A Family of Block Preconditioners for Block Systems

Raymond H. CHAN, Xiao-Qing JIN

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


The solution of block system Amnx = b by the preconditioned conjugate gradient method where Amn is an m-by-m block matrix with n-by-n Toeplitz blocks is studied. The preconditioner cF(1) (Amn) is a matrix that preserves the block structure of Amn. Specifically, it is defined as the minimizer of ||Amn - Cmn ||F over all m-by-m block matrices Cmn with n-by-n circulant blocks. We prove that if Amn is positive definite, then cF(1)(Amn) is positive definite too. We also show that cF(1)(Amn) is a good preconditioner for solving separable block systems with Toeplitz blocks and quadrantally symmetric block Toeplitz systems. We then discuss some of the spectral properties of the operator cF(1). In particular, we show that the operator norms ||cF(1)||2 = ||cF(1)||F = 1.
Original languageEnglish
Pages (from-to)1218-1235
Number of pages18
JournalSIAM Journal on Scientific Computing
Issue number5
Publication statusPublished - Sept 1992
Externally publishedYes


  • Toeplitz matrix
  • circulant matrix
  • circulant operator
  • preconditioned conjugate gradient method


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