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.

COMPUTING THE STRUCTURED PSEUDOSPECTRUM OF A TOEPLITZ MATRIX AND ITS EXTREME POINTS / Butta', Paolo; N., Guglielmi; Noschese, Silvia. - In: SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS. - ISSN 0895-4798. - STAMPA. - 33:4(2012), pp. 1300-1319. [10.1137/120864349]

COMPUTING THE STRUCTURED PSEUDOSPECTRUM OF A TOEPLITZ MATRIX AND ITS EXTREME POINTS

BUTTA', Paolo;NOSCHESE, Silvia
2012

Abstract

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.
2012
eigenvalue; pseudospectrum; spectral abscissa; spectral radius; structured pseudospectrum; toeplitz structure
01 Pubblicazione su rivista::01a Articolo in rivista
COMPUTING THE STRUCTURED PSEUDOSPECTRUM OF A TOEPLITZ MATRIX AND ITS EXTREME POINTS / Butta', Paolo; N., Guglielmi; Noschese, Silvia. - In: SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS. - ISSN 0895-4798. - STAMPA. - 33:4(2012), pp. 1300-1319. [10.1137/120864349]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/492320
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