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    * Licensed to the Apache Software Foundation (ASF) under one or more
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    * this work for additional information regarding copyright ownership.
    * The ASF licenses this file to You under the Apache License, Version 2.0
    * (the "License"); you may not use this file except in compliance with
    * the License.  You may obtain a copy of the License at
    *
    *      https://www.apache.org/licenses/LICENSE-2.0
   *
   * Unless required by applicable law or agreed to in writing, software
   * distributed under the License is distributed on an "AS IS" BASIS,
   * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
   * See the License for the specific language governing permissions and
   * limitations under the License.
   */
  
  /*
   * This is not the original file distributed by the Apache Software Foundation
   * It has been modified by the Hipparchus project
   */
  
  package org.hipparchus.ode;
  
  import org.hipparchus.exception.MathIllegalArgumentException;
  import org.hipparchus.exception.MathIllegalStateException;
  import org.hipparchus.linear.Array2DRowRealMatrix;
  import org.hipparchus.ode.nonstiff.AdaptiveStepsizeIntegrator;
  import org.hipparchus.ode.nonstiff.DormandPrince853Integrator;
  import org.hipparchus.ode.sampling.ODEStateInterpolator;
  import org.hipparchus.ode.sampling.ODEStepHandler;
  import org.hipparchus.util.FastMath;
  
  /**
   * This class is the base class for multistep integrators for Ordinary
   * Differential Equations.
   * <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>Rather than storing several previous steps separately, this 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:
   * \[
   * r_n = [ s_2(n), s_3(n) \ldots s_k(n) ]^T
   * \]
   * (we omit the k index in the notation for clarity)</p>
   * <p>
   * Multistep integrators with Nordsieck representation are highly sensitive to
   * large step changes because when the step is multiplied by factor a, the
   * k<sup>th</sup> component of the Nordsieck vector is multiplied by a<sup>k</sup>
   * and the last components are the least accurate ones. The default max growth
   * factor is therefore set to a quite low value: 2<sup>1/order</sup>.
   * </p>
   *
   * @see org.hipparchus.ode.nonstiff.AdamsBashforthIntegrator
   * @see org.hipparchus.ode.nonstiff.AdamsMoultonIntegrator
   */
  public abstract class MultistepIntegrator extends AdaptiveStepsizeIntegrator {
  
      /** First scaled derivative (h y'). */
      protected double[] scaled;
  
      /** Nordsieck matrix of the higher scaled derivatives.
       * <p>(h<sup>2</sup>/2 y'', h<sup>3</sup>/6 y''' ..., h<sup>k</sup>/k! y<sup>(k)</sup>)</p>
       */
      protected Array2DRowRealMatrix nordsieck;
  
      /** Starter integrator. */
      private ODEIntegrator starter;
  
      /** Number of steps of the multistep method (excluding the one being computed). */
      private final int nSteps;
  
      /** Stepsize control exponent. */
      private double exp;
  
      /** Safety factor for stepsize control. */
      private double safety;
  
      /** Minimal reduction factor for stepsize control. */
      private double minReduction;
  
      /** Maximal growth factor for stepsize control. */
      private double maxGrowth;
  
      /**
       * Build a multistep integrator with the given stepsize bounds.
       * <p>The default starter integrator is set to the {@link
       * DormandPrince853Integrator Dormand-Prince 8(5,3)} integrator with
       * some defaults settings.</p>
       * <p>
       * The default max growth factor is set to a quite low value: 2<sup>1/order</sup>.
      * </p>
      * @param name name of the method
      * @param nSteps number of steps of the multistep method
      * (excluding the one being computed)
      * @param order order of the method
      * @param minStep minimal step (must be positive even for backward
      * integration), the last step can be smaller than this
      * @param maxStep maximal step (must be positive even for backward
      * integration)
      * @param scalAbsoluteTolerance allowed absolute error
      * @param scalRelativeTolerance allowed relative error
      * @exception MathIllegalArgumentException if number of steps is smaller than 2
      */
     protected MultistepIntegrator(final String name, final int nSteps,
                                   final int order,
                                   final double minStep, final double maxStep,
                                   final double scalAbsoluteTolerance,
                                   final double scalRelativeTolerance)
         throws MathIllegalArgumentException {
 
         super(name, minStep, maxStep, scalAbsoluteTolerance, scalRelativeTolerance);
 
         if (nSteps < 2) {
             throw new MathIllegalArgumentException(LocalizedODEFormats.INTEGRATION_METHOD_NEEDS_AT_LEAST_TWO_PREVIOUS_POINTS,
                                                    nSteps, 2, true);
         }
 
         starter = new DormandPrince853Integrator(minStep, maxStep,
                                                  scalAbsoluteTolerance,
                                                  scalRelativeTolerance);
         this.nSteps = nSteps;
 
         exp = -1.0 / order;
 
