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Chapter 9
H 2 Model-order reduction from order
N to N-1
In the co-order k = 1 case, the polynomial ρ(s) in (8.44) is of degree 0. Then it
holds that ρ(s) ≡ 1 and also ρ(δ i ) = 1 for i =1,...,N in (8.44). Then the system
of equations (8.44) yields the following N polynomial equations in the N unknowns
x 1 ,...,x N :
⎛ ⎞
⎛ ⎞ ⎛ ⎞
1
e(δ 1) x2 1
x 1 0
1
e(δ 2) x2 2
x 2
0
⎜
− M(δ ⎝ . ⎟ 1 ,...,δ N )
⎜
=
⎠
⎝ .
⎟ ⎜
(9.1)
⎠ ⎝ .
⎟
⎠
0
1
e(δ N ) x2 N
where the matrix M is again V (−δ 1 ,..., −δ N ) V (δ 1 ,..., δ N ) −1 .
x N
9.1 Solving the system of quadratic equations
Upon choosing the polynomial r(x 1 ,x 2 ,...,x N ) in Section (8.5) equal to each of the
monomials x i , the 2 N × 2 N matrices A T x i
, for i =1, 2,...,N, are constructed by
applying the Stetter-Möller matrix method. Here we consider the quotient space
C[x 1 ,...,x N ]/I, where I is the ideal generated by the polynomials involved in the
system of equations (9.1).
The eigenvalues of the matrix A T x i
are the values of x i at the solutions of the
corresponding system of equations. Moreover, once one eigenpair of some matrix A T x i
for some particular i is known, the values of x 1 ,...,x N can be read off from the
eigenvector because this vector exhibits the Stetter-structure.
Any solution (x 1 ,...,x N ) to the system of equations (9.1) gives rise to a corresponding
set of coefficients ã 0 , ã 1 ,...,ã N−1 from which q 0 =(−1) N−1 ã N−1 and,
subsequently, the coefficients a 1 ,a 2 ,...,a N−2 can be computed as follows: If q 0 =0,
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