14 research outputs found
Eigenvalues of even very nice Toeplitz matrices can be unexpectedly erratic
It was shown in a series of recent publications that the eigenvalues of
Toeplitz matrices generated by so-called simple-loop symbols admit
certain regular asymptotic expansions into negative powers of . On the
other hand, recently two of the authors considered the pentadiagonal Toeplitz
matrices generated by the symbol , which does not satisfy
the simple-loop conditions, and derived asymptotic expansions of a more
complicated form. We here use these results to show that the eigenvalues of the
pentadiagonal Toeplitz matrices do not admit the expected regular asymptotic
expansion. This also delivers a counter-example to a conjecture by Ekstr\"{o}m,
Garoni, and Serra-Capizzano and reveals that the simple-loop condition is
essential for the existence of the regular asymptotic expansion.Comment: 28 pages, 7 figure
Eigenvalues of the laplacian matrices of the cycles with one weighted edge
In this paper we study the eigenvalues of the laplacian matrices of the
cyclic graphs with one edge of weight and the others of weight . We
denote by the order of the graph and suppose that tends to infinity. We
notice that the characteristic polynomial and the eigenvalues depend only on
. After that, through the rest of the paper we
suppose that . It is easy to see that the eigenvalues belong to
and are asymptotically distributed as the function
on . We obtain a series of results about the individual behavior of
the eigenvalues. First, we describe more precisely their localization in
subintervals of . Second, we transform the characteristic equation to a
form convenient to solve by numerical methods. In particular, we prove that
Newton's method converges for every . Third, we derive asymptotic
formulas for all eigenvalues, where the errors are uniformly bounded with
respect to the number of the eigenvalue.Comment: 29 pages, 5 figure