org.apache.commons.math3.ode.nonstiff
Class AdamsMoultonIntegrator

java.lang.Object
  org.apache.commons.math3.ode.AbstractIntegrator
      org.apache.commons.math3.ode.nonstiff.AdaptiveStepsizeIntegrator
          org.apache.commons.math3.ode.MultistepIntegrator
              org.apache.commons.math3.ode.nonstiff.AdamsIntegrator
                  org.apache.commons.math3.ode.nonstiff.AdamsMoultonIntegrator

All Implemented Interfaces:

FirstOrderIntegrator, ODEIntegrator

public class AdamsMoultonIntegrator
extends AdamsIntegrator

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 y_n+1, another method must be used to compute a first estimate of y_n+1, then compute y'_n+1, then compute a final estimate of y_n+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: y_n+1 = y_n + h y'_n+1
• k = 2: y_n+1 = y_n + h (y'_n+1+y'_n)/2
• k = 3: y_n+1 = y_n + h (5y'_n+1+8y'_n-y'_n-1)/12
• k = 4: y_n+1 = y_n + h (9y'_n+1+19y'_n-5y'_n-1+y'_n-2)/24
• ...

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

Implementation details

We define scaled derivatives s_i(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:

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

Instead of using the classical representation with first derivatives only (y_n, s₁(n+1) and q_n+1), our implementation uses the Nordsieck vector with higher degrees scaled derivatives all taken at the same step (y_n, s₁(n) and r_n) where r_n 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) ... s_k(n), the formula being exact for degree k polynomials.

 s1(n-i) = s1(n) + ∑j j (-i)j-1 sj(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 r_n and q_n 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 (-i)^j-1 terms:

        [  -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 predicted Nordsieck vector at step n+1 is computed from the Nordsieck vector at step n as follows:

• Y_n+1 = y_n + s₁(n) + u^T r_n
• S₁(n+1) = h f(t_n+1, Y_n+1)
• R_n+1 = (s₁(n) - S₁(n+1)) P^-1 u + P^-1 A P r_n

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:

• y_n+1 = y_n + S₁(n+1) + [ -1 +1 -1 +1 ... ±1 ] r_n+1
• s₁(n+1) = h f(t_n+1, y_n+1)
• r_n+1 = R_n+1 + (s₁(n+1) - S₁(n+1)) P^-1 u

where the upper case Y_n+1, S₁(n+1) and R_n+1 represent the predicted states whereas the lower case y_n+1, s_n+1 and r_n+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.

Since:

2.0

Version:

$Id: AdamsMoultonIntegrator.java 7721 2013-02-14 14:07:13Z CardosoP $

Nested Class Summary

Nested classes/interfaces inherited from class org.apache.commons.math3.ode.MultistepIntegrator

MultistepIntegrator.NordsieckTransformer

Field Summary

Fields inherited from class org.apache.commons.math3.ode.MultistepIntegrator

nordsieck, scaled

Fields inherited from class org.apache.commons.math3.ode.nonstiff.AdaptiveStepsizeIntegrator

mainSetDimension, scalAbsoluteTolerance, scalRelativeTolerance, vecAbsoluteTolerance, vecRelativeTolerance

Fields inherited from class org.apache.commons.math3.ode.AbstractIntegrator

isLastStep, resetOccurred, stepHandlers, stepSize, stepStart

Constructor Summary

AdamsMoultonIntegrator(int nSteps, double minStep, double maxStep, double[] vecAbsoluteTolerance, double[] vecRelativeTolerance)
Build an Adams-Moulton integrator with the given order and error control parameters.

AdamsMoultonIntegrator(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

void	integrate(ExpandableStatefulODE equations, double t) Integrate a set of differential equations up to the given time.

Methods inherited from class org.apache.commons.math3.ode.nonstiff.AdamsIntegrator

initializeHighOrderDerivatives, updateHighOrderDerivativesPhase1, updateHighOrderDerivativesPhase2

Methods inherited from class org.apache.commons.math3.ode.MultistepIntegrator

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

Methods inherited from class org.apache.commons.math3.ode.nonstiff.AdaptiveStepsizeIntegrator

filterStep, getCurrentStepStart, getMaxStep, getMinStep, initializeStep, resetInternalState, sanityChecks, setInitialStepSize, setStepSizeControl, setStepSizeControl

Methods inherited from class org.apache.commons.math3.ode.AbstractIntegrator

acceptStep, addEventHandler, addEventHandler, addStepHandler, clearEventHandlers, clearStepHandlers, computeDerivatives, getCurrentSignedStepsize, getEvaluations, getEventHandlers, getMaxEvaluations, getName, getStepHandlers, initIntegration, integrate, setEquations, setMaxEvaluations, setStateInitialized

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Detail

AdamsMoultonIntegrator

public AdamsMoultonIntegrator(int nSteps,
                              double minStep,
                              double maxStep,
                              double scalAbsoluteTolerance,
                              double scalRelativeTolerance)
                       throws NumberIsTooSmallException

Build an Adams-Moulton integrator with the given order and error 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:

NumberIsTooSmallException - if order is 1 or less

AdamsMoultonIntegrator

public AdamsMoultonIntegrator(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:

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

integrate

public void integrate(ExpandableStatefulODE equations,
                      double t)
               throws NumberIsTooSmallException,
                      DimensionMismatchException,
                      MaxCountExceededException,
                      NoBracketingException

Integrate a set of differential equations up to the given time.

This method solves an Initial Value Problem (IVP).

The set of differential equations is composed of a main set, which can be extended by some sets of secondary equations. The set of equations must be already set up with initial time and partial states. At integration completion, the final time and partial states will be available in the same object.

Since this method stores some internal state variables made available in its public interface during integration (AbstractIntegrator.getCurrentSignedStepsize()), it is not thread-safe.

Specified by:

integrate in class AdamsIntegrator

Parameters:

equations - complete set of differential equations to integrate

t - target time for the integration (can be set to a value smaller than t0 for backward integration)

Throws:

NumberIsTooSmallException - if integration step is too small

DimensionMismatchException - if the dimension of the complete state does not match the complete equations sets dimension

MaxCountExceededException - if the number of functions evaluations is exceeded

NoBracketingException - if the location of an event cannot be bracketed