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9.2. COMPUTING THE APPROXIMATION G(S) 153
the approximation G(s) of order N − 1 with the smallest distance between all the
real approximations G(s) and the original system H(s) of order N. If this solution
happens to be feasible it constitutes the globally optimal approximation G(s) of order
N − 1. If this is not the case, we discard this solution and proceed with the second
smallest real eigenvalue λ of the matrix A T V H
.
This approach avoids the computation of all the 2 N eigenvalues of the matrix A T V H
or of the matrices A T x i
for i =1,...,N. Computing the smallest real eigenvalue(s) of
a sparse matrix directly can be achieved by employing iterative eigenvalue solvers, as
discussed in Chapter 6. The goal here is to achieve a substantial gain in computational
efficiency.
Remark 9.1. When computing the global minimum of a Minkowski dominated polynomial
p, as discussed in Part II of this thesis, one may construct the matrix A T p
using the Stetter-Möller matrix method. The ideal I which is used to construct this
matrix is generated by the polynomials in the system of first-order conditions of the
polynomial p. The smallest real eigenvalue of this matrix yields the value and the
location of the global minimum of p. The polynomials which generate the ideal I are
directly related with the ‘criterion’ p since they are the partial derivatives of p.
Here we are working with the matrix A T V H
and an ideal I which is generated by
polynomials involved in the system of equations we want to solve. Note that the third
order criterion V H is not related to these polynomials. Note furthermore that it is still
valid to employ the nD-systems approach in this case. We have similar structures
in the nD-system but the degrees of the structures equal m = 2 instead of an odd
degree m =2d − 1 (see Equations (4.2) and (5.3)).
9.2 Computing the approximation G(s)
Let x 1 ,...,x N be a solution to the system of equations (9.1) which is read off from
the Stetter eigenvector corresponding to the eigenvalue λ min . This eigenvalue λ min
is the smallest real eigenvalue obtained by an eigenvalue computation on the matrix
A T V H
as in the previous section. Then the globally optimal approximation G(s) of
order N − 1 is computed as described in Theorem 8.3 for k =1.
The method described in this chapter is an extension of the model order reduction
approach developed by Hanzon, Maciejowski and Chou as described in [50]. In that
paper approximations G(s) of order N − 1 are constructed by computing all the
eigenvalues of all the matrices A T x i
for i =1,...,N. In this chapter this method
is extended and improved by working with the single operator A T V H
as discussed
above. Working with this operator enables the ability to zoom in on the smallest real
eigenvalue(s). Therefore iterative eigenvalue solvers as described in Chapter 6 are
used, which furthermore make it possible to use the nD-systems approach of Chapter
5. In this way one can compute only some eigenvalues of the operator in a matrix-free
fashion, which increases the efficiency of the approach. Co-order one model reduction
using the approach from this chapter is also discussed in [18]. The approach of this