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10.6. EXAMPLES 181
5. Compute the non-zero finite eigenvalues ρ 1 and the corresponding eigenvectors
w of the regular generalized eigenvalue problem (F + ρ 1 G)w =0.
6. Determine the corresponding solutions (x 1 ,...,x N ) by computing solutions v
to the polynomial eigenvalue problem Ãã N−1
(ρ 1 ) T v = 0 and by reading off
the values of x 1 ,...,x N from the entries of the eigenvectors v at the specified
positions. These solutions (x 1 ,...,x N ) are constructed to satisfy the constraint
ã N−1 = 0 in (10.3).
7. Extend the already computed solution set (for q 0 0) with any solutions to
(10.5), for q 0 = 0, which also satisfy the linear constraint (10.3).
8. Compute all the corresponding H 2 -criterion values, i.e., by substituting (x 1 ,
..., x N ) and the corresponding ρ 1 into the third order polynomial specified in
Equation (10.42).
9. Select the feasible solution (ˆx 1 ,...,ˆx N ) and ˆρ 1 that produces the smallest (positive)
real criterion value.
10. Use the selected solution (ˆx 1 ,...,ˆx N ) and ˆρ 1 to compute the real stable polynomial
ã(s) from Equation (8.27). This is a polynomial of order N − 2 since
the coefficient ã N−1 is zero. Furthermore, read off q 0 from the coefficient ã N−2
as q 0 =(−1) N−2 ã N−2 .
11. Compute a(s) from ã(s) and q 0 .
12. Compute b(s) using Equation (8.49).
13. The globally optimal approximation G(s) of order N − 2 is given by G(s) =
b(s)/a(s).
10.6 Examples
In this section three worked examples are presented to illustrate the co-order k =2
techniques discussed in this chapter.
In Subsection 10.6.1 a system of order 6 is reduced to order 4. The approximation
of order 4 is computed by applying the co-order k = 1 technique of Chapter 9 twice
and the co-order k = 2 technique of the present chapter once. The co-order k =2
technique is applied here without computing the Kronecker canonical form. Instead,
a balancing method is used to improve the ill-conditioning of the singular matrix
pencil. The results for both approaches are very similar.
Subsection 10.6.2 describes the reduction of a system of order 7 to a globally
optimal H 2 approximation of order 5. Computing the Kronecker canonical form is
necessary in this example, because otherwise all the eigenvalues of the singular matrix
pencil are incorrectly specified to be 0, indeterminate or infinite by all the numerical
methods we have available, due to ill-conditioning of the singular pencil.