Inverse Toeplitz preconditioners for Hermitian Toeplitz systems
Fu‐Rong Lin, Wai‐Ki Ching · Numerical Linear Algebra with Applications · 2004
Abstract In this paper we consider solving Hermitian Toeplitz systems Tnx=b by using the preconditioned conjugate gradient (PCG) method. Here the Toeplitz matrices Tn are assumed to be generated by a non‐negative continuous 2π‐periodic function ƒ, i.e. Tn=𝒯n[ƒ]. It was proved in (Linear Algebra Appl. 1993; 190:181) that if ƒ is positive then the spectrum of 𝒯n[1/ƒ]𝒯n[ƒ] is clustered around 1. We prove that the trigonometric polynomial q (s⩾2, cf. (2) and (3)) converges to 1/ƒ uniformly as n→∞ under the condition that 1/ƒ is in Wiener class. It follows that the computational cost of the PCG method can be reduced by replacing 1/ƒ with q, where N