Convergence of the Robbins-Monro method for linear problems in a Banach space
type: Item
instance of: scholarly article
title: Convergence of the Robbins-Monro method for linear problems in a Banach space
zbMATH Open document ID: 0683.62041
DOI: 10.1016/0022-247X(89)90235-7
published in: Journal of Mathematical Analysis and Applications
publication date: 1989-01-01T00:00:00Z
review text: Consider the equation \(Ax=b\), where b belongs to some Banach space X and A is a bounded linear operator from X into itself. It has been noticed by several people under different restrictions that in order to obtain an iterative solution one can use the Robbins-Monro method. The recursive formula is given by \(x_{n+1}=x_ n-\alpha_ n\quad (A_ nx_ n-b_ n),\) common assumptions are \(\alpha_ n\to 0,\) \(\sum \alpha_ n=\infty\) and the weighted means of \(A_ n\) and \(b_ n\) converge to A and b, respectively. The aim of the authors is to give a unified treatment of the convergence of \((x_ n)\) to a solution of \(Ax=b\) under mild conditions.
Mathematics Subject Classification ID: 62L20
Mathematics Subject Classification ID: 60B12
Mathematics Subject Classification ID: 60B11
zbMATH DE Number: 4119437
zbMATH Keywords: almost sure convergence
zbMATH Keywords: Banach space
zbMATH Keywords: bounded linear operator
zbMATH Keywords: iterative solution
zbMATH Keywords: Robbins-Monro method
zbMATH Keywords: recursive formula
author: Harro Walk
author: László Zsidó
MaRDI profile type: Publication
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full work available at URL: 0022-247x(89)90235-7
OpenAlex ID: W2079867248
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description: scientific article; zbMATH DE number 4119437