org.apache.commons.math.ode.nonstiff

Class GraggBulirschStoerIntegrator

java.lang.Object

org.apache.commons.math.ode.AbstractIntegrator

org.apache.commons.math.ode.nonstiff.AdaptiveStepsizeIntegrator

org.apache.commons.math.ode.nonstiff.GraggBulirschStoerIntegrator

All Implemented Interfaces:

FirstOrderIntegrator, ODEIntegrator

public class GraggBulirschStoerIntegrator
extends AdaptiveStepsizeIntegrator

This class implements a Gragg-Bulirsch-Stoer integrator for Ordinary Differential Equations.

The Gragg-Bulirsch-Stoer algorithm is one of the most efficient ones currently available for smooth problems. It uses Richardson extrapolation to estimate what would be the solution if the step size could be decreased down to zero.

This method changes both the step size and the order during integration, in order to minimize computation cost. It is particularly well suited when a very high precision is needed. The limit where this method becomes more efficient than high-order embedded Runge-Kutta methods like Dormand-Prince 8(5,3) depends on the problem. Results given in the Hairer, Norsett and Wanner book show for example that this limit occurs for accuracy around 1e-6 when integrating Saltzam-Lorenz equations (the authors note this problem is extremely sensitive to the errors in the first integration steps), and around 1e-11 for a two dimensional celestial mechanics problems with seven bodies (pleiades problem, involving quasi-collisions for which automatic step size control is essential).

This implementation is basically a reimplementation in Java of the odex fortran code by E. Hairer and G. Wanner. The redistribution policy for this code is available here, for convenience, it is reproduced below.

Copyright (c) 2004, Ernst Hairer

Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met: Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer. Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution.

THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.

Since:

1.2

Version:

$Revision: 1073158 $ $Date: 2011-02-21 22:46:52 +0100 (lun. 21 févr. 2011) $

Field Summary

Fields

Modifier and Type	Field and Description
private double[][]	coeff extrapolation coefficients.
private int[]	costPerStep overall cost of applying step reduction up to iteration k+1, in number of calls.
private double[]	costPerTimeUnit cost per unit step.
private int	maxChecks maximal number of checks for each iteration.
private int	maxIter maximal number of iterations for which checks are performed.
private int	maxOrder maximal order.
private static java.lang.String	METHOD_NAME Integrator method name.
private int	mudif interpolation order control parameter.
private double[]	optimalStep optimal steps for each order.
private double	orderControl1 first order control factor.
private double	orderControl2 second order control factor.
private boolean	performTest stability check enabling parameter.
private int[]	sequence step size sequence.
private double	stabilityReduction stepsize reduction factor in case of stability check failure.
private double	stepControl1 first stepsize control factor.
private double	stepControl2 second stepsize control factor.
private double	stepControl3 third stepsize control factor.
private double	stepControl4 fourth stepsize control factor.
private boolean	useInterpolationError use interpolation error in stepsize control.

Fields inherited from class org.apache.commons.math.ode.nonstiff.AdaptiveStepsizeIntegrator

mainSetDimension, scalAbsoluteTolerance, scalRelativeTolerance, vecAbsoluteTolerance, vecRelativeTolerance

Fields inherited from class org.apache.commons.math.ode.AbstractIntegrator

isLastStep, resetOccurred, stepHandlers, stepSize, stepStart

Constructor Summary

Constructors

Constructor and Description
GraggBulirschStoerIntegrator(double minStep, double maxStep, double[] vecAbsoluteTolerance, double[] vecRelativeTolerance) Simple constructor.
GraggBulirschStoerIntegrator(double minStep, double maxStep, double scalAbsoluteTolerance, double scalRelativeTolerance) Simple constructor.

Method Summary

Modifier and Type	Method and Description
void	addEventHandler(EventHandler function, double maxCheckInterval, double convergence, int maxIterationCount) Add an event handler to the integrator.
void	addStepHandler(StepHandler handler) Add a step handler to this integrator.
private void	extrapolate(int offset, int k, double[][] diag, double[] last) Extrapolate a vector.
private void	initializeArrays() Initialize the integrator internal arrays.
double	integrate(FirstOrderDifferentialEquations equations, double t0, double[] y0, double t, double[] y) Integrate the differential equations up to the given time.
private void	rescale(double[] y1, double[] y2, double[] scale) Update scaling array.
void	setInterpolationControl(boolean useInterpolationErrorForControl, int mudifControlParameter) Set the interpolation order control parameter.
void	setOrderControl(int maximalOrder, double control1, double control2) Set the order control parameters.
void	setStabilityCheck(boolean performStabilityCheck, int maxNumIter, int maxNumChecks, double stepsizeReductionFactor) Set the stability check controls.
void	setStepsizeControl(double control1, double control2, double control3, double control4) Set the step size control factors.
private boolean	tryStep(double t0, double[] y0, double step, int k, double[] scale, double[][] f, double[] yMiddle, double[] yEnd, double[] yTmp) Perform integration over one step using substeps of a modified midpoint method.

