A note on best conditioned preconditioners
type: Item
instance of: scholarly article
title: A note on best conditioned preconditioners
zbMATH Open document ID: 0812.65030
DOI: 10.1007/BF01955879
author: V. Pereyra
published in: BIT
publication date: 1995-05-14T00:00:00Z
review text: In this short note, the author gives a characterization of best conditioned preconditioners \(M\) for the conjugate gradient method applied to Hermitian positive definite linear systems \(Ax= b\). Denoting by \({\mathcal M}_ Q\) and \({\mathcal M}^ +_ Q\) the sets of matrices being similar to a complex diagonal matrix and a positive definite diagonal matrix with respect to a unitary transformation \(Q\), the best conditioned preconditioner \(M_ b\) minimizes the spectral condition number \(\kappa(M^{-1/2} AM^{-1/2})\) over \({\mathcal M}^ +_ Q\). It is shown that \(M_ b\) also minimizes the Frobenius norm \(\| M- A\|_ F\) over \({\mathcal M}_ Q\) provided the matrix \(QAQ^*\) has Young's property A.
Mathematics Subject Classification ID: 65F10
Mathematics Subject Classification ID: 65F35
zbMATH DE Number: 644799
zbMATH Keywords: best conditioned preconditioners
zbMATH Keywords: conjugate gradient method
zbMATH Keywords: spectral condition number
zbMATH Keywords: Young's property \(A\)
reviewed by: R. H. W. Hoppe
MaRDI profile type: Publication
cites work: A Family of Block Preconditioners for Block Systems
cites work: Best-conditioned circulant preconditioners
cites work: An Optimal Circulant Preconditioner for Toeplitz Systems
cites work: On Best Conditioned Matrices
cites work: Circulant and Skewcirculant Matrices for Solving Toeplitz Matrix Problems
description: scientific article; zbMATH DE number 644799