Faculty Publications
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Item Generalized Three-Step Numerical Methods for Solving Equations in Banach Spaces(MDPI, 2022) Argyros, M.I.; Argyros, I.K.; Regmi, S.; George, S.In this article, we propose a new methodology to construct and study generalized three-step numerical methods for solving nonlinear equations in Banach spaces. These methods are very general and include other methods already in the literature as special cases. The convergence analysis of the specialized methods is been given by assuming the existence of high-order derivatives which are not shown in these methods. Therefore, these constraints limit the applicability of the methods to equations involving operators that are sufficiently many times differentiable although the methods may converge. Moreover, the convergence is shown under a different set of conditions. Motivated by the optimization considerations and the above concerns, we present a unified convergence analysis for the generalized numerical methods relying on conditions involving only the operators appearing in the method. This is the novelty of the article. Special cases and examples are presented to conclude this article. © 2022 by the authors.Item 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.Item 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.Item 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.Item Hybrid Chebyshev-Type Methods for Solving Nonlinear Equations(Multidisciplinary Digital Publishing Institute (MDPI), 2025) Argyros, I.K.; George, S.Chebyshev-type methods have replaced the Chebyshev method in practice for solving nonlinear equations in abstract spaces. These methods are of the same R-order of three. However, they are easier to deal with, since the computationally expensive second derivative of the operator involved does not appear on these methods. However, the invertibility of the first derivative is still required at each step of the iteration. In this article, the inverse is replaced by a finite sum of linear operators. The convergence of the new Hybrid Chebyshev-Type Method (HCTM) is established under relaxed generalized continuity assumptions on the derivative and majorizing sequences. The iterates of the new methods converge to the original ones, but they are easier to find. Moreover, the numerical examples demonstrate that the new iterates converge essentially as fast to the solution. The methodology of this article can be used on other methods with inverses along the same lines due to its generality. © 2024 by the authors.