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Band-times-circulant preconditioners for non-symmetric Toeplitz systems

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Abstract

In this paper we study \(n\times n\) non-symmetric, real Toeplitz systems of the form \(T_n(f)x = b\), where the generating function of the Toeplitz matrix f is known a priori. We study the behavior of a specific circulant preconditioner and we also propose a preconditioner arising from the combination of a band Toeplitz matrix and circulant matrices, for ill-conditioned Toeplitz systems. For the solution of the system we use Krylov subspace methods and more specifically the Conjugate Gradient method for the corresponding preconditioned system of the normal equations and the Preconditioned Generalized Minimal Residual method. We prove theoretical results, which guarantee the efficiency of the proposed preconditioner either when f is continuous or it is piecewise continuous. We emphasize that f may have zeros in \((-\pi ,\pi ]\). Finally, we present various numerical experiments, where the efficiency of the proposed preconditioning technique is shown, since we get the solution of the preconditioned system in a small number of iterations.

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Acknowledgements

The authors would like to thank the anonymous referees for the constructive comments that improved the paper.

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Authors and Affiliations

  1. Department of Mathematics, University of Ioannina, Ioannina, GR45110, Greece

    Dimitrios Noutsos & Grigorios Tachyridis

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  1. Dimitrios Noutsos
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  2. Grigorios Tachyridis
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Correspondence to Dimitrios Noutsos.

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Communicated by Michiel E. Hochstenbach.

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Part of this research is co-financed by Greece and the European Union (European Social Fund-ESF) through the Operational Programme “Human Resources Development, Education and Lifelong Learning 2014–2020” in the context of the project “Krylov subspace methods and Perron-Frobenius theory” (MIS 5047643).

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Noutsos, D., Tachyridis, G. Band-times-circulant preconditioners for non-symmetric Toeplitz systems. Bit Numer Math 62, 561–590 (2022). https://doi.org/10.1007/s10543-021-00883-y

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