Componentwise analysis of direct factorization of real symmetric and Hermitian matrices
type: Item
instance of: scholarly article
title: Componentwise analysis of direct factorization of real symmetric and Hermitian matrices
zbMATH Open document ID: 0894.65009
DOI: 10.1016/S0024-3795(97)00334-0
author: Ivan Slapničar
published in: Linear Algebra and its Applications
publication date: 1998-08-16T00:00:00Z
review text: A factorization algorithm for an \(n \times n\) symmetric matrix \(H\) is developed, \(H=GJG^{T}\), where \(G\) has full column rank and \(J=\text{diag}(\pm 1)\). This decomposition presents an extension of the Cholesky method for positive definite matrices and also a modification of the method of \textit{J. R. Bunch} and \textit{B. N. Parlett} [SIAM J. Numer. Anal. 8, 639-655 (1971; Zbl 0199.49802)]. Further in the paper, componentwise error bounds for the factorization are derived. The main result is the componentwise backward error bound: the computed \(G\) and \(J\) are the exact factors of the perturbed matrix \(H+E\), that is \[ GJG^{T} = H+E, \quad| E| \leq 91n(| H| + | G| | G| ^{T})\varepsilon, \] where \(\varepsilon\) is the machine precision. The obtained estimate reveals the error structure better and has a simpler form, which is more suitable for further applications. A componentwise forward error bound for the same symmetric factorization is also proved. This generalizes \textit{Jiguang Sun}'s studies to indefinite matrices [Linear Algebra Appl. 173, 77-97 (1992; Zbl 0808.65018), Theorem 2.2.1]. All results are extended to the case of Hermitian matrices \(H\) and to the Bunch-Parlett factorization. Finally, bounds for the scaled condition of the matrix \(G\) are found, i.e. the inequality \(\kappa(\text{scal}(G)) \leq O(n3.781^n)\) is proved irrespective of the condition and of the singularity of \(H\). All results are illustrated by a numerical example.
Mathematics Subject Classification ID: 65F05
zbMATH DE Number: 1116072
zbMATH Keywords: componentwise error analysis
zbMATH Keywords: factorization algorithm
zbMATH Keywords: Cholesky method
zbMATH Keywords: Hermitian matrices
zbMATH Keywords: error bounds
zbMATH Keywords: numerical example
reviewed by: Neli Dimitrova
describes a project that uses: LAPACK
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1116072