Browsing by Author "Argyros, Michael I."
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Item Asymptotically Newton-Type Methods without Inverses for Solving Equations(2024-04-02) Argyros, Ioannis K.; George, Santhosh; Shakhno, Stepan; Regmi, Samundra; Havdiak, Mykhailo; Argyros, Michael 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.Item On the Kantorovich Theory for Nonsingular and Singular Equations(2024-05-28) Argyros, Ioannis K.; George, Santhosh; Regmi, Samundra; Argyros, Michael I.We develop a new Kantorovich-like convergence analysis of Newton-type methods to solve nonsingular and singular nonlinear equations in Banach spaces. The outer or generalized inverses are exchanged by a finite sum of linear operators making the implementation of these methods easier than in earlier studies. The analysis uses relaxed generalized continuity of the derivatives of operators involved required to control the derivative and on real majorizing sequences. The same approach can also be implemented on other iterative methods with inverses. The examples complement the theory by verifying the convergence conditions and demonstrating the performance of the methods.Item Symmetric-Type Multi-Step Difference Methods for Solving Nonlinear Equations(2024-03-08) Argyros, Ioannis K.; Shakhno, Stepan; Regmi, Samundra; Yarmola, Halyna; Argyros, Michael I.Symmetric-type methods (STM) without derivatives have been used extensively to solve nonlinear equations in various spaces. In particular, multi-step STMs of a higher order of convergence are very useful. By freezing the divided differences in the methods and using a weight operator a method is generated using m steps (m a natural number) of convergence order 2 m. This method avoids a large increase in the number of operator evaluations. However, there are several problems with the conditions used to show the convergence: the existence of high order derivatives is assumed, which are not in the method; there are no a priori results for the error distances or information on the uniqueness of the solutions. Therefore, the earlier studies cannot guarantee the convergence of the method to solve nondifferentiable equations. However, the method may converge to the solution. Thus, the convergence conditions can be weakened. These problems arise since the convergence order is determined using the Taylor series which requires the existence of high-order derivatives which are not present in the method, and they may not even exist. These concerns are our motivation for authoring this article. Moreover, the novelty of this article is that all the aforementioned problems are addressed positively, and by using conditions only related to the divided differences in the method. Furthermore, a more challenging and important semi-local analysis of convergence is presented utilizing majorizing sequences in combination with the concept of the generalized continuity of the divided difference involved. The convergence is also extended from the Euclidean to the Banach space. We have chosen to demonstrate our technique in the present method. But it can be used in other studies using the Taylor series to show the convergence of the method. The applicability of other single- or multi-step methods using the inverses of linear operators with or without derivatives can also be extended with the same methodology along the same lines. Several examples are provided to test the theoretical results and validate the performance of the method.