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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:
   * \[
   *   \left\{\begin{align}
   *   s_1(n) &amp;= h y'_n \text{ for first derivative}\\
   *   s_2(n) &amp;= \frac{h^2}{2} y_n'' \text{ for second derivative}\\
   *   s_3(n) &amp;= \frac{h^3}{6} y_n''' \text{ for third derivative}\\
   *   &amp;\cdots\\
   *   s_k(n) &amp;= \frac{h^k}{k!} y_n^{(k)} \text{ for } k^\mathrm{th} \text{ derivative}
   *   \end{align}\right.
   * \]</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:
   * \[
   *   q_n = [ s_1(n-1) s_1(n-2) \ldots s_1(n-(k-1)) ]^T
   * \]
   * (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:
   * \[
   * r_n = [ s_2(n), s_3(n) \ldots s_k(n) ]^T
   * \]
   * (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.
   * \[
   * s_1(n-i) = s_1(n) + \sum_{j\gt 0} (j+1) (-i)^j s_{j+1}(n)
   * \]
   * 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:
   * \[
   * q_n = s_1(n) u + P r_n
   * \]
   * where u is the [ 1 1 ... 1 ]<sup>T</sup> vector and P is the (k-1)&times;(k-1) matrix built
   * with the \((j+1) (-i)^j\) terms with i being the row number starting from 1 and j being
   * the column number starting from 1:
   * \[
   *   P=\begin{bmatrix}
   *   -2  &amp;  3 &amp;   -4 &amp;    5 &amp; \ldots \\
   *   -4  &amp; 12 &amp;  -32 &amp;   80 &amp; \ldots \\
   *   -6  &amp; 27 &amp; -108 &amp;  405 &amp; \ldots \\
   *   -8  &amp; 48 &amp; -256 &amp; 1280 &amp; \ldots \\
   *       &amp;    &amp;  \ldots\\
   *    \end{bmatrix}
   * \]
   *
   * <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>
  * <p>where A is a rows shifting matrix (the lower left part is an identity matrix):</p>
  * <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>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>
  * <p>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 \((j+1) (-i)^j\) 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:
      * \[
      * r_{n+1} = (s_1(n) - s_1(n+1)) P^{-1} u + P^{-1} A P r_n
      * \]
      * 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:
      * \[
      * r_{n+1} = (s_1(n) - s_1(n+1)) P^{-1} u + P^{-1} A P r_n
      * \]
      * 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]);
             }
         }
     }
 
 }