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New preconditioners for systems of linear equations with Toeplitz structure

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Abstract

In this paper, we consider applying the preconditioned conjugate gradient (PCG) method to solve system of linear equations \(T x = \mathbf b \) where \(T\) is a block Toeplitz matrix with Toeplitz blocks (BTTB). We first consider Level-2 circulant preconditioners based on generalized Jackson kernels. Then, BTTB preconditioners based on a splitting of BTTB matrices are proposed. We show that the BTTB preconditioners based on splitting are special cases of embedding-based BTTB preconditioners, which are also good BTTB preconditioners. As an application, we apply the proposed preconditioners to solve BTTB least squares problems. Our preconditioners work for BTTB systems with nonnegative generating functions. The implementations of the construction of the preconditioners and the relevant matrix-vector multiplications are also presented. Finally, Numerical examples, including image restoration problems, are presented to demonstrate the efficiency of our proposed preconditioners.

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Notes

  1. The code can be found in http://www.math.cuhk.edu.hk/~rchan/paper/SIAM/toeplitz091106.rar.

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Authors and Affiliations

  1. Department of Mathematics, Shantou University, Shantou, 515063, Guangdong, China

    Yong-Jie Shi & Xue-Bo Pi

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  1. Yong-Jie Shi
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  2. Xue-Bo Pi
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Correspondence to Yong-Jie Shi.

Additional information

The research was supported in part by the Guangdong Provincial NSF under contract No. 10151503101000023.

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Shi, YJ., Pi, XB. New preconditioners for systems of linear equations with Toeplitz structure. Calcolo 51, 31–55 (2014). https://doi.org/10.1007/s10092-012-0074-0

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