### 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

**Abstract:**
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 (*d*D, *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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