         // set the default values of the algorithm control parameters
         setSafety(0.9);
         setMinReduction(0.2);
         setMaxGrowth(FastMath.pow(2.0, -exp));
 
     }
 
     /**
      * Build a multistep integrator with the given stepsize bounds.
      * <p>The default starter integrator is set to the {@link
      * DormandPrince853Integrator Dormand-Prince 8(5,3)} integrator with
      * some defaults settings.</p>
      * <p>
      * The default max growth factor is set to a quite low value: 2<sup>1/order</sup>.
      * </p>
      * @param name name of the method
      * @param nSteps number of steps of the multistep method
      * (excluding the one being computed)
      * @param order order of the method
      * @param minStep minimal step (must be positive even for backward
      * integration), the last step can be smaller than this
      * @param maxStep maximal step (must be positive even for backward
      * integration)
      * @param vecAbsoluteTolerance allowed absolute error
      * @param vecRelativeTolerance allowed relative error
      */
     protected MultistepIntegrator(final String name, final int nSteps,
                                   final int order,
                                   final double minStep, final double maxStep,
                                   final double[] vecAbsoluteTolerance,
                                   final double[] vecRelativeTolerance) {
         super(name, minStep, maxStep, vecAbsoluteTolerance, vecRelativeTolerance);
 
         if (nSteps < 2) {
             throw new MathIllegalArgumentException(LocalizedODEFormats.INTEGRATION_METHOD_NEEDS_AT_LEAST_TWO_PREVIOUS_POINTS,
                                                    nSteps, 2, true);
         }
 
         starter = new DormandPrince853Integrator(minStep, maxStep,
                                                  vecAbsoluteTolerance,
                                                  vecRelativeTolerance);
         this.nSteps = nSteps;
 
         exp = -1.0 / order;
 
         // set the default values of the algorithm control parameters
         setSafety(0.9);
         setMinReduction(0.2);
         setMaxGrowth(FastMath.pow(2.0, -exp));
 
     }
 
     /**
      * Get the starter integrator.
      * @return starter integrator
      */
     public ODEIntegrator getStarterIntegrator() {
         return starter;
     }
 
     /**
      * Set the starter integrator.
      * <p>The various step and event handlers for this starter integrator
      * will be managed automatically by the multi-step integrator. Any
      * user configuration for these elements will be cleared before use.</p>
      * @param starterIntegrator starter integrator
      */
     public void setStarterIntegrator(ODEIntegrator starterIntegrator) {
         this.starter = starterIntegrator;
     }
 
     /** Start the integration.
      * <p>This method computes one step using the underlying starter integrator,
      * and initializes the Nordsieck vector at step start. The starter integrator
      * purpose is only to establish initial conditions, it does not really change
      * time by itself. The top level multistep integrator remains in charge of
      * handling time propagation and events handling as it will starts its own
      * computation right from the beginning. In a sense, the starter integrator
      * can be seen as a dummy one and so it will never trigger any user event nor
      * call any user step handler.</p>
      * @param equations complete set of differential equations to integrate
      * @param initialState initial state (time, primary and secondary state vectors)
      * @param finalTime target time for the integration
      * (can be set to a value smaller than {@code initialState.getTime()} for backward integration)
      * @exception MathIllegalArgumentException if arrays dimension do not match equations settings
      * @exception MathIllegalArgumentException if integration step is too small
      * @exception MathIllegalStateException if the number of functions evaluations is exceeded
      * @exception MathIllegalArgumentException if the location of an event cannot be bracketed
      */
     protected void start(final ExpandableODE equations, final ODEState initialState, final double finalTime)
         throws MathIllegalArgumentException, MathIllegalStateException {
 
         // make sure NO user events nor user step handlers are triggered,
         // this is the task of the top level integrator, not the task of the starter integrator
         starter.clearEventDetectors();
         starter.clearStepHandlers();
 
         // set up one specific step handler to extract initial Nordsieck vector
         starter.addStepHandler(new NordsieckInitializer((nSteps + 3) / 2));
 
         // start integration, expecting a InitializationCompletedMarkerException
         try {
 
             starter.integrate(getEquations(), initialState, finalTime);
 
             // we should not reach this step
             throw new MathIllegalStateException(LocalizedODEFormats.MULTISTEP_STARTER_STOPPED_EARLY);
 
         } catch (InitializationCompletedMarkerException icme) { // NOPMD
             // this is the expected nominal interruption of the start integrator
 
             // count the evaluations used by the starter
             getEvaluationsCounter().increment(starter.getEvaluations());
 
         }
 
         // remove the specific step handler
         starter.clearStepHandlers();
 
     }
 
     /** 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 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>)
      */
     protected abstract Array2DRowRealMatrix initializeHighOrderDerivatives(double h, double[] t, double[][] y, double[][] yDot);
 
     /** Get the minimal reduction factor for stepsize control.
      * @return minimal reduction factor
      */
     public double getMinReduction() {
         return minReduction;
     }
 
