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12.1. CONCLUSIONS 233
solver as described in Chapter 6. The third order polynomial V H given in Section 9.1
offers an additional computational advantage. The globally optimal approximation
of order N − 1 is obtained by computing the smallest real eigenvalue of the matrix
A T V H
which avoids the computation of all the eigenvalues. This is achieved by using
an iterative Jacobi–Davidson solver. In this way it is furthermore possible to work in
a matrix-free fashion.
In [50] the highest possible order for a co-order k = 1 model reduction was 9.
In Section 9.3 the potential of the reduction technique in combination with an nDsystem
and an iterative solver on the matrix A VH is demonstrated by an example
which involves the reduction of a system of order 10 to a system of order 9 without
explicitly constructing any matrices.
In the Chapters 10 and 11 the more difficult problems of computing a globally
optimal real H 2 -approximation G(s) for the co-order k =2,andk = 3 case are
addressed. As shown in this thesis, repeated application of the co-order one technique
to achieve a larger reduction of the model-order is non-optimal, which shows the
practical relevance of the co-order two and three problem.
When taking the additional linear condition in the co-order k = 2 case in Chapter
10 into account, the Stetter-Möller matrix method yields a rational matrix in ρ.
Using the result of Theorem 10.1 in Section 10.1, this matrix is transformed into a
polynomial matrix in ρ, which yields a polynomial eigenvalue problem. The important
observation that the polynomial degree of ρ in this matrix is equal to N − 1 is given
in Corollary 10.2. To solve this polynomial eigenvalue problem it is cast into an
equivalent singular generalized eigenvalue problem in Section 10.2. To accurately
compute the meaningful eigenvalues of the singular matrix pencil, the singular part
and infinite eigenvalues are split off by computing the Kronecker canonical form, as
described in Section 10.3. In Section 10.5 all these techniques are combined in one
algorithm for the co-order k = 2 case.
In Section 10.6 three examples are given where the algorithm is successfully applied
to obtain a globally optimal approximation of order N − 2. In the second example
in Subsection 10.6.2 all the eigenvalues of the involved singular matrix pencil were
incorrectly specified to be 0, indeterminate or infinite by all the numerical methods
we have available, due to ill-conditioning of the singular pencil. By computing the
Kronecker canonical form the singular parts and the infinite eigenvalues are split off
which enables to reliably compute the eigenvalues by a numerical method. Subsection
10.6.3 of this chapter describes an example of a globally optimal H 2 model-order
reduction of order 4 to order 2. The co-order k = 1 and k = 2 techniques are applied
and their performance is compared which each other. The results for this example
turn out to be quite different. The co-order k = 2 technique turns out to exhibit the
best performance in terms of the H 2 -approximation criterion.
In Chapter 11 we show how the Stetter-Möller matrix method yields two rationally
parameterized matrices in ρ 1 and ρ 2 when taking the two additional linear condition
in the co-order k = 3 case into account. Using the result in Theorem 10.1 again, both
matrices are made polynomial in ρ 1 and ρ 2 , which yields a two-parameter polynomial