A fast method for solving a block tridiagonal quasi-Toeplitz linear system

Belhaj, Skander; Hcini, Fahd; Zhang, Yulin

doi:10.4171/PM/2036

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Portugaliae Mathematica

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Volume 76, Issue 3/4, 2019, pp. 287–299

DOI: 10.4171/PM/2036

Published online: 2020-07-15

A fast method for solving a block tridiagonal quasi-Toeplitz linear system

Skander Belhaj^[1], Fahd Hcini^[2] and Yulin Zhang^[3]

(1) University of Tunis El Manar, Tunisia
(2) University of Tunis El Manar, Tunisia
(3) Universidade do Minho, Braga, Portugal

This paper addresses the problem of solving block tridiagonal quasi-Toeplitz linear systems. Inspired by [10], we propose a more general algorithm for such systems. The algorithm is based on a block decomposition for block tridiagonal quasi-Toeplitz matrices and the Sherman–Morrison–Woodbury inversion formula. We also compare the proposed approach to the standard block $LU$ decomposition method and the Gauss algorithm. A theoretical error analysis is also presented. All algorithms have been implemented in Matlab. Numerical experiments performed on a wide variety of test problems show the effectiveness of our algorithm in terms of efficiency, stability and robustness.

Keywords: System of linear equations, block tridiagonal quasi-Toeplitz matrix, block LU decomposition, Sherman–Morrison–Woodbury inversion formula

Belhaj Skander, Hcini Fahd, Zhang Yulin: A fast method for solving a block tridiagonal quasi-Toeplitz linear system. Port. Math. 76 (2019), 287-299. doi: 10.4171/PM/2036