Jackson's theorem and circulant preconditioned Toeplitz systems

Raymond H. CHAN*, Man Chung YEUNG

*Corresponding author for this work

Research output: Journal PublicationsJournal Article (refereed)peer-review

6 Citations (Scopus)


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 f{hook} whose lth derivative f{hook}(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 Πk = 2q (log2 k k2(l + α)).

Original languageEnglish
Pages (from-to)191-205
Number of pages15
JournalJournal of Approximation Theory
Issue number2
Publication statusPublished - Aug 1992
Externally publishedYes


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