## Class MultistepIntegrator

• All Implemented Interfaces:
ODEIntegrator
Direct Known Subclasses:
AdamsIntegrator

public abstract class MultistepIntegrator
extends AdaptiveStepsizeIntegrator
This class is the base class for multistep integrators for Ordinary Differential Equations.

We define scaled derivatives si(n) at step n as: \left\{\begin{align} s_1(n) &= h y'_n \text{ for first derivative}\\ s_2(n) &= \frac{h^2}{2} y_n'' \text{ for second derivative}\\ s_3(n) &= \frac{h^3}{6} y_n''' \text{ for third derivative}\\ &\cdots\\ s_k(n) &= \frac{h^k}{k!} y_n^{(k)} \text{ for } k^\mathrm{th} \text{ derivative} \end{align}\right.

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: $r_n = [ s_2(n), s_3(n) \ldots s_k(n) ]^T$ (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.

AdamsBashforthIntegrator, AdamsMoultonIntegrator
• ### Field Summary

Fields
Modifier and Type Field Description
protected Array2DRowRealMatrix nordsieck
Nordsieck matrix of the higher scaled derivatives.
protected double[] scaled
First scaled derivative (h y').
• ### Constructor Summary

Constructors
Modifier Constructor 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.
• ### Method Summary

All Methods
Modifier and Type Method 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.
int getNSteps()
Get the number of steps of the multistep method (excluding the one being computed).
double getSafety()
Get the safety factor for stepsize control.
ODEIntegrator getStarterIntegrator()
Get the starter integrator.
protected abstract Array2DRowRealMatrix initializeHighOrderDerivatives​(double h, double[] t, double[][] y, double[][] yDot)
Initialize the high order scaled derivatives at step start.
protected void rescale​(double newStepSize)
Rescale the instance.
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​(ODEIntegrator starterIntegrator)
Set the starter integrator.
protected void start​(ExpandableODE equations, ODEState initialState, double finalTime)
Start the integration.
• ### Methods inherited from class org.hipparchus.ode.nonstiff.AdaptiveStepsizeIntegrator

getMaxStep, getMinStep, getStepSizeHelper, initializeStep, resetInternalState, sanityChecks, setInitialStepSize, setStepSizeControl, setStepSizeControl
• ### Methods inherited from class org.hipparchus.ode.AbstractIntegrator

acceptStep, addEventDetector, addStepEndHandler, addStepHandler, clearEventDetectors, clearStepEndHandlers, clearStepHandlers, computeDerivatives, getCurrentSignedStepsize, getEquations, getEvaluations, getEvaluationsCounter, getEventDetectors, getMaxEvaluations, getName, getStepEndHandlers, getStepHandlers, getStepSize, getStepStart, initIntegration, isLastStep, resetOccurred, setIsLastStep, setMaxEvaluations, setStateInitialized, setStepSize, setStepStart
• ### Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait
• ### Methods inherited from interface org.hipparchus.ode.ODEIntegrator

integrate, integrate
• ### Field Detail

• #### scaled

protected double[] scaled
First scaled derivative (h y').
• #### nordsieck

protected Array2DRowRealMatrix nordsieck
Nordsieck matrix of the higher scaled derivatives.

(h2/2 y'', h3/6 y''' ..., hk/k! y(k))

• ### Constructor Detail

• #### MultistepIntegrator

protected MultistepIntegrator​(String name,
int nSteps,
int order,
double minStep,
double maxStep,
double scalAbsoluteTolerance,
double scalRelativeTolerance)
throws MathIllegalArgumentException
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.

Parameters:
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:
MathIllegalArgumentException - if number of steps is smaller than 2
• #### MultistepIntegrator

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.

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.

Parameters:
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
• ### Method Detail

• #### getStarterIntegrator

public ODEIntegrator getStarterIntegrator()
Get the starter integrator.
Returns:
starter integrator
• #### setStarterIntegrator

public void setStarterIntegrator​(ODEIntegrator starterIntegrator)
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.

Parameters:
starterIntegrator - starter integrator
• #### start

protected void start​(ExpandableODE equations,
ODEState initialState,
double finalTime)
throws MathIllegalArgumentException,
MathIllegalStateException
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.

Parameters:
equations - complete set of differential equations to integrate
initialState - initial state (time, primary and secondary state vectors)
finalTime - target time for the integration (can be set to a value smaller than initialState.getTime() for backward integration)
Throws:
MathIllegalArgumentException - if arrays dimension do not match equations settings
MathIllegalArgumentException - if integration step is too small
MathIllegalStateException - if the number of functions evaluations is exceeded
MathIllegalArgumentException - if the location of an event cannot be bracketed
• #### initializeHighOrderDerivatives

protected abstract Array2DRowRealMatrix initializeHighOrderDerivatives​(double h,
double[] t,
double[][] y,
double[][] yDot)
Initialize the high order scaled derivatives at step start.
Parameters:
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)
• #### getMinReduction

public double getMinReduction()
Get the minimal reduction factor for stepsize control.
Returns:
minimal reduction factor
• #### setMinReduction

public void setMinReduction​(double minReduction)
Set the minimal reduction factor for stepsize control.
Parameters:
minReduction - minimal reduction factor
• #### getMaxGrowth

public double getMaxGrowth()
Get the maximal growth factor for stepsize control.
Returns:
maximal growth factor
• #### setMaxGrowth

public void setMaxGrowth​(double maxGrowth)
Set the maximal growth factor for stepsize control.
Parameters:
maxGrowth - maximal growth factor
• #### getSafety

public double getSafety()
Get the safety factor for stepsize control.
Returns:
safety factor
• #### setSafety

public void setSafety​(double safety)
Set the safety factor for stepsize control.
Parameters:
safety - safety factor
• #### getNSteps

public int getNSteps()
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)
• #### rescale

protected void rescale​(double newStepSize)
Rescale the instance.

Since the scaled and Nordsieck arrays are shared with the caller, this method has the side effect of rescaling this arrays in the caller too.

Parameters:
newStepSize - new step size to use in the scaled and Nordsieck arrays
• #### computeStepGrowShrinkFactor

protected double computeStepGrowShrinkFactor​(double error)
Compute step grow/shrink factor according to normalized error.
Parameters:
error - normalized error of the current step
Returns:
grow/shrink factor for next step