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The Shifted Classical Circulant and Skew Circulant Splitting Iterative Methods for Toeplitz Matrices

Published online by Cambridge University Press:  20 November 2018

Zhongyun Liu ,
Xiaorong Qin ,
Nianci Wu  and
Yulin Zhang
Zhongyun Liu
Affiliation:
School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha 410076, P.R. China e-mail: [email protected]
Xiaorong Qin
Affiliation:
Centro de Matemática, Universidade do Minho, 4710-057 Braga, Portugal e-mail: [email protected]@[email protected]
Nianci Wu
Affiliation:
Centro de Matemática, Universidade do Minho, 4710-057 Braga, Portugal e-mail: [email protected]@[email protected]
Yulin Zhang
Affiliation:
Centro de Matemática, Universidade do Minho, 4710-057 Braga, Portugal e-mail: [email protected]@[email protected]
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Abstract

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It is known that every Toeplitz matrix $T$ enjoys a circulant and skew circulant splitting (denoted $\text{CSCS}$) i.e., $T=C-S$ a circulant matrix and $S$ a skew circulant matrix. Based on the variant of such a splitting (also referred to as $\text{CSCS}$), we first develop classical $\text{CSCS}$ iterative methods and then introduce shifted $\text{CSCS}$ iterative methods for solving hermitian positive definite Toeplitz systems in this paper. The convergence of each method is analyzed. Numerical experiments show that the classical $\text{CSCS}$ iterative methods work slightly better than the Gauss–Seidel $(\text{GS})$ iterative methods if the $\text{CSCS}$ is convergent, and that there is always a constant $\alpha $ such that the shifted $\text{CSCS}$ iteration converges much faster than the Gauss–Seidel iteration, no matter whether the $\text{CSCS}$ itself is convergent or not.

Keywords

15A2365F1065F15Hermitian positive definiteCSCS splittingGauss-Seidel splittingiterative methodToeplitz matrix
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 60 , Issue 4 , 01 December 2017 , pp. 807 - 815
Copyright
Copyright © Canadian Mathematical Society 2017

References

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