We consider the solutions of Hermitian Toeplitz systems where the Toeplitz matrices are generated by nonnegative functions f. The preconditioned conjugate gradient method with well-known circulant preconditioners fails in the case when f has zeros. In this paper, we employ Toeplitz matrices of xed band-width as preconditioners. Their generating functions g are trigonometric poly-nomials of xed ...

The Leverrier-Faddeev algorithm, as modified by Gower (1980), is little-known but is useful for deriving the algebraic, rather than numerical, spectral structure of matrices occurring in statistical methodology. An example is given of deriving explicit forms for the singular value decomposition of any block-circulant matrix and the spectral decomposition of any symmetric block-circulant matrix....

We study the solutions of Hermitian positive definite Toeplitz systems Tnx = b by the preconditioned conjugate gradient method. For preconditioner An the convergence rate is known to be governed by the distribution of the eigenvalues of the preconditioned matrix A−1 n Tn . New properties of the circulant preconditioners introduced by Strang, R. Chan, T. Chan, Szegö/Grenander and Tyrtyshnikov ar...

The spectrum of the eigenvalues, the conditioning, and other related properties of circulant-like matrices used to build up block preconditioners for the nonsymmetric algebraic linear equations of time-step integrators for linear boundary value problems are analyzed. Moreover, results concerning the entries of a class of Toeplitz matrices related to the latter are proposed. Generalizations of i...

Given an ensemble of N × N random matrices, a natural question to ask is whether or not the empirical spectral measures of typical matrices converge to a limiting spectral measure as N → ∞. While this has been proved for many thin patterned ensembles sitting inside all real symmetric matrices, frequently there is no nice closed form expression for the limiting measure. Further, current theorems...

Consider the powers of a square matrix A of order n in bottleneck algebra, where addition and multiplication are replaced by the max and min operations. The powers are periodic, starting from a certain power AK . The smallest such K is called the exponent of A and the length of the period is called the index of A. Cechlárová has characterized the matrices of index 1. Here we consider the case w...