src/main/java/org/apache/commons/math/ode/nonstiff/AdamsNordsieckTransformer.java

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package org.apache.commons.math.ode.nonstiff;

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

import org.apache.commons.math.fraction.BigFraction;
import org.apache.commons.math.linear.Array2DRowFieldMatrix;
import org.apache.commons.math.linear.Array2DRowRealMatrix;
import org.apache.commons.math.linear.DefaultFieldMatrixChangingVisitor;
import org.apache.commons.math.linear.FieldDecompositionSolver;
import org.apache.commons.math.linear.FieldLUDecompositionImpl;
import org.apache.commons.math.linear.FieldMatrix;
import org.apache.commons.math.linear.MatrixUtils;

/** Transformer to Nordsieck vectors for Adams integrators.
 * <p>This class i 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(k)<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</sub> j (-i)<sup>j-1</sup> s<sub>j</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 (-i)<sup>j-1</sup> terms:
 * <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>
 *
 * @version $Revision: 810196 $ $Date: 2009-09-01 21:47:46 +0200 (mar. 01 sept. 2009) $
 * @since 2.0
 */
public class AdamsNordsieckTransformer {

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

    /** Initialization matrix for the higher order derivatives wrt y'', y''' ... */
    private final Array2DRowRealMatrix initialization;

    /** Update matrix for the higher order derivatives h<sup>2</sup>/2y'', 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 nSteps number of steps of the multistep method
     * (excluding the one being computed)
     */
    private AdamsNordsieckTransformer(final int nSteps) {

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

        BigFraction[] u = new BigFraction[nSteps];
        Arrays.fill(u, BigFraction.ONE);
        BigFraction[] bigC1 = pSolver.solve(u);

        // 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[nSteps];
        Arrays.fill(shiftedP[0], BigFraction.ZERO);
        FieldMatrix<BigFraction> bigMSupdate =
            pSolver.solve(new Array2DRowFieldMatrix<BigFraction>(shiftedP, false));

        // initialization coefficients, computed from a R matrix = abs(P)
        bigP.walkInOptimizedOrder(new DefaultFieldMatrixChangingVisitor<BigFraction>(BigFraction.ZERO) {
            /** {@inheritDoc} */
            @Override
            public BigFraction visit(int row, int column, BigFraction value) {
                return ((column & 0x1) == 0x1) ? value : value.negate();
            }
        });
        FieldMatrix<BigFraction> bigRInverse =
            new FieldLUDecompositionImpl<BigFraction>(bigP).getSolver().getInverse();

        // convert coefficients to double
        initialization = MatrixUtils.bigFractionMatrixToRealMatrix(bigRInverse);
        update         = MatrixUtils.bigFractionMatrixToRealMatrix(bigMSupdate);
        c1             = new double[nSteps];
        for (int i = 0; i < nSteps; ++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) {
        synchronized(CACHE) {
            AdamsNordsieckTransformer t = CACHE.get(nSteps);
            if (t == null) {
                t = new AdamsNordsieckTransformer(nSteps);
                CACHE.put(nSteps, t);
            }
            return t;
        }
    }

    /** Get the number of steps of the method
     * (excluding the one being computed).
     * @return number of steps of the method
     * (excluding the one being computed)
     */
    public int getNSteps() {
        return c1.length;
    }

    /** Build the P matrix.
     * <p>The P matrix general terms are shifted j (-i)<sup>j-1</sup> terms:
     * <pre>
     *        [  -2   3   -4    5  ... ]
     *        [  -4  12  -32   80  ... ]
     *   P =  [  -6  27 -108  405  ... ]
     *        [  -8  48 -256 1280  ... ]
     *        [          ...           ]
     * </pre></p>
     * @param nSteps number of steps of the multistep method
     * (excluding the one being computed)
     * @return P matrix
     */
    private FieldMatrix<BigFraction> buildP(final int nSteps) {

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

        for (int i = 0; i < pData.length; ++i) {
            // build the P matrix elements from Taylor series formulas
            final BigFraction[] pI = pData[i];
            final int factor = -(i + 1);
            int aj = factor;
            for (int j = 0; j < pI.length; ++j) {
                pI[j] = new BigFraction(aj * (j + 2));
                aj *= factor;
            }
        }

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

    }

    /** Initialize the high order scaled derivatives at step start.
     * @param first first scaled derivative at step start
     * @param multistep scaled derivatives after step start (hy'1, ..., hy'k-1)
     * will be modified
     * @return high order derivatives at step start
     */
    public Array2DRowRealMatrix initializeHighOrderDerivatives(final double[] first,
                                                     final double[][] multistep) {
        for (int i = 0; i < multistep.length; ++i) {
            final double[] msI = multistep[i];
            for (int j = 0; j < first.length; ++j) {
                msI[j] -= first[j];
            }
        }
        return initialization.multiply(new Array2DRowRealMatrix(multistep, false));
    }

    /** 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]);
            }
        }
    }

}