26 research outputs found
Fast non-Hermitian Toeplitz eigenvalue computations, joining matrix-less algorithms and FDE approximation matrices
The present work is devoted to the eigenvalue asymptotic expansion of the
Toeplitz matrix whose generating function is complex valued and
has a power singularity at one point. As a consequence, is
non-Hermitian and we know that the eigenvalue computation is a non-trivial task
in the non-Hermitian setting for large sizes. We follow the work of Bogoya,
B\"ottcher, Grudsky, and Maximenko and deduce a complete asymptotic expansion
for the eigenvalues. After that, we apply matrix-less algorithms, in the spirit
of the work by Ekstr\"om, Furci, Garoni, Serra-Capizzano et al, for computing
those eigenvalues. Since the inner and extreme eigenvalues have different
asymptotic behaviors, we worked on them independently, and combined the results
to produce a high precision global numerical and matrix-less algorithm.
The numerical results are very precise and the computational cost of the
proposed algorithms is independent of the size of the considered matrices for
each eigenvalue, which implies a linear cost when all the spectrum is computed.
From the viewpoint of real world applications, we emphasize that the matrix
class under consideration includes the matrices stemming from the numerical
approximation of fractional diffusion equations. In the final conclusion
section a concise discussion on the matter and few open problems are presented.Comment: 21 page
Eigenvalues of laplacian matrices of the cycles with one negative-weighted edge
We study the individual behavior of the eigenvalues of the laplacian matrices
of the cyclic graph of order , where one edge has weight
, with , and all the others
have weights . This paper is a sequel of a previous one where we considered
(Eigenvalues of laplacian matrices of the
cycles with one weighted edge, Linear Algebra Appl. 653, 2022, 86--115). We
prove that for and
, one eigenvalue is
negative while the others belong to and are distributed as the function
. Additionally, we prove that as tends to ,
the outlier eigenvalue converges exponentially to
. We give exact
formulas for the half of the inner eigenvalues, while for the others we justify
the convergence of Newton's method and fixed-point iteration method. We find
asymptotic expansions, as tends to , both for the eigenvalues
belonging to and the outlier. We also compute the eigenvectors and
their norms.Comment: 28 pages, 8 figure
Generalized Electromagnetic fields in Chiral Medium
The time dependent Dirac-Maxwell's Equations in presence of electric and
magnetic sources are written in chiral media and the solutions for the
classical problem are obtained in unique simple and consistent manner. The
quaternion reformulation of generalized electromagnetic fields in chiral media
has also been developed in compact, simple and consistent manner
Toeplitz operators on the domain with -invariant symbols
Let be the irreducible bounded symmetric domain of complex
matrices that satisfy . The biholomorphism group of is realized
by with isotropy at the origin given by
. Denote by the subgroup of
diagonal matrices in . We prove that the set of
-invariant essentially bounded symbols yield
Toeplitz operators that generate commutative -algebras on all weighted
Bergman spaces over . Using tools from representation theory, we also
provide an integral formula for the spectra of these Toeplitz operators
Quaternion Analysis for Generalized Electromagnetic Fields of Dyons in Isotropic Medium
Quaternion analysis of time dependent Maxwell's equations in presence of
electric and magnetic charges has been developed and the solutions for the
classical problem of moving charges (electric and magnetic) are obtained in
unique, simple and consistent manner
Algebras of Toeplitz operators on the n-dimensional unit ball
We study -algebras generated by Toeplitz operators acting on the standard weighted Bergman space over the unit ball in . The symbols of generating operators are assumed to be of a certain product type, see (\ref{Introduction_form_of_the_symbol}). By choosing and in different function algebras and over lower dimensional unit balls and , respectively, and by assuming the invariance of under some torus action we obtain -algebras whose structural properties can be described. In the case of -quasi-radial functions and bounded uniformly continuous
or vanishing oscillation symbols we describe the structure of elements from the algebra , derive a list of irreducible
representations of , and prove completeness of this list in some cases. Some of these representations originate from a ``quantization effect'', induced
by the representation of as the direct sum of Bergman spaces over a lower dimensional unit ball with growing weight parameter. As an application we derive the
essential spectrum and index formulas for matrix-valued operators
Estimates for the condition numbers of large semi-definite Toeplitz matrices
This paper is devoted to asymptotic estimates for the condition numbers
of large n\cross n Toeplitz matrices in the case where
\alpha \element L^\infinity and . We describe several classes
of symbols for which increases like ,
or even . The consequences of the results for singular values, eigenvalues,
and the finite section method are discussed. We also consider Wiener-Hopf integral
operators and multidimensional Toeplitz operators
On the extreme eigenvalues and asymptotic conditioning of a class of Toeplitz matrix-sequences arising from fractional problems
The analysis of the spectral features of a Toeplitz matrix-sequence (Formula presented.), generated by the function (Formula presented.), real-valued almost everywhere (a.e.), has been provided in great detail in the last century, as well as the study of the conditioning, when f is nonnegative a.e. Here we consider a novel type of problem arising in the numerical approximation of distributed-order fractional differential equations (FDEs), where the matrices under consideration take the form (Formula presented.) (Formula presented.) belong to the interval (Formula presented.) with (Formula presented.) independent of n, (Formula presented.), (Formula presented.), and (Formula presented.) for every (Formula presented.). For nonnegative functions or sequences, the notation (Formula presented.) means that there exist positive constants c, d, independent of the variable x in the definition domain such that (Formula presented.) for any x. Since the resulting generating function depends on n, the standard theory cannot be applied and the analysis has to be performed using new ideas. Few selected numerical experiments are presented, also in connection with matrices that come from distributed-order FDE problems, and the adherence with the theoretical analysis is discussed, together with open questions and future investigations
Norm of inverses, spectra, and pseudospectra of large truncated Wiener-Hopf operators and Toeplitz matrices
Abstract. This paper is concerned with Wiener-Hopf integral operators on L p and with Toeplitz operators (or matrices) on l p. The symbols of the operators are assumed to be continuous matrix functions. It is well known that the invertibility of the operator itself and of its associated operator imply the invertibility of all sufficiently large truncations and the uniform boundedness of the norms of their inverses. Quantitative statements, such as results on the limit of the norms of the inverses, can be proved in the case p = 2 by means of C ∗-algebra techniques. In this paper we replace C ∗-algebra methods by more direct arguments to determine the limit of the norms of the inverses and thus also of the pseudospectra of large truncations in the case of general p