The best circulant preconditioners for hermitian toeplitz systems II: The multiple-zero case

In [10, 14], circulant-type preconditioners have been proposed for ill-conditioned Hermitian Toeplitz systems that are generated by nonnegative continuous functions with a zero of even order. The proposed circulant preconditioners can be constructed without requiring explicit knowledge of the generating functions. It was shown that the spectra of the preconditioned matrices are uniformly bounded except for a fixed number of outliers and that all eigenvalues are uniformly bounded away from zero. Therefore the conjugate gradient method converges linearly when applied to solving the circulant preconditioned systems. In [10, 14], it was claimed that this result can be extended to the case where the generating functions have multiple zeros. The main aim of this paper is to give a complete convergence proof of the method in [10, 14] for this class of generating functions.