Browsing by Author "Argyros, I.K."

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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 Broyden-type Banach to Hilbert space scheme for solving equations
(International Publications internationalpubls@yahoo.com, 2019) Argyros, I.K.; George, S.
We present a new semi-local convergence analysis for an inverse free Broyden-type Banach to Hilbert space scheme (BTS) in order to approximate a locally unique solution of an equation. The analysis is based on a center-Lipschitz-type condition and our idea of the restricted convergence region. The operators involved have regularly continuous divided differences. This way we provide, weaker sufficient semi-local convergence conditions, tighter error bounds, and a more precise information on the location of the solution. Hence, our approach extends the applicability of BTS under the same hypotheses as before. © 2019, International Publications. All rights reserved.
A class of derivative free schemes for solving nondifferentiable Banach space valued equations
(Springer Science and Business Media B.V., 2024) Argyros, I.K.; George, S.
In this study, we have extended the applicability of two-step methods with non-differentiable parts for solving nonlinear equations defined in Banach spaces. The convergence analysis uses conditions weaker than the ones in earlier studies. Other advantages include computable a priori error distances based on generalized conditions, an extended region of convergence as well as a better knowledge of the isolation for the solutions. The earlier results use assumptions on the eighth derivative of the main operator. But there are no derivatives on the schemes. Moreover, the previous results cannot be used for nondifferentiable equations although the schemes may converge. Numerical examples validate further our approach. © The Author(s), under exclusive licence to The Forum D’Analystes 2024.
A comparison between two competing sixth convergence order algorithms under the same set of conditions
(SINUS Association, 2021) Argyros, G.; Argyros, M.; Argyros, I.K.; George, S.
There is a plethora of algorithms of the same convergence order for generating a sequence approximating a solution of an equation involving Banach space operators. But the set of convergence criteria is not the same in general. This makes the comparison between them hard and only numerically. Moreover, the convergence is established using Taylor series and by assuming the existence of high order derivatives not even appearing on these algorithms. Furthermore, no computable error estimates, uniqueness for the solution results or a ball of convergence is given. We address all these problems by using only the first derivative that actually appears on these algorithms and under the same set of convergence conditions. Our technique is so general that it can be used to extend the applicability of other algorithms along the same lines. © 2021, SINUS Association. All rights reserved.
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.
A Comparison Study of the Classical and Modern Results of Semi-Local Convergence of Newton–Kantorovich Iterations-II
(MDPI, 2022) Regmi, S.; Argyros, I.K.; George, S.; Argyros, M.I.
This article is an independently written continuation of an earlier study with the same title [Mathematics, 2022, 10, 1225] on the Newton Process (NP). This process is applied to solve nonlinear equations. The complementing features are: the smallness of the initial approximation is expressed explicitly in turns of the Lipschitz or Hölder constants and the convergence order 1 + p is shown for p ∈ (0, 1]. The first feature becomes attainable by further simplifying proofs of convergence criteria. The second feature is possible by choosing a bit larger upper bound on the smallness of the initial approximation. This way, the completed convergence analysis is finer and can replace the classical one by Kantorovich and others. A two-point boundary value problem (TPBVP) is solved to complement this article. © 2022 by the authors. Licensee MDPI, Basel, Switzerland.
A New Parameter Choice Strategy for Lavrentiev Regularization Method for Nonlinear Ill-Posed Equations
(MDPI, 2022) George, S.; Padikkal, J.; Remesh, K.; Argyros, I.K.
In this paper, we introduced a new source condition and a new parameter-choice strategy which also gives the known best error estimate. To obtain the results we used the assumptions used in earlier studies. Further, we studied the proposed new parameter-choice strategy and applied it to the method (in the finite-dimensional setting) considered in George and Nair (2017). © 2022 by the authors.
A procedure for increasing the convergence order of iterative methods from p to 5p for solving nonlinear system
(Academic Press Inc., 2025) George, S.; M, M.; Gopal, M.; Godavarma, C.; Argyros, I.K.
In this paper, we propose a procedure to obtain an iterative method that increases its convergence order from p to 5p for solving nonlinear systems. Our analysis is given in more general Banach space settings and uses assumptions on the derivative of the involved operator only up to order max?{k,2}. Here, k is the order of the highest derivative used in the convergence analysis of the iterative method with convergence order p. A particular case of our analysis includes an existing fifth-order method and improves its applicability to more problems than the problems covered by the method's analysis in earlier study. © 2024
A study on the local convergence of a Steffensen-King-type iterative method
(Cambridge Scientific Publishers jonathan.mckenna@touchbriefings.com, 2017) Argyros, I.K.; George, S.
We study the local convergence of a Steffensen-King-type method to approximate a locally unique solution of a nonlinear equation. Earlier studies such as [14, 15, 17] show convergence under hypotheses on the third derivative or even higher. The convergence in this study is shown under hypotheses on the first derivative. Hence, the applicability of the method is expanded. Finally, numerical examples are also provided in this study. © CSP - Cambridge, UK; I & S - Florida, USA, 2017.
A unified local convergence for jarratt-type methods in banach space under weak conditions
(Chiang Mai University, 2015) Argyros, I.K.; George, S.
