### BANDED TOEPLITZ PRECONDITIONERS FOR TOEPLITZMATRICES FROM SINC METHODS

Zhi-Ru Ren

1. LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
• Received:2011-06-24 Revised:2012-03-12 Online:2012-09-15 Published:2012-09-24
• Supported by:

This work is supported by State Key Laboratory of Scientific/Engineering Computing, Chinese Academy of Sciences. The author is very much grateful to the anonymousreferees for their constructive suggestions and helpful comments which have led to significant improvement of the original manuscript of this paper.

Zhi-Ru Ren. BANDED TOEPLITZ PRECONDITIONERS FOR TOEPLITZMATRICES FROM SINC METHODS[J]. Journal of Computational Mathematics, 2012, 30(5): 533-543.

We give general expressions, analyze algebraic properties and derive eigenvalue boundsfor a sequence of Toeplitz matrices associated with the sinc discretizations of various ordersof differential operators. We demonstrate that these Toeplitz matrices can be satisfactorilypreconditioned by certain banded Toeplitz matrices through showing that the spectra ofthe preconditioned matrices are uniformly bounded. In particular, we also derive eigenvaluebounds for the banded Toeplitz preconditioners. These results are elementary inconstructing high-quality structured preconditioners for the systems of linear equationsarising from the sinc discretizations of ordinary and partial differential equations, and areuseful in analyzing algebraic properties and deriving eigenvalue bounds for the correspondingpreconditioned matrices. Numerical examples are given to show effectiveness of thebanded Toeplitz preconditioners.

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