Best-conditioned circulant preconditioners

Raymond H. CHAN*, C. K. WONG

*Corresponding author for this work

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

5 Citations (Scopus)


We discuss the solutions to a class of Hermitian positive definite systems Ax = b by the preconditioned conjugate gradient method with circulant preconditioner C. In general, the smaller the condition number κ(C−1/2 AC−1/2) is, the faster the convergence of the method will be. The circulant matrix Cb that minimizes κ(C−1/2 AC−1/2) is called the best-conditioned circulant preconditioner for the matrix A. We prove that if F AF∗ has Property A, where F is the Fourier matrix, then Cb minimizes ∥C − A∥F over all circulant matrices C. Here ∥ · ∥F denotes the Frobenius norm. We also show that there exists a noncirculant Toeplitz matrix A such that F AF∗ has Property A.

Original languageEnglish
Pages (from-to)205-211
Number of pages7
JournalLinear Algebra and Its Applications
Publication statusPublished - 15 Mar 1995
Externally publishedYes


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