The hyperpower iteration revisited
type: Item
instance of: scholarly article
title: The hyperpower iteration revisited
zbMATH Open document ID: 0848.65021
DOI: 10.1016/0024-3795(94)00076-X
author: Xuzhou Chen
author: R. E. Hartwig
published in: Linear Algebra and its Applications
publication date: 1996-02-25T00:00:00Z
review text: This paper discusses an extension of the hyperpower method [cf. \textit{A. Ben-Israel}, Math. Comput. 19, 452-455 (1965; Zbl 0136.12703)], which may be used for iterative computation of generalised inverses for example. The hyperpower method uses a basic iteration \(X_{k+1} = X_k (I + R_k + \dots + R^{q-1}_k)\), \(q \geq 2\), where \(A\) and \(X_0\) are arbitrary complex matrices and \(R_k\) is the residual \(I - AX_k\). The authors examine the method with residual modified to \(P(I - AX_k)\), with \(P\) idempotent. The main thrust of the paper is analysis of the convergence of \(B^{q^k}\) for some \(B \in \mathbb{C}^{n \times n}\), where \(B\) will be related to the matrices defined previously. If the basic iteration converges, an appropriate \(P\) and limit \(L\) have to be found and the paper discusses such possibilities.
Mathematics Subject Classification ID: 65F20
Mathematics Subject Classification ID: 65F10
Mathematics Subject Classification ID: 15A09
zbMATH DE Number: 841748
zbMATH Keywords: matrix inverse
zbMATH Keywords: idempotent matrices
zbMATH Keywords: hyperpower method
zbMATH Keywords: generalised inverses
zbMATH Keywords: convergence
reviewed by: Adrian Swift
MaRDI profile type: Publication
cites work: An Iterative Method for Computing the Generalized Inverse of an Arbitrary Matrix
cites work: A Note on an Iterative Method for Generalized Inversion of Matrices
cites work: On Iterative Computation of Generalized Inverses and Associated Projections
cites work: Q4061081
cites work: Q4743660
cites work: Q4149846
cites work: Anwendung des Schulz‐Verfahrens zur Nachkorrektur einer näherungsweise berechneten verallgemeinerten Inversen einer Matrix
cites work: Numerical algorithms for the Moore-Penrose inverse of a matrix: iterative methods
cites work: On the Numerical Properties of an Iterative Method for Computing the Moore–Penrose Generalized Inverse
cites work: Neumann-type expansion of reflexive generalized inverses of a matrix and the hyperpower iterative method
cites work: On generalized inverses and on the uniform convergence of \((I-\beta K)_ n\) with application to iterative methods
cites work: Q5528955
description: scientific article; zbMATH DE number 841748