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210 CHAPTER 11. H 2 MODEL-ORDER REDUCTION FROM ORDER N TO N-3
The two-parameter polynomial eigenvalue problem (11.3) involving the two matrices
Ãã N−1
(ρ 1 ,ρ 2 ) T and Ãã N−2
(ρ 1 ,ρ 2 ) T can be jointly written as:
[ ]
ÃãN−1 (ρ 1 ,ρ 2 ) T
ÃãN−2 (ρ 1 ,ρ 2 ) T
v = A(ρ 1 ,ρ 2 ) v = 0 (11.39)
where the matrix A(ρ 1 ,ρ 2 ) is rectangular and polynomial in both ρ 1 and ρ 2 . This
matrix has 2 N columns and at most 2 N+1 (unique) rows. The total degree of the terms
in ρ 1 and ρ 2 in the matrix A(ρ 1 ,ρ 2 ) is denoted by Ñ and, as stated in Corollary 10.2,
is equal to N −1 (where N is the order of the original model H(s) to reduce). Because
we are searching for blocks L T η 1
to split off, we continue working with the transposed
version of the matrix A(ρ 1 ,ρ 2 ). Thus, the matrix A(ρ 1 ,ρ 2 ) T can be written as:
A(ρ 1 ,ρ 2 ) T =
(
)
M 0 (ρ 2 )+ρ 1 M 1 (ρ 2 )+...+ ρÑ1 MÑ(ρ 2 )
(11.40)
where the matrices M i (ρ 2 ) are expanded as:
M i (ρ 2 )=M i,0 + ρ 2 M i,1 + ρ 2 2 M i,2 + ...+ ρÑ−i 2 M i, Ñ−i , (11.41)
for i =0,...,Ñ.
The matrix A(ρ 1 ,ρ 2 ) T is rectangular of dimension m×n. Note that the dimensions
of the matrix A(ρ 1 ,ρ 2 ) T are much smaller than the dimensions of the (linearized)
matrices P (ρ 2 ) T and Q(ρ 2 ) T in the previous subsection. Therefore m and n denote
different, smaller values here. Also in this non-linear case one should be able to split
off singular parts. Assuming that there is dependency between the columns of the
matrix A(ρ 1 ,ρ 2 ) T means that:
A(ρ 1 ,ρ 2 ) T z(ρ 1 ,ρ 2 ) = 0 (11.42)
has a non-zero solution z(ρ 1 ,ρ 2 ) depending on ρ 1 and ρ 2 . Take such a solution z in
which the degree of ρ 1 is η 1 , the degree of ρ 2 is η 2 and where η 1 is as small as possible.
Then z(ρ 1 ,ρ 2 ) can be written as:
z(ρ 1 ,ρ 2 )=z 0 (ρ 2 ) − ρ 1 z 1 (ρ 2 )+ρ 2 1 z 2 (ρ 2 )+...+(−1) η1 ρ η1
1 z η 1
(ρ 2 ) (11.43)
where z 0 (ρ 2 ),z 1 (ρ 2 ),...,z η1 (ρ 2 ) all polynomial in ρ 2 .
Substituting such a solution z(ρ 1 ,ρ 2 ) into (11.42) where the matrix A(ρ 1 ,ρ 2 ) T is
written as in (11.40), yields the following when the relationships for the coefficients of
the powers of ρ 1 are worked out and the resulting expressions are written in matrix-