Spectral Analysis of Nonsymmetric Quasi-Toeplitz matrices with Applications to Preconditioned Multistep Formulas

Abstract

Abstract. The eigenvalue spectrum of the nonsymmetric preconditioned matrices arising in time-dependent partial differential equations is analyzed and discussed. The matrices generated by the underlying numerical integrators are small rank perturbations of block Toeplitz matrices; preconditioners based on small rank approximations for those are considered. The eigenvalue distribution of the preconditioned matrix influences often crucially the convergence of Krylov iterative accelerators.Due to the lack of symmetry, the classical approach based on generating functions gives very little insight here. Therefore, to characterize the eigenvalues, a difference equation approach exploiting the band Toeplitz and circulant patterns generalizing well known Trench's results is proposed.Key words. circulant preconditioners, nonsymmetric Toeplitz matrices, eigenvalues, difference equations, linear systems of time-step integrators, linear multistep formulas in boundary value form, boundary value problems.

On the exponential of semi-infinite quasi-Toeplitz matrices

“…The decomposition A = T (a) + E is quite natural and widely used in the analysis of spectral properties of sequences {A n } of n × n Toeplitz-like matrices where the additive decomposition is given as the sum of a Toeplitz part, plus a matrix of small rank plus a matrix of small norm. We refer the reader to [4], [24] for the basic properties and for a list of references in this regard. The same kind of decomposition is described in [9,Example 2.28] where, unlike in this paper, it is assumed that the correction E is a compact operator in ℓ 2 .…”

Fast numerical solution of nonlinear nonlocal cochlear models

“…The localization technique based on the field of values is not useful here as well because the symmetric part of F, i.e., (F + F T )/2, is indefinite. By using the difference equation approach proposed in [3], we get that matrix F as in (14) is nonsingular and its eigenvalues are in the right half plane. Moreover, the condition numbers of F in the Euclidean norm are very much the same with respect of those of the matrix F in [7] and in [13]; see Table 1.…”

Generalized Pascal matrices generate classes closed under multiplication

“…Block-Toeplitz matrices, with Toeplitz or otherwise structured blocks, have been extensively studied [6]. Toeplitz-like or quasi-Toeplitz matrices are Toeplitz but for a low rank perturbation [3,2].…”