The best circulant preconditioners for Hermitian Toeplitz systems II: The multiple-zero case |
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| Authors: | Raymond H Chan Michael K Ng Andy M Yip |
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| Institution: | (1) Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong, China; e-mail: rchan@math.cuhk.edu.hk , HK;(2) Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong, China; e-mail: mng@maths.hku.hk , HK;(3) Department of Mathematics, University of California, Los Angeles, 405 Hilgard Avenue, Los Angeles, CA 90095-1555, USA; e-mail: mhyip@math.ucla.edu , US |
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| Abstract: | Summary. 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 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.
Received October 19, 1999 / Revised version received May 2, 2001 / Published online October 17, 2001 |
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| Keywords: | Mathematics Subject Classification: 65F10 65F15 65T10 |
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