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 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.