Spectrum of certain tridiagonal matrices when their dimension goes to infinity

Grassmann, Winfried K.; Tavakoli, Javad

doi:10.1016/j.laa.2009.04.013

Cited by 4 publications

(3 citation statements)

References 8 publications

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“…Numerical experiments show that (64) is more precise than (14), especially for α close to 1/2, but the errors are almost the same for α close to 1. Moreover, η α is more complicated than η α ( η α has two intervals of monotonicity), and the denominator 1/n naturally appears in the formula (5) for λ α,n,j with odd j.…”

Section: Solving the Main Equation By Newton's Methodsmentioning

confidence: 93%

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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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Section: Solving the Main Equation By Newton's Methodsmentioning

confidence: 93%

“…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

View full text Add to dashboard Cite

show abstract

“…Matrices L α,n can be considered as tridiagonal Toeplitz matrices with perturbations in the corners (1,1), (1, n), (n, 1) and (n, n). Several investigations in this area and some of its applications have been recently developed, see for example [2][3][4]6,7,10,12,13,15,16,[20][21][22]. These matrices can also be considered as periodic Jacobi matrices.…”

Section: Introductionmentioning

confidence: 99%