scholarly journals Ball convergence for combined three-step methods under generalized conditions in Banach space

2020 ◽  
Vol 65 (1) ◽  
pp. 127-137
Author(s):  
Ioannis K. Argyros ◽  
◽  
Ramandeep Behl ◽  
Daniel Gonzalez ◽  
Sandile S. Motsa ◽  
...  
Keyword(s):  
Banach Space ◽  
Ball Convergence

Ball convergence of the Newton–Gauss method in Banach space

SeMA Journal ◽  
2016 ◽  
Vol 74 (4) ◽  
pp. 429-439 ◽  
Author(s):  
Ioannis K. Argyros ◽  
Janak Raj Sharma ◽  
Deepak Kumar
Keyword(s):  
Banach Space ◽  
Gauss Method ◽  
Ball Convergence

scholarly journals Ball convergence for Traub-Steffensen like methods in Banach space

2015 ◽  
Vol 53 (2) ◽  
pp. 3-16
Author(s):  
Ioannis K. Argyros ◽  
Santhosh George
Keyword(s):  
Banach Space ◽  
Fréchet Derivative ◽  
Not Given ◽  
Frechet Derivative ◽  
The Third ◽  
Taylor Expansions ◽  
Lipschitz Constants ◽  
Ball Convergence ◽  
Computable Error Bounds ◽  
Local Convergence Analysis

Abstract We present a local convergence analysis for two Traub-Steffensen-like methods in order to approximate a locally unique solution of an equation in a Banach space setting. In earlier studies such as [16, 23] Taylor expansions and hypotheses up to the third Fréchet-derivative are used. We expand the applicability of these methods using only hypotheses on the first Fréchet derivative. Moreover, we obtain a radius of convergence and computable error bounds using Lipschitz constants not given before. Numerical examples are also presented in this study.

Ball Convergence for a Family of Quadrature-Based Methods for Solving Equations in Banach Space

2017 ◽  
Vol 14 (02) ◽  
pp. 1750017 ◽  
Author(s):  
Ioannis K. Argyros ◽  
Ramandeep Behl ◽  
S. S. Motsa
Keyword(s):  
Banach Space ◽  
Taylor Series Expansion ◽  
Fréchet Derivative ◽  
Dimensional Euclidean Space ◽  
Frechet Derivative ◽  
Space Setting ◽  
Lipschitz Constants ◽  
Solving Equations ◽  
Ball Convergence ◽  
Predictor Corrector

We present a local convergence analysis for a family of quadrature-based predictor–corrector methods in order to approximate a locally unique solution of an equation in a Banach space setting. In earlier studies such as Howk [2016] [Howk, C. L. [2016] “A classs of efficient quadrature-based predictor–corrector methods for solving nonlinear systems, Appl. Math. Comput. 276, 394–406] the [Formula: see text] order of convergence was shown on the [Formula: see text]-dimensional Euclidean space using Taylor series expansion and hypotheses reaching up to the third-order Fréchet-derivative of the operator involved although only the first-order Fréchet-derivative appears in these methods, which restrict the applicability of these methods. In this paper, we expand the applicability of these methods in a Banach space setting and using hypotheses only on the first Fréchet-derivative. Moreover, we provide computable radii of convergence as well as error bounds on the distances involved using Lipschitz constants. Numerical examples are also presented to solve equations in cases where earlier results cannot apply.

scholarly journals Ball Convergence for Combined Three-Step Methods Under Generalized Conditions in Banach Space

Symmetry ◽  
2019 ◽  
Vol 11 (8) ◽  
pp. 1002
Author(s):  
R. A. Alharbey ◽  
Ioannis K. Argyros ◽  
Ramandeep Behl
Keyword(s):  
Applied Sciences ◽  
Banach Space ◽  
Iterative Method ◽  
Applied Mathematics ◽  
Higher Order ◽  
Order Derivative ◽  
Numerical Examples ◽  
First Order ◽  
Seventh Order ◽  
Ball Convergence

Problems from numerous disciplines such as applied sciences, scientific computing, applied mathematics, engineering to mention some can be converted to solving an equation. That is why, we suggest higher-order iterative method to solve equations with Banach space valued operators. Researchers used the suppositions involving seventh-order derivative by Chen, S.P. and Qian, Y.H. But, here, we only use suppositions on the first-order derivative and Lipschitz constrains. In addition, we do not only enlarge the applicability region of them but also suggest computable radii. Finally, we consider a good mixture of numerical examples in order to demonstrate the applicability of our results in cases not covered before.

Ball Convergence for a Multi-Step Harmonic Mean Newton-Like Method in Banach Space

2019 ◽  
Vol 17 (05) ◽  
pp. 1940018 ◽  
Author(s):  
Ramandeep Behl ◽  
Ali Saleh Alshormani ◽  
Ioannis K. Argyros
Keyword(s):  
Banach Space ◽  
Nonlinear Equations ◽  
Local Convergence ◽  
Harmonic Mean ◽  
Order Derivative ◽  
Systems Of Nonlinear Equations ◽  
Numerical Examples ◽  
First Order ◽  
Ball Convergence ◽  
Local Convergence Analysis

In this paper, we present a local convergence analysis of some iterative methods to approximate a locally unique solution of nonlinear equations in a Banach space setting. In the earlier study [Babajee et al. (2015) “On some improved harmonic mean Newton-like methods for solving systems of nonlinear equations,” Algorithms 8(4), 895–909], demonstrate convergence of their methods under hypotheses on the fourth-order derivative or even higher. However, only first-order derivative of the function appears in their proposed scheme. In this study, we have shown that the local convergence of these methods depends on hypotheses only on the first-order derivative and the Lipschitz condition. In this way, we not only expand the applicability of these methods but also proposed the theoretical radius of convergence of these methods. Finally, a variety of concrete numerical examples demonstrate that our results even apply to solve those nonlinear equations where earlier studies cannot apply.

scholarly journals Ball convergence theorem for inexact Newton methods in Banach space

2020 ◽  
Vol 29 (2) ◽  
pp. 113-120
Author(s):  
IOANNIS K. ARGYROS ◽  
GEORGE SANTHOSH
Keyword(s):  
Banach Space ◽  
Nonlinear Equation ◽  
Convergence Theorem ◽  
Local Convergence ◽  
Newton Methods ◽  
Convergence Conditions ◽  
Inexact Newton Methods ◽  
Numerical Examples ◽  
Ball Convergence ◽  
Local Convergence Analysis

We present a local convergence analysis for inexact Newton methods in order to approximate a solution of a nonlinear equation in a Banach space. Our sufficient convergence conditions involve only hypotheses on the first Frèchet-derivative of the operator involved. Numerical examples are also provided in this study.

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