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5.1. THE ND-SYSTEM 55
Example 5.1. Let us again consider the polynomial:
p 1 (x 1 ,x 2 )=(x 4 1 + x 4 2)+x 3 1 +2x 2 1 +3x 1 x 2 (5.12)
and the associated first-order conditions for minimality:
⎧
⎨ d (1) (x 1 ,x 2 ) = x 3 1 + 3 4 x2 1 +x 1 + 3 4 x 2 = 0
⎩
d (2) (x 1 ,x 2 ) = x 3 2 + 3 4 x 1 = 0
(5.13)
as studied before in Example 4.3. The polynomials f (1) (x 1 ,x 2 ) and f (2) (x 1 ,x 2 )in
this example are:
⎧
⎨ f (1) (x 1 ,x 2 ) = − 3 4 x2 1 −x 1 − 3 4 x 2
(5.14)
⎩
f (2) (x 1 ,x 2 ) = − 3 4 x 1
Now we are not directly interested in computing the global minimum of p 1 (x 1 ,x 2 )
but rather in (i) computing the action of the matrices A T x 1
and A T x 2
and, more generally,
(ii) computing the action of the matrix A T r(x 1,x 2)
on a given vector v without
constructing these matrices explicitly. Recall that the action on a given vector serves
as the input for an iterative eigenproblem solver to compute some of the eigenvalues
of the corresponding matrix.
(i) To focus attention on the computation of the actions of the matrices A T x 1
and
A T x 2
directly from f (1) (x 1 ,x 2 ) and f (2) (x 1 ,x 2 ), without constructing these matrices
explicitly, the following associated system of two-dimensional difference equations is
considered: ⎧
⎨ y t1+3,t 2
= (− 3 4 σ2 1 −σ 1 − 3 4 σ 2) y t1,t 2
(5.15)
⎩
y t1,t 2+3 = (− 3 4 σ 1) y t1,t 2
which is written in the following equivalent form:
⎧
⎨ y t1+3,t 2
= − 3 4 y t 1+2,t 2
−y t1+1,t 2
− 3 4 y t 1,t 2+1
(5.16)
⎩
y t1,t 2+3 = − 3 4 y t 1+1,t 2
An initial state for this autonomous 2-dimensional system is constituted by w 0,0 =
(y 0,0 ,y 1,0 ,y 2,0 ,y 0,1 ,y 1,1 ,y 2,1 ,y 0,2 ,y 1,2 ,y 2,2 ) T which corresponds to a 3×3 square array
of values at integer time instants (t 1 ,t 2 ) in the 2-dimensional time plane. The relations
between the various values of the two-dimensional time series in the equations in (5.16)
for t 1 = t 2 = 0 are displayed in Figure 5.1 by the arrows. The red square denotes the
initial state w 0,0 .
The action of the matrix A T x 1
on w 0,0 yields the vector A T x 1
w 0,0 = w 1,0 =(y 1,0 ,
y 2,0 ,y 3,0 ,y 1,1 ,y 2,1 ,y 3,1 ,y 1,2 ,y 2,2 ,y 3,2 ) T , which also corresponds to a 3 × 3 square
array of integer time instants (t 1 ,t 2 ) in the 2-dimensional time plane. Compared
to the location of the initial state w 0,0 , this array is shifted by one unit along