Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Spectral behavior of preconditioned non-Hermitian multilevel block
  Toeplitz matrices with matrix-valued symbol   This note is devoted to preconditioning strategies for non-Hermitian
multilevel block Toeplitz linear systems associated with a multivariate
Lebesgue integrable matrix-valued symbol. In particular, we consider special
preconditioned matrices, where the preconditioner has a band multilevel block
Toeplitz structure, and we complement known results on the localization of the
spectrum with global distribution results for the eigenvalues of the
preconditioned matrices. In this respect, our main result is as follows. Let
$I_k:=(-\pi,\pi)^k$, let $\mathcal M_s$ be the linear space of complex $s\times
s$ matrices, and let $f,g:I_k\to\mathcal M_s$ be functions whose components
$f_{ij},\,g_{ij}:I_k\to\mathbb C,\ i,j=1,\ldots,s,$ belong to $L^\infty$.
Consider the matrices $T_n^{-1}(g)T_n(f)$, where $n:=(n_1,\ldots,n_k)$ varies
in $\mathbb N^k$ and $T_n(f),T_n(g)$ are the multilevel block Toeplitz matrices
of size $n_1\cdots n_ks$ generated by $f,g$. Then
$\{T_n^{-1}(g)T_n(f)\}_{n\in\mathbb N^k}\sim_\lambda g^{-1}f$, i.e. the family
of matrices $\{T_n^{-1}(g)T_n(f)\}_{n\in\mathbb N^k}$ has a global (asymptotic)
spectral distribution described by the function $g^{-1}f$, provided $g$
possesses certain properties (which ensure in particular the invertibility of
$T_n^{-1}(g)$ for all $n$) and the following topological conditions are met:
the essential range of $g^{-1}f$, defined as the union of the essential ranges
of the eigenvalue functions $\lambda_j(g^{-1}f),\ j=1,\ldots,s$, does not
disconnect the complex plane and has empty interior. This result generalizes
the one obtained by Donatelli, Neytcheva, Serra-Capizzano in a previous work,
concerning the non-preconditioned case $g=1$. The last part of this note is
devoted to numerical experiments, which confirm the theoretical analysis and
suggest the choice of optimal GMRES preconditioning techniques to be used for
the considered linear systems.

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