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226 CHAPTER 11. H 2 MODEL-ORDER REDUCTION FROM ORDER N TO N-3
Table 11.2: Results of reduction process of two-parameter pencil A 1 (ρ 1 ,ρ 2 ) T using
the approach of Subsection 11.3.4.
Matrix Dimensions η 1 η 2 Dimensions of L η1 Variables
A 1 (ρ 1 ,ρ 2 ) T 15 × 30 0 0 4(0× 1) ρ 1 ,ρ 2
A 2 (ρ 1 ,ρ 2 ) T 15 × 26 1 2 1 × 2 ρ 1 ,ρ 2
A 3 (ρ 1 ,ρ 2 ) T 14 × 24 3 3 3 × 4 ρ 1 ,ρ 2
A 4 (ρ 1 ,ρ 2 ) T 11 × 20 1 8 1 × 2 ρ 1 ,ρ 2
A 5 (ρ 1 ,ρ 2 ) T 10 × 18 3 15 2(3× 4) ρ 1 ,ρ 2
A 6 (ρ 1 ,ρ 2 ) T 4 × 10 0 0 0 × 3 ρ 2
A 7 (ρ 1 ,ρ 2 ) T 4 × 7 0 3 3(0× 1) ρ 2
A 8 (ρ 1 ,ρ 2 ) T 4 × 4 − − − −
When the non-linear approach of Subsection 11.3.4 is used, the eigenvalues ρ 2
of the pencil A 1 (ρ 1 ,ρ 2 ) T are among the values of ρ 2 which make the transformation
matrices V (ρ 2 ) and W (ρ 2 ) singular. For this small example it was possible to compute
the values of ρ 2 exactly as the zeros of the determinants of the transformation matrices
V and W . Once the values for ρ 2 are known, these can be substituted into the matrices
ÃãN−1 (ρ 1 ,ρ 2 ) T and Ãã N−2
(ρ 1 ,ρ 2 ) T . Then the corresponding values for ρ 1 are the
values which make the matrices Ãã N−1
(ρ 1 ,ρ 2 ) T and Ãã N−2
(ρ 1 ,ρ 2 ) T simultaneously
singular for this known value of ρ 2 . These values for ρ 1 can also be determined by
computing the zeros of the determinants of Ãã N−1
(ρ 1 ,ρ 2 ) T and Ãã N−2
(ρ 1 ,ρ 2 ) T .
Only real values of ρ 1 and ρ 2 yield solutions which lead to feasible approximations
of order 1. After determining all the pairs (ρ 1 ,ρ 2 ) which make the matrices V (ρ 2 ),
W (ρ 2 ), ÃãN−1 (ρ 1 ,ρ 2 ) T and Ãã N−2
(ρ 1 ,ρ 2 ) T simultaneously singular, it turned out
that the matrices V (ρ 2 ) and W (ρ 2 ) used for the reduction of dimension 10 × 18 to
4 × 10 (see Table 11.2) contained five real-valued pairs (ρ 1 ,ρ 2 ). These values are
computed as zeros from some factor of the determinants. The five real values of ρ 1
are the real zeros of:
−73724881053644800000 + 394401417758222784000ρ 1 −
543650121764883196800ρ 2 1 + 343281065062589182600ρ 3 1−
117176950555078573308ρ 4 1 + 22474612752967348983ρ 5 1−
2288807613803985228ρ 6 1 + 96606717088729788ρ 7 1,
(11.83)
and the five real values of ρ 2 are the real zeros of:
55748192717662688 − 608378684643388064ρ 2 +
1093069815842928726ρ 2 2 − 737547208075861557ρ 3 2+
188328490866932812ρ 4 2 + 2970655440662889ρ 5 2−
10165163659763936ρ 6 2 + 1789013279420922ρ 7 2.
(11.84)
The numerical values of these five pairs (ρ 1 ,ρ 2 ) are: (3.95361, −4.61399), (0.274237,
0.112808), (1.76776, 1.21776), (5.25849, 2.05994), and (4.39521, 1.70287).
Once the values for ρ 1 and ρ 2 are available, the values for x 1 ,x 2 ,x 3 ,x 4 can be read
off from the corresponding (normalized) Stetter eigenvectors. Using these five tuples