Semilocal Convergence for a Fifth-Order Newton's Method Using Recurrence Relations in Banach Spaces

We study a modified Newton's method with fifth-order convergence for nonlinear equations in Banach spaces. We make an attempt to establish the semilocal convergence of this method by using recurrence relations. The recurrence relations for the method are derived, and then an existence-uniqueness theorem is given to establish the R-order of the method to be five and a priori error bounds. Finally, a numerical application is presented to demonstrate our approach

Convergence of an Iteration of Fifth-Order Using Weaker Conditions on First Order Fréchet Derivative in Banach Spaces

[EN] The convergence analysis both local under weaker Argyros-type conditions and semilocal under. omega-condition is established using first order Frechet derivative for an iteration of fifth order in Banach spaces. This avoids derivatives of higher orders which are either difficult to compute or do not exist at times. The Lipchitz and the Holder conditions are particular cases of the omega-condition. Examples can be constructed for which the Lipchitz and Holder conditions fail but the omega-condition holds. Recurrence relations are used for the semilocal convergence analysis. Existence and uniqueness theorems and the error bounds for the solution are provided. Different examples are solved and convergence balls for each of them are obtained. These examples include Hammerstein-type integrals to demonstrate the applicability of our approach.Singh, S.; Gupta, D.; Singh, R.; Singh, M.; Martínez Molada, E. (2018). Convergence of an Iteration of Fifth-Order Using Weaker Conditions on First Order Fréchet Derivative in Banach Spaces. International Journal of Computational Methods. 15(6):1-18. https://doi.org/10.1142/S0219876218500482S11815

Semilocal convergence by using recurrence relations for fifth-order method in Banach spaces

In this paper, a semilocal convergence result in Banach spaces of an efficient fifth-order method is analyzed. Recurrence relations are used in order to prove this convergence, and some a priori error bounds are found. This scheme is finally used to estimate the solution of an integral equation and so, the theoretical results are numerically checked. We use this example to show the better efficiency of the current method compared with other existing ones, including Newton's scheme.This research was supported by Ministerio de Ciencia y Tecnologia MTM2011-28636-C02-{01,02}.Cordero Barbero, A.; Hernandez-Veron, MA.; Romero, N.; Torregrosa Sánchez, JR. (2015). Semilocal convergence by using recurrence relations for fifth-order method in Banach spaces. Journal of Computational and Applied Mathematics. 273:205-213. https://doi.org/10.1016/j.cam.2014.06.008S20521327

On the convergence of a higher order family of methods and its dynamics

[EN] In this paper, we present the study of the local convergence of a higher-order family of methods. Moreover, the dynamical behavior of this family of iterative methods applied to quadratic polynomials is studied. Some anomalies are found in this family by means of studying the associated rational function. Parameter spaces are shown and the study of the stability of all the fixed points is presented. (C) 2016 Elsevier B.V. All rights reserved.This research was supported by Universidad Internacional de La Rioja (UNIR, http://www.unir.net), under the Plan Propio de Investigación, Desarrollo e Innovación 3 [2015–2017]. Research group: Modelación matemática aplicada a la ingeniería(MOMAIN), by the grant SENECA 19374/PI/14 and by Ministerio de Ciencia y Tecnología MTM2014-52016-C2-{01,02}-P.Argyros, IK.; Cordero Barbero, A.; Alberto Magreñán, A.; Torregrosa Sánchez, JR. (2017). On the convergence of a higher order family of methods and its dynamics. Journal of Computational and Applied Mathematics. 309:542-562. https://doi.org/10.1016/j.cam.2016.04.022S54256230

Estudio sobre convergencia y dinámica de los métodos de Newton, Stirling y alto orden

Las matemáticas, desde el origen de esta ciencia, han estado al servicio de la sociedad tratando de dar respuesta a los problemas que surgían. Hoy en día sigue siendo así, el desarrollo de las matemáticas está ligado a la demanda de otras ciencias que necesitan dar solución a situaciones concretas y reales. La mayoría de los problemas de ciencia e ingeniería no pueden resolverse usando ecuaciones lineales, es por tanto que hay que recurrir a las ecuaciones no lineales para modelizar dichos problemas (Amat, 2008; véase también Argyros y Magreñán, 2017, 2018), entre otros. El conflicto que presentan las ecuaciones no lineales es que solo en unos pocos casos es posible encontrar una solución única, por tanto, en la mayor parte de los casos, para resolverlas hay que recurrir a los métodos iterativos. Los métodos iterativos generan, a partir de un punto inicial, una sucesión que puede converger o no a la solución