Fast Algorithms for the Approximation of the Pseudospectral Abscissa and Pseudospectral Radius of a Matrix

Abstract

The ε-pseudospectral abscissa and radius of an n × n matrix are, respectively, the maximal real part and the maximal modulus of points in its ε-pseudospectrum, defined using the spectral norm. Existing techniques compute these quantities accurately, but the cost is multiple singular value decompositions and eigenvalue decompositions of order n, making them impractical when n is large. We present new algorithms based on computing only the spectral abscissa or radius of a sequence of matrices, generating a sequence of lower bounds for the pseudospectral abscissa or radius. We characterize fixed points of the iterations, and we discuss conditions under which the sequence of lower bounds converges to local maximizers of the real part or modulus over the pseudospectrum, proving a locally linear rate of convergence for ε sufficiently small. The convergence results depend on a perturbation theorem for the normalized eigenprojection of a matrix as well as a characterization of the group inverse (reduced resolvent) of a singular matrix defined by a rankone perturbation. The total cost of the algorithms is typically only a constant times the cost of computing the spectral abscissa or radius, where the value of this constant usually increases with ε, and may be less than 10 in many practical cases of interest.

Numerical computation of structured complex stability radii of large-scale matrices and pencils

“…Thus, our strategy consists of computing a sequence of suitable structured rank-1 perturbed pencils λE − (A + εBuv H C) such that one of the perturbed eigenvalues converges to the rightmost pseudoeigenvalue of λE−A. A similar technique has already been successfully applied to compute the pseudospectral abscissa of a matrix, see [17]. We need the following result for the first order perturbation theory of matrix pencils.…”

“…In this paper we have introduced a new iterative scheme for computing the structured complex stability radius of a matrix or a pencil, based on the method introduced in [17]. The algorithm computes a sequence of structured pseudospectral abscissae.…”

ℋ∞‐Norm Computation for Large and Sparse Descriptor Systems

“…Thus, our strategy consists of computing a sequence of suitable structured rank-1 perturbed pencils λE − (A + εBuv H C) such that one of the perturbed eigenvalues converges to the rightmost structured pseudopole of G(s), similarly as in [2]. We need the following result for the first order perturbation theory of matrix pencils.…”

“…Furthermore, we call G(s) proper if lim ω→∞ G(iω) 2 < ∞, otherwise we call it improper. By RH p×m ∞ we denote the rational Banach space of all stable and proper functions of the form (2). For this space we define the H ∞ -norm, given by…”

“…This routine uses the relationship between the H ∞ -norm the structured complex stability radius of a matrix or a pencil. Based on the method introduced in [2], the algorithm computes a sequence of structured pseudospectral abscissae. This is done by computing an optimal rank-1 perturbation of the system such that one of the eigenvalues of the perturbed matrix or pencil converges to the rightmost structured pseudopole of the transfer function.…”