Are the eigenvalues of preconditioned banded symmetric Toeplitz matrices known in almost closed form?
Fayyaz Ahmad, Eman Salem Alaidarous, Dina Abdullah Alrehaili, Sven‐Erik Ekström, Isabella Furci, Stefano Serra‐Capizzano · Numerical Algorithms · 2017
Bogoya, Böttcher, Grudsky, and Maximenko have recently obtained the precise asymptotic expansion for the eigenvalues of a sequence of Toeplitz matrices {T n (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 {T n−1(g)T n (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.