Relaxing convergence conditions for an inverse-free Jarratt type approximation

We consider an inverse-free Jarratt-type approximation of order four in a Banach space (Argyros et al., 1996). We establish a convergence theorem by using recurrence relations. The purpose of this note is to relax convergence conditions and give an example where our convergence theorem can be applied but not the other ones

Local Convergence for an Improved Jarratt-type Method in Banach Space

We present a local convergence analysis for an improved Jarratt-type methods of order at least five to approximate a solution of a nonlinear equation in a Banach space setting. The convergence ball and error estimates are given using hypotheses up to the first Fréchet derivative in contrast to earlier studies using hypotheses up to the third Fréchet derivative. Numerical examples are also provided in this study, where the older hypotheses are not satisfied to solve equations but the new hypotheses are satisfied

Convergence Theorem for a Family of New Modified Halley’s Method in Banach Space

We establish convergence theorems of Newton-Kantorovich type for a family of new modified Halley’s method in Banach space to solve nonlinear operator equations. We present the corresponding error estimate. To show the application of our theorems, two numerical examples are given

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

Semilocal convergence analysis of an eighth order iterative method for solving nonlinear systems

In this paper, the semilocal convergence of the eighth order iterative method is proved in Banach space by using the recursive relation, and the proof process does not need high order derivative. By selecting the appropriate initial point and applying the Lipschitz condition to the first order Fréchet derivative in the whole region, the existence and uniqueness domain are obtained. In addition, the theoretical results of semilocal convergence are applied to two nonlinear systems, and satisfactory results are obtained