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Block band Toeplitz preconditioners derived from generating function approximations: analysis and applications

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Abstract

We are concerned with the study and the design of optimal preconditioners for ill-conditioned Toeplitz systems that arise from a priori known real-valued nonnegative generating functions f(x,y) having roots of even multiplicities. Our preconditioned matrix is constructed by using a trigonometric polynomial θ(x,y) obtained from Fourier/kernel approximations or from the use of a proper interpolation scheme. Both of the above techniques produce a trigonometric polynomial θ(x,y) which approximates the generating function f(x,y), and hence the preconditioned matrix is forced to have clustered spectrum. As θ(x,y) is chosen to be a trigonometric polynomial, the preconditioner is a block band Toeplitz matrix with Toeplitz blocks, and therefore its inversion does not increase the total complexity of the PCG method. Preconditioning by block Toeplitz matrices has been treated in the literature in several papers. We compare our method with their results and we show the efficiency of our proposal through various numerical experiments.

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Authors and Affiliations

  1. Department of Mathematics, University of Ioannina, 45110, Ioannina, Greece

    D. Noutsos & P. Vassalos

  2. Dipartimento di Fisica e Matematica, Università dell’Insubria-Sede di Como, via Valleggio 11, 22100, Como, Italy

    S. Serra Capizzano

Authors
  1. D. Noutsos
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  2. S. Serra Capizzano
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  3. P. Vassalos
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Correspondence to D. Noutsos.

Additional information

This research was co-funded by the European Union in the framework of the program “Pythagoras I” of the “Operational Program for Education and Initial Vocational Training” of the 3rd Community Support Framework of the Hellenic Ministry of Education, funded by national sources (25%) and by the European Social Fund - ESF (75%). The work of the second and of the third author was partially supported by MIUR (Italian Ministry of University and Research), grant number 2004015437.

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Noutsos, D., Capizzano, S.S. & Vassalos, P. Block band Toeplitz preconditioners derived from generating function approximations: analysis and applications. Numer. Math. 104, 339–376 (2006). https://doi.org/10.1007/s00211-006-0020-7

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