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Volume 30, Issue 5
Banded Toeplitz Preconditioners for Toeplitz Matrices from Sinc Methods

Zhi-Ru Ren

DOI: 10.4208/jcm.1203-m3761

J. Comp. Math., 30 (2012), pp. 533-543.

Published online: 2012-10

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  • Abstract

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

  • Keywords

Toeplitz matrix, Banded Toeplitz preconditioner, Generating function, Sinc method, Eigenvalue bounds.

  • AMS Subject Headings

65F10, 65F50.

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-30-533, author = {Zhi-Ru Ren}, title = {Banded Toeplitz Preconditioners for Toeplitz Matrices from Sinc Methods}, journal = {Journal of Computational Mathematics}, year = {2012}, volume = {30}, number = {5}, pages = {533--543}, abstract = {

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

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1203-m3761}, url = {http://global-sci.org/intro/article_detail/jcm/8448.html} }
TY - JOUR T1 - Banded Toeplitz Preconditioners for Toeplitz Matrices from Sinc Methods AU - Zhi-Ru Ren JO - Journal of Computational Mathematics VL - 5 SP - 533 EP - 543 PY - 2012 DA - 2012/10 SN - 30 DO - http://doi.org/10.4208/jcm.1203-m3761 UR - https://global-sci.org/intro/article_detail/jcm/8448.html KW - Toeplitz matrix, Banded Toeplitz preconditioner, Generating function, Sinc method, Eigenvalue bounds. AB -

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

Zhi-Ru Ren. (2012). Banded Toeplitz Preconditioners for Toeplitz Matrices from Sinc Methods. Journal of Computational Mathematics. 30 (5). 533-543. doi:10.4208/jcm.1203-m3761
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