@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.}
}