The Spectrum of a Family of Circulant Preconditioned Toeplitz Systems

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


The solutions of symmetric positive definite Toeplitz systems Ax = b are studied by the preconditioned conjugate gradient method. The preconditioner is the circulant matrix C that minimizes the Frobenius norm ‖C - A‖F [T. Chan, “An Optimal Circulant Preconditioner for Toeplitz Systems,” UCLA Department of Mathematics, CAM Report 87-06, June 1987]. The convergence rate of these iterative methods is known to depend on the distribution of the eigenvalues of C-1A. For Toeplitz matrix A with entries which are Fourier coefficients of a positive function in the Wiener class, this paper establishes the invertibility of C, finds the asymptotic behaviour of the eigenvalues of the preconditioned matrix C-1A as the dimension increases and proves that they are clustered around 1.
Original languageEnglish
Pages (from-to)503-506
Number of pages4
JournalSIAM Journal on Numerical Analysis
Issue number2
Publication statusPublished - Apr 1989
Externally publishedYes


  • Toeplitz matrix
  • circulant matrix
  • preconditioned conjugate gradient method


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