public class AdamsNordsieckTransformer extends Object
This class is used by AdamsBashforth
and
AdamsMoulton
integrators to convert between
classical representation with several previous first derivatives and Nordsieck
representation with higher order scaled derivatives.
We define scaled derivatives s_{i}(n) at step n as:
s_{1}(n) = h y'_{n} for first derivative s_{2}(n) = h^{2}/2 y''_{n} for second derivative s_{3}(n) = h^{3}/6 y'''_{n} for third derivative ... s_{k}(n) = h^{k}/k! y^{(k)}_{n} for k^{th} derivative
With the previous definition, the classical representation of multistep methods uses first derivatives only, i.e. it handles y_{n}, s_{1}(n) and q_{n} where q_{n} is defined as:
q_{n} = [ s_{1}(n1) s_{1}(n2) ... s_{1}(n(k1)) ]^{T}(we omit the k index in the notation for clarity).
Another possible representation uses the Nordsieck vector with higher degrees scaled derivatives all taken at the same step, i.e it handles y_{n}, s_{1}(n) and r_{n}) where r_{n} is defined as:
r_{n} = [ s_{2}(n), s_{3}(n) ... 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, s_{1}(ni) can be computed from s_{1}(n), s_{2}(n) ... s_{k}(n), the formula being exact for degree k polynomials.
s_{1}(ni) = s_{1}(n) + ∑_{j>1} j (i)^{j1} s_{j}(n)The previous formula can be used with several values for i to compute the transform between classical representation and Nordsieck vector at step end. The transform between r_{n} and q_{n} 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 (k1)×(k1) matrix built with the j (i)^{j1} terms:
[ 2 3 4 5 ... ] [ 4 12 32 80 ... ] P = [ 6 27 108 405 ... ] [ 8 48 256 1280 ... ] [ ... ]
Changing i into +i in the formula above can be used to compute a similar transform between classical representation and Nordsieck vector at step start. The resulting matrix is simply the absolute value of matrix P.
For AdamsBashforth
method, the 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 ]
For AdamsMoulton
method, the predicted Nordsieck vector
at step n+1 is computed from the Nordsieck vector at step n as follows:
We observe that both methods use similar update formulas. In both cases a P^{1}u vector and a P^{1} A P matrix are used that do not depend on the state, they only depend on k. This class handles these transformations.
Modifier and Type  Method and Description 

static AdamsNordsieckTransformer 
getInstance(int nSteps)
Get the Nordsieck transformer for a given number of steps.

int 
getNSteps()
Get the number of steps of the method
(excluding the one being computed).

Array2DRowRealMatrix 
initializeHighOrderDerivatives(double h,
double[] t,
double[][] y,
double[][] yDot)
Initialize the high order scaled derivatives at step start.

Array2DRowRealMatrix 
updateHighOrderDerivativesPhase1(Array2DRowRealMatrix highOrder)
Update the high order scaled derivatives for Adams integrators (phase 1).

void 
updateHighOrderDerivativesPhase2(double[] start,
double[] end,
Array2DRowRealMatrix highOrder)
Update the high order scaled derivatives Adams integrators (phase 2).

public static AdamsNordsieckTransformer getInstance(int nSteps)
nSteps
 number of steps of the multistep method
(excluding the one being computed)public int getNSteps()
public Array2DRowRealMatrix initializeHighOrderDerivatives(double h, double[] t, double[][] y, double[][] yDot)
h
 step size to use for scalingt
 first steps timesy
 first steps statesyDot
 first steps derivativespublic Array2DRowRealMatrix updateHighOrderDerivativesPhase1(Array2DRowRealMatrix highOrder)
The complete update of high order derivatives has a form similar to:
r_{n+1} = (s_{1}(n)  s_{1}(n+1)) P^{1} u + P^{1} A P r_{n}this method computes the P^{1} A P r_{n} part.
highOrder
 high order scaled derivatives
(h^{2}/2 y'', ... h^{k}/k! y(k))updateHighOrderDerivativesPhase2(double[], double[], Array2DRowRealMatrix)
public void updateHighOrderDerivativesPhase2(double[] start, double[] end, Array2DRowRealMatrix highOrder)
The complete update of high order derivatives has a form similar to:
r_{n+1} = (s_{1}(n)  s_{1}(n+1)) P^{1} u + P^{1} A P r_{n}this method computes the (s_{1}(n)  s_{1}(n+1)) P^{1} u part.
Phase 1 of the update must already have been performed.
start
 first order scaled derivatives at step startend
 first order scaled derivatives at step endhighOrder
 high order scaled derivatives, will be modified
(h^{2}/2 y'', ... h^{k}/k! y(k))updateHighOrderDerivativesPhase1(Array2DRowRealMatrix)
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