Hartley preconditioners for Toeplitz systems generated by positive continuous functions

In this paper, we consider the solution ofn-by-n symmetric positive definite Toeplitz systemsT_nx=b by the preconditioned conjugate gradient (PCG) method. The preconditionerM_n is defined to be the minimizer of T_n–B_nF over allB_n H_n whereH_n is the Hartley algebra. We show that if the generating functionf ofT_n is a positive 2-periodic continuous even function, then the spectrum of the preconditioned systemM_n^–1T_n will be clustered around 1. Thus, if the PCG method is applied to solve the preconditioned system, the convergence rate will be superlinear.