org.apache.commons.math.ode.nonstiff

• All Implemented Interfaces:
FirstOrderIntegrator, ODEIntegrator

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.

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-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:

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 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 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 (-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:

• 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.

Since:
2.0
• Constructor Detail

double minStep,
double maxStep,
double scalAbsoluteTolerance,
double scalRelativeTolerance)
throws java.lang.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 (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:
java.lang.IllegalArgumentException - if order is 1 or less

double minStep,
double maxStep,
double[] vecAbsoluteTolerance,
double[] vecRelativeTolerance)
throws java.lang.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 (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
Throws:
java.lang.IllegalArgumentException - if order is 1 or less
• Method Detail

• integrate

public double integrate(FirstOrderDifferentialEquations equations,
double t0,
double[] y0,
double t,
double[] y)
throws DerivativeException,
IntegratorException
Integrate the differential equations up to the given time.

This method solves an Initial Value Problem (IVP).

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

Specified by:
integrate in interface FirstOrderIntegrator
Specified by:
integrate in class AdamsIntegrator
Parameters:
equations - differential equations to integrate
t0 - initial time
y0 - initial value of the state vector at t0
t - target time for the integration (can be set to a value smaller than t0 for backward integration)
y - placeholder where to put the state vector at each successful step (and hence at the end of integration), can be the same object as y0
Returns:
stop time, will be the same as target time if integration reached its target, but may be different if some EventHandler stops it at some point.
Throws:
DerivativeException - this exception is propagated to the caller if the underlying user function triggers one
IntegratorException - if the integrator cannot perform integration