Ball convergence for an eighth order efficient method under weak conditions in Banach spaces

SeMA Journal ◽  
2016 ◽  
Vol 74 (4) ◽  
pp. 513-521 ◽  
Author(s):  
Ioannis K. Argyros ◽  
Santhosh George ◽  
Shobha M. Erappa
Keyword(s):  
Banach Spaces ◽  
Efficient Method ◽  
Eighth Order ◽  
Ball Convergence

scholarly journals Ball Convergence of an Efficient Eighth Order Iterative Method Under Weak Conditions

Mathematics ◽  
2018 ◽  
Vol 6 (11) ◽  
pp. 260 ◽  
Author(s):  
Janak Sharma ◽  
Ioannis Argyros ◽  
Sunil Kumar
Keyword(s):  
Iterative Method ◽  
Iterative Methods ◽  
Higher Order ◽  
High Order ◽  
Convergence Order ◽  
Eighth Order ◽  
First Derivative ◽  
Ball Convergence ◽  
Derivatives Of

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.

scholarly journals Convergence Analysis of Weighted-Newton Methods of Optimal Eighth Order in Banach Spaces

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 198 ◽  
Author(s):  
Janak Sharma ◽  
Ioannis Argyros ◽  
Sunil Kumar
Keyword(s):  
Banach Spaces ◽  
Order Of Convergence ◽  
Higher Order ◽  
Newton Methods ◽  
Eighth Order ◽  
Good Convergence ◽  
Convergence Error ◽  
Uniqueness Of The Solution ◽  
Derivatives Of ◽  
Theoretical Results

We generalize a family of optimal eighth order weighted-Newton methods to Banach spaces and study their local convergence. In a previous study, the Taylor expansion of higher order derivatives is employed which may not exist or may be very expensive to compute. However, the hypotheses of the present study are based on the first Fréchet-derivative only, thereby the application of methods is expanded. New analysis also provides the radius of convergence, error bounds and estimates on the uniqueness of the solution. Such estimates are not provided in the approaches that use Taylor expansions of derivatives of higher order. Moreover, the order of convergence for the methods is verified by using computational order of convergence or approximate computational order of convergence without using higher order derivatives. Numerical examples are provided to verify the theoretical results and to show the good convergence behavior.

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

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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