A New Preconditioner for Toeplitz Matrices

A New Preconditioner for Toeplitz Matrices

In this paper we introduce and analyze a new preconditioner for Toeplitz matrices that exhibits excellent spectral properties: the eigenvalues of the preconditioned matrix are highly clustered around the unity. As a result, it yields very rapid convergence when used to solve Toeplitz equations via t...

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Veröffentlicht in:IEEE signal processing letters 2009-09, Vol.16 (9), p.758-761
Hauptverfasser: Dominguez-Jimenez, M.E., Ferreira, P.J.S.G.
Format: Artikel
Sprache:eng
Schlagworte:
Computational complexity
Conjugate gradient method
Convergence
Eigenvalues
Eigenvalues and eigenfunctions
Equations
Gradient methods
Iterative methods
Mathematical analysis
Matrices
Matrix methods
PCG
preconditioners
Signal processing
Spectra
Stability
Sufficient conditions
Toeplitz matrices
Unity
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Beschreibung
Zusammenfassung:In this paper we introduce and analyze a new preconditioner for Toeplitz matrices that exhibits excellent spectral properties: the eigenvalues of the preconditioned matrix are highly clustered around the unity. As a result, it yields very rapid convergence when used to solve Toeplitz equations via the preconditioned conjugate gradient method. The new preconditioner can be regarded as a refinement of preconditioners built by embedding the Toeplitz matrix in a positive definite circulant. Necessary and sufficient conditions that ensure that the positive definite embedding is possible are given.
ISSN:1070-9908
1558-2361
DOI:10.1109/LSP.2009.2024735