Construction of Some New Iterative Families for Solving System of Nonlinear Equations

Abstract

Nonlinear systems of equations appear in many disciplines such as engineering, mathematics, robotics and computer sciences because majority of physical systems are nonlinear in nature. One of the most basic numerical problems encountered in computational economics is to find the solution to a nonlinear equation or a whole system of nonlinear equations. In this thesis we introduce a technique for solving nonlinear system of equations that improve the order of convergence for any given iterative method. The family of new iterative methods are built up and analyzed. A development of an inverse first-order divided difference operator for functions of several variables is presented. The main advantage of these methods is that, they have higher - order of convergence and they do not require the evaluation of any second or higher order Fr´echet derivatives. This thesis consists of three chapters CHAPTER 1 gives a brief explanation about the need of iterative methods in scientific and engineering problems. Some basic concepts and definitions regarding system of nonlinear equations are introduced. Some fundamental concepts are also explained in this chapter. CHAPTER 2 gives a brief survey of literature. It gives a review of methods with cubic as well higher- order convergent iterative methods. In CHAPTER 3 the family of iterative methods for computing the solution of system of nonlinear equations has been introduced. The proposed methods do not require the evaluation of second or higher order Fr´echet derivatives per iteration to proceed and reach fourth and sixth order of convergence. An improvement of the local order of convergence is presented to increase the efficiency of the iterative methods with an appropriate number of evaluations of the function and its derivative. First-order divided difference operator for functions of several variables are used to prove the local convergence order. A discussion on the efficiency index of the contribution with comparison to the other iterative methods is also given. Finally, numerical tests illustrate the theoretical aspects using the programming package Mathematica 7:1.