On tropical eigenvalues of tridiagonal Toeplitz matrices

Tavakolipour, Hanieh; Shakeri, Fatemeh

doi:10.1016/j.laa.2017.11.009

Cited by 10 publications

(7 citation statements)

References 8 publications

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“…In this article, we illustrate how to take advantage of the weighted digraphs associated with pentadiagonal Toeplitz matrices to facilitate finding their characteristic maxpolynomials and therefore their tropical eigenvalues. Note that the structure of the aforementioned graph here is essentially different from that of tridiagonal matrices studied in Reference 31 and hence, the methods and proofs are quite different. In this article, we explicitly compute all the n + 1 terms of the characteristic maxpolynomial of

\tilde{P}

by using O ( n ) arithmetic operations.…”

Section: Some Tropical and Graph‐theoretical Conceptsmentioning

confidence: 98%

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Asymptotics of the eigenvalues for exponentially parameterized pentadiagonal matrices

Tavakolipour

Shakeri

2020

Numerical Linear Algebra App

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Summary Let P(t) be an n × n (complex) exponentially parameterized pentadiagonal matrix. In this article, using a theorem of Akian, Bapat, and Gaubert, we present explicit formulas for asymptotics of the moduli of the eigenvalues of P(t) as t → ∞. Our approach is based on exploiting the relation with tropical algebra and the weighted digraphs of matrices. We prove that this asymptotics tends to a unique limit or two limits. Also, for n − 2 largest magnitude eigenvalues of P(t) we compute the asymptotics as n → ∞, in addition to t. When P(t) is also symmetric, these formulas allow us to compute the asymptotics of the 2‐norm condition number. The number of arithmetic operations involved, does not depend on n. We illustrate our results by some numerical tests.

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\tilde{P}

by using O ( n ) arithmetic operations.…”

Section: Some Tropical and Graph‐theoretical Conceptsmentioning

confidence: 98%

“…In Reference 31, we proposed explicit formulas involving O (1) arithmetic operations for essential terms and also involving O (1) arithmetic operations for tropical eigenvalues of tridiagonal Toeplitz matrices. We proved that such tropical matrices have at most two distinct tropical eigenvalues.…”

Section: Some Tropical and Graph‐theoretical Conceptsmentioning

confidence: 99%

Asymptotics of the eigenvalues for exponentially parameterized pentadiagonal matrices

Tavakolipour

Shakeri

2020

Numerical Linear Algebra App

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“…Note that matrix M is a block tridiagonal matrix, i.e., a matrix whose blocks are all E except for those in the main diagonal and in the first diagonals above and below the main diagonal. We refer to [43,44] for some work related to tridiagonal matrices in the tropical (i.e., either max-plus or min-plus) algebra.…”

Section: A Proof Of Theoremmentioning

confidence: 99%

Switched Max-Plus Linear-Dual Inequalities: Application in Scheduling of Multi-Product Processing Networks

Zorzenon

Komenda²,

Raisch³

2022

IFAC-PapersOnLine

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“…Over the past decade, there has been an increasing interest in Toeplitz matrices with certain perturbations, see [3,7,8,11,12,14,21,22,27,28,32], or [17,20,23,29] for more general researches. In [11,12] the authors find the characteristic polynomial for some cases of Toeplitz matrices with corner perturbations.…”

Section: Introductionmentioning

confidence: 99%

Eigenvalues of the laplacian matrices of the cycles with one weighted edge

Grudsky¹,

Maximenko²,

Alejandro³

2022

Preprint

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In this paper we study the eigenvalues of the laplacian matrices of the cyclic graphs with one edge of weight α and the others of weight 1. We denote by n the order of the graph and suppose that n tends to infinity. We notice that the characteristic polynomial and the eigenvalues depend only on Re(α). After that, through the rest of the paper we suppose that 0 < α < 1. It is easy to see that the eigenvalues belong to [0, 4] and are asymptotically distributed as the function g(x) = 4 sin 2 (x/2) on [0, π]. We obtain a series of results about the individual behavior of the eigenvalues. First, we describe more precisely their localization in subintervals of [0,4]. Second, we transform the characteristic equation to a form convenient to solve by numerical methods. In particular, we prove that Newton's method converges for every n ≥ 3. Third, we derive asymptotic formulas for all eigenvalues, where the errors are uniformly bounded with respect to the number of the eigenvalue.

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On tropical eigenvalues of tridiagonal Toeplitz matrices

Cited by 10 publications

References 8 publications

Asymptotics of the eigenvalues for exponentially parameterized pentadiagonal matrices

Asymptotics of the eigenvalues for exponentially parameterized pentadiagonal matrices

Switched Max-Plus Linear-Dual Inequalities: Application in Scheduling of Multi-Product Processing Networks

Eigenvalues of the laplacian matrices of the cycles with one weighted edge

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