Circulant preconditioners for indefinite Toeplitz systems

In recent papers circulant preconditioners were proposed for ill-conditioned Hermitian Toeplitz matrices generated by 2pi-periodic continuous functions with zeros of even order. It was show that the spectra of the preconditioned matrices are uniformly bounded except for a finite number of outliers and therefore the conjugate gradient method, when applied to solving these circulant preconditioned systems, converges very quickly. I this paper, we consider indefinite Toeplitz matrices generated by 2pi-periodic continuous functions with zeros of odd order. In particular, we show that the singular values of the preconditioned matrices are essentially bounded. Numerical results are presented to illustrate the fast convergence of CGNE, MINRES and QMR methods.