org.apache.commons.math.ode
public abstract class MultistepIntegrator extends AdaptiveStepsizeIntegrator
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 a 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.
AdamsBashforthIntegrator
,
AdamsMoultonIntegrator
Modifier and Type | Class and Description |
---|---|
private class |
MultistepIntegrator.CountingDifferentialEquations
Wrapper for differential equations, ensuring start evaluations are counted.
|
private static class |
MultistepIntegrator.InitializationCompletedMarkerException
Marker exception used ONLY to stop the starter integrator after first step.
|
private class |
MultistepIntegrator.NordsieckInitializer
Specialized step handler storing the first step.
|
static interface |
MultistepIntegrator.NordsieckTransformer
Transformer used to convert the first step to Nordsieck representation.
|
Modifier and Type | Field and Description |
---|---|
private double |
exp
Stepsize control exponent.
|
private double |
maxGrowth
Maximal growth factor for stepsize control.
|
private double |
minReduction
Minimal reduction factor for stepsize control.
|
protected Array2DRowRealMatrix |
nordsieck
Nordsieck matrix of the higher scaled derivatives.
|
private int |
nSteps
Number of steps of the multistep method (excluding the one being computed).
|
private double |
safety
Safety factor for stepsize control.
|
protected double[] |
scaled
First scaled derivative (h y').
|
private FirstOrderIntegrator |
starter
Starter integrator.
|
mainSetDimension, scalAbsoluteTolerance, scalRelativeTolerance, vecAbsoluteTolerance, vecRelativeTolerance
isLastStep, resetOccurred, stepHandlers, stepSize, stepStart
Modifier | Constructor and Description |
---|---|
protected |
MultistepIntegrator(String name,
int nSteps,
int order,
double minStep,
double maxStep,
double[] vecAbsoluteTolerance,
double[] vecRelativeTolerance)
Build a multistep integrator with the given stepsize bounds.
|
protected |
MultistepIntegrator(String name,
int nSteps,
int order,
double minStep,
double maxStep,
double scalAbsoluteTolerance,
double scalRelativeTolerance)
Build a multistep integrator with the given stepsize bounds.
|
Modifier and Type | Method and Description |
---|---|
protected double |
computeStepGrowShrinkFactor(double error)
Compute step grow/shrink factor according to normalized error.
|
double |
getMaxGrowth()
Get the maximal growth factor for stepsize control.
|
double |
getMinReduction()
Get the minimal reduction factor for stepsize control.
|
double |
getSafety()
Get the safety factor for stepsize control.
|
ODEIntegrator |
getStarterIntegrator()
Get the starter integrator.
|
protected abstract Array2DRowRealMatrix |
initializeHighOrderDerivatives(double[] first,
double[][] multistep)
Initialize the high order scaled derivatives at step start.
|
void |
setMaxGrowth(double maxGrowth)
Set the maximal growth factor for stepsize control.
|
void |
setMinReduction(double minReduction)
Set the minimal reduction factor for stepsize control.
|
void |
setSafety(double safety)
Set the safety factor for stepsize control.
|
void |
setStarterIntegrator(FirstOrderIntegrator starterIntegrator)
Set the starter integrator.
|
protected void |
start(double t0,
double[] y0,
double t)
Start the integration.
|
filterStep, getCurrentStepStart, getMaxStep, getMinStep, initializeStep, integrate, resetInternalState, sanityChecks, setInitialStepSize
acceptStep, addEndTimeChecker, addEventHandler, addStepHandler, clearEventHandlers, clearStepHandlers, computeDerivatives, getCurrentSignedStepsize, getEvaluations, getEventHandlers, getMaxEvaluations, getName, getStepHandlers, requiresDenseOutput, resetEvaluations, setEquations, setMaxEvaluations, setStateInitialized
protected double[] scaled
protected Array2DRowRealMatrix nordsieck
(h2/2 y'', h3/6 y''' ..., hk/k! y(k))
private FirstOrderIntegrator starter
private final int nSteps
private double exp
private double safety
private double minReduction
private double maxGrowth
protected MultistepIntegrator(String name, int nSteps, int order, double minStep, double maxStep, double scalAbsoluteTolerance, double scalRelativeTolerance)
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.
name
- name of the methodnSteps
- number of steps of the multistep method
(excluding the one being computed)order
- order of the methodminStep
- minimal step (must be positive even for backward
integration), the last step can be smaller than thismaxStep
- maximal step (must be positive even for backward
integration)scalAbsoluteTolerance
- allowed absolute errorscalRelativeTolerance
- allowed relative errorprotected MultistepIntegrator(String name, int nSteps, int order, double minStep, double maxStep, double[] vecAbsoluteTolerance, double[] vecRelativeTolerance)
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.
name
- name of the methodnSteps
- number of steps of the multistep method
(excluding the one being computed)order
- order of the methodminStep
- minimal step (must be positive even for backward
integration), the last step can be smaller than thismaxStep
- maximal step (must be positive even for backward
integration)vecAbsoluteTolerance
- allowed absolute errorvecRelativeTolerance
- allowed relative errorpublic ODEIntegrator getStarterIntegrator()
public void setStarterIntegrator(FirstOrderIntegrator starterIntegrator)
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.
starterIntegrator
- starter integratorprotected void start(double t0, double[] y0, double t) throws DerivativeException, IntegratorException
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.
t0
- initial timey0
- initial value of the state vector at t0t
- target time for the integration
(can be set to a value smaller than t0
for backward integration)IntegratorException
- if the integrator cannot perform integrationDerivativeException
- this exception is propagated to the caller if
the underlying user function triggers oneprotected abstract Array2DRowRealMatrix initializeHighOrderDerivatives(double[] first, double[][] multistep)
first
- first scaled derivative at step startmultistep
- scaled derivatives after step start (hy'1, ..., hy'k-1)
will be modifiedpublic double getMinReduction()
public void setMinReduction(double minReduction)
minReduction
- minimal reduction factorpublic double getMaxGrowth()
public void setMaxGrowth(double maxGrowth)
maxGrowth
- maximal growth factorpublic double getSafety()
public void setSafety(double safety)
safety
- safety factorprotected double computeStepGrowShrinkFactor(double error)
error
- normalized error of the current stepCopyright (c) 2003-2013 Apache Software Foundation