773 research outputs found
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
, let be the linear space of complex matrices, and let be functions whose components
belong to .
Consider the matrices , where varies
in and are the multilevel block Toeplitz matrices
of size generated by . Then
, i.e. the family
of matrices has a global (asymptotic)
spectral distribution described by the function , provided
possesses certain properties (which ensure in particular the invertibility of
for all ) and the following topological conditions are met:
the essential range of , defined as the union of the essential ranges
of the eigenvalue functions , 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 . 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.Comment: 18 pages, 26 figure
Chain models and the spectra of tridiagonal k-Toeplitz matrices
Chain models can be represented by a tridiagonal matrix with periodic entries
along its diagonals. Eigenmodes of open chains are represented by spectra of
such tridiagonal -Toeplitz matrices, where represents length of the
repeated unit, allowing for a maximum of distinct types of elements in the
chain. We present an analysis that allows for generality in and values in
representing elements of the chain, including non-Hermitian
systems. Numerical results of spectra of some special -Toeplitz matrices are
presented as a motivation. This is followed by analysis of a general
tridiagonal -Toeplitz matrix of increasing dimensions, beginning with 3-term
recurrence relations between their characteristic polynomials involving a
order coefficient polynomial, with the variables and coefficients in
. The existence of limiting zeros for these polynomials and their
convergence are established, and the conditioned order coefficient
polynomial is shown to provide a continuous support for the limiting spectra
representing modes of the chain. This analysis also includes the at most
eigenvalues outside this continuous set. It is shown that this continuous
support can as well be derived using Widom's conditional theorems (and its
recent extensions) for the existence of limiting spectra for block-Toeplitz
operators, except in special cases. Numerical examples are used to graphically
demonstrate theorems. As an addendum, we derive expressions for
computation of the determinant of tridiagonal -Toeplitz matrices of any
dimension
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