java.lang.Object
Type Parameters:
T - the type of the field elements

public class AdamsNordsieckFieldTransformer<T extends CalculusFieldElement<T>> extends Object
Transformer to Nordsieck vectors for Adams integrators.

This class is used by Adams-Bashforth and Adams-Moulton integrators to convert between classical representation with several previous first derivatives and Nordsieck representation with higher order scaled derivatives.

We define scaled derivatives si(n) at step n as: \left\{\begin{align} s_1(n) &= h y'_n \text{ for first derivative}\\ s_2(n) &= \frac{h^2}{2} y_n'' \text{ for second derivative}\\ s_3(n) &= \frac{h^3}{6} y_n''' \text{ for third derivative}\\ &\cdots\\ s_k(n) &= \frac{h^k}{k!} y_n^{(k)} \text{ for } k^\mathrm{th} \text{ derivative} \end{align}\right.

With the previous definition, the classical representation of multistep methods uses first derivatives only, i.e. it handles yn, s1(n) and qn where qn is defined as: $q_n = [ s_1(n-1) s_1(n-2) \ldots s_1(n-(k-1)) ]^T$ (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 yn, s1(n) and rn) where rn is defined as: $r_n = [ s_2(n), s_3(n) \ldots s_k(n) ]^T$ (here again we omit the k index in the notation for clarity)

Taylor series formulas show that for any index offset i, s1(n-i) can be computed from s1(n), s2(n) ... sk(n), the formula being exact for degree k polynomials. $s_1(n-i) = s_1(n) + \sum_{j\gt 0} (j+1) (-i)^j s_{j+1}(n)$ 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 rn and qn 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 (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: $P=\begin{bmatrix} -2 & 3 & -4 & 5 & \ldots \\ -4 & 12 & -32 & 80 & \ldots \\ -6 & 27 & -108 & 405 & \ldots \\ -8 & 48 & -256 & 1280 & \ldots \\ & & \ldots\\ \end{bmatrix}$

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 Adams-Bashforth method, 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 ]


For Adams-Moulton method, 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
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.

We observe that both methods use similar update formulas. In both cases a P-1u 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.

• ## Method Summary

Modifier and Type
Method
Description
static <T extends CalculusFieldElement<T>>AdamsNordsieckFieldTransformer<T>
getInstance(Field<T> field, int nSteps)
Get the Nordsieck transformer for a given field and number of steps.
Array2DRowFieldMatrix<T>
initializeHighOrderDerivatives(T h, T[] t, T[][] y, T[][] yDot)
Initialize the high order scaled derivatives at step start.
Array2DRowFieldMatrix<T>
updateHighOrderDerivativesPhase1(Array2DRowFieldMatrix<T> highOrder)
Update the high order scaled derivatives for Adams integrators (phase 1).
void
updateHighOrderDerivativesPhase2(T[] start, T[] end, Array2DRowFieldMatrix<T> highOrder)
Update the high order scaled derivatives Adams integrators (phase 2).

### Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait
• ## Method Details

• ### getInstance

public static <T extends CalculusFieldElement<T>>  getInstance(Field<T> field, int nSteps)
Get the Nordsieck transformer for a given field and number of steps.
Type Parameters:
T - the type of the field elements
Parameters:
field - field to which the time and state vector elements belong
nSteps - number of steps of the multistep method (excluding the one being computed)
Returns:
Nordsieck transformer for the specified field and number of steps
• ### initializeHighOrderDerivatives

public  initializeHighOrderDerivatives(T h, T[] t, T[][] y, T[][] yDot)
Initialize the high order scaled derivatives at step start.
Parameters:
h - step size to use for scaling
t - first steps times
y - first steps states
yDot - first steps derivatives
Returns:
Nordieck vector at start of first step (h2/2 y''n, h3/6 y'''n ... hk/k! y(k)n)
• ### updateHighOrderDerivativesPhase1

public  updateHighOrderDerivativesPhase1(Array2DRowFieldMatrix<T> highOrder)
Update the high order scaled derivatives for Adams integrators (phase 1).

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 rn part.

Parameters:
highOrder - high order scaled derivatives (h2/2 y'', ... hk/k! y(k))
Returns:
updated high order derivatives
• ### updateHighOrderDerivativesPhase2

public void updateHighOrderDerivativesPhase2(T[] start, T[] end, Array2DRowFieldMatrix<T> highOrder)
Update the high order scaled derivatives Adams integrators (phase 2).

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 (s1(n) - s1(n+1)) P-1 u part.

Phase 1 of the update must already have been performed.

Parameters:
start - first order scaled derivatives at step start
end - first order scaled derivatives at step end
highOrder - high order scaled derivatives, will be modified (h2/2 y'', ... hk/k! y(k))