The circulant operator in the banach algebra of matrices

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)41-53
Journal / PublicationLinear Algebra and Its Applications
Volume149
Publication statusPublished - 15 Apr 1991
Externally publishedYes

Abstract

We study an operator c which maps every n-by-n matrix An to a circulant matrix c(An) that minimizes the Frobenius norm ‖AnCnF over all n-by-n circulant matrices Cn. The circulant matrix c(An), called the optimal circulant preconditioner, has proved to be a good preconditioner for a general class of Toeplitz systems. In this paper, we give different formulations of the operator, discuss its algebraic and geometric properties, and compute its operator norms in different Banach algebras of matrices. Using these results, we are able to give an efficient algorithm for finding the superoptimal circulant preconditioner which is defined to be the minimizer of ‖ICn-1AnF over all nonsingular circulant matrices Cn.

Citation Format(s)

The circulant operator in the banach algebra of matrices. / Chan, Raymond H.; Jin, Xiao-Qing; Yeung, Man-Chung.
In: Linear Algebra and Its Applications, Vol. 149, 15.04.1991, p. 41-53.

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