ANALYTICAL INVERSE FOR THE SYMMETRIC CIRCULANT TRIDIAGONAL MATRIX (original) (raw)

Exact analytical formulations for the of symmetric circulant tridiagonal matrices

2017

This study presents a time efficient, exact analytical approach for finding the inverse, decomposition, and solving linear systems of equations where symmetric and symmetric circulant matrices appear is presented. A set of matrices are introduced that any symmetric circulant matrix could be decomposed into them as well as their straightforward inverse. After that, it will be shown how they could be used to find the inverse of the matrix. Moreover, solving related linear equations can be carried out using implemented decomposition for these special, prevalent matrices. Keywords—: Circulant matrix, Decomposition method, Linear equations, Matrix inversion, Toeplitz matrix, Sparse Matrix.

An inversion algorithm for general tridiagonal matrix

Applied Mathematics and Mechanics, 2009

An algorithm for the inverse of a general tridiagonal matrix is presented. For a tridiagonal matrix having the Doolittle factorization, an inversion algorithm is established. The algorithm is then generalized to deal with a general tridiagonal matrix without any restriction. Comparison with other methods is provided, indicating low computational complexity of the proposed algorithm, and its applicability to general tridiagonal matrices.

Inversion of general tridiagonal matrices

Applied Mathematics Letters, 2006

In the current work, the authors present a symbolic algorithm for finding the inverse of any general nonsingular tridiagonal matrix. The algorithm is mainly based on the work presented in [Y. Huang, W.F. McColl, Analytic inversion of general tridiagonal matrices,

A Breakdown Free Numerical Algorithm for Inverting General Tridiagonal Matrices

arXiv (Cornell University), 2022

In the current paper the authors linked two methods in order to evaluate general n-th order tridiagonal determinants. A breakdown free numerical algorithm is developed for computing the inverse of any n×n general nonsingular tridiagonal matrix without imposing any constrains. The algorithm is suited for implementation using any computer language such as FORTRAN, PYTHON, MATLAB, MAPLE, C, C++, MACSYMA, ALGOL, PASCAL and JAVA. Some illustrative examples are presented.

On the inverses of general tridiagonal matrices

Linear Algebra and Its Applications, 2010

In this work, the sign distribution for all inverse elements of general tridiagonal H-matrices is presented. In addition, some computable upper and lower bounds for the entries of the inverses of diagonally dominant tridiagonal matrices are obtained. Based on the sign distribution, these bounds greatly improve some well-known results due to [23], [26], Nabben (1999) and recently given by [24], Peluso and Popolizio [25] and so forth. It is also stated that the inverse of a general tridiagonal matrix may be described by 2n − 2 parameters ({θ k } n k=2 and {ϕ k } n−1 k=1 ) instead of 2n + 2 ones as given by [3], El-Mikkawy and Karawia (2006) [4] and [10]. According to these results, a new symbolic algorithm for finding the inverse of a tridiagonal matrix without imposing any restrictive conditions is presented, which improves some recent results. Finally, several applications to the preconditioning technology, the numerical solution of differential equations and the birth-death processes together with numerical tests are given.

A Note on the New Method of Computing the Inverse of 3 x 3 Square Matrices and Its Applications

Asian Research Journal of Mathematics

Computing the inverse of 3 x 3 square matrices using known methods such as Gauss-Jordon Method needs more time. During examination, candidates need to go through long process to find the inverse of 3 x 3 square matrices. Some students also find it very difficult and confusing when using Gauss-Jordon and Adjugate methods to find the inverse of the matrix. In this note, we present a new method that is simple and easy way of finding the inverse of 3 x 3 square matrices. We further give some applications of matrices in the real world phenomenon. Many other challenging problems can be addressed surprisingly by applying this strategy.

Explicit Form of the Inverse Matrices of Tribonacci Circulant Type Matrices

Abstract and Applied Analysis, 2015

It is a hot topic that circulant type matrices are applied to networks engineering. The determinants and inverses of Tribonacci circulant type matrices are discussed in the paper. Firstly, Tribonacci circulant type matrices are defined. In addition, we show the invertibility of Tribonacci circulant matrix and present the determinant and the inverse matrix based on constructing the transformation matrices. By utilizing the relation between left circulant, -circulant matrices and circulant matrix, the invertibility of Tribonacci left circulant and Tribonacci -circulant matrices is also discussed. Finally, the determinants and inverse matrices of these matrices are given, respectively.

On the inverse of a general pentadiagonal matrix

Applied Mathematics and Computation, 2008

In this paper, employing the general Doolittle factorization, an efficient algorithm is developed to find the inverse of a general pentadiagonal matrix which is suitable for implementation using computer algebra systems software such as Matlab and Maple. Examples are given to illustrate the efficiency of the algorithm.