Best-conditioned circulant preconditioners
type: Item
instance of: scholarly article
title: Best-conditioned circulant preconditioners
zbMATH Open document ID: 0826.65020
DOI: 10.1016/0024-3795(93)00171-U
author: Yanyan Li
published in: Linear Algebra and its Applications
publication date: 1995-11-26T00:00:00Z
review text: The solutions to a class of Hermitian positive definite systems \(Ax = b\) by the preconditioned conjugate gradient method (PCG) with circulant preconditioner \(C\) are discussed. The rate of convergence of the PCG method depends on how small the condition number \(\kappa (C^{-{1\over 2}} AC^{-{1\over 2}})\) is. It is shown that if the matrix \(FAF^*\) has property A, then \(C_F = F^*\delta (FAF^*)F\) minimizes \(\kappa (C^{-{1\over 2}} AC^{-{1\over 2}})\) over all Hermitian positive definite circulant matrices \(C\), where \(F\) is the Fourier matrix and \(\delta(B)\) is the diagonal matrix such that \(\delta (B)_{i,i} = (B)_{i,i}\), \(1 \leq i \leq n\). It is also shown that there exists a noncirculant Toeplitz matrix \(A\) such that \(FAF^*\) has property A.
Mathematics Subject Classification ID: 65F10
Mathematics Subject Classification ID: 65F35
zbMATH DE Number: 753869
zbMATH Keywords: circulant matrix
zbMATH Keywords: Hermitian positive definite systems
zbMATH Keywords: preconditioned conjugate gradient method
zbMATH Keywords: circulant preconditioner
zbMATH Keywords: convergence
zbMATH Keywords: condition number
zbMATH Keywords: Fourier matrix
zbMATH Keywords: Toeplitz matrix
reviewed by: Shenquan Xie
MaRDI profile type: Publication
cites work: Q3323187
cites work: Circulant Preconditioners for Hermitian Toeplitz Systems
cites work: The Spectrum of a Family of Circulant Preconditioned Toeplitz Systems
cites work: The Spectra of Super-Optimal Circulant Preconditioned Toeplitz Systems
cites work: The circulant operator in the Banach algebra of matrices
cites work: Circulant Preconditioners for Toeplitz Matrices with Positive continuous Generating Functions
cites work: An Optimal Circulant Preconditioner for Toeplitz Systems
cites work: On Best Conditioned Matrices
cites work: Some Aspects of Circulant Preconditioners
cites work: A Proposal for Toeplitz Matrix Calculations
cites work: Optimal and Superoptimal Circulant Preconditioners
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description: scientific article; zbMATH DE number 753869