Explicit inverses of some tridiagonal matrices

Fonseca, Carlos M. da; Petronilho, J.

doi:10.1016/s0024-3795(00)00289-5

Cited by 143 publications

(84 citation statements)

References 10 publications

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“…Closed-form expressions for the other structural polynomials can be derived using the explicit inverse for 2-Toeplitz matrices derived by da Fonseca and Petronilho [2], and using Eqs. (1.1) and (1.2).…”

Section: Structural Polynomials Of Alternating Pathsmentioning

confidence: 99%

Spectra and structural polynomials of graphs of relevance to the theory of molecular conduction

et al. 2017

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In chemistry and physics, distortivity of π-systems (stabilisation of bond-alternated structures) is an important factor in the calculation of geometric, energetic, and electronic properties of molecules via graph theoretical methods. We use the spectra of paths and cycles with alternating vertex and edge weights to obtain the eigenvalues and eigenvectors for a class of linear and cyclic ladders with alternating rung and backbone edge weights. We derive characteristic polynomials and other structural polynomials formed from the cofactors of the characteristic matrix for these graphs. We also obtain spectra and structural polynomials for ladders with flipped weights and/or Möbius topology. In all cases, the structural polynomials for the composite graphs are expressed in terms of products of polynomials for graphs of half order. This form of the expressions allows global deductions about the transmission spectra of molecular devices in the graph-theoretical theory of ballistic molecular conduction.

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Section: Structural Polynomials Of Alternating Pathsmentioning

confidence: 99%

Spectra and structural polynomials of graphs of relevance to the theory of molecular conduction

et al. 2017

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“…It is known [5,10,43] that the solution to (22) is given by (20) and may be written down in the form…”

Section: Inversion Of the Operator ∇mentioning

confidence: 99%

A highly scalable parallel algorithm for solving Toeplitz tridiagonal systems of linear equations

Terekhov

2016

Journal of Parallel and Distributed Computing

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Based on a modification of the "Dichotomy Algorithm" (Terekhov, 2010), we propose a parallel procedure for solving tridiagonal systems of equations with Toeplitz matrices. Taking the structure of the Toeplitz matrices, we may substantially reduce the number of the "preliminary calculations" of the Dichotomy Algorithm, which makes it possible to effectively solve a series as well as a single system of equations. On the example of solving of elliptic equations by the Separation Variable Method, we show that the computation accuracy is comparable with the sequential version of the Thomas method, and the dependence of the speedup on the number of processors is almost linear. The proposed modification is aimed at parallel realization of a broad class of numerical methods including the inversion of Toeplitz and quasi-Toeplitz tridiagonal matrices.

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“…Alternatively, we apply the usual techniques for discrete boundary value problems, see [1,7]. Specifically, we have the following result expressing the entries of R −1 + D 2 in terms of two solutions of a difference equations.…”

Section: Polyominoes With Link Number Greater Than Onementioning

confidence: 99%

Green matrices associated with generalized linear polyominoes

Carmona

Encinas

Riera

2015

Linear Algebra and its Applications

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A Polyomino is an edge-connected union of cells in the planar square lattice. Here we consider generalized linear polyominoes; that is, the polyominoes supported by a n × 2 lattice. In this paper, we obtain the Green function and the Kirchhoff index of a generalized linear polyomino as a perturbation of a 2n-path by adding weighted edges between opposite vertices. This approach deeply links generalized linear polyomino Green functions with the inverse M-matrix problem, and especially with the so-called Green matrices.

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Explicit inverses of some tridiagonal matrices

Abstract: We give explicit inverses of tridiagonal 2-Toeplitz and 3-Toeplitz matrices which generalize some well-known results concerning the inverse of a tridiagonal Toeplitz matrix.

Cited by 143 publications

References 10 publications

Spectra and structural polynomials of graphs of relevance to the theory of molecular conduction

Spectra and structural polynomials of graphs of relevance to the theory of molecular conduction

A highly scalable parallel algorithm for solving Toeplitz tridiagonal systems of linear equations

Green matrices associated with generalized linear polyominoes

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