Browsing by Author "Argyros, C.I."

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A BALL COMPARISON BETWEEN EXTENDED MODIFIED JARRATT METHODS UNDER THE SAME SET OF CONDITIONS FOR SOLVING EQUATIONS AND SYSTEMS OF EQUATIONS
(Petrozavodsk State University, 2022) Argyros, I.K.; George, S.; Argyros, C.I.
In this paper, we compare the radii of convergence of Jarratt-type methods under the same set of conditions for solving nonlinear equations and systems of equations. Our convergence analysis is based on the first Fréchet derivative that only appears on the method. Numerical examples where the theoretical results are tested complete the paper © Petrozavodsk State University, 2022
A Comparison Between Two Ostrowski-type Fourth Order Methods for Solving Equations Under the Same Set of Conditions
(International Publications, 2022) Argyros, I.K.; George, S.; Argyros, C.I.
In this study, we compare two Ostrowski-type fourth order methods for solving equations under the same set of conditions Our convergence analysis is based on the first Fréchet derivative that only appears on the method. Earlier studies use up to the fifth derivative to show convergence. The conditions limit their usage, especially since these derivatives are not on these methods. Numerical examples where the theoretical results are tested complete the paper. © 2022, International Publications. All rights reserved.
A Comparison Study of the Classical and Modern Results of Semi-Local Convergence of Newton-Kantorovich Iterations
(MDPI, 2022) Regmi, S.; Argyros, I.K.; George, S.; Argyros, C.I.
There are a plethora of semi-local convergence results for Newton’s method (NM). These results rely on the Newton–Kantorovich criterion. However, this condition may not be satisfied even in the case of scalar equations. For this reason, we first present a comparative study of established classical and modern results. Moreover, using recurrent functions and at least as small constants or majorant functions, a finer convergence analysis for NM can be provided. The new constants and functions are specializations of earlier ones; hence, no new conditions are required to show convergence of NM. The technique is useful on other iterative methods as well. Numerical examples complement the theoretical results. © 2022 by the authors. Licensee MDPI, Basel, Switzerland.
Advances in Nonlinear Variational Inequalities Volume 25 (2022), Number 1, 49-58 Comparing and Extending Two Fourth Order Methods Under the Same Hypotheses for Equations
(International Publications, 2022) Argyros, I.K.; George, S.; Argyros, C.I.
We compare and extend two fourth order methods for nonlinear equations. Our convergence analysis used assumptions only on the first derivative. Earlier studies have used hypotheses up to the fifth derivative, limiting the applicability of the method. Numerical examples complete the article. © 2022, International Publications. All rights reserved.
An extended radius of convergence comparison between two sixth order methods under general continuity for solving equations
(DergiPark, 2022) Regmi, S.; Argyros, I.K.; George, S.; Argyros, C.I.
In this paper we compare the radius of convergence of two sixth convergence order methods for solving nonlinear equation. We present the local convergence analysis not given before, which is based on the first Frechet derivative that only appears on the method. Numerical examples where the theoretical results are tested complete the paper. © 2022, DergiPark. All rights reserved.
Contemporary algorithms: Theory and applications
(Nova Science Publishers, Inc., 2023) Argyros, C.I.; Regmi, S.; Argyros, I.K.; George, S.
The book provides different avenues to study algorithms. It also brings new techniques and methodologies to problem solving in computational Sciences, Engineering, Scientific Computing and Medicine (imaging, radiation therapy) to mention a few. A plethora of algorithms which are universally applicable is presented on a sound analytical way. The chapters are written independently of each other, so they can be understood without reading earlier chapters. But some knowledge of Analysis, Linear Algebra and some Computing experience is required. The organization and content of the book cater to senior undergraduate, graduate students, researchers, practitioners, professionals and academicians in the aforementioned disciplines. It can also be used as a reference book and includes numerous references and open problems. Â© 2023 by Nova Science Publishers, Inc. All rights reserved.
