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BTTB preconditioners for BTTB systems

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Abstract

In this paper, we consider solving the BTTB system \({\cal T}_{m,n}[f] {\bf{x}} = {\bf{b}}\) by the preconditioned conjugate gradient (PCG) method, where \({\cal T}_{m,n}[f]\) denotes the m × m block Toeplitz matrix with n × n Toeplitz blocks (BTTB) generated by a (2π, 2π)-periodic continuous function f(x, y). We propose using the BTTB matrix \({\cal T}_{m,n}[1/f]\) to precondition the BTTB system and prove that only O(m) + O(n) eigenvalues of the preconditioned matrix \({\cal T}_{m,n}[1/f] {\cal T}_{m,n}[f]\) are not around 1 under the condition that f(x, y) > 0. We then approximate 1/f(x, y) by a bivariate trigonometric polynomial, which can be obtained in O(m n log(m n)) operations by using the fast Fourier transform technique. Numerical results show that our BTTB preconditioner is more efficient than block circulant preconditioners.

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Authors and Affiliations

  1. Department of Mathematics, Shantou University, Shantou, Guangdong, 515063, People’s Republic of China

    Fu-Rong Lin

  2. Department of Information and Electronic Engineering, Xuzhou College of Industrial Technology, Xuzhou, Jiangsu, 221006, People’s Republic of China

    Chi-Xi Wang

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  1. Fu-Rong Lin
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  2. Chi-Xi Wang
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Correspondence to Fu-Rong Lin.

Additional information

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

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Lin, FR., Wang, CX. BTTB preconditioners for BTTB systems. Numer Algor 60, 153–167 (2012). https://doi.org/10.1007/s11075-011-9516-z

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  • DOI: https://doi.org/10.1007/s11075-011-9516-z

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