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Fast solvers for tridiagonal Toeplitz linear systems

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Abstract

Let A be a tridiagonal Toeplitz matrix denoted by \(A = {\text {Tritoep}} (\beta , \alpha , \gamma )\). The matrix A is said to be: strictly diagonally dominant if \(|\alpha |>|\beta |+|\gamma |\), weakly diagonally dominant if \(|\alpha |\ge |\beta |+|\gamma |\), subdiagonally dominant if \(|\beta |\ge |\alpha |+|\gamma |\), and superdiagonally dominant if \(|\gamma |\ge |\alpha |+|\beta |\). In this paper, we consider the solution of a tridiagonal Toeplitz system \(A\mathbf {x}= \mathbf {b}\), where A is subdiagonally dominant, superdiagonally dominant, or weakly diagonally dominant, respectively. We first consider the case of A being subdiagonally dominant. We transform A into a block \(2\times 2\) matrix by an elementary transformation and then solve such a linear system using the block LU factorization. Compared with the LU factorization method with pivoting, our algorithm takes less flops, and needs less memory storage and data transmission. In particular, our algorithm outperforms the LU factorization method with pivoting in terms of computing efficiency. Then, we deal with superdiagonally dominant and weakly diagonally dominant cases, respectively. Numerical experiments are finally given to illustrate the effectiveness of our algorithms.

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Acknowledgements

The authors would like to thank the supports of the National Natural Science Foundation of China under Grant no. 11371075, the Hunan Key Laboratory of mathematical modeling and analysis in engineering, the research innovation program of Changsha University of Science and Technology for postgraduate students under Grant (CX2019SS34), and the Portuguese Funds through FCT-Fundacão para a Ciência, within the Project UIDB/00013/2020 and UIDP/00013/2020.

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Correspondence to Yulin Zhang.

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Communicated by Jinyun Yuan.

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Liu, Z., Li, S., Yin, Y. et al. Fast solvers for tridiagonal Toeplitz linear systems. Comp. Appl. Math. 39, 315 (2020). https://doi.org/10.1007/s40314-020-01369-3

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  • DOI: https://doi.org/10.1007/s40314-020-01369-3

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