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Volume 31, Issue 1
On Structured Variants of Modified HSS Iteration Methods for Complex Toeplitz Linear Systems

Fang Chen, Yaolin Jiang & Qingquan Liu

DOI: 10.4208/jcm.1208-m4022

J. Comp. Math., 31 (2013), pp. 57-67.

Published online: 2013-02

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  • Abstract

The $Modified$ $Hermitian$ $and$ $skew$-$Hermitian$ $splitting$ (MHSS) iteration method was presented and studied by Bai, Benzi and Chen (Computing, 87(2010), 93-111) for solving a class of complex symmetric linear systems. In this paper, using the properties of Toeplitz matrix, we propose a class of structured MHSS iteration methods for solving the complex Toeplitz linear system. Theoretical analysis shows that the structured MHSS iteration method is unconditionally convergent to the exact solution. When the MHSS iteration method is used directly to complex symmetric Toeplitz linear systems, the computational costs can be considerately reduced by use of Toeplitz structure. Finally, numerical experiments show that the structured MHSS iteration method and the structured MHSS preconditioner are efficient for solving the complex Toeplitz linear system.

  • Keywords

Toeplitz matrix, MHSS iteration method, Complex symmetric linear system.

  • AMS Subject Headings

65F10, 65F50.

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-31-57, author = {Fang Chen, Yaolin Jiang and Qingquan Liu}, title = {On Structured Variants of Modified HSS Iteration Methods for Complex Toeplitz Linear Systems}, journal = {Journal of Computational Mathematics}, year = {2013}, volume = {31}, number = {1}, pages = {57--67}, abstract = {

The $Modified$ $Hermitian$ $and$ $skew$-$Hermitian$ $splitting$ (MHSS) iteration method was presented and studied by Bai, Benzi and Chen (Computing, 87(2010), 93-111) for solving a class of complex symmetric linear systems. In this paper, using the properties of Toeplitz matrix, we propose a class of structured MHSS iteration methods for solving the complex Toeplitz linear system. Theoretical analysis shows that the structured MHSS iteration method is unconditionally convergent to the exact solution. When the MHSS iteration method is used directly to complex symmetric Toeplitz linear systems, the computational costs can be considerately reduced by use of Toeplitz structure. Finally, numerical experiments show that the structured MHSS iteration method and the structured MHSS preconditioner are efficient for solving the complex Toeplitz linear system.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1208-m4022}, url = {http://global-sci.org/intro/article_detail/jcm/9721.html} }
TY - JOUR T1 - On Structured Variants of Modified HSS Iteration Methods for Complex Toeplitz Linear Systems AU - Fang Chen, Yaolin Jiang & Qingquan Liu JO - Journal of Computational Mathematics VL - 1 SP - 57 EP - 67 PY - 2013 DA - 2013/02 SN - 31 DO - http://doi.org/10.4208/jcm.1208-m4022 UR - https://global-sci.org/intro/article_detail/jcm/9721.html KW - Toeplitz matrix, MHSS iteration method, Complex symmetric linear system. AB -

The $Modified$ $Hermitian$ $and$ $skew$-$Hermitian$ $splitting$ (MHSS) iteration method was presented and studied by Bai, Benzi and Chen (Computing, 87(2010), 93-111) for solving a class of complex symmetric linear systems. In this paper, using the properties of Toeplitz matrix, we propose a class of structured MHSS iteration methods for solving the complex Toeplitz linear system. Theoretical analysis shows that the structured MHSS iteration method is unconditionally convergent to the exact solution. When the MHSS iteration method is used directly to complex symmetric Toeplitz linear systems, the computational costs can be considerately reduced by use of Toeplitz structure. Finally, numerical experiments show that the structured MHSS iteration method and the structured MHSS preconditioner are efficient for solving the complex Toeplitz linear system.

Fang Chen, Yaolin Jiang and Qingquan Liu. (2013). On Structured Variants of Modified HSS Iteration Methods for Complex Toeplitz Linear Systems. Journal of Computational Mathematics. 31 (1). 57-67. doi:10.4208/jcm.1208-m4022
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