org.hipparchus.ode.nonstiff

• All Implemented Interfaces:
ODEIntegrator

public class AdamsBashforthIntegrator
extends AdamsIntegrator
This class implements explicit Adams-Bashforth integrators for Ordinary Differential Equations.

Adams-Bashforth methods (in fact due to Adams alone) are explicit multistep ODE solvers. This implementation is a variation of the classical one: it uses adaptive stepsize to implement error control, whereas classical implementations are fixed step size. The value of state vector at step n+1 is a simple combination of the value at step n and of the derivatives at steps n, n-1, n-2 ... Depending on the number k of previous steps one wants to use for computing the next value, different formulas are available:

• k = 1: yn+1 = yn + h y'n
• k = 2: yn+1 = yn + h (3y'n-y'n-1)/2
• k = 3: yn+1 = yn + h (23y'n-16y'n-1+5y'n-2)/12
• k = 4: yn+1 = yn + h (55y'n-59y'n-1+37y'n-2-9y'n-3)/24
• ...

A k-steps Adams-Bashforth method is of order k.

There must be sufficient time for the starter integrator to take several steps between the the last reset event, and the end of integration, otherwise an exception may be thrown during integration. The user can adjust the end date of integration, or the step size of the starter integrator to ensure a sufficient number of steps can be completed before the end of integration.

### Implementation details

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


The definitions above use the classical representation with several previous first derivatives. Lets define

   qn = [ s1(n-1) s1(n-2) ... s1(n-(k-1)) ]T

(we omit the k index in the notation for clarity). With these definitions, Adams-Bashforth methods can be written:
• k = 1: yn+1 = yn + s1(n)
• k = 2: yn+1 = yn + 3/2 s1(n) + [ -1/2 ] qn
• k = 3: yn+1 = yn + 23/12 s1(n) + [ -16/12 5/12 ] qn
• k = 4: yn+1 = yn + 55/24 s1(n) + [ -59/24 37/24 -9/24 ] qn
• ...

Instead of using the classical representation with first derivatives only (yn, s1(n) and qn), our 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

(here again we omit the k index in the notation for clarity)

Taylor series formulas show that for any index offset i, s1(n-i) can be computed from s1(n), s2(n) ... sk(n), the formula being exact for degree k polynomials.

 s1(n-i) = s1(n) + ∑j>0 (j+1) (-i)j sj+1(n)

The previous formula can be used with several values for i to compute the transform between classical representation and Nordsieck vector. The transform between rn and qn resulting from the Taylor series formulas above is:
 qn = s1(n) u + P rn

where u is the [ 1 1 ... 1 ]T vector and P is the (k-1)×(k-1) matrix built with the (j+1) (-i)j terms with i being the row number starting from 1 and j being the column number starting from 1:
        [  -2   3   -4    5  ... ]
[  -4  12  -32   80  ... ]
P =  [  -6  27 -108  405  ... ]
[  -8  48 -256 1280  ... ]
[          ...           ]


Using the Nordsieck vector has several advantages:

• it greatly simplifies step interpolation as the interpolator mainly applies Taylor series formulas,
• it simplifies step changes that occur when discrete events that truncate the step are triggered,
• it allows to extend the methods in order to support adaptive stepsize.

The Nordsieck vector at step n+1 is computed from the Nordsieck vector at step n as follows:

• yn+1 = yn + s1(n) + uT rn
• s1(n+1) = h f(tn+1, yn+1)
• rn+1 = (s1(n) - s1(n+1)) P-1 u + P-1 A P rn
where A is a rows shifting matrix (the lower left part is an identity matrix):
        [ 0 0   ...  0 0 | 0 ]
[ ---------------+---]
[ 1 0   ...  0 0 | 0 ]
A = [ 0 1   ...  0 0 | 0 ]
[       ...      | 0 ]
[ 0 0   ...  1 0 | 0 ]
[ 0 0   ...  0 1 | 0 ]


The P-1u vector and the P-1 A P matrix do not depend on the state, they only depend on k and therefore are precomputed once for all.

