Hartley preconditioners for Toeplitz systems generated by positive continuous functions

In this paper, we consider the solution ofn-by-n symmetric positive definite Toeplitz systemsTnx=b by the preconditioned conjugate gradient (PCG) method. The preconditionerMn is defined to be the minimizer of ‖Tn−Bn‖F over allBn εHn whereHn is the Hartley algebra. We show that if the generating functionf ofTn is a positive 2π-periodic continuous even function, then the spectrum of the preconditioned systemMn−1Tn will be clustered around 1. Thus, if the PCG method is applied to solve the preconditioned system, the convergence rate will be superlinear.