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* contributor license agreements. See the NOTICE file distributed with
* 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
*
* http://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,
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* See the License for the specific language governing permissions and
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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.Cloner;
/**
* 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'). */
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 FirstOrderIntegrator 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 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(Math.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);
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(Math.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(FirstOrderIntegrator 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 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} */
public int getDimension() {
return getExpandable().getTotalDimension();
}
/** {@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.
* @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.
* @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;
}
/** 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 Math.min(maxGrowth, Math.max(minReduction, safety * Math.pow(error, exp)));
}
/** 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.
* @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. */
private class NordsieckInitializer implements StepHandler {
/** Steps counter. */
private int count;
/** 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)
* @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} */
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 = Cloner.clone(yDot[0]);
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} */
public void init(double t0, double[] y0, double time) {
// nothing to do
}
}
/** 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);
}
}
}