Convergence criteria of three step schemes for solving equations
(MDPI, 2021) Regmi, S.; Argyros, C.I.; Argyros, I.K.; George, S.
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. © 2021 by the authors. Licensee MDPI, Basel, Switzerland.
Extended Convergence for m−step Iterative Methods and Applications
(International Publications, 2022) Argyros, I.K.; George, S.; Argyros, C.I.
We present a semi-local convergence analysis of m−step iterative methods in order to approximate a locally unique solution for Banach space valued equations. Our analysis extends the applicability of these methods. Using the center-Lipschitz con-dition, we determine a more precise domain containing the iterates leading to at least as tight Lipschitz constants as well as a finer semi-local convergence analysis than in earlier studies. Numerical examples are also presented, where the convergence criteria are tested and compared favorably to existing ones. © 2022, International Publications. All rights reserved.
Extended convergence of a sixth order scheme for solving equations under ω-continuity conditions
(Sciendo, 2022) Regmi, S.; Argyros, C.I.; Argyros, I.K.; George, S.
The applicability of an efficient sixth convergence order scheme is extended for solving Banach space valued equations. In previous works, the seventh derivative has been used not appearing on the scheme. But we use only the first derivative that appears on the scheme. Moreover, bounds on the error distances and results on the uniqueness of the solution are provided (not given in earlier works) based on ω-continuity conditions. Numerical examples complete this article. © 2021 Samundra Regmi et al., published by Sciendo.
EXTENDED LOCAL CONVERGENCE AND COMPARISONS FOR TWO THREE-STEP JARRATT-TYPE METHODS under THE SAME CONDITIONS
(Institute of Mathematics. Polish Academy of Sciences, 2022) Argyros, I.K.; George, S.; Argyros, C.I.
We extend and compare two three-step Jarratt-type methods for solving a nonlinear equation under the same conditions. Our convergence analysis is based on the first Fréchet derivative that only appears in the method. Numerical examples where the theoretical results are tested complete the paper. © Instytut Matematyczny PAN, 2022.
Extended Newton’s Method With Applications To Interior Point Algorithms Of Mathematical Programming∗
(Tsing Hua University, 2022) Regmi, S.; Argyros, C.I.; Argyros, I.K.; George, S.
We use a weaker Newton—Kantorovich theorem for solving equations, introduced in [3] to analyze interior point methods. This way our approach extends earlier works in [6] on Newton’s method and interior point algorithms. © 2022, Tsing Hua University. All rights reserved.
Extended Semilocal Convergence for Chebyshev-Halley-Type Schemes for Solving Nonlinear Equations under Weak Conditions
(Universal Wiser Publisher, 2023) Regmi, S.; Argyros, I.K.; George, S.; Argyros, C.I.
The application of the Chebyshev-Halley type scheme for nonlinear equations is extended with no additional conditions. In particular, the purpose of this study is two folds. The proof of the semi-local convergence analysis is based on the recurrence relation technique in the first fold. In the second fold, the proof relies on majorizing sequences. Iterates are shown to belong to a larger domain resulting in tighter Lipschitz constants and a finer convergence analysis than in earlier works. The convergence order of these methods is at least three. The numerical example further validates the theoretical results. © 2023 Samundra Regmi, et al.
EXTENDING THE APPLICABILITY OF A SEVENTH-ORDER METHOD FOR EQUATIONS UNDER GENERALIZED CONDITIONS
(Institute of Mathematics. Polish Academy of Sciences, 2023) Regmi, S.; Argyros, I.K.; George, S.; Argyros, C.I.
We extend the applicability of a seventh-order method for solving Banach space-valued equations. This is achieved by using generalized conditions on the first derivative which only appears in the method. Earlier works use conditions up to the eighth derivative to establish convergence. Our technique is very general and can be used to extend the applicability of other methods along the same lines. © Instytut Matematyczny PAN, 2023.
Extending the Convergence of Two Similar Sixth Order Schemes for Solving Equations under Generalized Conditions
(Universal Wiser Publisher, 2021) Argyros, I.K.; George, S.; Argyros, C.I.
