Technical Report 2016-017

Eigenvalues of Banded Symmetric Toeplitz Matrices are Known Almost in Close Form?

S.-E. Ekström and S. Serra-Capizzano

September 2016

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(theta), 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_j^(h)-f(theta_j^(h))=sum_k c_k(theta_j^(h))  h^k,      theta_j^(h)=jpi h, j=1,...,n,

with c_k(theta) 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, dge 2) with blocks included, so opening the door to a fast computation of the spectrum of matrices approximating partial differential operators.

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