On an improved local convergence analysis for the Secant method
type: Item
instance of: scholarly article
title: On an improved local convergence analysis for the Secant method
zbMATH Open document ID: 1176.65068
author: Ioannis Konstantinos Argyros
author: Hongmin Ren
published in: Numerical Algorithms
publication date: 2009-10-14T00:00:00Z
review text: The authors are concerned with the problem of approximating the locally unique solution of a nonlinear operator equation in Banach spaces. The operator F involving in the operator equation is supposed Fréchet-differentiable on an open convex subset of the given Banach space \(X\). The classical secant method has been generalized for equations in Banach spaces and there has been an extensive literature on the local and semilocal convergence of the secant method under various Lipschitz-type conditions. In the case of the local convergence of the secant method, the idea is to find a ball (the convergence ball), as large as possible, such that the convergence of the sequence of iterations to the solution of the equation can be achieved for initial guesses chosen from the convergence ball. Under a special Hölder continuous-type condition for the divided differences, the radius of convergence given by \textit{H. Ren} and \textit{Q. Wu} [J. Comput. Appl. Math. 194, No.~2, 284--293 (2006; Zbl 1101.65057)], is enlarged and the local convergence of the secant method is proved. One numerical example is performed to illustrate the performance of the obtained result.
reviewed by: Iulian Coroian
Mathematics Subject Classification ID: 65J15
Mathematics Subject Classification ID: 47J25
zbMATH DE Number: 5615009
zbMATH Keywords: Banach spaces
zbMATH Keywords: secant method
zbMATH Keywords: Fréchet-derivative
zbMATH Keywords: local convergence
zbMATH Keywords: radius of convergence
zbMATH Keywords: Newton's method
zbMATH Keywords: nonlinear operator equation
zbMATH Keywords: numerical example
MaRDI profile type: Publication
full work available at URL: s11075-009-9271-6
OpenAlex ID: W2083300135
cites work: The secant method and fixed points of nonlinear operators
cites work: A unifying local-semilocal convergence analysis and applications for two-point Newton-like methods in Banach space
cites work: Convergence and Applications of Newton-type Iterations
cites work: Q4240123
cites work: Q4294177
cites work: Q3605655
cites work: The secant method and divided differences Hölder continuous
cites work: The secant method for nondifferentiable operators
cites work: Semilocal convergence of the secant method under mild convergence conditions of differentiability
cites work: The convergence ball of Newton's method and the uniqueness ball of equations under Hölder-type continuous derivatives
cites work: The convergence ball of the secant method under Hölder continuous divided differences
cites work: Mysovskii-type theorem for the Secant method under Hölder continuous Fréchet derivative
cites work: A new semilocal convergence theorem for the secant method under Hölder continuous divided differences
cites work: Regula-falsi-Verfahren mit konsistenter Steigung und Majorantenprinzip
DOI: 10.1007/S11075-009-9271-6
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description: scientific article; zbMATH DE number 5615009