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 * 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
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package org.apache.commons.math3.ode;

import org.apache.commons.math3.exception.DimensionMismatchException;
import org.apache.commons.math3.exception.MathIllegalStateException;
import org.apache.commons.math3.exception.MaxCountExceededException;
import org.apache.commons.math3.exception.NoBracketingException;
import org.apache.commons.math3.exception.NumberIsTooSmallException;
import org.apache.commons.math3.exception.util.LocalizedFormats;
import org.apache.commons.math3.linear.Array2DRowRealMatrix;
import org.apache.commons.math3.ode.nonstiff.AdaptiveStepsizeIntegrator;
import org.apache.commons.math3.ode.nonstiff.DormandPrince853Integrator;
import org.apache.commons.math3.ode.sampling.StepHandler;
import org.apache.commons.math3.ode.sampling.StepInterpolator;
import org.apache.commons.math3.util.FastMath;

This class is the base class for multistep integrators for Ordinary
Differential Equations.
We define scaled derivatives si(n) at step n as:
s1(n) = h y'n for first derivative
s2(n) = h2/2 y''n for second derivative
s3(n) = h3/6 y'''n for third derivative
...
sk(n) = hk/k! y(k)n for kth derivative

Rather than storing several previous steps separately, this implementation uses
the Nordsieck vector with higher degrees scaled derivatives all taken at the same
step (yn, s1(n) and rn) where rn is defined as:
rn = [ s2(n), s3(n) ... sk(n) ]T

(we omit the k index in the notation for clarity)


Multistep integrators with Nordsieck representation are highly sensitive to
large step changes because when the step is multiplied by factor a, the
kth component of the Nordsieck vector is multiplied by ak
and the last components are the least accurate ones. The default max growth
factor is therefore set to a quite low value: 21/order.

See Also:
AdamsBashforthIntegrator
AdamsMoultonIntegrator
Since:
2.0/**
 * 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:
 * <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>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:
 * <pre>
 * r<sub>n</sub> = [ s<sub>2</sub>(n), s<sub>3</sub>(n) ... s<sub>k</sub>(n) ]<sup>T</sup>
 * </pre>
 * (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.apache.commons.math3.ode.nonstiff.AdamsBashforthIntegrator
 * @see org.apache.commons.math3.ode.nonstiff.AdamsMoultonIntegrator
 * @since 2.0
 */
public abstract class MultistepIntegrator extends AdaptiveStepsizeIntegrator {

    
First scaled derivative (h y'). /** First scaled derivative (h y'). */
    protected double[] scaled;

    
Nordsieck matrix of the higher scaled derivatives.
(h2/2 y'', h3/6 y''' ..., hk/k! y(k))
/** 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. /** Starter integrator. */
    private FirstOrderIntegrator starter;

    
Number of steps of the multistep method (excluding the one being computed). /** Number of steps of the multistep method (excluding the one being computed). */
    private final int nSteps;

    
Stepsize control exponent. /** Stepsize control exponent. */
    private double exp;

    
Safety factor for stepsize control. /** Safety factor for stepsize control. */
    private double safety;

    
Minimal reduction factor for stepsize control. /** Minimal reduction factor for stepsize control. */
    private double minReduction;

    
Maximal growth factor for stepsize control. /** Maximal growth factor for stepsize control. */
    private double maxGrowth;

    
Build a multistep integrator with the given stepsize bounds.
The default starter integrator is set to the Dormand-Prince 8(5,3) integrator with some defaults settings.

The default max growth factor is set to a quite low value: 21/order.

Params:
name – name of the method
nSteps – number of steps of the multistep method
(excluding the one being computed)
order – order of the method
minStep – minimal step (must be positive even for backward
integration), the last step can be smaller than this
maxStep – maximal step (must be positive even for backward
integration)
scalAbsoluteTolerance – allowed absolute error
scalRelativeTolerance – allowed relative error
Throws:
NumberIsTooSmallException – if number of steps is smaller than 2/**
     * 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 NumberIsTooSmallException 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 NumberIsTooSmallException {

        super(name, minStep, maxStep, scalAbsoluteTolerance, scalRelativeTolerance);

        if (nSteps < 2) {
            throw new NumberIsTooSmallException(
                  LocalizedFormats.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.
The default starter integrator is set to the Dormand-Prince 8(5,3) integrator with some defaults settings.

The default max growth factor is set to a quite low value: 21/order.

