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    *      https://www.apache.org/licenses/LICENSE-2.0
   *
   * Unless required by applicable law or agreed to in writing, software
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   * See the License for the specific language governing permissions and
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  package org.hipparchus.ode.nonstiff;
  
  import java.util.Arrays;
  
  import org.hipparchus.exception.MathIllegalArgumentException;
  import org.hipparchus.exception.MathIllegalStateException;
  import org.hipparchus.linear.Array2DRowRealMatrix;
  import org.hipparchus.linear.RealMatrix;
  import org.hipparchus.linear.RealMatrixPreservingVisitor;
  import org.hipparchus.ode.EquationsMapper;
  import org.hipparchus.ode.LocalizedODEFormats;
  import org.hipparchus.ode.ODEStateAndDerivative;
  import org.hipparchus.util.FastMath;
  
  
  /**
   * This class implements implicit Adams-Moulton integrators for Ordinary
   * Differential Equations.
   *
   * <p>Adams-Moulton methods (in fact due to Adams alone) are implicit
   * multistep ODE solvers. This implementation is a variation of the classical
   * one: it uses adaptive stepsize to implement error control, whereas
   * classical implementations are fixed step size. The value of state vector
   * at step n+1 is a simple combination of the value at step n and of the
   * derivatives at steps n+1, n, n-1 ... Since y'<sub>n+1</sub> is needed to
   * compute y<sub>n+1</sub>, another method must be used to compute a first
   * estimate of y<sub>n+1</sub>, then compute y'<sub>n+1</sub>, then compute
   * a final estimate of y<sub>n+1</sub> using the following formulas. Depending
   * on the number k of previous steps one wants to use for computing the next
   * value, different formulas are available for the final estimate:</p>
   * <ul>
   *   <li>k = 1: y<sub>n+1</sub> = y<sub>n</sub> + h y'<sub>n+1</sub></li>
   *   <li>k = 2: y<sub>n+1</sub> = y<sub>n</sub> + h (y'<sub>n+1</sub>+y'<sub>n</sub>)/2</li>
   *   <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>
   *   <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>
   *   <li>...</li>
   * </ul>
   *
   * <p>A k-steps Adams-Moulton method is of order k+1.</p>
   *
   * <p> There must be sufficient time for the {@link #setStarterIntegrator(org.hipparchus.ode.ODEIntegrator)
   * starter integrator} to take several steps between the the last reset event, and the end
   * of integration, otherwise an exception may be thrown during integration. The user can
   * adjust the end date of integration, or the step size of the starter integrator to
   * ensure a sufficient number of steps can be completed before the end of integration.
   * </p>
   *
   * <p><strong>Implementation details</strong></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>The definitions above use the classical representation with several previous first
   * derivatives. Lets define
   * \[
   *   q_n = [ s_1(n-1) s_1(n-2) \ldots s_1(n-(k-1)) ]^T
   * \]
   * (we omit the k index in the notation for clarity). With these definitions,
   * Adams-Moulton methods can be written:</p>
   * <ul>
   *   <li>k = 1: y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n+1)</li>
   *   <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>
   *   <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>
   *   <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>
   *   <li>...</li>
   * </ul>
   *
   * <p>Instead of using the classical representation with first derivatives only (y<sub>n</sub>,
   * s<sub>1</sub>(n+1) and q<sub>n+1</sub>), our implementation uses the Nordsieck vector with
   * higher degrees scaled derivatives all taken at the same step (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. 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>Using the Nordsieck vector has several advantages:</p>
  * <ul>
  *   <li>it greatly simplifies step interpolation as the interpolator mainly applies
  *   Taylor series formulas,</li>
  *   <li>it simplifies step changes that occur when discrete events that truncate
  *   the step are triggered,</li>
  *   <li>it allows to extend the methods in order to support adaptive stepsize.</li>
  * </ul>
  *
  * <p>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>
  * 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>
  * 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>The P<sup>-1</sup>u vector and the P<sup>-1</sup> A P matrix do not depend on the state,
  * they only depend on k and therefore are precomputed once for all.</p>
  *
  */
 public class AdamsMoultonIntegrator extends AdamsIntegrator {
 
     /** Name of integration scheme. */
     public static final String METHOD_NAME = "Adams-Moulton";
 
     /**
      * Build an Adams-Moulton integrator with the given order and error control parameters.
      * @param nSteps number of steps of the method excluding the one being computed
      * @param minStep minimal step (sign is irrelevant, regardless of
      * integration direction, forward or backward), the last step can
      * be smaller than this
      * @param maxStep maximal step (sign is irrelevant, regardless of
      * integration direction, forward or backward), the last step can
      * be smaller than this
      * @param scalAbsoluteTolerance allowed absolute error
      * @param scalRelativeTolerance allowed relative error
      * @exception MathIllegalArgumentException if order is 1 or less
      */
     public AdamsMoultonIntegrator(final int nSteps,
                                   final double minStep, final double maxStep,
                                   final double scalAbsoluteTolerance,
                                   final double scalRelativeTolerance)
         throws MathIllegalArgumentException {
         super(METHOD_NAME, nSteps, nSteps + 1, minStep, maxStep,
               scalAbsoluteTolerance, scalRelativeTolerance);
     }
 