Methods inherited from class org.apache.commons.math.ode.nonstiff.AdaptiveStepsizeIntegrator

filterStep, getCurrentStepStart, getMaxStep, getMinStep, initializeStep, resetInternalState, sanityChecks, setInitialStepSize

Methods inherited from class org.apache.commons.math.ode.AbstractIntegrator

acceptStep, addEndTimeChecker, clearEventHandlers, clearStepHandlers, computeDerivatives, getCurrentSignedStepsize, getEvaluations, getEventHandlers, getMaxEvaluations, getName, getStepHandlers, requiresDenseOutput, resetEvaluations, setEquations, setMaxEvaluations, setStateInitialized

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Field Detail

METHOD_NAME

private static final java.lang.String METHOD_NAME

Integrator method name.

See Also:

Constant Field Values

maxOrder

private int maxOrder

maximal order.

sequence

private int[] sequence

step size sequence.

costPerStep

private int[] costPerStep

overall cost of applying step reduction up to iteration k+1, in number of calls.

costPerTimeUnit

private double[] costPerTimeUnit

cost per unit step.

optimalStep

private double[] optimalStep

optimal steps for each order.

coeff

private double[][] coeff

extrapolation coefficients.

performTest

private boolean performTest

stability check enabling parameter.

maxChecks

private int maxChecks

maximal number of checks for each iteration.

maxIter

private int maxIter

maximal number of iterations for which checks are performed.

stabilityReduction

private double stabilityReduction

stepsize reduction factor in case of stability check failure.

stepControl1

private double stepControl1

first stepsize control factor.

stepControl2

private double stepControl2

second stepsize control factor.

stepControl3

private double stepControl3

third stepsize control factor.

stepControl4

private double stepControl4

fourth stepsize control factor.

orderControl1

private double orderControl1

first order control factor.

orderControl2

private double orderControl2

second order control factor.

useInterpolationError

private boolean useInterpolationError

use interpolation error in stepsize control.

mudif

private int mudif

interpolation order control parameter.

Constructor Detail

GraggBulirschStoerIntegrator

public GraggBulirschStoerIntegrator(double minStep,
                                    double maxStep,
                                    double scalAbsoluteTolerance,
                                    double scalRelativeTolerance)

Simple constructor. Build a Gragg-Bulirsch-Stoer integrator with the given step bounds. All tuning parameters are set to their default values. The default step handler does nothing.

Parameters:

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

GraggBulirschStoerIntegrator

public GraggBulirschStoerIntegrator(double minStep,
                                    double maxStep,
                                    double[] vecAbsoluteTolerance,
                                    double[] vecRelativeTolerance)

Simple constructor. Build a Gragg-Bulirsch-Stoer integrator with the given step bounds. All tuning parameters are set to their default values. The default step handler does nothing.

Parameters:

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

Method Detail

setStabilityCheck

public void setStabilityCheck(boolean performStabilityCheck,
                              int maxNumIter,
                              int maxNumChecks,
                              double stepsizeReductionFactor)

Set the stability check controls.

The stability check is performed on the first few iterations of the extrapolation scheme. If this test fails, the step is rejected and the stepsize is reduced.

By default, the test is performed, at most during two iterations at each step, and at most once for each of these iterations. The default stepsize reduction factor is 0.5.

Parameters:

performStabilityCheck - if true, stability check will be performed, if false, the check will be skipped

maxNumIter - maximal number of iterations for which checks are performed (the number of iterations is reset to default if negative or null)

maxNumChecks - maximal number of checks for each iteration (the number of checks is reset to default if negative or null)

stepsizeReductionFactor - stepsize reduction factor in case of failure (the factor is reset to default if lower than 0.0001 or greater than 0.9999)

setStepsizeControl

public void setStepsizeControl(double control1,
                               double control2,
                               double control3,
                               double control4)

Set the step size control factors.

The new step size hNew is computed from the old one h by:

 hNew = h * stepControl2 / (err/stepControl1)^(1/(2k+1))
 

where err is the scaled error and k the iteration number of the extrapolation scheme (counting from 0). The default values are 0.65 for stepControl1 and 0.94 for stepControl2.

The step size is subject to the restriction:

 stepControl3^(1/(2k+1))/stepControl4 <= hNew/h <= 1/stepControl3^(1/(2k+1))
 

The default values are 0.02 for stepControl3 and 4.0 for stepControl4.

Parameters:

control1 - first stepsize control factor (the factor is reset to default if lower than 0.0001 or greater than 0.9999)

control2 - second stepsize control factor (the factor is reset to default if lower than 0.0001 or greater than 0.9999)

control3 - third stepsize control factor (the factor is reset to default if lower than 0.0001 or greater than 0.9999)

control4 - fourth stepsize control factor (the factor is reset to default if lower than 1.0001 or greater than 999.9)

setOrderControl

public void setOrderControl(int maximalOrder,
                            double control1,
                            double control2)

Set the order control parameters.

