Browsing by Author "Regmi, S."

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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.
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 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.
Asymptotically Newton-Type Methods without Inverses for Solving Equations
(Multidisciplinary Digital Publishing Institute (MDPI), 2024) Argyros, I.K.; George, S.; Shakhno, S.; Regmi, S.; Havdiak, M.; Argyros, M.I.
The implementation of Newton’s method for solving nonlinear equations in abstract domains requires the inversion of a linear operator at each step. Such an inversion may be computationally very expensive or impossible to find. That is why alternative iterative methods are developed in this article that require no inversion or only one inversion of a linear operator at each step. The inverse of the operator is replaced by a frozen sum of linear operators depending on the Fréchet derivative of an operator. The numerical examples illustrate that for all practical purposes, the new methods are as effective as Newton’s but much cheaper to implement. The same methodology can be used to create similar alternatives to other methods using inversions of linear operators such as divided differences or other linear operators. © 2024 by the authors.
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.
Contemporary Algorithms: Theory and Applications, Volume V
(Nova Science Publishers, Inc., 2025) Argyros, M.I.; Regmi, S.; Argyros, I.K.; George, S.
Due to the explosion of technology as well as scientific and parallel computing, faster computers have become available. This development simply means that new optimization algorithms should be introduced to take advantage of these developments. This book provides different avenues for studying algorithms. It also brings new techniques and methodologies to problem solving in Computational Sciences, Engineering, Scientific Computing and Medicine (imaging, radiation therapy). A plethora of problems from diverse disciplines can be converted using mathematical modeling to an equation defined on suitable abstract spaces usually involving the n-dimensional Euclidean space, Hilbert space, Banach Space or even more general spaces. The solution to these equations is sought in a closed form. But this is only possible in special cases. That is why researchers and practitioners must use algorithms as an alternative. Â© 2025 by Nova Science Publishers, Inc.
Contemporary Algorithms: Theory and Applications. Volume I
(Nova Science Publishers, Inc., 2022) Argyros, C.; Regmi, S.; Argyros, I.K.; George, S.
This 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 are presented in 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 this 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© 2022 by Nova Science Publishers, Inc. All rights reserved.
Contemporary algorithms: Theory and applications. Volume IV
(Nova Science Publishers, Inc., 2024) Argyros, G.I.; Regmi, S.; Argyros, I.K.; George, S.
Due to the explosion of technology, scientific and parallel computing, faster computers have become available. This development simply means that new optimized algorithms should be developed to take advantage of these improvements. There is where this book containing such algorithms comes in handy, with applications in economics, mathematics, biology, chemistry, physics, parallel computing, engineering, and also numerical solution of differential and integral equations. A plethora of problems from diverse disciplines can be converted using mathematical modeling to an equation defined on suitable abstract spaces usually involving the n-dimensional Euclidean space or Hilbert space or Banach Space or even more general spaces. The solution of these equations is sought in closed form. But this is possible only in special cases. That is why researchers and practitioners use algorithms which seem to be the only alternative. This book can be used by senior undergraduate students, graduate students, researchers and practitioners in the aforementioned areas in the classroom or as a reference material. Readers should know the fundamentals of numerical functional analysis, economic theory, and Newtonian physics. Some knowledge of computers and contemporary programming shall be very helpful to the readers. Â© 2024 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.
Convergence of High-Order Derivative-Free Algorithms for the Iterative Solution of Systems of Not Necessarily Differentiable Equations
(Multidisciplinary Digital Publishing Institute (MDPI), 2024) Regmi, S.; Argyros, I.K.; George, S.
In this study, we extended the applicability of a derivative-free algorithm to encompass the solution of operators that may be either differentiable or non-differentiable. Conditions weaker than the ones in earlier studies are employed for the convergence analysis. The earlier results considered assumptions up to the existence of the ninth order derivative of the main operator, even though there are no derivatives in the algorithm, and the Taylor series on the finite Euclidian space restricts the applicability of the algorithm. Moreover, the previous results could not be used for non-differentiable equations, although the algorithm could converge. The new local result used only conditions on the divided difference in the algorithm to show the convergence. Moreover, the more challenging semi-local convergence that had not previously been studied was considered using majorizing sequences. The paper included results on the upper bounds of the error estimates and domains where there was only one solution for the equation. The methodology of this paper is applicable to other algorithms using inverses and in the setting of a Banach space. Numerical examples further validate our approach. © 2024 by the authors.
