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 −1 n (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 Numer Algor 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.

We consider the ergodic (or additive eigenvalue) problem for the Neumann-type boundary-value problem for Hamilton–Jacobi equations and the corresponding discounted problems. Denoting by uλ the solution of the discounted problem with discount factor λ > 0, we establish the convergence of the whole family to a solution of the ergodic problem as λ → 0, and give a representation formula for the limit function via the Mather measures and Peierls function. As an interesting by-product, we introduce Mather measures associated with Hamilton–Jacobi equations with the Neumann-type boundary conditions. These results are variants of the main results in a recent paper by Davini et al., who study the same convergence problem on smooth compact manifolds without boundary.

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