Computing the Structured Pseudospectrum of a Toeplitz Matrix and Its Extreme Points

The computation of the structured pseudospectral abscissa and radius (with respect to the Frobenius norm) of a Toeplitz matrix is discussed and two algorithms based on a low-rank property to construct extremal perturbations are presented. The algorithms are inspired by those considered in [N. Guglie...

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Bibliographic Details
Published in:SIAM journal on matrix analysis and applications 2012-01, Vol.33 (4), p.1300-1319
Main Authors: Butta, P, Guglielmi, N, Noschese, S
Format: Article
Language:English
Subjects:
Algorithms
Computation
Computer programs
Convergence
Eigenvalues
Eigenvectors
Hierarchies
Mathematical analysis
Norms
Perturbation methods
Representations
Software
Sparsity
Symmetry
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Summary:The computation of the structured pseudospectral abscissa and radius (with respect to the Frobenius norm) of a Toeplitz matrix is discussed and two algorithms based on a low-rank property to construct extremal perturbations are presented. The algorithms are inspired by those considered in [N. Guglielmi and M. Overton, SIAM J. Matrix Anal. Appl., 32 (2011), pp. 1166--1192] for the unstructured case, but their extension to structured pseudospectra and analysis presents several difficulties. Natural generalizations of the algorithms, allowing us to draw significant sections of the structured pseudospectra in proximity of extremal points, are also discussed. Since no algorithms are available in the literature to draw such structured pseudospectra, the approach we present seems promising to extend existing software tools (Eigtool, Seigtool) to structured pseudospectra representation for Toeplitz matrices. We discuss local convergence properties of the algorithms and show some applications to a few illustrative examples. [PUBLICATION ABSTRACT]
ISSN:0895-4798
1095-7162
DOI:10.1137/120864349