We present a unified local convergence analysis for Jarratt-type methods in order to approximate a solution of a nonlinear equation in a Banach space setting. Our methods include the Jarratt;Inverse free Jarratt; super-Halley and other high order methods. The convergence ball and error estimates are given for these methods under the same conditions. Numerical examples are also provided in this study. © 2015 by the Mathematical Association of Thailand. All rights reserved.
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 algorithm with feasible inexact projections for solving constrained generalized equations
(John Wiley and Sons Ltd, 2025) Regmi, S.; Argyros, I.K.; George, S.
The goal of this article is to design a more flexible algorithm than the ones used previously for solving constrained generalized equations. It turns out that the new algorithm even if specialized provides a finer error analysis with advantages: larger radius of convergence; tighter upper error bounds on the distances; and a more precise information on the isolation of the solution. Moreover, the same advantages exist even if the generalized equation reduces to a nonlinear equation. These advantages are obtained under the same computational cost, since the new parameters and majorant functions are special cases of the ones used in earlier studies. Applications complement the theoretical results. © 2024 John Wiley & Sons Ltd.
An apriori parameter choice strategy and a fifth order iterative scheme for Lavrentiev regularization method
(Institute for Ionics, 2023) George, S.; Saeed, M.; Argyros, I.K.; Padikkal, J.
In this paper, we propose a new source condition and introduce a new apriori parameter choice strategy for Lavrentiev regularization method for nonlinear ill-posed operator equation involving a monotone operator in the setting of a Hilbert space. Also, a fifth order iterative method is being proposed for approximately solving Lavrentiev regularized equation. A numerical example is illustrated to demonstrate the performance of the method. © 2022, The Author(s) under exclusive licence to Korean Society for Informatics and Computational Applied Mathematics.
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.
An Improved Convergence Analysis of a Multi-Step Method with High-Efficiency Indices
(Multidisciplinary Digital Publishing Institute (MDPI), 2025) George, S.; Gopal, M.; Bhide, S.; Argyros, I.K.
A multi-step method introduced by Raziyeh and Masoud for solving nonlinear systems with convergence order five has been considered in this paper. The convergence of the method was studied using Taylor series expansion, which requires the function to be six times differentiable. However, our convergence study does not depend on the Taylor series. We use the derivative of F up to two only in our convergence analysis, which is presented in a more general Banach space setting. Semi-local analysis is also discussed, which was not given in earlier studies. Unlike in earlier studies (where two sets of assumptions were used), we used the same set of assumptions for semi-local analysis and local convergence analysis. We discussed the dynamics of the method and also gave some numerical examples to illustrate theoretical findings. © 2025 by the authors.
An improved semi-local convergence analysis for a three point method of order 1.839 in banach space
(International Publications internationalpubls@yahoo.com, 2015) Argyros, I.K.; Padikkal, P.; George, S.
We present a new semi-local convergence analysis for a three point method of order 1.839 in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. The advantages of the new approach over earlier ones such as [18] are: weaker and easier to verify convergence conditions. Moreover the radius of convergence is given in an explicit form. Furthermore, uniqueness results are also presented for the first time as far as we know in this paper. Finally, numerical example illustrating the theoretical results is given.
An improved semilocal convergence analysis for the Halley's method
(International Publications internationalpubls@yahoo.com, 2018) Argyros, I.K.; Khattri, S.K.; George, S.
We expand the applicability of the Halley's method for approximating a locally unique solution of nonlinear equations in a Banach space setting. Our majorizing sequences are finer than in earlier studies such as [1, 3, 4, 5, 6, 7, 8, 9, 10, 11, 19, 20, 21, 23] and furthermore developed convergence criteria can be weaker. Finally numerical work is reported that compares favorably to the existing approaches in the literature [6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 24, 25, 26, 28]. © 2018 International Publications. All rights reserved.
An inverse free broyden’s method for solving equations
(International Publications internationalpubls@yahoo.com, 2020) Argyros, I.K.; George, S.
Based on a center-Lipschitz-type condition and our idea of the restricted convergence domain, we present a new semi-local convergence analysis for an inverse free Broyden’s method (BM) in order to approximate a locally unique solution of an equation in a Hilbert space setting. The operators involved have regularly continuous divided differences. This way we provide, weaker sufficient semi-local convergence conditions, tighter error bounds, and a more precise information on the location of the solution are provided in this study. Hence, our approach extends the applicability of BM under the same hypotheses as before. Finally, we consider some special cases. © 2020, International Publications. All rights reserved.
The asymptotic mesh independence principle of Newton's method under weaker conditions
(2016) Argyros, I.K.; Sheth, S.M.; Younis, R.M.; George, S.
We present a new asymptotic mesh independence principle of Newton's method for discretized nonlinear operator equations. Our hypotheses are weaker than in earlier studies such as [1], [8]-[12]. This way we extend the applicability of the mesh independence principle which asserts that the behavior of the discretized version is asymptotically the same as that of the original iteration and consequently, the number of steps required by the two processes to converge within a given tolerance is essentially the same. The results apply to solve a boundary value problem that cannot be solved with the earlier hypotheses given in [12]. 2016 International Publications. All rights reserved.