Universal iterative methods for computing generalized inverses
type: Item
instance of: scholarly article
title: Universal iterative methods for computing generalized inverses
zbMATH Open document ID: 0907.65041
DOI: 10.1023/A:1006514125818
author: P. S. Stanimitrović
author: Dragan S. Djordjević
published in: Acta Mathematica Hungarica
publication date: 1998-11-22T00:00:00Z
review text: Let \(H_1,H_2\) be two finite-dimensional complex Hilbert spaces and \(A:H_1\to H_2\) a linear operator. Consider the following conditions on an operator \(X\) from \(H_2\) into \(H_1\): (1) \(AXA=A\); (2) \(XAX=X\); (3) \((AX)^\ast=AX\); (4) \((XA)^\ast=XA\); (5) \(AX=XA\). For \(\mathcal S\) a subset of \(\{1,2,3,4,5\}\), let us call an operator \(X\) satisfying the conditions in \(\mathcal S\) an \(\mathcal S\)-inverse of \(A\). If \(\mathcal S=\{1,2,3,4\}\) or \(\mathcal S=\{1,2,5\}\), the corresponding \(\mathcal S\)-inverses are unique and coincide with the Moore-Penrose inverse and the group inverse of \(A\), respectively. The authors construct iterative processes, based on hyper-power iterative method or on Neumann-type series expansions, which converge to a \(\{1,2\}\), a \(\{1,2,3\}\), a \(\{1,2,4\}\), the Moore-Penrose, the group, and the weighted Moore-Penrose inverse, respectively (the last is defined by replacing (3) and (4) with \((MAX)^\ast=MAX\) and \((XAN)^\ast=XAN\), where \(M,N\) are given positive definite ``weight'' matrices). In all cases, it is assumed that certain operators \(W_1,W_2\) are known such that \(W_2 A W_1\) is invertible. Error bounds are also established and it is shown that the methods are self-correcting. The paper concludes with a few examples illustrating the theory.
reviewed by: Miroslav Engliš
Mathematics Subject Classification ID: 65F20
Mathematics Subject Classification ID: 65F10
Mathematics Subject Classification ID: 15A09
zbMATH DE Number: 1225610
zbMATH Keywords: generalized inverses
zbMATH Keywords: Moore-Penrose inverse
zbMATH Keywords: finite-dimensional complex Hilbert spaces
zbMATH Keywords: hyper-power iterative method
zbMATH Keywords: Neumann-type series expansions
zbMATH Keywords: error bounds
zbMATH Keywords: group inverse
MaRDI profile type: Publication
full work available at URL: a:1006514125818
OpenAlex ID: W55013433
description: scientific article; zbMATH DE number 1225610