Circulant Preconditioners for Hermitian Toeplitz Systems

Abstract

The solutions of Hermitian positive definite Toeplitz systems Ax b by the preconditioned conjugate gradient method for three families of circulant preconditioners C is studied. The convergence rates of these iterative methods depend on the spectrum of C-A. For a Toeplitz matrix A with entries that are Fourier coetficients of a positive function f in the Wiener class, the invertibility of C is established, as well as that the spectrum ofthe preconditioned matrix C-A clusters around one. It is proved that iffis (l + )-times differentiable, with > 0, then the error after 2q conjugate gradient steps will decrease like ((q also shown that ifC copies the central diagonals ofA, then C minimizes C AIla and C AII