     /**
      * Build an Adams-Moulton integrator with the given order and error control parameters.
      * @param nSteps number of steps of the method excluding the one being computed
      * @param minStep minimal step (sign is irrelevant, regardless of
      * integration direction, forward or backward), the last step can
      * be smaller than this
      * @param maxStep maximal step (sign is irrelevant, regardless of
      * integration direction, forward or backward), the last step can
      * be smaller than this
      * @param vecAbsoluteTolerance allowed absolute error
      * @param vecRelativeTolerance allowed relative error
      * @exception IllegalArgumentException if order is 1 or less
      */
     public AdamsMoultonIntegrator(final int nSteps,
                                   final double minStep, final double maxStep,
                                   final double[] vecAbsoluteTolerance,
                                   final double[] vecRelativeTolerance)
         throws IllegalArgumentException {
         super(METHOD_NAME, nSteps, nSteps + 1, minStep, maxStep,
               vecAbsoluteTolerance, vecRelativeTolerance);
     }
 
     /** {@inheritDoc} */
     @Override
     protected double errorEstimation(final double[] previousState, final double predictedTime,
                                      final double[] predictedState,
                                      final double[] predictedScaled,
                                      final RealMatrix predictedNordsieck) {
         final double error = predictedNordsieck.walkInOptimizedOrder(new Corrector(previousState, predictedScaled, predictedState));
         if (Double.isNaN(error)) {
             throw new MathIllegalStateException(LocalizedODEFormats.NAN_APPEARING_DURING_INTEGRATION,
                                                 predictedTime);
         }
         return error;
     }
 
     /** {@inheritDoc} */
     @Override
     protected AdamsStateInterpolator finalizeStep(final double stepSize, final double[] predictedState,
                                                   final double[] predictedScaled, final Array2DRowRealMatrix predictedNordsieck,
                                                   final boolean isForward,
                                                   final ODEStateAndDerivative globalPreviousState,
                                                   final ODEStateAndDerivative globalCurrentState,
                                                   final EquationsMapper equationsMapper) {
 
         final double[] correctedYDot = computeDerivatives(globalCurrentState.getTime(), predictedState);
 
         // update Nordsieck vector
         final double[] correctedScaled = new double[predictedState.length];
         for (int j = 0; j < correctedScaled.length; ++j) {
             correctedScaled[j] = getStepSize() * correctedYDot[j];
         }
         updateHighOrderDerivativesPhase2(predictedScaled, correctedScaled, predictedNordsieck);
 
         final ODEStateAndDerivative updatedStepEnd =
                         equationsMapper.mapStateAndDerivative(globalCurrentState.getTime(),
                                                               predictedState, correctedYDot);
         return new AdamsStateInterpolator(getStepSize(), updatedStepEnd,
                                           correctedScaled, predictedNordsieck, isForward,
                                           getStepStart(), updatedStepEnd,
                                           equationsMapper);
 
     }
 
     /** Corrector for current state in Adams-Moulton method.
      * <p>
      * This visitor implements the Taylor series formula:
      * <pre>
      * 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>
      * </pre>
      * </p>
      */
     private class Corrector implements RealMatrixPreservingVisitor {
 
         /** Previous state. */
         private final double[] previous;
 
         /** Current scaled first derivative. */
         private final double[] scaled;
 
         /** Current state before correction. */
         private final double[] before;
 
         /** Current state after correction. */
         private final double[] after;
 
         /** Simple constructor.
          * <p>
          * All arrays will be stored by reference to caller arrays.
          * </p>
          * @param previous previous state
          * @param scaled current scaled first derivative
          * @param state state to correct (will be overwritten after visit)
          */
         Corrector(final double[] previous, final double[] scaled, final double[] state) {
             this.previous = previous; // NOPMD - array reference storage is intentional and documented here
             this.scaled   = scaled;   // NOPMD - array reference storage is intentional and documented here
             this.after    = state;    // NOPMD - array reference storage is intentional and documented here
             this.before   = state.clone();
         }
 
         /** {@inheritDoc} */
         @Override
         public void start(int rows, int columns,
                           int startRow, int endRow, int startColumn, int endColumn) {
             Arrays.fill(after, 0.0);
         }
 
         /** {@inheritDoc} */
         @Override
         public void visit(int row, int column, double value) {
             if ((row & 0x1) == 0) {
                 after[column] -= value;
             } else {
                 after[column] += value;
             }
         }
 
         /**
          * End visiting the Nordsieck vector.
          * <p>The correction is used to control stepsize. So its amplitude is
          * considered to be an error, which must be normalized according to
          * error control settings. If the normalized value is greater than 1,
          * the correction was too large and the step must be rejected.</p>
          * @return the normalized correction, if greater than 1, the step
          * must be rejected
          */
         @Override
         public double end() {
 
             final StepsizeHelper helper = getStepSizeHelper();
             double error = 0;
             for (int i = 0; i < after.length; ++i) {
                 after[i] += previous[i] + scaled[i];
                 if (i < helper.getMainSetDimension()) {
                     final double tol   = helper.getTolerance(i, FastMath.max(FastMath.abs(previous[i]), FastMath.abs(after[i])));
                     final double ratio = (after[i] - before[i]) / tol; // (corrected-predicted)/tol
                     error += ratio * ratio;
                 }
             }
 
             return FastMath.sqrt(error / helper.getMainSetDimension());
 
         }
     }
 
 }