@TechReport{	  it:2017-015,
  author	= {Sven-Erik Ekstr{\"o}m and Carlo Garoni},
  title		= {An Interpolation-Extrapolation Algorithm for Computing the
		  Eigenvalues of Preconditioned Banded Symmetric Toeplitz
		  Matrices},
  institution	= {Department of Information Technology, Uppsala University},
  department	= {Division of Scientific Computing},
  year		= {2017},
  number	= {2017-015},
  month		= aug,
  abstract	= {In the past few years, Bogoya, B{\"o}ttcher, Grudsky, and
		  Maximenko obtained for the eigenvalues of a Toeplitz matrix
		  $T_n(f)$, under suitable assumptions on the generating
		  function $f$, the precise asymptotic expansion as the
		  matrix size $n$ goes to infinity. On the basis of several
		  numerical experiments, it was conjectured by
		  Serra-Capizzano that a completely analogous expansion also
		  holds for the eigenvalues of the preconditioned Toeplitz
		  matrix $T_n(u)^{-1}T_n(v)$, provided $f=v/u$ is monotone
		  and further conditions on $u$ and $v$ are satisfied. Based
		  on this expansion, we here propose and analyze an
		  interpolation--extrapolation algorithm for computing the
		  eigenvalues of $T_n(u)^{-1}T_n(v)$. We illustrate the
		  performance of the algorithm through numerical experiments
		  and we also present its generalization to the case where
		  $f=v/u$ is non-monotone.}
}