The applicability of two competing efficient sixth convergence order schemes is extended for solving Banach space valued equations. In previous works, the seventh derivative has been used not appearing on the schemes. But we use only the first derivative that appears on the scheme. Moreover, bounds on the error distances and results on the uniqueness of the solution are provided not given in the earlier works based on ?-continuity conditions. Our technique extends other schemes analogously, since it is so general. Numerical examples complete this work. © 2021 Ioannis K. Argyros, et al.
Extending the Traub Theory for Solving Nonlinear Equations
(Universal Wiser Publisher, 2022) Regmi, S.; Argyros, I.K.; George, S.; Argyros, C.I.
In this paper, we extend the Traub theory for solving nonlinear equation. The extension is based on recurrent functions, the center Lipschitz condition and the notion of the restricted convergence domain. Numerical examples indicate that the new results can be utilized to solve nonlinear equations, but not earlier ones. © 2022 Ioannis K. Argyros, et al.
Hybrid Newton-like Inverse Free Algorithms for Solving Nonlinear Equations
(Multidisciplinary Digital Publishing Institute (MDPI), 2024) Argyros, I.K.; George, S.; Regmi, S.; Argyros, C.I.
Iterative algorithms requiring the computationally expensive in general inversion of linear operators are difficult to implement. This is the reason why hybrid Newton-like algorithms without inverses are developed in this paper to solve Banach space-valued nonlinear equations. The inverses of the linear operator are exchanged by a finite sum of fixed linear operators. Two types of convergence analysis are presented for these algorithms: the semilocal and the local. The Fréchet derivative of the operator on the equation is controlled by a majorant function. The semi-local analysis also relies on majorizing sequences. The celebrated contraction mapping principle is utilized to study the convergence of the Krasnoselskij-like algorithm. The numerical experimentation demonstrates that the new algorithms are essentially as effective but less expensive to implement. Although the new approach is demonstrated for Newton-like algorithms, it can be applied to other single-step, multistep, or multipoint algorithms using inverses of linear operators along the same lines. © 2024 by the authors.
Kantorovich-type results for generalized equations with applications
(Springer Science and Business Media B.V., 2022) Regmi, S.; Argyros, I.K.; George, S.; Argyros, C.I.
Kantorovich-type results for generalized equations are extended with no additional conditions using Newton procedures. Iterates are shown to belong in a smaller domain resulting to tighter Lipschitz constants and a finer convergence analysis than in earlier works. © 2022, The Author(s), under exclusive licence to The Forum D’Analystes.
On the local convergence and comparison between two novel eighth convergence order schemes for solving nonlinear equations
(Cambridge Scientific Publishers, 2021) Regmi, S.; Argyros, I.K.; George, S.; Argyros, C.I.
We compare two eighth order schemes for solving nonlinear equations involving Banach space valued equations. This is done by using assumptions only on the first derivative that does appear on the schemes, whereas in earlier works up to the ninth derivative (not on the scheme) are used to establish the convergence. Our technique is so general that it can be used to extend the usage of other schemes along the same lines. © 2021. All Rights Reserved.
On the local convergence of two novel schemes of convergence order eight for solving equations: An extension
(International Publications, 2021) Argyros, I.K.; George, S.; Argyros, C.I.
We extend the applicability of two eighth order schemes for solving nonlinear equations for Banach space valued equations.This is done by using assumptions only on the first derivative that does appear on the schemes, whereas in earlier works up to the ninth derivative (not on the scheme) are used to establish the convergence. Our technique is so general that it can be used to extend the usage of other schemes along the same lines. © 2021, International Publications. All rights reserved.
On the semi-local convergence of an ostrowski-type method for solving equations
(MDPI, 2021) Argyros, C.I.; Argyros, I.K.; Joshi, J.; Regmi, S.; George, S.
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. © 2021 by the authors. Licensee MDPI, Basel, Switzerland.