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188 CHAPTER 10. H 2 MODEL-ORDER REDUCTION FROM ORDER N TO N-2
The system of corresponding quadratic equations in x 1 ,...,x 7 and ρ 1 is:
⎛
1+ρ 1δ 1
⎞
⎛ ⎞ ⎛ ⎞
e(δ 1) x2 1
x 1 0
1+ρ 1δ 2
e(δ 2) x2 2
x 2
0
1+ρ 1δ 3
e(δ 3) x2 3
x 3
0
1+ρ 1δ 4
e(δ 4) x2 4
− M(δ 1 ,...,δ 7 )
x 4
=
0
, (10.51)
1+ρ 1δ 5
e(δ 5) x2 5
x 5
0
⎜
1+ρ 1δ 6
⎝ e(δ 6) x2 6
⎟
⎜
⎠
⎝
x 6
⎟ ⎜
⎠ ⎝
0 ⎟
⎠
0
1+ρ 1δ 7
e(δ 7) x2 7
where M(δ 1 ,...,δ 7 ) is the complex matrix given by V (−δ 1 ,...,−δ 7 ) V (δ 1 ,...,δ 7 ) −1 .
The linear constraint, defined by Equation (10.3), is given by:
ã N−1 = 2304
5 x 1 + ( 18144
1625 − 45792
1625 i)x 2 + ( 18144
1625 + 45792
1625 i)x 3
− 23328
25 x 4 + ( 688176
5785 + 559872
5785 i)x 5 + ( 688176
5785 − 559872
5785 i)x 6
+ 2359296
11125 x 7 = 0.
(10.52)
It is used to construct the matrix AãN−1 (ρ 1 ) T . This matrix is rational in ρ 1 , and
after diagonal rescaling (according to Theorem 10.1) to make it polynomial in ρ 1 ,a
matrix Ãã N−1
(ρ 1 ) T is obtained. The total degree of ρ 1 in this matrix is N − 1=6
(see Corollary 10.2). The size of the matrix Ãã N−1
(ρ 1 ) T is 2 N × 2 N = 128 × 128. The
sparsity structure of this matrix is shown in Figure 10.7.
To find the solutions of the quadratic system of polynomial equations (10.51),
which also satisfy the linear constraint (10.52), one has to find the solutions of the
polynomial eigenvalue problem Ãã N−1
(ρ 1 ) T v = 0. This is done by writing the polynomial
matrix Ãã N−1
(ρ 1 ) T as A 0 + ρ 1 A 1 + ...+ ρ 6 1A 6 and by using a linearization
technique. This yields the generalized eigenvalue problem (B + ρ 1 C)ṽ = 0, with the
matrices B and C defined as:
⎛
⎞
0 I 0 0 0 0
0 0 I 0 0 0
0 0 0 I 0 0
B =
,
0 0 0 0 I 0
⎜
⎟
⎝ 0 0 0 0 0 I ⎠
A 0 A 1 A 2 A 3 A 4 A 5
⎛
C =
⎜
⎝
x 7
⎞
−I 0 0 0 0 0
0 −I 0 0 0 0
0 0 −I 0 0 0
0 0 0 −I 0 0
⎟
0 0 0 0 −I 0 ⎠
0 0 0 0 0 A 6
.
(10.53)