document: The best circulant preconditioners for Hermitian Toeplitz systems.II . The multiple-zero case 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
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