieck vector at step n as follows:
 * <ul>
 *   <li>y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n) + u<sup>T</sup> r<sub>n</sub></li>
 *   <li>s<sub>1</sub>(n+1) = h f(t<sub>n+1</sub>, y<sub>n+1</sub>)</li>
 *   <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>
 * </ul>
 * where A is a rows shifting matrix (the lower left part is an identity matrix):
 * <pre>
 *        [ 0 0   ...  0 0 | 0 ]
 *        [ ---------------+---]
 *        [ 1 0   ...  0 0 | 0 ]
 *    A = [ 0 1   ...  0 0 | 0 ]
 *        [       ...      | 0 ]
 *        [ 0 0   ...  1 0 | 0 ]
 *        [ 0 0   ...  0 1 | 0 ]
 * </pre></p>
 *
 * <p>For {@link AdamsMoultonIntegrator Adams-Moulton} method, the predicted Nordsieck vector
 * at step n+1 is computed from the Nordsieck vector at step n as follows:
 * <ul>
 *   <li>Y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n) + u<sup>T</sup> r<sub>n</sub></li>
 *   <li>S<sub>1</sub>(n+1) = h f(t<sub>n+1</sub>, Y<sub>n+1</sub>)</li>
 *   <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>
 * </ul>
 * From this predicted vector, the corrected vector is computed as follows:
 * <ul>
 *   <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>
 *   <li>s<sub>1</sub>(n+1) = h f(t<sub>n+1</sub>, y<sub>n+1</sub>)</li>
 *   <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>
 * </ul>
 * where the upper case Y<sub>n+1</sub>, S<sub>1</sub>(n+1) and R<sub>n+1</sub> represent the
 * predicted states whereas the lower case y<sub>n+1</sub>, s<sub>n+1</sub> and r<sub>n+1</sub>
 * represent the corrected states.</p>
 *
 * <p>We observe that both methods use similar update formulas. In both cases a P<sup>-1</sup>u
 * vector and a P<sup>-1</sup> A P matrix are used that do not depend on the state,
 * they only depend on k. This class handles these transformations.</p>
 *
 */
public class AdamsNordsieckTransformer {

    /** Cache for already computed coefficients. */
    private static final Map<Integer, AdamsNordsieckTransformer> CACHE = new HashMap<>();

    /** Update matrix for the higher order derivatives h<sup>2</sup>/2 y'', h<sup>3</sup>/6 y''' ... */
    private final Array2DRowRealMatrix update;

    /** Update coefficients of the higher order derivatives wrt y'. */
    private final double[] c1;

    /** Simple constructor.
     * @param n number of steps of the multistep method
     * (excluding the one being computed)
     */
    private AdamsNordsieckTransformer(final int n) {

        final int rows = n - 1;

        // compute exact coefficients
        FieldMatrix<BigFraction> bigP = buildP(rows);
        FieldDecompositionSolver<BigFraction> pSolver =
            new FieldLUDecomposition<BigFraction>(bigP).getSolver();

        BigFraction[] u = new BigFraction[rows];
        Arrays.fill(u, BigFraction.ONE);
        BigFraction[] bigC1 = pSolver.solve(new ArrayFieldVector<BigFraction>(u, false)).toArray();

        // update coefficients are computed by combining transform from
        // Nordsieck to multistep, then shifting rows to represent step advance
        // then applying inverse transform
        BigFraction[][] shiftedP = bigP.getData();
        for (int i = shiftedP.length - 1; i > 0; --i) {
            // shift rows
            shiftedP[i] = shiftedP[i - 1];
        }
        shiftedP[0] = new BigFraction[rows];
        Arrays.fill(shiftedP[0], BigFraction.ZERO);
        FieldMatrix<BigFraction> bigMSupdate =
            pSolver.solve(new Array2DRowFieldMatrix<BigFraction>(shiftedP, false));

        // convert coefficients to double
        update         = MatrixUtils.bigFractionMatrixToRealMatrix(bigMSupdate);
        c1             = new double[rows];
        for (int i = 0; i < rows; ++i) {
            c1[i] = bigC1[i].doubleValue();
        }

    }

    /** Get the Nordsieck transformer for a given number of steps.
     * @param nSteps number of steps of the multistep method
     * (excluding the one being computed)
     * @return Nordsieck transformer for the specified number of steps
     */
    public static AdamsNordsieckTransformer getInstance(final int nSteps) { // NOPMD - PMD false positive
        synchronized(CACHE) {
            AdamsNordsieckTransformer t = CACHE.get(nSteps);
            if (t == null) {
                t = new AdamsNordsieckTransformer(nSteps);
                CACHE.put(nSteps, t);
            }
            return t;
        }
    }

    /** Build the P matrix.
     * <p>The P matrix general terms are shifted (j+1) (-i)<sup>j</sup> terms
     * with i being the row number starting from 1 and j being the column
     * number starting from 1:
     * <pre>
     *        [  -2   3   -4    5  ... ]
     *        [  -4  12  -32   80  ... ]
     *   P =  [  -6  27 -108  405  ... ]
     *        [  -8  48 -256 1280  ... ]
     *        [          ...           ]
     * </pre></p>
     * @param rows number of rows of the matrix
     * @return P matrix
     */
    private FieldMatrix<BigFraction> buildP(final int rows) {

        final BigFraction[][] pData = new BigFraction[rows][rows];

        for (int i = 1; i <= pData.length; ++i) {
            // build the P matrix elements from Taylor series formulas
            final BigFraction[] pI = pData[i - 1];
            final int factor = -i;
            int aj = factor;
            for (int j = 1; j <= pI.length; ++j) {
                pI[j - 1] = new BigFraction(aj * (j + 1));
                aj *= factor;
            }
        }

        return new Array2DRowFieldMatrix<BigFraction>(pData, false);

