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148 CHAPTER 8. H 2 MODEL-ORDER REDUCTION
Γ(δ 1 ,...,δ N )=
⎛
⎜
⎝
0 ... 0 0 1
.
. . ..
0 ... 0 1 0
⎞
⎟
⎠ V (δ 1 ,...,δ N ) −1 . (8.46)
This extends the approach of [50] to co-orders k>1 and from a conventional eigenvalue
problem for k = 1 to a multi-parameter polynomial eigenvalue problem formulation
as worked out in the next chapters.
For fixed values of ρ(s), the equations in system (8.44) are in Gröbner basis form
with respect to any total degree monomial ordering. The polynomial expressions
associated with these equations generate an ideal I. Now the idea is to choose a
suitable polynomial r(x 1 ,x 2 ,...,x N ) and to address the linear operator A r which
performs multiplication by the polynomial r modulo the ideal I in the quotient space C
[x 1 ,x 2 ,...,x N ]/I. This quotient space is finite-dimensional and therefore constitutes
a finite set of 2 N (possibly complex) solutions (x 1 ,...,x N ). For each solution a
corresponding approximation G(s) of order N − k can be computed. Note that G(s)
is only a feasible solution if it is real and stable.
A monomial basis here is given by the following set B:
B = {x α1
1 ···xα N
N
|α 1 ,...,α N ∈{0, 1}}. (8.47)
With respect to B, the operator A r can now be represented by a matrix A T r of size
2 N × 2 N as shown in Section 3.3.
Upon choosing r(x 1 ,x 2 ,...,x N ) as the polynomial x i the matrix A T x i
can be constructed,
for each i =1, 2,...,N. As discussed in Section 3.3, the tuple of matrices
(A T x 1
,A T x 2
, ..., A T x N
) constitutes a matrix solution to the system of equations (8.44)
we intend to solve. All the 2 N solutions of this system can be obtained by computing
the 2 N N-tuples of eigenvalues of the matrices A T x 1
,A T x 2
, ..., A T x N
.
The matrices A T r , for arbitrary polynomials r, all commute and thus they have
common eigenvectors. The eigenvalues of the matrix A T r correspond to the values of r
at the (possibly complex) solutions (x 1 ,x 2 ,...,x N ) to the system of equations (8.44).
Similar properties obviously hold for the matrices A r . If all the eigenvalues have multiplicity
one, the eigenvectors of a matrix A T r exhibit the Stetter vector structure and
therefore the values x 1 ,x 2 ,...,x N can be obtained from the entries of such a Stetter
vector (see Section 3.3 again). Thus, an eigenpair computation on a single matrix A T r
for a well chosen polynomial r can bring all the solutions to the system of equations
(8.44). All the 2 N scalar solutions can be obtained in this way by working with one
matrix A T r only.
The approach above can be used, to set up a classical eigenvalue problem for the
co-order k = 1 case to solve the system of equations, since the additional variables ρ i
do not show up here.
In the co-order k > 1, the additional variables ρ 1 ,...,ρ k−1 show up in the
quadratic system of equations (8.44). We now consider the linear operator which
performs multiplication by the polynomial r within the quotient space C (ρ 1 , ...,