On the computation of structured singular values and pseudospectra

Abstract

Structured singular values and pseudospectra play an important role in assessing the properties of a linear system under structured perturbations. This paper discusses computational aspects of structured pseudospectra for structures that admit an eigenvalue minimization characterization, including the classes of real, skew-symmetric, Hermitian, and Hamiltonian perturbations. For all these structures we develop algorithms that require O(n 2) operations per grid point, combining the Schur decomposition with a Lanczos method. These algorithms form the basis of a graphical Matlab interface for plotting structured pseudospectra.

Nearest common root of polynomials, approximate greatest common divisor and the structured singular value

“…The approach is not limited to"small" perturbations in the polynomials' coefficients and general vector norms can be used in principle to represent uncertainty (in the paper quadratic and l ∞ norms are considered). The resulting optimization is non-convex and computationally demanding for large problems, although tight convex bounds (and techniques for improving them) have been reported in the literature [16], [17], [14], [20], [22], [29], [33], [34], [35]. Algorithms combining the technique presented in this work with aspects of [24] and [25] also seem possible and will be reported in a future publication.…”

Computing Structured Singular Values for Delay and Polynomial Eigenvalue Problems

“…In the second column, it is presented the set of block diagonal matrices denoted by BLK. In the third, fourth and fifth columns, it is presented the upper and lower bounds By using the algorithm [13], one can obtain the perturbation * * ∆  with In Table 5, it is presented the comparison of the bounds of SSV computed by MUSSV and the algorithm [13] for the matrix 5 A given bellow. In the very first column, it is presented the dimension of the matrix 5 A .…”

Perturbation Theory for Hamiltonian Matrices and the Distance to Bounded-Realness

“…This can be done by computing the Hamiltonian pseudospectrum Λ τ ðH; CÞ with the method of [19] and testing whether or not Λ τ ðH; CÞ ∩ iR ¼ ∅. Alternatively, we compute the eigenvalues of H − τJ and H þ τJ .…”