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10.4. COMPUTING THE APPROXIMATION G(S) 179
They only admit the trivial solution w 1 =0,ifρ 1 α j , and for ρ 1 = α j
non-trivial solutions in the span of (1, 0,...,0) T .
we get
10.3.3 Deflation of a singular pencil
As shown in the previous two subsections, the singular pencil B + ρ 1 C in (10.13) can
be transformed into the Kronecker canonical form. For the application at hand it is
sufficient to split off the singular part. If desired, infinite and zero eigenvalues of the
regular part of the pencil may also be removed by exact arithmetic as they do not
have any meaning for this application either.
Then the remaining pencil is regular and only contains non-zero finite eigenvalues.
Moreover, the pencil is generally much better conditioned than the original pencil.
Let us denote the regular pencil D r + ρ 1 E r , with its infinite and zero eigenvalues
removed, as the pencil F −ρ 1 G. Then the matrices F and G are square and invertible
matrices and therefore this generalized eigenvalue problem (F − ρ 1 G)w = 0 can be
treated as a conventional eigenvalue problem by looking at (G −1 F − ρ 1 I)w =0.
The matrix involved here is a smaller, but still exact, matrix G −1 F containing only
non-zero finite eigenvalues which can be computed in a reliable way by numerical
methods. The eigenvalues ρ 1 and corresponding eigenvectors of the matrix G −1 F ,
lead to solutions (ρ 1 ,x 1 ,...,x N ) of the system of quadratic equations (10.2) which
satisfy the additional constraint (10.3).
10.4 Computing the approximation G(s)
Let x 1 ,...,x N and ρ 1 be a solution of the system of equations (10.2), which satisfies
the additional constraint ã N−1 = 0 in (10.3). Using this solution an approximation
G(s) of order N − 2 can be computed as described in Theorem 8.3 for k =2.
From the approximations obtained with the approach given in Theorem 8.3 for
k = 2, it is straightforward to select those that are feasible, i.e., which give rise to real
stable approximations G(s) of order N − 2. It is then possible to select the globally
optimal approximation by computing the H 2 -norm V H of the difference H(s) − G(s)
for every feasible solution G(s) and selecting the one for which this criterion value is
minimal.
The third order polynomial, given in Theorem 8.2, which represents the H 2 -
criterion of H(s) − G(s), attains for the co-order k = 2 case the form:
V H (x 1 ,x 2 ,...,x N ,ρ 1 ) =
e(s)
∣∣d(s) − b(s)
2
a(s) ∣∣
=
N∑
i=1
H 2
(1 + ρ 1 δ i ) 2 (1 − ρ 1 δ i )
e(δ i )d ′ (δ i )d(−δ i )
x 3 i .
(10.42)