Params:
name – name of the method
nSteps – number of steps of the multistep method
(excluding the one being computed)
order – order of the method
minStep – minimal step (must be positive even for backward
integration), the last step can be smaller than this
maxStep – maximal step (must be positive even for backward
integration)
vecAbsoluteTolerance – allowed absolute error
vecRelativeTolerance – allowed relative error/**
     * 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);
        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.
Returns:
starter integrator/**
     * Get the starter integrator.
     * @return starter integrator
     */
    public ODEIntegrator getStarterIntegrator() {
        return starter;
    }

    
Set the starter integrator.
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.
Params:
starterIntegrator – starter integrator/**
     * 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(FirstOrderIntegrator starterIntegrator) {
        this.starter = starterIntegrator;
    }

    
Start the integration.
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.
Params:
t0 – initial time
y0 – initial value of the state vector at t0
t – target time for the integration
(can be set to a value smaller than t0 for backward integration)
Throws:
DimensionMismatchException – if arrays dimension do not match equations settings
NumberIsTooSmallException – if integration step is too small
MaxCountExceededException – if the number of functions evaluations is exceeded
NoBracketingException – if the location of an event cannot be bracketed/** 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 t0 initial time
     * @param y0 initial value of the state vector at t0
     * @param t target time for the integration
     * (can be set to a value smaller than <code>t0</code> for backward integration)
     * @exception DimensionMismatchException if arrays dimension do not match equations settings
     * @exception NumberIsTooSmallException if integration step is too small
     * @exception MaxCountExceededException if the number of functions evaluations is exceeded
     * @exception NoBracketingException if the location of an event cannot be bracketed
     */
    protected void start(final double t0, final double[] y0, final double t)
        throws DimensionMismatchException, NumberIsTooSmallException,
               MaxCountExceededException, NoBracketingException {

        // make sure NO user event nor user step handler is triggered,
        // this is the task of the top level integrator, not the task
        // of the starter integrator
        starter.clearEventHandlers();
        starter.clearStepHandlers();

        // set up one specific step handler to extract initial Nordsieck vector
        starter.addStepHandler(new NordsieckInitializer((nSteps + 3) / 2, y0.length));

        // start integration, expecting a InitializationCompletedMarkerException
        try {

            if (starter instanceof AbstractIntegrator) {
                ((AbstractIntegrator) starter).integrate(getExpandable(), t);
            } else {
                starter.integrate(new FirstOrderDifferentialEquations() {

                    
{@inheritDoc} /** {@inheritDoc} */
                    public int getDimension() {
                        return getExpandable().getTotalDimension();
                    }

                    
{@inheritDoc} /** {@inheritDoc} */
                    public void computeDerivatives(double t, double[] y, double[] yDot) {
                        getExpandable().computeDerivatives(t, y, yDot);
                    }

                }, t0, y0, t, new double[y0.length]);
            }

            // we should not reach this step
            throw new MathIllegalStateException(LocalizedFormats.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
            getCounter().increment(starter.getEvaluations());

        }

        // remove the specific step handler
        starter.clearStepHandlers();

    }

    
Initialize the high order scaled derivatives at step start.
Params:
h – step size to use for scaling
t – first steps times
y – first steps states
yDot – first steps derivatives
Returns:
Nordieck vector at first step (h2/2 y''n,
h3/6 y'''n ... hk/k! y(k)n)/** 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(final double h, final double[] t,
                                                                           final double[][] y,
                                                                           final double[][] yDot);

    
Get the minimal reduction factor for stepsize control.
Returns:
minimal reduction factor/** 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.
Params:
minReduction – minimal reduction factor/** 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.
Returns:
maximal growth factor/** 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.
Params:
maxGrowth – maximal growth factor/** 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.
Returns:
safety factor/** Get the safety factor for stepsize control.
     * @return safety factor
     */
    public double getSafety() {
      return safety;
    }

    
Set the safety factor for stepsize control.
Params:
safety – safety factor/** 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).
Returns:
number of steps of the multistep method (excluding the one being computed)/** 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;
    }

    
Compute step grow/shrink factor according to normalized error.
Params:
error – normalized error of the current step
Returns:
grow/shrink factor for next step/** 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)));
    }

    
Transformer used to convert the first step to Nordsieck representation.
Deprecated:
as of 3.6 this unused interface is deprecated/** Transformer used to convert the first step to Nordsieck representation.
     * @deprecated as of 3.6 this unused interface is deprecated
     */
    @Deprecated
    public interface NordsieckTransformer {
        
