Skew cyclic displacements and decompositions of inverse matrix for an innovative structure matrix

Jiang, Xiaoyu; Hong, Kicheon; Fu, Zunwei

doi:10.22436/jnsa.010.08.02

Cited by 9 publications

(7 citation statements)

References 30 publications

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“…In [1][2][3][4], the authors have studied Toeplitz matrices, general Hessenberg matrices and opposite-bordered tridiagonal matrix and so on. These matices have been usually applied in various application areas ranging from the computation of special functions to number theory [5], as well as in engineering, economics and heat conduction [6][7][8].…”

Section: Introductionmentioning

confidence: 99%

Analytic determinants and inverses of Toeplitz and Hankel tridiagonal matrices with perturbed columns

Jiang

et al. 2020

Special Matrices

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In this paper, our main attention is paid to calculate the determinants and inverses of two types Toeplitz and Hankel tridiagonal matrices with perturbed columns. Specifically, the determinants of the n × n Toeplitz tridiagonal matrices with perturbed columns (type I, II) can be expressed by using the famous Fibonacci numbers, the inverses of Toeplitz tridiagonal matrices with perturbed columns can also be expressed by using the well-known Lucas numbers and four entries in matrix 𝔸. And the determinants of the n×n Hankel tridiagonal matrices with perturbed columns (type I, II) are (−1]) {\left( { - 1} \right)^{{{n\left( {n - 1} \right)} \over 2}}} times of the determinant of the Toeplitz tridiagonal matrix with perturbed columns type I, the entries of the inverses of the Hankel tridiagonal matrices with perturbed columns (type I, II) are the same as that of the inverse of Toeplitz tridiagonal matrix with perturbed columns type I, except the position. In addition, we present some algorithms based on the main theoretical results. Comparison of our new algorithms and some recent works is given. The numerical result confirms our new theoretical results. And we show the superiority of our method by comparing the CPU time of some existing algorithms studied recently.

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Section: Introductionmentioning

confidence: 99%

Analytic determinants and inverses of Toeplitz and Hankel tridiagonal matrices with perturbed columns

Jiang

et al. 2020

Special Matrices

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“…Compared to an equivalent CPU implementation that utilizes a single CPU core, PSCR method indicated up to 24-fold speedups. Many studies have been conducted for tridiagonal matrices [13][14][15][16][17][18][19]. Typical results for their inverses include Usmani's algorithm [20] based on rudimentary matrix analysis, El-Mikkawy and Atlan's two symbolic algorithms [21,22] based on the Doolittle LU factorization of the k-tridiagonal matrix, Jia et al 's algorithms [23,24] based on block diagonalization technique, and so on.…”

Section: Introductionmentioning

confidence: 99%

Determinants and inverses of perturbed periodic tridiagonal Toeplitz matrices

Wei

Jiang

et al. 2019

Adv Differ Equ

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In this paper, we deal mainly with a class of periodic tridiagonal Toeplitz matrices with perturbed corners. By matrix decomposition with the Sherman–Morrison–Woodbury formula and constructing the corresponding displacement of matrices we derive the formulas on representation of the determinants and inverses of the periodic tridiagonal Toeplitz matrices with perturbed corners of type I in the form of products of Fermat numbers and some initial values. Furthermore, the properties of type II matrix can be also obtained, which benefits from the relation between type I and II matrices. Finally, we propose two algorithms for computing these properties and make some analysis about them to illustrate our theoretical results.

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“…On the other hand, many studies have been conducted for tridiagonal matrices or periodic tridiagonal matrices, especially for their determinants and inverses [22][23][24][25][26][27][28][29][30]. Two decades ago, Wittenburg [31] studied the inverse of tridiagonal toeplitz and periodic matrices and applied them to elastostatics and vibration theory.…”

Section: Introductionmentioning

confidence: 99%

A Study of Determinants and Inverses for Periodic Tridiagonal Toeplitz Matrices with Perturbed Corners Involving Mersenne Numbers

et al. 2019

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In this paper, we study periodic tridiagonal Toeplitz matrices with perturbed corners. By using some matrix transformations, the Schur complement and matrix decompositions techniques, as well as the Sherman-Morrison-Woodbury formula, we derive explicit determinants and inverses of these matrices. One feature of these formulas is the connection with the famous Mersenne numbers. We also propose two algorithms to illustrate our formulas.

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Skew cyclic displacements and decompositions of inverse matrix for an innovative structure matrix

Cited by 9 publications

References 30 publications

Analytic determinants and inverses of Toeplitz and Hankel tridiagonal matrices with perturbed columns

Analytic determinants and inverses of Toeplitz and Hankel tridiagonal matrices with perturbed columns

Determinants and inverses of perturbed periodic tridiagonal Toeplitz matrices

A Study of Determinants and Inverses for Periodic Tridiagonal Toeplitz Matrices with Perturbed Corners Involving Mersenne Numbers

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