Relations between condition numbers and the convergence of the Jacobi method for real positive definite matrices
type: Item
instance of: scholarly article
title: Relations between condition numbers and the convergence of the Jacobi method for real positive definite matrices
zbMATH Open document ID: 0535.65015
DOI: 10.1007/BF01400253
published in: Numerische Mathematik
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/132980
review text: Let A be an \(n\times n\) real symmetric diagonal dominant matrix with positive diagonal part D, and let \(S^ 2=D^{-1}\) and \(H=SAS\). The following relation between the condition number \(k(H)=\| H^{- 1}\|\| H\|\) and the spectral radius r of the Jacobi matrix associated to A is proved: \((k(H)-1)/(k(H)+1)\leq r\leq(k(H)-1)/(1+k(H)/(n- 1)).\) Moreover, relations among k(H), k(A), the condition numbers \(C(A)=\|| A^{-1}| | A| \|\), and C(H) are investigated.
Mathematics Subject Classification ID: 65F10
Mathematics Subject Classification ID: 65F35
zbMATH DE Number: 3850367
zbMATH Keywords: symmetric diagonal dominant matrix
zbMATH Keywords: condition number
zbMATH Keywords: spectral radius
zbMATH Keywords: Jacobi matrix
author: M. Arioli
author: Francesco Romani
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3850367