scholarly journals On the Semi-Local Convergence of an Ostrowski-Type Method for Solving Equations

Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2281
Author(s):  
Christopher I. Argyros ◽  
Ioannis K. Argyros ◽  
Janak Joshi ◽  
Samundra Regmi ◽  
Santhosh George
Keyword(s):  
Banach Space ◽  
Convergence Analysis ◽  
Local Convergence ◽  
Crucial Role ◽  
Quantum Physics ◽  
General Nature ◽  
Type Method ◽  
Physical Systems ◽  
Solving Equations ◽  
Abstract Spaces

Symmetries play a crucial role in the dynamics of physical systems. As an example, microworld and quantum physics problems are modeled on principles of symmetry. These problems are then formulated as equations defined on suitable abstract spaces. Then, these equations can be solved using iterative methods. In this article, an Ostrowski-type method for solving equations in Banach space is extended. This is achieved by finding a stricter set than before containing the iterates. The convergence analysis becomes finer. Due to the general nature of our technique, it can be utilized to enlarge the utilization of other methods. Examples finish the paper.

Semi-local convergence of a Newton-like method for solving equations with a singular derivative

2018 ◽  
Vol 27 (1) ◽  
pp. 01-08
Author(s):  
IOANNIS K. ARGYROS ◽  
◽  
GEORGE SANTHOSH ◽  
Keyword(s):  
Banach Space ◽  
Convergence Analysis ◽  
Real Line ◽  
Local Convergence ◽  
Approximate Solutions ◽  
Convergence Domain ◽  
Solving Equations ◽  
Solutions Of Equations ◽  
Local Convergence Analysis ◽  
Special Case

We present a semi-local convergence analysis for a Newton-like method to approximate solutions of equations when the derivative is not necessarily non-singular in a Banach space setting. In the special case when the equation is defined on the real line the convergence domain is improved for this method when compared to earlier results. Numerical results where earlier results cannot apply but the new results can apply to solve nonlinear equations are also presented in this study.

scholarly journals On the Semi-Local Convergence of a Traub-Type Method for Solving Equations

Foundations ◽  
2022 ◽  
Vol 2 (1) ◽  
pp. 114-127
Author(s):  
Samundra Regmi ◽  
Christopher I. Argyros ◽  
Ioannis K. Argyros ◽  
Santhosh George
Keyword(s):  
Banach Space ◽  
Local Convergence ◽  
Type Method ◽  
Lipschitz Constants ◽  
Solving Equations ◽  
Original Domain ◽  
Theoretical Results

The celebrated Traub’s method involving Banach space-defined operators is extended. The main feature in this study involves the determination of a subset of the original domain that also contains the Traub iterates. In the smaller domain, the Lipschitz constants are smaller too. Hence, a finer analysis is developed without the usage of additional conditions. This methodology applies to other methods. The examples justify the theoretical results.

On the Ostrowski Method for Solving Equations

2021 ◽  
Vol 2 ◽  
pp. 3
Author(s):  
Ioannis K. Argyros ◽  
Santhosh George ◽  
Christopher I. Argyros
Keyword(s):  
Banach Space ◽  
Convergence Analysis ◽  
Local Convergence ◽  
Numerical Experiments ◽  
Convergence Domain ◽  
Lipschitz Constants ◽  
Solving Equations ◽  
Ostrowski’S Method

In this paper, we revisited the Ostrowski's method for solving Banach space valued equations. We developed a technique  to determine a subset of the original convergence domain and using this new Lipschitz constants derived. These constants are at least as tight as the earlier ones leading to a finer convergence analysis in both the semi-local and the local convergence case. These techniques are very general, so they can be used to extend the applicability of other methods without additional hypotheses. Numerical experiments complete this study.

scholarly journals Convergence Criteria of Three Step Schemes for Solving Equations

Mathematics ◽  
2021 ◽  
Vol 9 (23) ◽  
pp. 3106
Author(s):  
Samundra Regmi ◽  
Christopher I. Argyros ◽  
Ioannis K. Argyros ◽  
Santhosh George
Keyword(s):  
Banach Space ◽  
Euclidean Space ◽  
Convergence Analysis ◽  
Local Convergence ◽  
Dimensional Euclidean Space ◽  
Convergence Criteria ◽  
Iterative Schemes ◽  
Finite Dimensional ◽  
Taylor Expansions ◽  
Solving Equations

We develop a unified convergence analysis of three-step iterative schemes for solving nonlinear Banach space valued equations. The local convergence order has been shown before to be five on the finite dimensional Euclidean space assuming Taylor expansions and the existence of the sixth derivative not on these schemes. So, the usage of them is restricted six or higher differentiable mappings. But in our paper only the first Frèchet derivative is utilized to show convergence. Consequently, the scheme is expanded. Numerical applications are also given to test convergence.

