JACKSON'S THEOREM AND CIRCULANT PRECONDITIONED TOEPLITZ SYSTEMS

Research output: Journal Publications and ReviewsRGC 21 - Publication in refereed journalpeer-review

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Author(s)

Detail(s)

Original languageEnglish
Pages (from-to)191-205
Journal / PublicationJournal of Approximation Theory
Volume70
Issue number2
Publication statusPublished - Aug 1992
Externally publishedYes

Abstract

The preconditioned conjugate gradient method is used to solve n-by-n Hermitian Toeplitz systems Anx = b. The preconditioner Sn is the Strang's circulant preconditioner which is defined to be the circulant matrix that copies the central diagonals of An. The convergence rate of the method depends on the spectrum of Sn-1An. Using Jackson's theorem in approximation theory, we prove that if An has a positive generating fucntion ƒ whose lth derivative ƒ(l), l ≥ 0, is Lipschitz of order 0 < α ≤ 1, then the method converges superlinearly. We show moreover that the error after 2q conjugate gradient steps decreases like Πqk=2 (log2 k/k2(l + α)).

Citation Format(s)

JACKSON'S THEOREM AND CIRCULANT PRECONDITIONED TOEPLITZ SYSTEMS. / CHAN, Raymond H.; YEUNG, Man-Chung.
In: Journal of Approximation Theory, Vol. 70, No. 2, 08.1992, p. 191-205.

Research output: Journal Publications and ReviewsRGC 21 - Publication in refereed journalpeer-review