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10.6. EXAMPLES 189
Figure 10.7: Structure of the matrix Ãã N−1
(ρ 1 ) T
The eigenvector ṽ in the generalized eigenvalue problem (B − ρ 1 C)ṽ = 0 is structured
as (v T ,ρ 1 v T ,ρ 2 1v T ,ρ 3 1v T ,ρ 4 1v T ,ρ 5 1v T ) T and the matrices B and C are of dimensions
(N − 1) 2 N × (N − 1) 2 N = 768 × 768.
The matrix pencil B + ρ 1 C is singular and computing the eigenvalues of this
pencil using standard numerical methods fails: all the 768 computed eigenvalues are
incorrectly specified to be 0, indeterminate and infinite, due to ill-conditioning of the
singular pencil. In this case, a balancing technique as used in the previous example
does not work either. Therefore we resorted to compute the Kronecker canonical form
of this pencil using exact arithmetic.
Once the Kronecker canonical form is (partially) obtained the blocks corresponding
to the unwanted indeterminate, infinite and zero eigenvalues, which cause the
ill-conditioning, are split off. In this way we first split off six indeterminate eigenvalues,
which reduces the size of the pencil to 762 × 762. This 762 × 762 pencil is
the regular part of the matrix pencil B + ρ 1 C which contains zero, infinite and finite
non-zero eigenvalues. Furthermore, a total number of 441 zero and infinite eigenvalues
are determined by computing the associated Jordan blocks with exact arithmetic.
Also these values are split off, because they also have no meaning for the model-order
reduction problem. This yields a pencil of size 321 × 321 which only contains nonzero
finite eigenvalues. These are the eigenvalues which are of importance for the H 2
model-order reduction problem. This pencil is denoted by F + ρ 1 G.
The regular generalized eigenvalue problem (F + ρ 1 G)w = 0 can now easily be
solved by a numerical method, since it is no longer ill-conditioned and does not include
any unwanted eigenvalues which hamper the computation.