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In the last decade many efficient iterative solvers for n×nHermitian positive definite Toeplitz systems have been devised. Many of them are based on band Toeplitz preconditioners: they are optimal but require the knowledge of the zeros of the underlying generating function. In some cases this information is available and in some cases is not. In [27] an economic numerical procedure for finding these zeros within a given precision has been devised. Here we provide conditions on the approximation error of these zeros in order to maintain the optimality that is a convergence rate independent of the dimension nof the considered linear systems.

The zero distribution of orthogonal polynomials pn, N,n=0, 1,… generated by recurrence coefficients an, N and bn, N depending on a parameter N has been recently considered by Kuijlaars and Van Assche under the assumption that an, N and bn, N behave like a(n/N) and b(n/N), respectively, where a(·) and b(·) are continuous functions. Here, we extend this result by allowing a(·) and b(·) to be measurable functions so that the presence of possible jumps is included. The novelty is also in the sense of the mathematical tools since, instead of applying complex analysis arguments, we use recently developed results from asymptotic matrix theory due to Tyrtyshnikov, Serra Capizzano, and Tilli.

It is often asked why Toeplitz-like matrices with unbounded symbols are worth studying. This paper gives an answer by presenting several concrete problems that motivate such studies. It surveys the central results of the theory of Generalized Locally Toeplitz (GLT) sequences in a self-contained tool-kit fashion, and gives a new extension from bounded Riemann integrable functions to unbounded almost everywhere continuous functions. The emergence of unbounded symbols is illustrated by local grid refinements in finite difference and finite element discretizations and also by preconditioning strategies.