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11.4. COMPUTING THE APPROXIMATION G(S) 221
The difference here with respect to the previous approach lies in the construction
of the transformation matrices V and W . The first columns of the matrices V are
chosen, as usual, as the η 1 + 1 vectors z 0 (ρ 2 ),...,z η1 (ρ 2 ) which makes this a polynomial
matrix in ρ 2 . The first columns of the matrix W are chosen as the η 1 vectors
B(ρ 2 ) z 1 (ρ 2 ),...,B(ρ 2 ) z η1 (ρ 2 ). In this approach both the transformation matrices
are polynomial in the variable ρ 2 , whereas in the previous approach the matrix W
was polynomial in both ρ 1 and ρ 2 . This makes it easier to compute the inverse of the
transformation matrix W .
After all the singular parts of the matrix pencil B(ρ 2 )+ρ 1 C(ρ 1 ,ρ 2 ) are split off,
there should not exist a regular part which admits other solutions than the trivial
solution. Because the existence of a regular part is impossible (for the same reasons
mentioned in Section 11.3.1), the solutions for ρ 1 and ρ 2 correspond to the values
which make the transformation matrices V (ρ 2 ) and W (ρ 2 ) singular as in (11.20) of
Proposition 11.2.
Remark 11.7. The value of Ñ can change (increase or decrease), during the process
of splitting off singular parts of the matrix pencil. This has a direct effect on the
decomposition of the matrix pencil (11.59) in (11.60) and (11.61).
11.4 Computing the approximation G(s)
Let x 1 ,...,x N and ρ 1 and ρ 2 be a solution of the system of equations (11.1) which
also satisfies the additional constraints ã N−1 = 0 and ã N−2 = 0 in (11.2). Using such
a solution an approximation of order N −3 can be computed as described in Theorem
8.3 for k =3.
From the set of solutions to the quadratic system of equations (11.1) that satisfy
the constraints ã N−1 = 0 and ã N−2 = 0 in (11.2), it is straightforward to select those
that are feasible, i.e., which give rise to a real stable approximation G(s) of order
N − 3. It is then possible to select the globally optimal approximation by computing
the H 2 -norm of the difference H(s)−G(s) for every feasible solution G(s) and selecting
the one for which this criterion is minimal. This H 2 -norm of H(s)−G(s) is, according
to Theorem 8.2 for k = 3, given by:
V H (x 1 ,x 2 ,...,x N ,ρ 1 ,ρ 2 ) =
e(s)
∣∣d(s) − b(s)
a(s) ∣∣
=
N∑
i=1
2
H 2
(1 + ρ 1 δ i + ρ 2 δi 2)2 (1 − ρ 1 δ i + ρ 2 δi 2)
e(δ i )d ′ x 3 i .
(δ i )d(−δ i )
(11.74)