Semilocal convergence of a sixth-order method in Banach spaces

2012 ◽  
Vol 61 (3) ◽  
pp. 413-427 ◽  
Author(s):  
Lin Zheng ◽  
Chuanqing Gu
Keyword(s):  
Banach Spaces ◽  
Semilocal Convergence ◽  
Sixth Order ◽  
Order Method

On a sixth-order Jarratt-type method in Banach spaces

2015 ◽  
Vol 08 (04) ◽  
pp. 1550065 ◽  
Author(s):  
Ioannis K. Argyros ◽  
Santhosh George
Keyword(s):  
Banach Spaces ◽  
Semilocal Convergence ◽  
Fréchet Derivative ◽  
Sixth Order ◽  
Convergence Conditions ◽  
Type Method ◽  
Frechet Derivative ◽  
The Third ◽  
Local Convergence Analysis ◽  
Jarratt Method

We present a local convergence analysis of a sixth-order Jarratt-type method in order to approximate a solution of a nonlinear equation in a Banach space. Our sufficient convergence conditions involve only hypotheses on the first Fréchet-derivative of the operator involved. Earlier studies such as [X. Wang, J. Kou and C. Gu, Semilocal convergence of a sixth-order Jarratt method in Banach spaces, Numer. Algorithms 57 (2011) 441–456.] require hypotheses up to the third Fréchet-derivative. Numerical examples are also provided in this study.

scholarly journals Local convergence of a multi-step high order method with divided differences under hypotheses on the first derivative

2017 ◽  
Vol 16 (1) ◽  
pp. 41-50
Author(s):  
Ioannis K. Argyros ◽  
Santhosh George
Keyword(s):  
Banach Spaces ◽  
Local Convergence ◽  
Sixth Order ◽  
Divided Differences ◽  
Order Method ◽  
Step Method ◽  
Convergence Conditions ◽  
Numerical Examples ◽  
High Order Method ◽  
First Derivative

AbstractThis paper is devoted to the study of a multi-step method with divided differences for solving nonlinear equations in Banach spaces. In earlier studies, hypotheses on the Fréchet derivative up to the sixth order of the operator under consideration is used to prove the convergence of the method. That restricts the applicability of the method. In this paper we extended the applicability of the sixth-order multi-step method by using only hypotheses on the first derivative of the operator involved. Our convergence conditions are weaker than the conditions used in earlier studies. Numerical examples where earlier results cannot be applied to solve equations but our results can be applied are also given in this study.

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