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2 Scaling In the second stage, a diagonal matrix D is constructed so that B = D−1 A22 D is nearly balanced. 48). , eTj B = Bej for j = 1, . . , m. Under the assumption that A22 is irreducible, Osborne [257] showed that if B = D−1 A22 D is balanced in Euclidean norm, then B has minimal Frobenius norm among all diagonal similarity transformations of A22 . 1. The iterative procedure proposed by Osborne for computing the scaling matrix D is used in the balancing algorithm by Parlett and Reinsch [263].

38) satisfies γ ≤ (2/δ)m M ε. 39) and obtain some constant Cˆ so that 2 √ ˆ (2/δ)m √ ε ˆ (2/δ)m ε2 , ≤ 2CM d2 (Si , X1 ) < 2CM 1 − ε2 √ where the latter inequality holds for ε ≤ 1/ 2. 39) is prohibitely expensive; the explicit computation of pi (Ai−1 ) alone requires O(mn3 ) flops. The purpose of this section is to reduce the cost of an overall iteration down to O(mn2 ) flops. First, we recall the well-known result that shifted QR iterations preserve matrices in unreduced Hessenberg form. 21. A square matrix A is said to be in upper Hessenberg form if all its entries below the first subdiagonal are zero.

54) ⎥ ⎥ ⎥ ⎥. ⎥ ⎦ 0 . . 55) with state vector z(·) and input u(·). 54). 55) to σ1 , . . , σm . Since x(1 : m) is uniquely defined by this property, we obtain the following connection: Any pole placement algorithm for single-input systems is a suitable method for computing a multiple of the first column of the shift polynomial; and vice versa. To some extent, this connection has already been used by Varga for designing a multi-shift pole placement algorithm [334]. A not-so-serious application is the expression of the QL iteration [116], a permuted version of the QR iteration, in three lines of Matlab code, using functions of the Matlab Control Toolbox [235]: s = eig(H(1:m,1:m)); x = acker(H(n-m+1:n,n-m+1:n)’,H(n-m,n-m+1)*eye(m,1),s); H = ctrbf(H, [zeros(1,n-m-1) 1 x]’, []); An exceedingly more serious consequence is caused by the observation that placing a large number of poles in a single-input problem is often very ill-conditioned [157, 165] in the sense that the poles of the closed loop system are very sensitive to perturbations in the input data.

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