Direct comparison between two third convergence order schemes for solving equations
(MDPI AG, 2020) Regmi, S.; Argyros, I.K.; George, S.
We provide a comparison between two schemes for solving equations on Banach space. A comparison between same convergence order schemes has been given using numerical examples which can go in favor of either scheme. However, we do not know in advance and under the same set of conditions which scheme has the largest ball of convergence, tighter error bounds or best information on the location of the solution. We present a technique that allows us to achieve this objective. Numerical examples are also given to further justify the theoretical results. Our technique can be used to compare other schemes of the same convergence order. © 2020 by the authors. Licensee MDPI, Basel, Switzerland.
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 Convergence of Three Step Iterative Methods for Solving Equations in Banach Space with Applications
(MDPI, 2022) Regmi, S.; Argyros, I.K.; George, S.; Argyros, C.
Symmetries are vital in the study of physical phenomena such as quantum physics and the micro-world, among others. Then, these phenomena reduce to solving nonlinear equations in abstract spaces. These equations in turn are mostly solved iteratively. That is why the objective of this paper was to obtain a uniform way to study three-step iterative methods to solve equations defined on Banach spaces. The convergence is established by using information appearing in these methods. This is in contrast to earlier works which relied on derivatives of the higher order to establish the convergence. The numerical example completes this paper. © 2022 by the authors.
Extended Kantorovich theory for solving nonlinear equations with applications
(Springer Nature, 2023) Regmi, S.; Argyros, I.K.; George, S.; Argyros, M.
The Kantorovich theory plays an important role in the study of nonlinear equations. It is used to establish the existence of a solution for an equation defined in an abstract space. The solution is usually determined by using an iterative process such as Newton’s or its variants. A plethora of convergence results are available based mainly on Lipschitz-like conditions on the derivatives, and the celebrated Kantorovich convergence criterion. But there are even simple real equations for which this criterion is not satisfied. Consequently, the applicability of the theory is limited. The question there arises: is it possible to extend this theory without adding convergence conditions? The answer is, Yes! This is the novelty and motivation for this paper. Other extensions include the determination of better information about the solution, i.e. its uniqueness ball; the ratio of quadratic convergence as well as more precise error analysis. The numerical section contains a Hammerstein-type nonlinear equation and other examples as applications. © 2023, The Author(s) under exclusive licence to Sociedade Brasileira de Matemática Aplicada e Computacional.
Extended kung–traub methods for solving equations with applications
(MDPI, 2021) Regmi, S.; Argyros, I.K.; George, S.; Magreñán Ruiz, Á.A.; Argyros, M.
Kung and Traub (1974) proposed an iterative method for solving equations defined on the real line. The convergence order four was shown using Taylor expansions, requiring the existence of the fifth derivative not in this method. However, these hypotheses limit the utilization of it to functions that are at least five times differentiable, although the methods 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 difference, which appear in 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. © 2021 by the authors. Licensee MDPI, Basel, Switzerland.
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 applicability of newton’s algorithm with projections for solving generalized equations
(MDPI AG diversity@mdpi.com, 2020) Argyros, M.I.; Argyros, G.I.; Argyros, I.K.; Regmi, S.; George, S.
A new technique is developed to extend the convergence ball of Newton’s algorithm with projections for solving generalized equations with constraints on the multidimensional Euclidean space. This goal is achieved by locating a more precise region than in earlier studies containing the solution on which the Lipschitz constants are smaller than the ones used in previous studies. These advances are obtained without additional conditions. This technique can be used to extend the usage of other iterative algorithms. Numerical experiments are used to demonstrate the superiority of the new results. © 2020 by the authors. Licensee MDPI, Basel, Switzerland.