@TechReport{	  it:2016-017,
  author	= {S.-E. Ekstr{\"o}m and S. Serra-Capizzano},
  title		= {Eigenvalues of Banded Symmetric {T}oeplitz Matrices are
		  Known Almost in Close Form?},
  institution	= {Department of Information Technology, Uppsala University},
  department	= {Division of Scientific Computing},
  year		= {2016},
  number	= {2016-017},
  month		= sep,
  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
		  
		  \[ \lambda_{j}^{(h)}-f\left(\theta_{j}^{(h)}\right)=\sum_k
		  c_k\left(\theta_{j}^{(h)}\right)\, h^k,\quad \quad
		  \theta_j^{(h)}=j\pi h, j=1,\ldots,n, \]
		  
		  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, $d\ge
		  2$) with blocks included, so opening the door to a fast
		  computation of the spectrum of matrices approximating
		  partial differential operators.}
}