Superlinear PCG methods for symmetric Toeplitz systems

In this paper we deal with the solution, by means of preconditioned conjugate gradient (PCG) methods, of $n\times n$ symmetric Toeplitz systems $A_n(f) \mathbf{ x}= \mathbf{ b}$ with nonnegative generating function $f$. Here the function $f$ is assumed to be continuous and strictly positive, or is assumed to have isolated zeros of even order. In the first case we use as preconditioner the natural and the optimal $\tau$ approximation of $A_n(f)$ proposed by Bini and Di Benedetto, and we prove that the related PCG method has a superlinear rate of convergence and a total arithmetic cost of $O(n\log n)$ ops. Under the second hypothesis we cannot guarantee that the natural $\tau$ matrix is positive definite, while for the optimal we show that, in the ill-conditioned case, this can be really a bad choice. Consequently, we define a new $\tau$ matrix for preconditioning the given system; then, by applying the Sherman-Morrison-Woodbury inversion formula to the preconditioned system, we introduce a small, constant number of subsidiary systems which can be solved again by means of the previous PCG method. Finally, we perform some numerical experiments that show the effectiveness of the devised technique and the adherence with the theoretical analysis.