Circulant preconditioners for Toeplitz matrices with piecewise continuous generating functions

The authors consider the solution of n-by-n Toeplitz systems T[sub n]x = b by preconditioned conjugate gradient methods. The preconditioner C[sub n] is the T. Chan circulant preconditioner, which is defined to be the circulant matrix that minimizes [parallel]B[sub n] - T[sub n][parallel][sub F] over all circulant matrices B[sub n]. For Toeplitz matrices generated by positive 2[pi]-periodic continuous functions, they have shown earlier that the spectrum of the preconditioned system C[sup [minus]1][sub n]T[sub n] is clustered around 1 and hence the convergence rate of the preconditioned system is superlinear. However, in this paper, they show that if instead the generating function is only piecewise continuous, then for all [epsilon] sufficiently small, there are O(log n) eigenvalues of C[sup [minus]1][sub n]T[sub n] that lie outside the interval (1 - [epsilon], 1 + [epsilon]). In particular, the spectrum of C[sup [minus]1][sub n]T[sub n] cannot be clustered around 1. Numerical examples are given to verify that the convergence rate of the method is no longer superlinear in general. 20 refs.