     /** Set the minimal reduction factor for stepsize control.
      * @param minReduction minimal reduction factor
      */
     public void setMinReduction(final double minReduction) {
         this.minReduction = minReduction;
     }
 
     /** Get the maximal growth factor for stepsize control.
      * @return maximal growth factor
      */
     public double getMaxGrowth() {
         return maxGrowth;
     }
 
     /** Set the maximal growth factor for stepsize control.
      * @param maxGrowth maximal growth factor
      */
     public void setMaxGrowth(final double maxGrowth) {
         this.maxGrowth = maxGrowth;
     }
 
     /** Get the safety factor for stepsize control.
      * @return safety factor
      */
     public double getSafety() {
       return safety;
     }
 
     /** Set the safety factor for stepsize control.
      * @param safety safety factor
      */
     public void setSafety(final double safety) {
       this.safety = safety;
     }
 
     /** Get the number of steps of the multistep method (excluding the one being computed).
      * @return number of steps of the multistep method (excluding the one being computed)
      */
     public int getNSteps() {
       return nSteps;
     }
 
     /** Rescale the instance.
      * <p>Since the scaled and Nordsieck arrays are shared with the caller,
      * this method has the side effect of rescaling this arrays in the caller too.</p>
      * @param newStepSize new step size to use in the scaled and Nordsieck arrays
      */
     protected void rescale(final double newStepSize) {
 
         final double ratio = newStepSize / getStepSize();
         for (int i = 0; i < scaled.length; ++i) {
             scaled[i] = scaled[i] * ratio;
         }
 
         final double[][] nData = nordsieck.getDataRef();
         double power = ratio;
         for (int i = 0; i < nData.length; ++i) {
             power = power * ratio;
             final double[] nDataI = nData[i];
             for (int j = 0; j < nDataI.length; ++j) {
                 nDataI[j] = nDataI[j] * power;
             }
         }
 
         setStepSize(newStepSize);
 
     }
 
     /** Compute step grow/shrink factor according to normalized error.
      * @param error normalized error of the current step
      * @return grow/shrink factor for next step
      */
     protected double computeStepGrowShrinkFactor(final double error) {
         return FastMath.min(maxGrowth, FastMath.max(minReduction, safety * FastMath.pow(error, exp)));
     }
 
     /** Specialized step handler storing the first step. */
     private class NordsieckInitializer implements ODEStepHandler {
 
         /** Steps counter. */
         private int count;
 
         /** Start of the integration. */
         private ODEStateAndDerivative savedStart;
 
         /** First steps times. */
         private final double[] t;
 
         /** First steps states. */
         private final double[][] y;
 
         /** First steps derivatives. */
         private final double[][] yDot;
 
         /** Simple constructor.
          * @param nbStartPoints number of start points (including the initial point)
          */
         NordsieckInitializer(final int nbStartPoints) {
             this.count  = 0;
             this.t      = new double[nbStartPoints];
             this.y      = new double[nbStartPoints][];
             this.yDot   = new double[nbStartPoints][];
         }
 
         /** {@inheritDoc} */
         @Override
         public void handleStep(ODEStateInterpolator interpolator) {
 
             if (count == 0) {
                 // first step, we need to store also the point at the beginning of the step
                 savedStart   = interpolator.getPreviousState();
                 t[0]    = savedStart.getTime();
                 y[0]    = savedStart.getCompleteState();
                 yDot[0] = savedStart.getCompleteDerivative();
             }
 
             // store the point at the end of the step
             ++count;
             final ODEStateAndDerivative curr = interpolator.getCurrentState();
             t[count]    = curr.getTime();
             y[count]    = curr.getCompleteState();
             yDot[count] = curr.getCompleteDerivative();
 
             if (count == t.length - 1) {
 
                 // this was the last point we needed, we can compute the derivatives
                 setStepStart(savedStart);
                 final double rawStep = (t[t.length - 1] - t[0]) / (t.length - 1);
                 setStepSize(getStepSizeHelper().filterStep(rawStep, rawStep >= 0, true));
 
                 // first scaled derivative
                 scaled = yDot[0].clone();
                 for (int j = 0; j < scaled.length; ++j) {
                     scaled[j] *= getStepSize();
                 }
 
                 // higher order derivatives
                 nordsieck = initializeHighOrderDerivatives(getStepSize(), t, y, yDot);
 
                 // stop the integrator now that all needed steps have been handled
                 throw new InitializationCompletedMarkerException();
 
             }
 
         }
 
     }
 
     /** Marker exception used ONLY to stop the starter integrator after first step. */
     private static class InitializationCompletedMarkerException
         extends RuntimeException {
 
         /** Serializable version identifier. */
         private static final long serialVersionUID = -1914085471038046418L;
 
         /** Simple constructor. */
         InitializationCompletedMarkerException() {
             super((Throwable) null);
         }
 
     }
 
 }