Best-Conditioned Circulant Preconditioners

Research output: Journal Publications and Reviews (RGC: 21, 22, 62)21_Publication in refereed journalpeer-review

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Original languageEnglish
Pages (from-to)205-211
Journal / PublicationLinear Algebra and Its Applications
Publication statusPublished - 15 Mar 1995
Externally publishedYes


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 ∥CAF 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.