Are the eigenvalues of preconditioned banded symmetric Toeplitz matrices known in almost closed form?

Dina Abdullah Alrehaili,Eman Salem Al-Aidarous,Isabella Furci,Stefano Serra-Capizzano,Fayyaz Ahmad,Sven-Erik Ekström

doi:10.1007/s11075-017-0404-z

Dina Abdullah Alrehaili, Eman Salem Al-Aidarous + Show 4 more

Open Access

https://doi.org/10.1007/s11075-017-0404-z

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Abstract

Bogoya, Böttcher, Grudsky, and Maximenko have recently obtained the precise asymptotic expansion for the eigenvalues of a sequence of Toeplitz matrices {Tn(f)}, under suitable assumptions on the associated generating function f. In this paper, we provide numerical evidence that some of these assumptions can be relaxed and extended to the case of a sequence of preconditioned Toeplitz matrices {Tn−1(g)Tn(f)}, for f trigonometric polynomial, g nonnegative, not identically zero trigonometric polynomial, r = f/g, and where the ratio r plays the same role as f in the nonpreconditioned case. Moreover, based on the eigenvalue asymptotics, we devise an extrapolation algorithm for computing the eigenvalues of preconditioned banded symmetric Toeplitz matrices with a high level of accuracy, with a relatively low computational cost, and with potential application to the computation of the spectrum of differential operators.

Highlights

IntroductionA matrix of size n, having a fixed entry along each diagonal, is called Toeplitz and enjoys the expression
A matrix of size n, having a fixed entry along each diagonal, is called Toeplitz and enjoys the expression ⎡ a0 a−1 a−2 ⎤ a−(n−1) ai−j n i,j =1 = ⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ a1 a2 ... ... a−2 a−1 ⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ .an−1 · · · · · · a2 a1 a0
In the case where φ is real-valued, all the matrices Tn(φ) are Hermitian and much is known about their spectral properties, from the localization of the eigenvalues to the asymptotic spectral distribution in the Weyl sense: in particular φ is the spectral symbol of {Tn(φ)}n, see [7, 14] and the references therein

Summary

Introduction

A matrix of size n, having a fixed entry along each diagonal, is called Toeplitz and enjoys the expression. In the case where φ is real-valued, all the matrices Tn(φ) are Hermitian and much is known about their spectral properties, from the localization of the eigenvalues to the asymptotic spectral distribution in the Weyl sense: in particular φ is the spectral symbol of {Tn(φ)}n, see [7, 14] and the references therein. The algorithm is completely analogous to the extrapolation procedure, which is employed in the context of Romberg integration (to obtain high precision approximations of an integral from a few coarse trapezoidal approximations [20, Section 3.4], see [8] for more advanced algorithms) In this regard, the asymptotic expansion (1) plays here the same role as the Euler–Maclaurin summation formula [20, Section 3.3]. The third and last purpose of this paper is to formulate, on the basis of numerical experiments, a conjecture on the higher-order asymptotic of the eigenvalues if the monotonicity assumption on r = f/g is not in force. We illustrate how this conjecture can be used along with our extrapolation algorithm in order to compute some of the eigenvalues of Pn(f, g) in the case where r is nonmonotone

Error bounds for the coefficients ck in the asymptotic expansion

Error bounds for numerically approximated eigenvalues

Numerical tests

Conclusions