On a quadratically convergent method using divided differences of order one under the gamma condition
type: Item
instance of: scholarly article
title: On a quadratically convergent method using divided differences of order one under the gamma condition
zbMATH Open document ID: 1152.65063
author: Ioannis Konstantinos Argyros
author: Hongmin Ren
published in: Central European Journal of Mathematics
publication date: 2008-06-25T00:00:00Z
review text: The authors study the functional equation: \(F(x)=0\), where \(F\) is a twice Fréchet differentiable operator between a convex subset of a Banach space \(X\) and a Banach space \(Y\). They construct a sequence \(x_{n}\) approximating the unique solution \(x^*\) of the functional equation \[ x_{n+1}=x_{n}-[2x_{n}-x_{n-1},x_{n-1}]^{-1}F(x_{n}) ;\quad n>0 . \] Here \([x,y]\) denotes the divided difference of order 1 at the point \((x,y)\) and \[ [x,y](x-y)=F(x)-F(y). \] The first theorem of the paper gives conditions on the first and second Fréchet derivative of \(F\) under which the approximating sequence is well defined, remains in a certain neighborhood of \(x^*\) and converges to \(x^*\). The second theorem provides conditions under which semilocal convergence of the sequence takes place. Numerical examples illustrate the results.
Mathematics Subject Classification ID: 65J15
Mathematics Subject Classification ID: 47J25
zbMATH DE Number: 5292959
zbMATH Keywords: local and semilocal convergence
zbMATH Keywords: Fréchet derivative
zbMATH Keywords: Steffensen method
zbMATH Keywords: Newton-Kantorovich condition
zbMATH Keywords: nonlinear operator equation
zbMATH Keywords: majorizing sequence
zbMATH Keywords: two-point iterative method
zbMATH Keywords: gamma condition
MaRDI profile type: Publication
full work available at URL: s11533-008-0015-y
OpenAlex ID: W2078197337
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DOI: 10.2478/S11533-008-0015-Y
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description: scientific article; zbMATH DE number 5292959