On a Family of High‐Order Iterative Methods under Kantorovich Conditions and Some Applications

S Amat,M J Legaz,S Busquier,C Bermúdez,S Plaza

doi:10.1155/2012/782170

Abstract

This paper is devoted to the study of a class of high‐order iterative methods for nonlinear equations on Banach spaces. An analysis of the convergence under Kantorovich‐type conditions is proposed. Some numerical experiments, where the analyzed methods present better behavior than some classical schemes, are presented. These applications include the approximation of some quadratic and integral equations.

Highlights

Conditions are imposed on x0 and on F in order to assure the convergence of {xn}n to a solution x∗ of F x 0. This analysis, usually known as Kantorovich type, are based on a relationship between the problem in a Banach space and a single nonlinear scalar equation which leads the behavior of the problem
As was indicated in the introduction, we are interested in the study of the family of iterative methods as follows yn xn
Several techniques are usually considered to study the convergence of iterative methods, as we can see in the following papers 4, 17–20

Summary

Introduction

F x 0, 1.1 where F : Ω ⊆ X → Y is a nonlinear operator between Banach spaces, using the following family of high-order iterative methods:. Conditions are imposed on x0 and on F in order to assure the convergence of {xn}n to a solution x∗ of F x 0 This analysis, usually known as Kantorovich type, are based on a relationship between the problem in a Banach space and a single nonlinear scalar equation which leads the behavior of the problem. For a nonlinear system of m equations and m unknowns, the first Frechet derivative is a matrix with m2 entries, while the second Frechet derivative has m3 entries This implies a huge amount of operations in order to evaluate every iteration. The structure of this paper is as follows: in Section 2 we present some particular examples of methods included in the family and, we assert convergence and uniqueness theorems Kantorovich type. In all these problems the proposed methods seem more efficient than second-order methods

A Family of High-Order Iterative Methods

Semilocal Convergence

LF xn LF xn 2GF xn

Approximation of Riccati’s Equations

Approximation of Hammerstein Equations

Conclusions