@TechReport{	  it:2018-005,
  author	= {Sven-Erik Ekstr{\"o}m and Isabella Furci and Stefano
		  Serra-Capizzano},
  title		= {Exact Formulae and Matrix-Less Eigensolvers for Block
		  Banded Symmetric {T}oeplitz Matrices},
  institution	= {Department of Information Technology, Uppsala University},
  department	= {Division of Scientific Computing},
  year		= {2018},
  number	= {2018-005},
  month		= mar,
  abstract	= {Precise asymptotic expansions for the eigenvalues of a
		  Toeplitz matrix $T_n(f)$, as the matrix size $n$ tends to
		  infinity, have recently been obtained, under suitable
		  assumptions on the associated generating function $f$. A
		  restriction is that $f$ has to be polynomial, monotone, and
		  scalar-valued. In this paper we focus on the case where
		  $\mathbf{f}$ is an $s\times s$ matrix-valued trigonometric
		  polynomial with $s\ge 1$, and $T_n(\mathbf{f})$ is the
		  block Toeplitz matrix generated by $\mathbf{f}$, whose size
		  is $N(n,s)=sn$. The case $s=1$ corresponds to that already
		  treated in the literature. We numerically derive conditions
		  which ensure the existence of an asymptotic expansion for
		  the eigenvalues. Such conditions generalize those known for
		  the scalar-valued setting. Furthermore, following a
		  proposal in the scalar-valued case by the first author,
		  Garoni, and the third author, we devise an extrapolation
		  algorithm for computing the eigenvalues of banded symmetric
		  block Toeplitz matrices with a high level of accuracy and a
		  low computational cost. The resulting algorithm is an
		  eigensolver that does not need to store the original
		  matrix, does not need to perform matrix-vector products,
		  and for this reason is called {\em matrix-less}. We use the
		  asymptotic expansion for the efficient computation of the
		  spectrum of special block Toeplitz structures and we
		  provide exact formulae for the eigenvalues of the matrices
		  coming from the $\mathbb{Q}_p$ Lagrangian Finite Element
		  approximation of a second order elliptic differential
		  problem. Numerical results are presented and critically
		  discussed.}
}