T - the type of the field elementspublic class AdamsMoultonFieldIntegrator<T extends CalculusFieldElement<T>> extends AdamsFieldIntegrator<T>
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:
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.
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-Moulton methods can be written:
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:
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 rnwhere 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:
The predicted Nordsieck vector at step n+1 is computed from the Nordsieck vector at step n as follows:
[ 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:
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.
nordsieck, scaled| Constructor and 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.
|
| Modifier and Type | Method and 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.
|
initializeHighOrderDerivatives, integrate, updateHighOrderDerivativesPhase1, updateHighOrderDerivativesPhase2computeStepGrowShrinkFactor, getMaxGrowth, getMinReduction, getNSteps, getSafety, getStarterIntegrator, rescale, setMaxGrowth, setMinReduction, setSafety, setStarterIntegrator, startgetMaxStep, getMinStep, getStepSizeHelper, initializeStep, resetInternalState, sanityChecks, setInitialStepSize, setStepSizeControl, setStepSizeControlacceptStep, addEventHandler, addEventHandler, addStepHandler, clearEventHandlers, clearStepHandlers, computeDerivatives, getCurrentSignedStepsize, getEquations, getEvaluations, getEvaluationsCounter, getEventHandlers, getEventHandlersConfigurations, getField, getMaxEvaluations, getName, getStepHandlers, getStepSize, getStepStart, initIntegration, isLastStep, resetOccurred, setIsLastStep, setMaxEvaluations, setStateInitialized, setStepSize, setStepStartpublic AdamsMoultonFieldIntegrator(Field<T> field, int nSteps, double minStep, double maxStep, double scalAbsoluteTolerance, double scalRelativeTolerance) throws MathIllegalArgumentException
field - field to which the time and state vector elements belongnSteps - number of steps of the method excluding the one being computedminStep - minimal step (sign is irrelevant, regardless of
integration direction, forward or backward), the last step can
be smaller than thismaxStep - maximal step (sign is irrelevant, regardless of
integration direction, forward or backward), the last step can
be smaller than thisscalAbsoluteTolerance - allowed absolute errorscalRelativeTolerance - allowed relative errorMathIllegalArgumentException - if order is 1 or lesspublic AdamsMoultonFieldIntegrator(Field<T> field, int nSteps, double minStep, double maxStep, double[] vecAbsoluteTolerance, double[] vecRelativeTolerance) throws IllegalArgumentException
field - field to which the time and state vector elements belongnSteps - number of steps of the method excluding the one being computedminStep - minimal step (sign is irrelevant, regardless of
integration direction, forward or backward), the last step can
be smaller than thismaxStep - maximal step (sign is irrelevant, regardless of
integration direction, forward or backward), the last step can
be smaller than thisvecAbsoluteTolerance - allowed absolute errorvecRelativeTolerance - allowed relative errorIllegalArgumentException - if order is 1 or lessprotected double errorEstimation(T[] previousState, T predictedTime, T[] predictedState, T[] predictedScaled, FieldMatrix<T> predictedNordsieck)
errorEstimation in class AdamsFieldIntegrator<T extends CalculusFieldElement<T>>previousState - state vector at step startpredictedTime - time at step endpredictedState - predicted state vector at step endpredictedScaled - predicted value of the scaled derivatives at step endpredictedNordsieck - predicted value of the Nordsieck vector at step endprotected 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)
finalizeStep in class AdamsFieldIntegrator<T extends CalculusFieldElement<T>>stepSize - step size used in the scaled and Nordsieck arrayspredictedY - predicted state at end of steppredictedScaled - predicted first scaled derivativepredictedNordsieck - predicted Nordsieck vectorisForward - integration direction indicatorglobalPreviousState - start of the global stepglobalCurrentState - end of the global stepequationsMapper - mapper for ODE equations primary and secondary componentsCopyright © 2016-2021 CS GROUP. All rights reserved.