Superlinear convergence for PCG using band plus algebra preconditioners for Toeplitz systems

AbstractThe paper studies fast and efficient solution algorithms for n×n symmetric ill conditioned Toeplitz systems Tn(f)x=b where the generating function f is known a priori, real valued, nonnegative, and has isolated roots of even order. The preconditioner that we propose is a product of a band Toeplitz matrix and matrices that belong to a certain trigonometric algebra. The basic idea behind the proposed scheme is to combine the advantages of all components of the product that are well known when every component is used as a stand-alone preconditioner. As a result we obtain a flexible preconditioner which can be applied to the system Tn(f)x=b infusing superlinear convergence to the PCG method. The important feature of the proposed technique is that it can be extended to cover the 2D case, i.e. ill-conditioned block Toeplitz matrices with Toeplitz blocks. We perform many numerical experiments, whose results confirm the theoretical analysis and effectiveness of the proposed strategy

Unitary equivalence to a truncated Toeplitz operator: analytic symbols

Unlike Toeplitz operators on $H^2$, truncated Toeplitz operators do not have a natural matricial characterization. Consequently, these operators are difficult to study numerically. In this note we provide criteria for a matrix with distinct eigenvalues to be unitarily equivalent to a truncated Toeplitz operator having an analytic symbol. This test is constructive and we illustrate it with several examples. As a byproduct, we also prove that every complex symmetric operator on a Hilbert space of dimension $\leq 3$ is unitarily equivalent to a direct sum of truncated Toeplitz operators.Comment: 15 page

Recent progress on truncated Toeplitz operators

This paper is a survey on the emerging theory of truncated Toeplitz operators. We begin with a brief introduction to the subject and then highlight the many recent developments in the field since Sarason's seminal paper in 2007.Comment: 46 page