Circulant and skew-circulant preconditioners for skew-hermitian type Toeplitz systems

Raymond H. CHAN*, Xiao Qing JIN

*Corresponding author for this work

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

26 Citations (Scopus)


We study the solutions of Toeplitz systems Anx=b by the preconditioned conjugate gradient method. The n ×n matrix An is of the form a0I+Hn where a0 is a real number, I is the identity matrix and Hn is a skew-Hermitian Toeplitz matrix. Such matrices often appear in solving discretized hyperbolic differential equations. The preconditioners we considered here are the circulant matrix Cn and the skew-circulant matrix Sn where An=1/2(Cn+Sn). The convergence rate of the iterative method depends on the distribution of the singular values of the matrices C-1n An and S-1nAn. For Toeplitz matrices An with entries which are Fourier coefficients of functions in the Wiener class, we show the invertibility of Cn and Sn and prove that the singular values of C-1nAn and S-1nAn are clustered around 1 for large n. Hence, if the conjugate gradient method is applied to solve the preconditioned systems, we expect fast convergence.

Original languageEnglish
Pages (from-to)632-646
Number of pages15
Issue number4
Publication statusPublished - Dec 1991
Externally publishedYes


  • circulant matrix
  • preconditioned conjugate gradient method
  • skew-circulant matrix
  • Skew-Hermitian type Toeplitz matrix


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