• All Implemented Interfaces:
FirstOrderIntegrator, ODEIntegrator

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.

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 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 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.

Since:
2.0
• Constructor Detail

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

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 (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:
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: