Eigenvalues of banded symmetric Toeplitz matrices are known almost in close form ?

It is well-known that the eigenvalues of (real) symmetric banded Toeplitz matrices of size n are approximately given by an equispaced sampling of the symbol f(θ), up to an error which grows at most as h = (n + 1)−1, where the symbol is a real-valued cosine polynomial. Under the condition that f is monotone, we show that there is hierarchy of symbols so that λ (h) j − f ( θ (h) j ) = ∑ k ck ( θ (h) j ) h, θ (h) j = jπh, j = 1, . . . , n, with ck(θ) higher order symbols. In the general case, a more complicate expression holds but still we find a structural hierarchy of symbols. The latter asymptotic expansions constitute a starting point for computing the eigenvalues of large symmetric banded Toeplitz matrices by using classical extrapolation methods. Selected numerics are shown in 1D and a similar study is briefly discussed in the multilevel setting (dD, d ≥ 2) with blocks included, so opening the door to a fast computation of the spectrum of matrices approximating partial differential operators.