title: The best circulant preconditioners for Hermitian Toeplitz systems.II. The multiple-zero case
abstract: For pt.I. see SIAM J. Numer. Anal., vol. 38, p. 876-896. Circulant-type preconditioners have been proposed previously 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. Previously 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 for this class of generating functions
keyphrases: circulant preconditioners Hermitian Toeplitz systems multiple-zero case nonnegative continuous functions generating functions preconditioned matrices eigenvalues conjugate gradient method