A note on the growth factor in Gaussian elimination for accretive-dissipative matrices
type: Item
instance of: scholarly article
title: A note on the growth factor in Gaussian elimination for accretive-dissipative matrices
zbMATH Open document ID: 1320.65045
author: Minghua Lin
published in: Calcolo
publication date: 2015-03-23T00:00:00Z
review text: The \(n\times n\) complex matrix \textbf{A} is accretive-dissipative if it can be written as \(\mathbf{A}= \mathbf{B}+i \mathbf{C}\), where \(\mathbf{B} = \frac{1}{2}(\mathbf{A}+\mathbf{A}^\ast)\) and \(\mathbf{C} = \frac{1}{2i}(\mathbf{A}-\mathbf{A}^\ast)\) are both (Hermitian) positive definite. If \textbf{B}, \textbf{C} are real symmetric positive definite, then \textbf{A} is called a Higham matrix. The growth factor (in Gaussian elimination) of \textbf{A} is defined as a ratio of the maximum element (in absolute value) of any intermediate result of the Gaussian elimination process to the maximum element of the original matrix \textbf{A}. The presented short note proves that if \textbf{A} is accretive-dissipative, then its growth factor in Gaussian elimination is less than 4. If \textbf{A} is a Higham matrix then the growth factor is less than \(2\sqrt{2}\). This result improves those obtained recently and is one step closer to the final solution of Higham's conjecture.
Mathematics Subject Classification ID: 65F05
Mathematics Subject Classification ID: 15A45
zbMATH DE Number: 6418674
zbMATH Keywords: accretive-dissipative matrix
zbMATH Keywords: Higham matrix
zbMATH Keywords: growth factor
zbMATH Keywords: Gaussian elimination
MaRDI profile type: Publication
full work available at URL: s10092-013-0089-1
OpenAlex ID: W2161768507
cites work: On the growth factor in Gaussian elimination for matrices with sharp angular field of values
cites work: On the growth factor in Gaussian elimination for generalized Higham matrices
cites work: Factorizing complex symmetric matrices with positive definite real and imaginary parts
cites work: Norm inequalities for accretive-dissipative operator matrices
cites work: Fischer type determinantal inequalities for accretive-dissipative matrices
DOI: 10.1007/S10092-013-0089-1
Recommended article: Q3463204
Recommended article: On the growth factor in Gaussian elimination for matrices with sharp angular field of values
Recommended article: On the growth factor in Gaussian elimination for generalized Higham matrices
Recommended article: Growth in Gaussian Elimination, Orthogonal Matrices, and the 2-Norm
Recommended article: Growth in Gaussian elimination for weighing matrices, \(W (n, n-1)\)
Recommended article: On Growth in Gaussian Elimination with Complete Pivoting
Recommended article: An upper bound on the growth ratio of Gaussian elimination
Recommended article: Bounding the growth factor in Gaussian elimination for Buckley's class of complex symmetric matrices
Recommended article: On the properties of accretive-dissipative matrices
Recommended article: On some inequalities for accretive-dissipative matrices
description: scientific article; zbMATH DE number 6418674