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

java.lang.Object
  extended by org.apache.commons.math.ode.AbstractIntegrator
      extended by org.apache.commons.math.ode.nonstiff.AdaptiveStepsizeIntegrator
          extended by 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 f??vr. 2011) $

Field Summary
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 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
GraggBulirschStoerIntegrator(double minStep, double maxStep, double[] vecAbsoluteTolerance, double[] vecRelativeTolerance)
          Simple constructor.
GraggBulirschStoerIntegrator(double minStep, double maxStep, double scalAbsoluteTolerance, double scalRelativeTolerance)
          Simple constructor.
 
Method Summary
 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 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


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