Convergence of Newton-like methods for singular operator equations using outer inverses
type: Item
instance of: scholarly article
title: Convergence of Newton-like methods for singular operator equations using outer inverses
zbMATH Open document ID: 0797.65047
DOI: 10.1007/BF01385696
author: Xiaojun Chen
author: M. Zuhair Nashed
published in: Numerische Mathematik
publication date: 1994-10-13T00:00:00Z
full work available at URL: https://eudml.org/doc/133761
review text: For singular \(F\)-differentiable nonlinear operators between Banach spaces convergence theorems of (semilocal) Kantorovich-type are proved for Newton-like methods \(x_{k+1} = x_ k - A(x_ k)^ \#F(x_ k)\), \(k \geq 0\). Here \(A(x)\) is a bounded linear operator approximating \(F'(x)\) and \(A(x)^ \#\) a bounded outer inverse of \(A(x)\); that is, \(A(x)^ \# A(x) A(x)^ \# = A(x)^ \#\). The results reduce precisely to the well- known Kantorovich and Mysovskii theorems for the case of invertible derivatives. The analysis utilizes Banach-type lemmas and perturbation bounds for outer inverses which show that (for bounded linear operators) the set of outer inverses contains elements that behave similar to bounded linear inverses. Several examples compare the results with related known results and also addresses some computational issues. Various interesting remarks relevant to this setting conclude the paper.
Mathematics Subject Classification ID: 65J15
Mathematics Subject Classification ID: 47J25
zbMATH DE Number: 569183
zbMATH Keywords: singular derivatives
zbMATH Keywords: semilocal convergence
zbMATH Keywords: semilocal Kantorovich-type convergence theorems
zbMATH Keywords: Banach spaces
zbMATH Keywords: Newton-like methods
zbMATH Keywords: outer inverse
zbMATH Keywords: Mysovskii theorems
reviewed by: Werner C. Rheinboldt
MaRDI profile type: Publication
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OpenAlex ID: W2054204520
description: scientific article; zbMATH DE number 569183