Un estudio unificado de la convergencia semilocal de métodos tipo Newton de dos puntos en espacios de Banach

En este trabajo consideramos procesos iterativos tipo Newton de dos puntos para aproximar una solución de una ecuación no lineal en espacios de Banach. A partir de una expresión general que caracteriza estos procesos iterativos, obtenemos resultados de convergencia semilocal, estableciendo una teoría unificada para el an´alisis de este tipo de procesos.Ministerio de Educación y Cienci

Improved Iterative Solution of Linear Fredholm Integral Equations of Second Kind via Inverse-Free Iterative Schemes

[EN] This work is devoted to Fredholm integral equations of second kind with non-separable kernels. Our strategy is to approximate the non-separable kernel by using an adequate Taylor's development. Then, we adapt an already known technique used for separable kernels to our case. First, we study the local convergence of the proposed iterative scheme, so we obtain a ball of starting points around the solution. Then, we complete the theoretical study with the semilocal convergence analysis, that allow us to obtain the domain of existence for the solution in terms of the starting point. In this case, the existence of a solution is deduced. Finally, we illustrate this study with some numerical experiments.This research was partially supported by a grant of the Spanish Ministerio de Ciencia, Innovacion y Universidades (Ref. PGC2018-095896-B-C21-C22).Gutiérrez, JM.; Hernández-Verón, MÁ.; Martínez Molada, E. (2020). Improved Iterative Solution of Linear Fredholm Integral Equations of Second Kind via Inverse-Free Iterative Schemes. Mathematics. 8(10):1-13. https://doi.org/10.3390/math8101747S113810Argyros, I. K. (1988). On a class of nonlinear integral equations arising in neutron transport. Aequationes Mathematicae, 36(1), 99-111. doi:10.1007/bf01837974Bruns, D. D., & Bailey, J. E. (1977). Nonlinear feedback control for operating a nonisothermal CSTR near an unstable steady state. Chemical Engineering Science, 32(3), 257-264. doi:10.1016/0009-2509(77)80203-0GANESH, M., & JOSHI, M. C. (1991). Numerical Solvability of Hammerstein Integral Equations of Mixed Type. IMA Journal of Numerical Analysis, 11(1), 21-31. doi:10.1093/imanum/11.1.21Anderson, B. D. O., & Kailath, T. (1971). Some Integral Equations with Nonsymmetric Separable Kernels. SIAM Journal on Applied Mathematics, 20(4), 659-669. doi:10.1137/0120065Ezquerro, J. A., & Hernández, M. A. (2004). A modification of the convergence conditions for Picard’s iteration. Computational & Applied Mathematics, 23(1). doi:10.1590/s0101-82052004000100003Amat, S., Ezquerro, J. A., & Hernández-Verón, M. A. (2013). Approximation of inverse operators by a new family of high-order iterative methods. Numerical Linear Algebra with Applications, 21(5), 629-644. doi:10.1002/nla.1917Barikbin, M. S., Vahidi, A. R., Damercheli, T., & Babolian, E. (2020). An iterative shifted Chebyshev method for nonlinear stochastic Itô–Volterra integral equations. Journal of Computational and Applied Mathematics, 378, 112912. doi:10.1016/j.cam.2020.112912Rabbani, M., Das, A., Hazarika, B., & Arab, R. (2020). Existence of solution for two dimensional nonlinear fractional integral equation by measure of noncompactness and iterative algorithm to solve it. Journal of Computational and Applied Mathematics, 370, 112654. doi:10.1016/j.cam.2019.11265

Sobre la región de accesibilidad de ciertas iteraciones de tercer orden

La región de accesibilidad de los procesos iterativos cuando se aplican a la resolución de ecuaciones no lineales adquiere cierto interés a la hora de elegir un proceso iterativo. Sabemos, a priori, que cuanto mayor es el orden de convergencia de los procesos iterativos, menor es su región de accesibilidad. Nosotros aquí presentamos una simple modificación de las iteraciones clásicas de tercer orden de manera que podamos considerar, para cada una de ellas, la misma región de accesibilidad que para el método de segundo orden más conocido, el método de Newton

On high-order iterative schemes for the matrix pth root avoiding the use of inverses

This paper is devoted to the approximation of matrix pth roots. We present and analyze a family of algorithms free of inverses. The method is a combination of two families of iterative methods. The first one gives an approximation of the matrix inverse. The second family computes, using the first method, an approximation of the matrix pth root. We analyze the computational cost and the convergence of this family of methods. Finally, we introduce several numerical examples in order to check the performance of this combination of schemes. We conclude that the method without inverse emerges as a good alternative since a similar numerical behavior with smaller computational cost is obtained.The research of the authors S.A. and S.B. was funded in part by Programa de Apoyo a la investigación de la Fundación Séneca-Agencia de Ciencia y Tecnología de la Región de Murcia 20928/PI/18 and by PID2019-108336GB-100 (MINECO/FEDER). The research of the author M.Á.H.-V. was supported in part by Spanish MCINN PGC2018-095896-B-C21. The research of the author Á.A.M. was funded in part by Programa de Apoyo a la investigación de la Fundación Séneca-Agencia de Ciencia y Tecnología de la Región de Murcia 20928/PI/18 and by Spanish MCINN PGC2018-095896-B-C21

On a Moser–Steffensen type method for nonlinear systems of equations

This paper is devoted to the construction and analysis of a Moser–Steffensen iterative scheme. The method has quadratic convergence without evaluating any derivative nor inverse operator. We present a complete study of the order of convergence for systems of equations, hypotheses ensuring the local convergence, and finally, we focus our attention to its numerical behavior. The conclusion is that the method improves the applicability of both Newton and Steffensen methods having the same order of convergence

A Unified Convergence Analysis for Some Two-Point Type Methods for Nonsmooth Operators

The aim of this paper is the approximation of nonlinear equations using iterative methods. We present a unified convergence analysis for some two-point type methods. This way we compare specializations of our method using not necessarily the same convergence criteria. We consider both semilocal and local analysis. In the first one, the hypotheses are imposed on the initial guess and in the second on the solution. The results can be applied for smooth and nonsmooth operators.Research of the first and third authors supported in part by Programa de Apoyo a la investigación de la fundación Séneca-Agencia de Ciencia y Tecnología de la Región de Murcia 20928/PI/18 and by MTM2015-64382-P. Research of the fourth and fifth authors supported by Ministerio de Economía y Competitividad under grant MTM2014-52016-C2-1P. This research received no external funding

How to Obtain Global Convergence Domains via Newton’s Method for Nonlinear Integral Equations

We use the theoretical significance of Newton’s method to draw conclusions about the existence and uniqueness of solution of a particular type of nonlinear integral equations of Fredholm. In addition, we obtain a domain of global convergence for Newton’s method

Newton’s method: an updated approach of Kantorovich’s theory

This book shows the importance of studying semilocal convergence in iterative methods through Newton's method and addresses the most important aspects of the Kantorovich's theory including implicated studies. Kantorovich's theory for Newton's method used techniques of functional analysis to prove the semilocal convergence of the method by means of the well-known majorant principle. To gain a deeper understanding of these techniques the authors return to the beginning and present a deep-detailed approach of Kantorovich's theory for Newton's method, where they include old results, for a historical perspective and for comparisons with new results, refine old results, and prove their most relevant results, where alternative approaches leading to new sufficient semilocal convergence criteria for Newton's method are given. The book contains many numerical examples involving nonlinear integral equations, two boundary value problems and systems of nonlinear equations related to numerous physical phenomena. The book is addressed to researchers in computational sciences, in general, and in approximation of solutions of nonlinear problems, in particular