A unifying approach to the construction of circulant preconditioners

Ivan Oseledets, Eugene Tyrtyshnikov

Research output: Contribution to journalArticlepeer-review

12 Citations (Scopus)

Abstract

The main result is the "black dot algorithm" and its fast version for the construction of a new circulant preconditioner for Toeplitz matrices. This new preconditioner C is sought directly as a solution to one of possible settings of the approximation problem A ≈ C + R, where A is a given matrix and R should be a "low-rank" matrix. This very problem is a key to the analysis of superlinear convergence properties of already established circulant and other matrix-algebra preconditioners. In this regard, our new preconditioner is likely to be the best of all possible circulant preconditioners. Moreover, in contrast to several "function-based" circulant preconditioners used for "bad" symbols, it is constructed entirely from the entries of a given matrix and performs equally as the best of the known or better than those for the same symbols.

Original languageEnglish
Pages (from-to)435-449
Number of pages15
JournalLinear Algebra and Its Applications
Volume418
Issue number2-3
DOIs
Publication statusPublished - 15 Oct 2006
Externally publishedYes

Keywords

  • Circulants
  • Low-rank matrices
  • Matrix approximations
  • Preconditioners
  • Skeleton decomposition
  • Spectral clusters
  • Spectral distributions
  • Superlinear convergence
  • Toeplitz matrices

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