The Gragg-Bulirsch-Stoer method changes both the step size and the order during integration, in order to minimize computation cost. Each extrapolation step increases the order by 2, so the maximal order that will be used is always even, it is twice the maximal number of columns in the extrapolation table.

 order is decreased if w(k-1) <= w(k)   * orderControl1
 order is increased if w(k)   <= w(k-1) * orderControl2
 

where w is the table of work per unit step for each order (number of function calls divided by the step length), and k is the current order.

The default maximal order after construction is 18 (i.e. the maximal number of columns is 9). The default values are 0.8 for orderControl1 and 0.9 for orderControl2.

Parameters:

maximalOrder - maximal order in the extrapolation table (the maximal order is reset to default if order <= 6 or odd)

control1 - first order control factor (the factor is reset to default if lower than 0.0001 or greater than 0.9999)

control2 - second order control factor (the factor is reset to default if lower than 0.0001 or greater than 0.9999)

addStepHandler

public void addStepHandler(StepHandler handler)

Add a step handler to this integrator.

The handler will be called by the integrator for each accepted step.

Specified by:

addStepHandler in interface ODEIntegrator

Overrides:

addStepHandler in class AbstractIntegrator

Parameters:

handler - handler for the accepted steps

See Also:

ODEIntegrator.getStepHandlers(), ODEIntegrator.clearStepHandlers()

addEventHandler

public void addEventHandler(EventHandler function,
                            double maxCheckInterval,
                            double convergence,
                            int maxIterationCount)

Add an event handler to the integrator.

Specified by:

addEventHandler in interface ODEIntegrator

Overrides:

addEventHandler in class AbstractIntegrator

Parameters:

function - event handler

maxCheckInterval - maximal time interval between switching function checks (this interval prevents missing sign changes in case the integration steps becomes very large)

convergence - convergence threshold in the event time search

maxIterationCount - upper limit of the iteration count in the event time search

See Also:

ODEIntegrator.getEventHandlers(), ODEIntegrator.clearEventHandlers()

initializeArrays

private void initializeArrays()

Initialize the integrator internal arrays.

setInterpolationControl

public void setInterpolationControl(boolean useInterpolationErrorForControl,
                                    int mudifControlParameter)

Set the interpolation order control parameter. The interpolation order for dense output is 2k - mudif + 1. The default value for mudif is 4 and the interpolation error is used in stepsize control by default.

Parameters:

useInterpolationErrorForControl - if true, interpolation error is used for stepsize control

mudifControlParameter - interpolation order control parameter (the parameter is reset to default if <= 0 or >= 7)

rescale

private void rescale(double[] y1,
                     double[] y2,
                     double[] scale)

Update scaling array.

Parameters:

y1 - first state vector to use for scaling

y2 - second state vector to use for scaling

scale - scaling array to update (can be shorter than state)

tryStep

private boolean tryStep(double t0,
                        double[] y0,
                        double step,
                        int k,
                        double[] scale,
                        double[][] f,
                        double[] yMiddle,
                        double[] yEnd,
                        double[] yTmp)
                 throws DerivativeException

Perform integration over one step using substeps of a modified midpoint method.

Parameters:

t0 - initial time

y0 - initial value of the state vector at t0

step - global step

k - iteration number (from 0 to sequence.length - 1)

scale - scaling array (can be shorter than state)

f - placeholder where to put the state vector derivatives at each substep (element 0 already contains initial derivative)

yMiddle - placeholder where to put the state vector at the middle of the step

yEnd - placeholder where to put the state vector at the end

yTmp - placeholder for one state vector

Returns:

true if computation was done properly, false if stability check failed before end of computation

Throws:

DerivativeException - this exception is propagated to the caller if the underlying user function triggers one

extrapolate

private void extrapolate(int offset,
                         int k,
                         double[][] diag,
                         double[] last)

Extrapolate a vector.

Parameters:

offset - offset to use in the coefficients table

k - index of the last updated point

diag - working diagonal of the Aitken-Neville's triangle, without the last element

last - last element

integrate

public double integrate(FirstOrderDifferentialEquations equations,
                        double t0,
                        double[] y0,
                        double t,
                        double[] y)
                 throws DerivativeException,
                        IntegratorException

Integrate the differential equations up to the given time.

This method solves an Initial Value Problem (IVP).

Since this method stores some internal state variables made available in its public interface during integration (ODEIntegrator.getCurrentSignedStepsize()), it is not thread-safe.

Specified by:

integrate in interface FirstOrderIntegrator

Specified by:

integrate in class AdaptiveStepsizeIntegrator

Parameters:

equations - differential equations to integrate

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)

y - placeholder where to put the state vector at each successful step (and hence at the end of integration), can be the same object as y0

Returns:

stop time, will be the same as target time if integration reached its target, but may be different if some EventHandler stops it at some point.

Throws:

DerivativeException - this exception is propagated to the caller if the underlying user function triggers one

IntegratorException - if the integrator cannot perform integration