Circulant Preconditioners for Hermitian Toeplitz Systems

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


The solutions of Hermitian positive definite Toeplitz systems Ax = b by the preconditioned conjugate gradient method for three families of circulant preconditioners C is studied. The convergence rates of these iterative methods depend on the spectrum of C-1A. For a Toeplitz matrix A with entries that are Fourier coefficients of a positive function f in the Wiener class, the invertibility of C is established, as well as that the spectrum of the preconditioned matrix C-1A clusters around one. It is proved that  if f is ( + 1)-times differentiable, with l > 0, then the error after 2q conjugate gradient steps will decrease like  ((q - 1)!)-2l. It is also shown that if C copies the central diagonals of A, then C minimizes  ǁC - Aǁ1 and ǁC - Aǁ.
Original languageEnglish
Pages (from-to)542-550
Number of pages9
JournalSIAM Journal on Matrix Analysis and Applications
Issue number4
Publication statusPublished - Oct 1989
Externally publishedYes


  • Toeplitz matrix
  • circulant matrix
  • preconditioned conjugate gradient method


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