A norm bound for projections with complex weights
type: Item
instance of: scholarly article
title: A norm bound for projections with complex weights
zbMATH Open document ID: 0993.15027
DOI: 10.1016/S0024-3795(00)00017-3
author: Elena Y. Bobrovnikova
author: Stephen A. Vavasis
published in: Linear Algebra and its Applications
publication date: 2002-09-17T00:00:00Z
review text: Let \(\mu\geqslant 0\), \(C_\mu=\{z\in{\mathbb C}\mid |\text{Im }z|\leqslant\text{Re }z\), \(z\neq 0\}\), and let \({\mathcal D}_\mu=\{\text{diag}(a_1,\dots,a_n)\in M_m({\mathbb C})\mid a_1,\dots,a_m\in C_\mu\}\). Then for a rank \(n\) matrix \(A\in M_{m,n}({\mathbb R})\) there exist numbers \(\chi_{A,\mu},\overline\chi_{A,\mu}\in{\mathbb R}\) such that \(\|(A^tDA)^{-1}A^tD\|\leqslant \chi_{A,\mu}\) and \(\|A(A^tDA)^{-1}A^tD\|\leqslant \overline \chi_{A,\mu}\). For \(\mu=0\) these estimates are known as Dikin-Stewart-Todd norm bounds for weighted projections in the case of an ill-conditioned real positive-definite weight matrix [see \textit{A. L. Forsgren}, SIAM J. Matrix Anal. Appl. 17, No. 4, 763-788 (1996; Zbl 0885.65046)].
Mathematics Subject Classification ID: 15A60
zbMATH DE Number: 1441331
zbMATH Keywords: norm bounds
zbMATH Keywords: weighted projections
zbMATH Keywords: ill-conditioned real positive-definite weight matrix
reviewed by: A. R. Alimov
MaRDI profile type: Publication
cites work: Accurate solution of weighted least squares by iterative methods
cites work: On Linear Least-Squares Problems with Diagonally Dominant Weight Matrices
cites work: Complete Orthogonal Decomposition for Weighted Least Squares
cites work: On scaled projections and pseudoinverses
cites work: A Dantzig-Wolfe-Like Variant of Karmarkar's Interior-Point Linear Programming Algorithm
cites work: Stable Numerical Algorithms for Equilibrium Systems
description: scientific article; zbMATH DE number 1441331