## 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 language | English |
---|---|

Pages (from-to) | 435-449 |

Number of pages | 15 |

Journal | Linear Algebra and Its Applications |

Volume | 418 |

Issue number | 2-3 |

DOIs | |

Publication status | Published - 15 Oct 2006 |

Externally published | Yes |

## Keywords

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