The Shifted Classical Circulant and Skew Circulant Splitting Iterative Methods for Toeplitz Matrices
Canadian mathematical bulletin, Tome 60 (2017) no. 4, pp. 807-815
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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.
Mots-clés :
15A23, 65F10, 65F15, Hermitian positive definite, CSCS splitting, Gauss-Seidel splitting, iterative method, Toeplitz matrix
Liu, Zhongyun; Qin, Xiaorong; Wu, Nianci; Zhang, Yulin. The Shifted Classical Circulant and Skew Circulant Splitting Iterative Methods for Toeplitz Matrices. Canadian mathematical bulletin, Tome 60 (2017) no. 4, pp. 807-815. doi: 10.4153/CMB-2016-077-5
@article{10_4153_CMB_2016_077_5,
author = {Liu, Zhongyun and Qin, Xiaorong and Wu, Nianci and Zhang, Yulin},
title = {The {Shifted} {Classical} {Circulant} and {Skew} {Circulant} {Splitting} {Iterative} {Methods} for {Toeplitz} {Matrices}},
journal = {Canadian mathematical bulletin},
pages = {807--815},
year = {2017},
volume = {60},
number = {4},
doi = {10.4153/CMB-2016-077-5},
url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2016-077-5/}
}
TY - JOUR AU - Liu, Zhongyun AU - Qin, Xiaorong AU - Wu, Nianci AU - Zhang, Yulin TI - The Shifted Classical Circulant and Skew Circulant Splitting Iterative Methods for Toeplitz Matrices JO - Canadian mathematical bulletin PY - 2017 SP - 807 EP - 815 VL - 60 IS - 4 UR - http://geodesic.mathdoc.fr/articles/10.4153/CMB-2016-077-5/ DO - 10.4153/CMB-2016-077-5 ID - 10_4153_CMB_2016_077_5 ER -
%0 Journal Article %A Liu, Zhongyun %A Qin, Xiaorong %A Wu, Nianci %A Zhang, Yulin %T The Shifted Classical Circulant and Skew Circulant Splitting Iterative Methods for Toeplitz Matrices %J Canadian mathematical bulletin %D 2017 %P 807-815 %V 60 %N 4 %U http://geodesic.mathdoc.fr/articles/10.4153/CMB-2016-077-5/ %R 10.4153/CMB-2016-077-5 %F 10_4153_CMB_2016_077_5
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