Familias paramétricas de procesos iterativos de alto orden de convergencia

Abstract

The goal of this memory is the numerical solution of nonlinear equations by iterative processes. We study the analysis of parametric families of Newton-type iterative processes in Banach spaces, so that we can take them on a wide range of problems, as integral equations, partial differential equations or boundary value problems. We obtain in Banach spaces a family of iterative processes with order of convergence at least three that includes the most known iterative processes with at least cubic convergence: Chebyshev's method, the Super-Halley method, the Halley method or the Euler method, as well as other families of iterations. We gradually relax the hypotheses of semilocal convergence that are usually used and obtain the domains where solutions are located and unique, together with some a priori and a posteriori error estimates. To realize the study of the semilocal convergence of the family, we use two different techniques: the majorant principle and one based in the construction of a system of recurrence relations. In the particular case of quadratic equations in Banach spaces, we establish a family of iterative processes with prefixed order of convergence. It is interesting to notice that in this case the parameters that appear in the family are defined from Catalan's numbers. In the real case, the iterative methods of the family are globally convergent if the order of convergence is even, and generally convergent if the order of convergence is odd. In the complex plane, we present a study of the convergence from a numerical and dynamical point of view. With the objective of generalizing the study done for quadratic equations, we analyze the convergence of the family when it is applied in the solution of a wider group of equations. We obtain in this way a new family of iterative processes with also prefixed order of convergence and establish results of semilocal and global convergence for these iterations.