Ball convergence theorems for Maheshwari-type eighth-order methods under weak conditions

The convergence order of numerous iterative methods is obtained using derivatives of a higher order, although these derivatives are not involved in the methods. Therefore, these methods cannot be used to solve equations with functions that do not have such high-order derivatives, since their convergence is not guaranteed. The convergence in this paper is shown, relying only on the first derivative. That is how we expand the applicability of some popular methods.

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Ball Convergence for Two Optimal Eighth-Order Methods Using Only the First Derivative

International Journal of Applied and Computational Mathematics ◽

10.1007/s40819-016-0196-1 ◽

2016 ◽

Vol 3 (3) ◽

pp. 2291-2301 ◽

Cited By ~ 1

Author(s):

Ioannis K. Argyros ◽

Munish Kansal ◽

V. Kanwar

Keyword(s):

Eighth Order ◽

First Derivative ◽

Ball Convergence

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Ball convergence theorems and the convergence planes of an iterative method for nonlinear equations

SeMA Journal ◽

10.1007/s40324-015-0047-8 ◽

2015 ◽

Vol 71 (1) ◽

pp. 39-55 ◽

Cited By ~ 11

Author(s):

Á. Alberto Magreñán ◽

Ioannis K. Argyros

Keyword(s):

Iterative Method ◽

Nonlinear Equations ◽

Convergence Theorems ◽

Ball Convergence

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Ball convergence for a three-point method with optimal convergence order eight under weak conditions

Asian-European Journal of Mathematics ◽

10.1142/s1793557120500485 ◽

2018 ◽

Vol 13 (02) ◽

pp. 2050048

Author(s):

Ioannis K. Argyros ◽

Munish Kansal ◽

V. Kanwar

Keyword(s):

Nonlinear Equation ◽

Error Bounds ◽

Eighth Order ◽

Numerical Examples ◽

Optimal Convergence ◽

New Class ◽

First Derivative ◽

Ball Convergence ◽

Computable Error Bounds ◽

Local Convergence Analysis

We present a local convergence analysis of an optimal eighth-order family of Ostrowski like methods for approximating a locally unique solution of a nonlinear equation. Earlier studies [T. Lotfi, S. Sharifi, M. Salimi and S. Siegmund, A new class of three-point methods with optimal convergence and its dynamics, Numer. Algorithms 68 (2015) 261–288.] have shown convergence of these methods under hypotheses up to the eighth derivative of the function although only the first derivative appears in the method. In this study, we expand the applicability of these methods using only hypotheses up to the first derivative of the function. By this way the applicability of these methods is extended under weaker hypotheses. Moreover, the radius of convergence and computable error bounds on the distances involved are also given in this study. Numerical examples are also presented in this study.

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