## Abstract

We consider the solution of n-by-n Toeplitz systems** A_{n}x = b** by the preconditioned conjugate gradient method. The preconditioner

*is the circulant matrix that minimizes ∥*

**C**_{n}**∥**

*B*_{n}− A_{n}_{F}over all circulant matrices

*. We show that if the generating function f is a positive 2π-periodic continuous function, then the spectrum of the preconditioned system*

**B**_{n}**will be clustered around one. In particular, if the preconditioned conjugate gradient method is applied to solve the preconditioned system, the convergence rate is superlinear.**

*C*_{n}^{−1}A_{n}Original language | English |
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Pages (from-to) | 233-240 |

Number of pages | 8 |

Journal | Mathematics of Computation |

Volume | 58 |

Issue number | 197 |

DOIs | |

Publication status | Published - Jan 1992 |

Externally published | Yes |