Initialize the high order scaled derivatives at step start.
Params:
h – step size to use for scaling
t – first steps times
y – first steps states
yDot – first steps derivatives
Returns:
Nordieck vector at first step (h2/2 y''n,
h3/6 y'''n ... hk/k! y(k)n)/** 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>)
         */
        Array2DRowRealMatrix initializeHighOrderDerivatives(final double h, final double[] t,
                                                            final double[][] y,
                                                            final double[][] yDot);
    }

    
Specialized step handler storing the first step. /** Specialized step handler storing the first step. */
    private class NordsieckInitializer implements StepHandler {

        
Steps counter. /** Steps counter. */
        private int count;

        
First steps times. /** First steps times. */
        private final double[] t;

        
First steps states. /** First steps states. */
        private final double[][] y;

        
First steps derivatives. /** First steps derivatives. */
        private final double[][] yDot;

        
Simple constructor.
Params:
nbStartPoints – number of start points (including the initial point)
n – problem dimension/** Simple constructor.
         * @param nbStartPoints number of start points (including the initial point)
         * @param n problem dimension
         */
        NordsieckInitializer(final int nbStartPoints, final int n) {
            this.count = 0;
            this.t     = new double[nbStartPoints];
            this.y     = new double[nbStartPoints][n];
            this.yDot  = new double[nbStartPoints][n];
        }

        
{@inheritDoc} /** {@inheritDoc} */
        public void handleStep(StepInterpolator interpolator, boolean isLast)
            throws MaxCountExceededException {

            final double prev = interpolator.getPreviousTime();
            final double curr = interpolator.getCurrentTime();

            if (count == 0) {
                // first step, we need to store also the point at the beginning of the step
                interpolator.setInterpolatedTime(prev);
                t[0] = prev;
                final ExpandableStatefulODE expandable = getExpandable();
                final EquationsMapper primary = expandable.getPrimaryMapper();
                primary.insertEquationData(interpolator.getInterpolatedState(), y[count]);
                primary.insertEquationData(interpolator.getInterpolatedDerivatives(), yDot[count]);
                int index = 0;
                for (final EquationsMapper secondary : expandable.getSecondaryMappers()) {
                    secondary.insertEquationData(interpolator.getInterpolatedSecondaryState(index), y[count]);
                    secondary.insertEquationData(interpolator.getInterpolatedSecondaryDerivatives(index), yDot[count]);
                    ++index;
                }
            }

            // store the point at the end of the step
            ++count;
            interpolator.setInterpolatedTime(curr);
            t[count] = curr;

            final ExpandableStatefulODE expandable = getExpandable();
            final EquationsMapper primary = expandable.getPrimaryMapper();
            primary.insertEquationData(interpolator.getInterpolatedState(), y[count]);
            primary.insertEquationData(interpolator.getInterpolatedDerivatives(), yDot[count]);
            int index = 0;
            for (final EquationsMapper secondary : expandable.getSecondaryMappers()) {
                secondary.insertEquationData(interpolator.getInterpolatedSecondaryState(index), y[count]);
                secondary.insertEquationData(interpolator.getInterpolatedSecondaryDerivatives(index), yDot[count]);
                ++index;
            }

            if (count == t.length - 1) {

                // this was the last point we needed, we can compute the derivatives
                stepStart = t[0];
                stepSize  = (t[t.length - 1] - t[0]) / (t.length - 1);

                // first scaled derivative
                scaled = yDot[0].clone();
                for (int j = 0; j < scaled.length; ++j) {
                    scaled[j] *= stepSize;
                }

                // higher order derivatives
                nordsieck = initializeHighOrderDerivatives(stepSize, t, y, yDot);

                // stop the integrator now that all needed steps have been handled
                throw new InitializationCompletedMarkerException();

            }

        }

        
{@inheritDoc} /** {@inheritDoc} */
        public void init(double t0, double[] y0, double time) {
            // nothing to do
        }

    }

    
Marker exception used ONLY to stop the starter integrator after first step. /** Marker exception used ONLY to stop the starter integrator after first step. */
    private static class InitializationCompletedMarkerException
        extends RuntimeException {

        
Serializable version identifier. /** Serializable version identifier. */
        private static final long serialVersionUID = -1914085471038046418L;

        
Simple constructor. /** Simple constructor. */
        InitializationCompletedMarkerException() {
            super((Throwable) null);
        }

    }

}