## Abstract

We discuss the solutions to a class of Hermitian positive definite systems Ax = b by the preconditioned conjugate gradient method with circulant preconditioner C. In general, the smaller the condition number κ(C^{−1/2} AC^{−1/2}) is, the faster the convergence of the method will be. The circulant matrix C_{b} that minimizes κ(C^{−1/2} AC^{−1/2}) is called the best-conditioned circulant preconditioner for the matrix A. We prove that if F AF∗ has Property A, where F is the Fourier matrix, then C_{b} minimizes ∥C − A∥_{F} over all circulant matrices C. Here ∥ · ∥_{F} denotes the Frobenius norm. We also show that there exists a noncirculant Toeplitz matrix A such that F AF∗ has Property A.

Original language | English |
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Pages (from-to) | 205-211 |

Number of pages | 7 |

Journal | Linear Algebra and Its Applications |

Volume | 218 |

DOIs | |

Publication status | Published - 15 Mar 1995 |

Externally published | Yes |