The spectra of preconditioned Toeplitz matrix sequences can have gaps

Thomas Huckle; Stefano Serra-Capizzano

doi:10.1137/S0895479802419269

The spectra of preconditioned Toeplitz matrix sequences can have gaps

Thomas Huckle
, Stefano Serra-Capizzano

Informatics 5 - Chair of Scientific Computing

University of Insubria

Research output: Contribution to journal › Article › peer-review

Abstract

Different than for the case of Toeplitz matrix sequences {T _n(f)}, f ∈ L¹, we can prove that the closure of the union of all the spectra of preconditioned matrix sequences of the form {T ^n-1 (g)T_n(f)}, f, g ∈ L¹, g ≥ 0, can have gaps if the essential range of f/g is not connected. The result has important consequences on the practical use of band Toeplitz precouditioners widely used in the literature both for (multilevel) ill-conditioned positive definite and (multilevel) indefinite Toeplitz linear systems.

Original language	English
Pages (from-to)	930-946
Number of pages	17
Journal	SIAM Journal on Matrix Analysis and Applications
Volume	25
Issue number	4
DOIs	https://doi.org/10.1137/S0895479802419269
State	Published - 2004

Keywords

Generating function
Preconditioning
Spectral distribution and localization results
Toeplitz matrix

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10.1137/S0895479802419269

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abstract = "Different than for the case of Toeplitz matrix sequences \{T n(f)\}, f ∈ L1, we can prove that the closure of the union of all the spectra of preconditioned matrix sequences of the form \{T n-1 (g)Tn(f)\}, f, g ∈ L1, g ≥ 0, can have gaps if the essential range of f/g is not connected. The result has important consequences on the practical use of band Toeplitz precouditioners widely used in the literature both for (multilevel) ill-conditioned positive definite and (multilevel) indefinite Toeplitz linear systems.",

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AU - Serra-Capizzano, Stefano

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N2 - Different than for the case of Toeplitz matrix sequences {T n(f)}, f ∈ L1, we can prove that the closure of the union of all the spectra of preconditioned matrix sequences of the form {T n-1 (g)Tn(f)}, f, g ∈ L1, g ≥ 0, can have gaps if the essential range of f/g is not connected. The result has important consequences on the practical use of band Toeplitz precouditioners widely used in the literature both for (multilevel) ill-conditioned positive definite and (multilevel) indefinite Toeplitz linear systems.

AB - Different than for the case of Toeplitz matrix sequences {T n(f)}, f ∈ L1, we can prove that the closure of the union of all the spectra of preconditioned matrix sequences of the form {T n-1 (g)Tn(f)}, f, g ∈ L1, g ≥ 0, can have gaps if the essential range of f/g is not connected. The result has important consequences on the practical use of band Toeplitz precouditioners widely used in the literature both for (multilevel) ill-conditioned positive definite and (multilevel) indefinite Toeplitz linear systems.

KW - Generating function

KW - Preconditioning

KW - Spectral distribution and localization results

KW - Toeplitz matrix

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JO - SIAM Journal on Matrix Analysis and Applications

JF - SIAM Journal on Matrix Analysis and Applications

IS - 4

ER -

The spectra of preconditioned Toeplitz matrix sequences can have gaps

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Keywords

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