Properties of a class of perturbed Toeplitz periodic tridiagonal matrices
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Properties of a class of perturbed Toeplitz periodic tridiagonal matrices Yaru Fu1,2 · Xiaoyu Jiang1,3 · Zhaolin Jiang1
· Seongtae Jhang2
Received: 20 January 2020 / Revised: 3 April 2020 / Accepted: 16 April 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract In this paper, for a class of perturbed Toeplitz periodic tridiagonal (PTPT) matrices, some properties, including the determinant, the inverse matrix, the eigenvalues and the eigenvectors, are studied in detail. Specifically, the determinant of the PTPT matrix can be explicitly expressed using the well-known Fibonacci numbers; the inverse of the PTPT matrix can also be explicitly expressed using the Lucas number and only four elements in the PTPT matrix. Eigenvalues and eigenvectors can be obtained under certain conditions. In addition, some algorithms are presented based on these theoretical results. Comparison of our new algorithms and some recent works is given. Numerical results confirm our new theoretical results and show that the new algorithms not only can obtain accurate results but also have much better computing efficiency than some existing algorithms studied recently. Keywords Perturbed Toeplitz periodic tridiagonal matrix · Determinant · Inverse · Eigenvalue · Eigenvector Mathematics Subject Classification 15A09 · 15A15 · 15A18 · 65F50
Communicated by Yimin wei. The research was supported by National Natural Science Foundation of China (Grant No.11671187) and the PhD Research Foundation of Linyi University (Grant No. LYDX2018BS052).
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Xiaoyu Jiang [email protected] Zhaolin Jiang [email protected] Yaru Fu [email protected] Seongtae Jhang [email protected]
1
School of Mathematics and Statistics, Linyi University, Linyi 276000, China
2
College of Information Technology, The University of Suwon, Hwaseong-si 445-743, South Korea
3
School of Information Science and Technology, Linyi University, Linyi 276000, China 0123456789().: V,-vol
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1 Introduction Tridiagonal matrices and periodic (or cyclic) tridiagonal matrices frequently appear in mathematical chemistry and computational physics as well as scientific and engineering investigations. Some applications of these matrices can be found in quantum chemistry, Hückel theory, boundary value problems (BVPs), fluid mechanics, spline approximation, parallel computing, vision, image and signal processing (VISP), etc. (Znojil 2000; Björck and Golub 1977; Iachello and Del Sol Mesa 1999; Ye and Ladik 1993; Verkaik and Lin 2005; Chawla et al. 1992; Alfaro and Montaner 1995). For a general periodic tridiagonal matrix ⎞ ⎛ a 1 c1 0 · · · · · · 0 δ ⎟ ⎜ ⎜ d 1 a 2 c2 . . . 0 ⎟ ⎟ ⎜ ⎜ .. ⎟ ⎜ 0 d a c ... . ⎟ ⎟ ⎜ 2 3 3 ⎜ . . . . . .. ⎟ .. ⎟ . . . . . J (δ, ; dk , ak , ck ; a1 , an ) = ⎜ , (1) . ⎜ . . . . . . ⎟ ⎟ ⎜ . ⎟ ⎜ . .. .. .. .. ⎜ . . 0 ⎟ . . . ⎟ ⎜ ⎟ ⎜ .. .. .. ⎝0 . . . cn−1 ⎠ 0 · · · · · · 0 dn−1 an n×n the inversion of J (δ, ; dk , ak , ck ; a1 , an ) has been studied extensively and found with simple analytic expression
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