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

Wei, Yunlan; Zheng, Yanpeng; Jiang, Zhaolin; Shon, Sugoog

doi:10.3390/math7100893

Cited by 9 publications

(4 citation statements)

References 43 publications

(46 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…On the other hand, the importance of determinants, inverses, norms and spread in special matrix analysis, several authors [4,6,10,11,13,17,20,23,[28][29][30][31][32] have done some research on these special matrices. Recently, Jiang and Hong [12] studied the explicit form of determinants and inverses of Tribonacci r-circulant type matrices, while Zheng and Shon [36] gave the exact determinants and inverses of generalized Lucas skew circulant type matrices.…”

Section: Introductionmentioning

confidence: 99%

On determinants, inverses, norms, and spread of skew circulant matrices involving the product of Pell and Pell-Lucas numbers

Fan¹,

Wei²,

Zheng³

et al. 2023

J. Math. Computer Sci.

View full text Add to dashboard Cite

In this paper, we discuss skew circulant matrices involving the product of Pell and Pell-Lucas numbers. The invertibility of the skew circulant matrices is investigated, while the fundamental theorems on the determinants and inverses of such matrices are derived by simple construction matrices. Specifically, the determinant and inverse of n × n skew circulant matrices can be expressed by the (n − 1)th, nth, (n + 1)th, (n + 2)th product of Pell and Pell-Lucas numbers. Some norms and bounds for spread of these matrices are given, respectively. In addition, we generalized these results to skew left circulant matrix involving the product of Pell and Pell-Lucas numbers. Finally, several numerical examples are illustrated to show the effectiveness of our theoretical results.

show abstract

Section: Introductionmentioning

confidence: 99%

On determinants, inverses, norms, and spread of skew circulant matrices involving the product of Pell and Pell-Lucas numbers

Fan¹,

Wei²,

Zheng³

et al. 2023

J. Math. Computer Sci.

View full text Add to dashboard Cite

show abstract

“…The symmetric tridiagonal matrices often arise as primary data in many computational quantum physical [1,2], mathematical [3][4][5], dynamical [6,7], computational quantum chemical [8,9], signal processing [10], or even medical [11] problems and hence are important. The current software reduces the generalized and the standard symmetric eigenproblems to a symmetric tridiagonal eigenproblem as a common practice [10,12,13].…”

Section: Introductionmentioning

confidence: 99%

A Novel Divisional Bisection Method for the Symmetric Tridiagonal Eigenvalue Problem

2022

View full text Add to dashboard Cite

The embarrassingly parallel nature of the Bisection Algorithm makes it easy and efficient to program on a parallel computer, but with an expensive time cost when all symmetric tridiagonal eigenvalues are wanted. In addition, few methods can calculate a single eigenvalue in parallel for now, especially in a specific order. This paper solves the issue with a new approach that can parallelize the Bisection iteration. Some pseudocodes and numerical results are presented. It shows our algorithm reduces the time cost by more than 35–70% compared to the Bisection algorithm while maintaining its accuracy and flexibility.

show abstract

“…RT method needs eigenvalues of a tridiagonal matrix to represent the potential formula. At present, there have been many results on tridiagonal matrices [45][46][47][48][49][50][51] , which are also widely used. It can be said that it is a powerful tool to solve the resistor network [33][34][35][36][37][38][39][40][41][42][43] .…”

mentioning

confidence: 99%

Fast algorithm and new potential formula represented by Chebyshev polynomials for an $$m\times n$$ globe network

Zhou

Zheng

Jiang

et al. 2022

Sci Rep

Self Cite

View full text Add to dashboard Cite

Resistor network is widely used. Many potential formulae of resistor networks have been solved accurately, but the scale of data is limited by manual calculation, and numerical simulation has become the trend of large-scale operation. This paper improves the general solution of potential formula for an $$m\times n$$ m × n globe network. Chebyshev polynomials are introduced to represent new potential formula of a globe network. Compared with the original potential formula, it saves time to calculate the potential. In addition, an algorithm for computing potential by the famous second type of discrete cosine transform (DCT-II) is also proposed. It is the first time to be used for machine calculation. Moreover, it greatly increases the efficiency of computing potential. In the application of this new potential formula, the equivalent resistance formulae in special cases are given and displayed by three-dimensional dynamic view. The new potential formulae and the proposed fast algorithm realize large-scale operation for resistor networks.

show abstract

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

Cited by 9 publications

References 43 publications

On determinants, inverses, norms, and spread of skew circulant matrices involving the product of Pell and Pell-Lucas numbers

On determinants, inverses, norms, and spread of skew circulant matrices involving the product of Pell and Pell-Lucas numbers

A Novel Divisional Bisection Method for the Symmetric Tridiagonal Eigenvalue Problem

Fast algorithm and new potential formula represented by Chebyshev polynomials for an $$m\times n$$ globe network

Contact Info

Product

Resources

About