Type Parameters:
T - the type of the field elements
All Implemented Interfaces:
FieldODEIntegrator<T>

This class implements implicit Adams-Moulton integrators for Ordinary Differential Equations.

Adams-Moulton methods (in fact due to Adams alone) are implicit 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+1, n, n-1 ... Since y'n+1 is needed to compute yn+1, another method must be used to compute a first estimate of yn+1, then compute y'n+1, then compute a final estimate of yn+1 using the following formulas. Depending on the number k of previous steps one wants to use for computing the next value, different formulas are available for the final estimate:

• k = 1: yn+1 = yn + h y'n+1
• k = 2: yn+1 = yn + h (y'n+1+y'n)/2
• k = 3: yn+1 = yn + h (5y'n+1+8y'n-y'n-1)/12
• k = 4: yn+1 = yn + h (9y'n+1+19y'n-5y'n-1+y'n-2)/24
• ...

A k-steps Adams-Moulton method is of order k+1.

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: \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.

The definitions above use the classical representation with several previous first derivatives. Lets define $q_n = [ s_1(n-1) s_1(n-2) \ldots s_1(n-(k-1)) ]^T$ (we omit the k index in the notation for clarity). With these definitions, Adams-Moulton methods can be written:

• k = 1: yn+1 = yn + s1(n+1)
• k = 2: yn+1 = yn + 1/2 s1(n+1) + [ 1/2 ] qn+1
• k = 3: yn+1 = yn + 5/12 s1(n+1) + [ 8/12 -1/12 ] qn+1
• k = 4: yn+1 = yn + 9/24 s1(n+1) + [ 19/24 -5/24 1/24 ] qn+1
• ...

Instead of using the classical representation with first derivatives only (yn, s1(n+1) and qn+1), 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: $r_n = [ s_2(n), s_3(n) \ldots s_k(n) ]^T$ (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. $s_1(n-i) = s_1(n) + \sum_{j\gt 0} (j+1) (-i)^j s_{j+1}(n)$ 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: $q_n = s_1(n) u + P r_n$ 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: $P=\begin{bmatrix} -2 & 3 & -4 & 5 & \ldots \\ -4 & 12 & -32 & 80 & \ldots \\ -6 & 27 & -108 & 405 & \ldots \\ -8 & 48 & -256 & 1280 & \ldots \\ & & \ldots\\ \end{bmatrix}$

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 predicted 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 ]

From this predicted vector, the corrected vector is computed as follows:
• yn+1 = yn + S1(n+1) + [ -1 +1 -1 +1 ... ±1 ] rn+1
• s1(n+1) = h f(tn+1, yn+1)
• rn+1 = Rn+1 + (s1(n+1) - S1(n+1)) P-1 u

where the upper case Yn+1, S1(n+1) and Rn+1 represent the predicted states whereas the lower case yn+1, sn+1 and rn+1 represent the corrected states.

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.

• ## Field Summary

Fields
Modifier and Type
Field
Description
static final String
METHOD_NAME
Name of integration scheme.

### Fields inherited from class org.hipparchus.ode.MultistepFieldIntegrator

nordsieck, scaled
• ## Constructor Summary

Constructors
Constructor
Description
AdamsMoultonFieldIntegrator(Field<T> field, int nSteps, double minStep, double maxStep, double[] vecAbsoluteTolerance, double[] vecRelativeTolerance)
Build an Adams-Moulton integrator with the given order and error control parameters.
AdamsMoultonFieldIntegrator(Field<T> field, int nSteps, double minStep, double maxStep, double scalAbsoluteTolerance, double scalRelativeTolerance)
Build an Adams-Moulton integrator with the given order and error control parameters.
• ## Method Summary

Modifier and Type
Method
Description
protected double
errorEstimation(T[] previousState, T predictedTime, T[] predictedState, T[] predictedScaled, FieldMatrix<T> predictedNordsieck)
Estimate error.
protected org.hipparchus.ode.nonstiff.AdamsFieldStateInterpolator<T>
finalizeStep(T stepSize, T[] predictedY, T[] predictedScaled, Array2DRowFieldMatrix<T> predictedNordsieck, boolean isForward, FieldODEStateAndDerivative<T> globalPreviousState, FieldODEStateAndDerivative<T> globalCurrentState, FieldEquationsMapper<T> equationsMapper)
Finalize the step.

### Methods inherited from class org.hipparchus.ode.nonstiff.AdamsFieldIntegrator

initializeHighOrderDerivatives, integrate, updateHighOrderDerivativesPhase1, updateHighOrderDerivativesPhase2

### Methods inherited from class org.hipparchus.ode.MultistepFieldIntegrator

computeStepGrowShrinkFactor, getMaxGrowth, getMinReduction, getNSteps, getSafety, getStarterIntegrator, rescale, setMaxGrowth, setMinReduction, setSafety, setStarterIntegrator, start

### Methods inherited from class org.hipparchus.ode.nonstiff.AdaptiveStepsizeFieldIntegrator

getMaxStep, getMinStep, getStepSizeHelper, initializeStep, resetInternalState, sanityChecks, setInitialStepSize, setStepSizeControl, setStepSizeControl

### Methods inherited from class org.hipparchus.ode.AbstractFieldIntegrator

acceptStep, addEventDetector, addStepEndHandler, addStepHandler, clearEventDetectors, clearStepEndHandlers, clearStepHandlers, computeDerivatives, getCurrentSignedStepsize, getEquations, getEvaluations, getEvaluationsCounter, getEventDetectors, getField, getMaxEvaluations, getName, getStepEndHandlers, getStepHandlers, getStepSize, getStepStart, incrementEvaluations, 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
• ## Field Details

• ### METHOD_NAME

public static final String METHOD_NAME
Name of integration scheme.
• ## Constructor Details

public AdamsMoultonFieldIntegrator(Field<T> field, int nSteps, double minStep, double maxStep, double scalAbsoluteTolerance, double scalRelativeTolerance) throws MathIllegalArgumentException
Build an Adams-Moulton integrator with the given order and error control parameters.
Parameters:
field - field to which the time and state vector elements belong
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 AdamsMoultonFieldIntegrator(Field<T> field, int nSteps, double minStep, double maxStep, double[] vecAbsoluteTolerance, double[] vecRelativeTolerance) throws IllegalArgumentException
Build an Adams-Moulton integrator with the given order and error control parameters.
Parameters:
field - field to which the time and state vector elements belong
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 Details

• ### errorEstimation

protected double errorEstimation(T[] previousState, T predictedTime, T[] predictedState, T[] predictedScaled, FieldMatrix<T> predictedNordsieck)
Estimate error.
Specified by:
errorEstimation in class AdamsFieldIntegrator<T extends CalculusFieldElement<T>>
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.AdamsFieldStateInterpolator<T> finalizeStep(T stepSize, T[] predictedY, T[] predictedScaled, Array2DRowFieldMatrix<T> predictedNordsieck, boolean isForward, FieldODEStateAndDerivative<T> globalPreviousState, FieldODEStateAndDerivative<T> globalCurrentState, FieldEquationsMapper<T> equationsMapper)
Finalize the step.
Specified by:
finalizeStep in class AdamsFieldIntegrator<T extends CalculusFieldElement<T>>
Parameters:
stepSize - step size used in the scaled and Nordsieck arrays
predictedY - 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