• ### Fields inherited from class org.hipparchus.ode.MultistepIntegrator

nordsieck, scaled
• ### Constructor Summary

Constructors
Constructor and Description
AdamsBashforthIntegrator(int nSteps, double minStep, double maxStep, double[] vecAbsoluteTolerance, double[] vecRelativeTolerance)
Build an Adams-Bashforth integrator with the given order and step control parameters.
AdamsBashforthIntegrator(int nSteps, double minStep, double maxStep, double scalAbsoluteTolerance, double scalRelativeTolerance)
Build an Adams-Bashforth integrator with the given order and step control parameters.
• ### Method Summary

All Methods
Modifier and Type Method and Description
protected double errorEstimation(double[] previousState, double predictedTime, double[] predictedState, double[] predictedScaled, RealMatrix predictedNordsieck)
Estimate error.
protected org.hipparchus.ode.nonstiff.AdamsStateInterpolator finalizeStep(double stepSize, double[] predictedState, double[] predictedScaled, Array2DRowRealMatrix predictedNordsieck, boolean isForward, ODEStateAndDerivative globalPreviousState, ODEStateAndDerivative globalCurrentState, EquationsMapper equationsMapper)
Finalize the step.
• ### Methods inherited from class org.hipparchus.ode.nonstiff.AdamsIntegrator

initializeHighOrderDerivatives, integrate, updateHighOrderDerivativesPhase1, updateHighOrderDerivativesPhase2
• ### Methods inherited from class org.hipparchus.ode.MultistepIntegrator

computeStepGrowShrinkFactor, getMaxGrowth, getMinReduction, getNSteps, getSafety, getStarterIntegrator, rescale, setMaxGrowth, setMinReduction, setSafety, setStarterIntegrator, start
• ### 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, addEventHandler, addEventHandler, addStepHandler, clearEventHandlers, clearStepHandlers, computeDerivatives, getCurrentSignedStepsize, getEquations, getEvaluations, getEvaluationsCounter, getEventHandlers, getEventHandlersConfigurations, getMaxEvaluations, getName, 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
• ### Constructor Detail

public AdamsBashforthIntegrator(int nSteps,
double minStep,
double maxStep,
double scalAbsoluteTolerance,
double scalRelativeTolerance)
throws MathIllegalArgumentException
Build an Adams-Bashforth integrator with the given order and step control parameters.
Parameters:
nSteps - number of steps of the method excluding the one being computed
minStep - minimal step (sign is irrelevant, regardless of integration direction, forward or backward), the last step can be smaller than this
maxStep - maximal step (sign is irrelevant, regardless of integration direction, forward or backward), the last step can be smaller than this
scalAbsoluteTolerance - allowed absolute error
scalRelativeTolerance - allowed relative error
Throws:
MathIllegalArgumentException - if order is 1 or less

public AdamsBashforthIntegrator(int nSteps,
double minStep,
double maxStep,
double[] vecAbsoluteTolerance,
double[] vecRelativeTolerance)
throws IllegalArgumentException
Build an Adams-Bashforth integrator with the given order and step control parameters.
Parameters:
nSteps - number of steps of the method excluding the one being computed
minStep - minimal step (sign is irrelevant, regardless of integration direction, forward or backward), the last step can be smaller than this
maxStep - maximal step (sign is irrelevant, regardless of integration direction, forward or backward), the last step can be smaller than this
vecAbsoluteTolerance - allowed absolute error
vecRelativeTolerance - allowed relative error
Throws:
IllegalArgumentException - if order is 1 or less
• ### Method Detail

• #### errorEstimation

protected double errorEstimation(double[] previousState,
double predictedTime,
double[] predictedState,
double[] predictedScaled,
RealMatrix predictedNordsieck)
Estimate error.
Specified by:
errorEstimation in class AdamsIntegrator
Parameters:
previousState - state vector at step start
predictedTime - time at step end
predictedState - predicted state vector at step end
predictedScaled - predicted value of the scaled derivatives at step end
predictedNordsieck - predicted value of the Nordsieck vector at step end
Returns:
estimated normalized local discretization error
• #### finalizeStep

protected org.hipparchus.ode.nonstiff.AdamsStateInterpolator finalizeStep(double stepSize,
double[] predictedState,
double[] predictedScaled,
Array2DRowRealMatrix predictedNordsieck,
boolean isForward,
ODEStateAndDerivative globalPreviousState,
ODEStateAndDerivative globalCurrentState,
EquationsMapper equationsMapper)
Finalize the step.
Specified by:
finalizeStep in class AdamsIntegrator
Parameters:
stepSize - step size used in the scaled and Nordsieck arrays
predictedState - predicted state at end of step
predictedScaled - predicted first scaled derivative
predictedNordsieck - predicted Nordsieck vector
isForward - integration direction indicator
globalPreviousState - start of the global step
globalCurrentState - end of the global step
equationsMapper - mapper for ODE equations primary and secondary components
Returns:
step interpolator