Secant-type methods and nondiscrete induction
type: Item
instance of: scholarly article
title: Secant-type methods and nondiscrete induction
zbMATH Open document ID: 1267.65054
author: Ioannis Konstantinos Argyros
author: Saïd Hilout
published in: Numerical Algorithms
publication date: 2012-11-27T00:00:00Z
review text: For solving the equation \(f(x)= 0\), where \(f\) is a Fréchet-differentiable operator defined on a subset \(D\) of a Banach space \(X\) with values in a Banach space \(Y\), the authors consider secant-type methods based on the concept of nondiscrete mathematical induction. Using Lipschitz and center-Lipschitz conditions on the Fréchet-differential of the involved operator instead of the only Lipschitz condition, new sufficient convergence conditions are derived. The results obtained in this paper extend the applicability of the methods under discussion by providing a new semilocal convergence analysis with the advantages: (1) weaker sufficient convergence conditions; (2) tighter error bounds on the distances involved. The presented results of numerical experiments confirm the theoretical considerations.
reviewed by: Otu Vaarmann
Mathematics Subject Classification ID: 65J15
Mathematics Subject Classification ID: 47J15
zbMATH DE Number: 6110505
zbMATH Keywords: secant-type methods
zbMATH Keywords: Banach space
zbMATH Keywords: nondiscrete mathematical induction
zbMATH Keywords: semilocal convergence
zbMATH Keywords: Traub's method
zbMATH Keywords: consistent approximation
zbMATH Keywords: rate of convergence
zbMATH Keywords: nonlinear operator equation
zbMATH Keywords: error bounds
zbMATH Keywords: numerical experiments
describes a project that uses: NewtonLib
MaRDI profile type: Publication
full work available at URL: s11075-012-9540-7
OpenAlex ID: W2095476212
cites work: Convergence by nondiscrete mathematical induction of a two step secant's method
cites work: A unifying local-semilocal convergence analysis and applications for two-point Newton-like methods in Banach space
cites work: New sufficient convergence conditions for the secant method
cites work: Convergence and Applications of Newton-type Iterations
cites work: Extending the Newton-Kantorovich hypothesis for solving equations
cites work: Convergence conditions for secant-type methods
cites work: Convergence domains of certain iterative methods for solving nonlinear equations
cites work: Q4401507
cites work: Q4818644
cites work: Q4929918
cites work: Optimal Error Bounds for the Newton–Kantorovich Theorem
cites work: Solving a special case of conservative problems by secant-like methods
cites work: Q3943431
cites work: Q5588027
cites work: Q5652137
cites work: Q4060241
cites work: Q4158408
cites work: Sharp error bounds for Newton's process
cites work: Q3343091
cites work: New general convergence theory for iterative processes and its applications to Newton-Kantorovich type theorems
cites work: Convergence ball of a modified secant method with convergence order \(1.839\dots\)
cites work: A Unified Convergence Theory for a Class of Iterative Processes
cites work: Q3293635
cites work: Eine Übertragung der Regula Falsi auf Gleichungen in Banachräumen I
cites work: Ableitungsfreie Verfahren mit höherer Konvergenzgeschwindigkeit
cites work: The Kantorovich Theorem for Newton's Method
cites work: Q5732060
cites work: A convergence theorem for Newton-like methods in Banach spaces
cites work: The majorant method in the theory of newton-kantorovich approximations and the pták error estimates
DOI: 10.1007/S11075-012-9540-7
Recommended article: Extending the applicability of secant methods and nondiscrete induction
Recommended article: Q4729250
Recommended article: Q3605635
Recommended article: Convergence by nondiscrete mathematical induction of a two step secant's method
Recommended article: Q5505279
Recommended article: A new type of recurrence relations for the secant method∗
Recommended article: Relaxed secant-type methods
Recommended article: On a class of secant-like methods for solving nonlinear equations
Recommended article: Q4371320
Recommended article: A new method of secant-like for nonlinear equations
description: scientific article; zbMATH DE number 6110505