    }

    /** Initialize the high order scaled derivatives at step start.
     * @param h step size to use for scaling
     * @param t first steps times
     * @param y first steps states
     * @param yDot first steps derivatives
     * @return Nordieck vector at start of first step (h<sup>2</sup>/2 y''<sub>n</sub>,
     * h<sup>3</sup>/6 y'''<sub>n</sub> ... h<sup>k</sup>/k! y<sup>(k)</sup><sub>n</sub>)
     */

    public Array2DRowRealMatrix initializeHighOrderDerivatives(final double h, final double[] t,
                                                               final double[][] y,
                                                               final double[][] yDot) {

        // using Taylor series with di = ti - t0, we get:
        //  y(ti)  - y(t0)  - di y'(t0) =   di^2 / h^2 s2 + ... +   di^k     / h^k sk + O(h^k)
        //  y'(ti) - y'(t0)             = 2 di   / h^2 s2 + ... + k di^(k-1) / h^k sk + O(h^(k-1))
        // we write these relations for i = 1 to i= 1+n/2 as a set of n + 2 linear
        // equations depending on the Nordsieck vector [s2 ... sk rk], so s2 to sk correspond
        // to the appropriately truncated Taylor expansion, and rk is the Taylor remainder.
        // The goal is to have s2 to sk as accurate as possible considering the fact the sum is
        // truncated and we don't want the error terms to be included in s2 ... sk, so we need
        // to solve also for the remainder
        final double[][] a     = new double[c1.length + 1][c1.length + 1];
        final double[][] b     = new double[c1.length + 1][y[0].length];
        final double[]   y0    = y[0];
        final double[]   yDot0 = yDot[0];
        for (int i = 1; i < y.length; ++i) {

            final double di    = t[i] - t[0];
            final double ratio = di / h;
            double dikM1Ohk    =  1 / h;

            // linear coefficients of equations
            // y(ti) - y(t0) - di y'(t0) and y'(ti) - y'(t0)
            final double[] aI    = a[2 * i - 2];
            final double[] aDotI = (2 * i - 1) < a.length ? a[2 * i - 1] : null;
            for (int j = 0; j < aI.length; ++j) {
                dikM1Ohk *= ratio;
                aI[j]     = di      * dikM1Ohk;
                if (aDotI != null) {
                    aDotI[j]  = (j + 2) * dikM1Ohk;
                }
            }

            // expected value of the previous equations
            final double[] yI    = y[i];
            final double[] yDotI = yDot[i];
            final double[] bI    = b[2 * i - 2];
            final double[] bDotI = (2 * i - 1) < b.length ? b[2 * i - 1] : null;
            for (int j = 0; j < yI.length; ++j) {
                bI[j]    = yI[j] - y0[j] - di * yDot0[j];
                if (bDotI != null) {
                    bDotI[j] = yDotI[j] - yDot0[j];
                }
            }

        }

        // solve the linear system to get the best estimate of the Nordsieck vector [s2 ... sk],
        // with the additional terms s(k+1) and c grabbing the parts after the truncated Taylor expansion
        final QRDecomposition decomposition = new QRDecomposition(new Array2DRowRealMatrix(a, false));
        final RealMatrix x = decomposition.getSolver().solve(new Array2DRowRealMatrix(b, false));

        // extract just the Nordsieck vector [s2 ... sk]
        final Array2DRowRealMatrix truncatedX = new Array2DRowRealMatrix(x.getRowDimension() - 1, x.getColumnDimension());
        for (int i = 0; i < truncatedX.getRowDimension(); ++i) {
            for (int j = 0; j < truncatedX.getColumnDimension(); ++j) {
                truncatedX.setEntry(i, j, x.getEntry(i, j));
            }
        }
        return truncatedX;

    }

    /** Update the high order scaled derivatives for Adams integrators (phase 1).
     * <p>The complete update of high order derivatives has a form similar to:
     * <pre>
     * 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>
     * </pre>
     * this method computes the P<sup>-1</sup> A P r<sub>n</sub> part.</p>
     * @param highOrder high order scaled derivatives
     * (h<sup>2</sup>/2 y'', ... h<sup>k</sup>/k! y(k))
     * @return updated high order derivatives
     * @see #updateHighOrderDerivativesPhase2(double[], double[], Array2DRowRealMatrix)
     */
    public Array2DRowRealMatrix updateHighOrderDerivativesPhase1(final Array2DRowRealMatrix highOrder) {
        return update.multiply(highOrder);
    }

    /** Update the high order scaled derivatives Adams integrators (phase 2).
     * <p>The complete update of high order derivatives has a form similar to:
     * <pre>
     * 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>
     * </pre>
     * this method computes the (s<sub>1</sub>(n) - s<sub>1</sub>(n+1)) P<sup>-1</sup> u part.</p>
     * <p>Phase 1 of the update must already have been performed.</p>
     * @param start first order scaled derivatives at step start
     * @param end first order scaled derivatives at step end
     * @param highOrder high order scaled derivatives, will be modified
     * (h<sup>2</sup>/2 y'', ... h<sup>k</sup>/k! y(k))
     * @see #updateHighOrderDerivativesPhase1(Array2DRowRealMatrix)
     */
    public void updateHighOrderDerivativesPhase2(final double[] start,
                                                 final double[] end,
                                                 final Array2DRowRealMatrix highOrder) {
        final double[][] data = highOrder.getDataRef();
        for (int i = 0; i < data.length; ++i) {
            final double[] dataI = data[i];
            final double c1I = c1[i];
            for (int j = 0; j < dataI.length; ++j) {
                dataI[j] += c1I * (start[j] - end[j]);
            }
        }
    }

}