scholarly journals Local convergence comparison between two novel sixth order methods for solving equations

2019 ◽  
Vol 18 (1) ◽  
pp. 5-19
Author(s):  
Ioannis K. Argyros ◽  
Santhosh George
Keyword(s):  
Banach Space ◽  
Convergence Analysis ◽  
Local Convergence ◽  
Sixth Order ◽  
Convergence Order ◽  
Convergence Criteria ◽  
Numerical Examples ◽  
First Derivative ◽  
Solving Equations ◽  
Local Convergence Analysis

Abstract The aim of this article is to provide the local convergence analysis of two novel competing sixth convergence order methods for solving equations involving Banach space valued operators. Earlier studies have used hypotheses reaching up to the sixth derivative but only the first derivative appears in these methods. These hypotheses limit the applicability of the methods. That is why we are motivated to present convergence analysis based only on the first derivative. Numerical examples where the convergence criteria are tested are provided. It turns out that in these examples the criteria in the earlier works are not satisfied, so these results cannot be used to solve equations but our results can be used.

Local convergence of deformed Jarratt-type methods in Banach space without inverses

2016 ◽  
Vol 09 (01) ◽  
pp. 1650015
Author(s):  
Ioannis K. Argyros ◽  
Santhosh George
Keyword(s):  
Banach Space ◽  
Convergence Analysis ◽  
Local Convergence ◽  
Computational Cost ◽  
Fréchet Derivative ◽  
Convergence Conditions ◽  
Type Method ◽  
Frechet Derivative ◽  
The Third ◽  
Local Convergence Analysis

We present a local convergence analysis for the Jarratt-type method of high convergence order 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 use hypotheses up to the third Fréchet-derivative. Hence, the applicability of these methods is expanded under weaker hypotheses and less computational cost for the constants involved in the convergence analysis. Numerical examples are also provided in this study.

scholarly journals Unified Convergence Analysis of Two-Step Iterative Methods for Solving Equations

2021 ◽  
Vol 1 (2) ◽  
pp. 68-85
Author(s):  
Ioannis K. Argyros
Keyword(s):  
Banach Space ◽  
Convergence Analysis ◽  
Iterative Methods ◽  
Local Convergence ◽  
General Setting ◽  
Convergence Criteria ◽  
Recurrent Functions ◽  
Solving Equations ◽  
Order Four ◽  
Methods Numerical

In this paper we consider unified convergence analysis of two-step iterative methods for solving equations in the Banach space setting. The convergence order four was shown using Taylor expansions requiring the existence of the fifth derivative not on this method. But these hypotheses limit the utilization of it to functions which are at least five times differentiable although the method may converge. As far as we know no semi-local convergence has been given in this setting. Our goal is to extend the applicability of this method in both the local and semi-local convergence case and in the more general setting of Banach space valued operators. Moreover, we use our idea of recurrent functions and conditions only on the first derivative and divided differences which appear on the method. This idea can be used to extend other high convergence multipoint and multistep methods. Numerical experiments testing the convergence criteria complement this study.

scholarly journals On the Convergence Region of Multi-step Chebyshev-Halley-type Schemes for Solving Equations

2019 ◽  
pp. 187-207
Author(s):  
Ioannis K. Argyros ◽  
Santhosh George
Keyword(s):  
Banach Space ◽  
Convergence Analysis ◽  
Local Convergence ◽  
Small Region ◽  
Computational Effort ◽  
Convergence Region ◽  
Numerical Examples ◽  
Solving Equations ◽  
Original Region ◽  
Local Convergence Analysis

The aim of this article is to extend the convergence region of certain multi-step Chebyshev-Halley-type schemes for solving Banach space valued nonlinear equations. In particular, we find an at least as small region as the region of the operator involved containing the iterates. This way the majorant functions are tighter than the ones related to the original region, leading to a finer local as well as a semi-local convergence analysis under the same computational effort. Numerical examples complete this article.

scholarly journals Extending the Applicability of a Two-Step Chord-Type Method for Non-Differentiable Operators

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 804
Author(s):  
Ioannis K. Argyros ◽  
Neha Gupta ◽  
J. P. Jaiswal
Keyword(s):  
Approximate Solution ◽  
Convergence Analysis ◽  
Banach Spaces ◽  
Recurrence Relation ◽  
Convergence Theorem ◽  
Local Convergence ◽  
Existence And Uniqueness ◽  
Numerical Illustration ◽  
Type Method ◽  
Local Convergence Analysis

The semi-local convergence analysis of a well defined and efficient two-step Chord-type method in Banach spaces is presented in this study. The recurrence relation technique is used under some weak assumptions. The pertinency of the assumed method is extended for nonlinear non-differentiable operators. The convergence theorem is also established to show the existence and uniqueness of the approximate solution. A numerical illustration is quoted to certify the theoretical part which shows that earlier studies fail if the function is non-differentiable.

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