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11.3. KRONECKER CANONICAL FORM OF A TWO-PARAMETER PENCIL 209
which is equivalent with (11.31) and where the upper left block is the transposed of
L T η 1
and is of dimension η 1 × (η 1 +1)=1× 2:
L η1 = ( ρ 1 1 ) T
. (11.37)
Finally, this singular L T η 1
part can be split off from the matrix pencil.
Once all the singular parts of the pencil P (ρ 2 ) T + ρ 1 Q(ρ 2 ) T are split off, all the
transformation matrices V (ρ 2 ) and W (ρ 2 ) are available. Using the result in Proposition
11.2, one has to compute the values which make each of the transformation
matrices V (ρ 2 ) and W (ρ 2 ) singular as in (11.20). When the solutions ρ 2 of (11.20)
are known, they can be substituted into the original problem (11.3), which yields the
square polynomial eigenvalue problems in the only parameter ρ 1 :
⎧
⎨ ÃãN−1 (ρ 1 ) T v =0
(11.38)
⎩
ÃãN−2 (ρ 1 ) T v =0
From an eigenvector v of (11.38), the values for x 1 ,...,x N can be read off, since
these vectors exhibit the Stetter structure. All the tuples of solutions (ρ 1 ,ρ 2 ,x 1 ,...,
x N ) found in this way are the solutions of the quadratic system of equations (11.1)
which satisfy both the constraints (11.2). The solutions of which the values for ρ 1
and ρ 2 are real, may give feasible approximations G(s) of order N − 3. The real
and stable approximation G(s) with the smallest H 2 -criterion value is the globally
optimal approximation of order N − 3.
Remark 11.2. Using this approach, it is possible to split off singular parts of(
a matrix
pencil which is linear in ρ 1 and ρ 2 . After applying a transformation W −1 (ρ 2 ) P (ρ 2 ) T +
ρ 1 Q(ρ 2 )
)V T (ρ 2 ), the lower right block ˜P (ρ 2 )+ρ 1 ˜Q(ρ2 ) of the equivalent pencil in
(11.31) is not necessarily linear in ρ 2 anymore. Therefore it is possible that an extra
linearization step (as described in Section 11.2) is needed after splitting off a singular
part. With such a linearization the matrix dimensions may grow rapidly.
11.3.3 Computing the transformation matrices for a non-linear
two-parameter matrix pencil
In Equation (11.22) and in the steps thereafter, it does not matter whether the matrices
P (ρ 2 ) and Q(ρ 2 ) are linear in ρ 2 or not. Moreover, even the variable ρ 1 is allowed
to show up in a non-linear fashion when following a different approach than in the
previous section. Without the linearization step, which brings (11.3) into the form of
(11.4), it is possible to work with smaller sized matrices. This is the subject of this
subsection.