link to my thesis

214 CHAPTER 11. H 2 MODEL-ORDER REDUCTION FROM ORDER N TO N-3
expression:
w c = M 0 (ρ 2 ) z c+1 (ρ 2 ) −
Ñ∑
i=2
ρ (i−1)
1
min(i + c, Ñ)
∑
j=i
M j (ρ 2 ) z i+c−j (ρ 2 ), (11.52)
for c =0,...,η 1 − 1. Both the matrices V (ρ 2 ) and W (ρ 1 ,ρ 2 ) are completed with
standard basis vectors to make them square and invertible.
The transformation:
( )
W (ρ 1 ,ρ 2 ) −1 M 0 (ρ 2 )+ρ 1 M 1 (ρ 2 )+...+ ρÑ1 MÑ(ρ 2 ) V (ρ 2 ) (11.53)
yields an equivalent pencil which has the same structure, up to the upper right zero
block, as the pencil in Equation (11.31) and where the block L T η 1
has the same structure
and dimensions of (η 1 +1)× η 1 as the block in (11.14). To make the upper right
block also zero, the standard basis vectors of the matrices W and V should be chosen
in a special way.
Using this technique one is able to split off singular parts of a non-linear matrix
pencil in two parameters: search for solutions z(ρ 1 ,ρ 2 ) of minimal but increasing
degrees η 1 for ρ 1 and degrees η 2 for ρ 2 (which are as small as possible for the given
degree η 1 ) and split off the singular parts of dimensions (η 1 +1)× η 1 until no more
singular parts exist or until the matrix size has become 0 × 0. Because the existence
of a regular part is impossible (because of the same reasons mentioned in the
previous section), the solutions ρ 1 and ρ 2 of (11.39) are the values which make the
transformation matrices V (ρ 2 ) and W (ρ 1 ,ρ 2 ) singular.
Remark 11.3. In the approach of Subsection 11.3.2, it is possible that the pencil is
non-linear after splitting off a singular part. Therefore an additional linearization step
is required. In the approach presented in this subsection, the pencil remains non-linear
after splitting off a singular part. Therefore the dimensions of the matrices involved
in this case are always smaller than in the linear case of Subsection 11.3.2.
Remark 11.4. A drawback of this approach is that the solutions ρ 1 and ρ 2 may remain
among the values which make the matrix W (ρ 1 ,ρ 2 ) singular. In that case one has to
solve a square polynomial eigenvalue problem in two variables, which can be difficult,
especially for large matrices or high powers of ρ 1 and ρ 2 . Moreover, computing
the inverse of W (ρ 1 ,ρ 2 ) to perform the transformation of the matrix pencil, can be
computationally challenging when using exact arithmetic because of the existence of
two variables with possibly high degrees.
Remark 11.5. The value of Ñ can change (increase or decrease), during the process
of splitting off singular parts of the matrix pencil. This has a direct effect on the
decomposition of the matrix pencil in (11.40) and (11.41).