Iterative methods for the solution of nonlinear equations

Abstract

One of the most important and challenging problems in scientific and engineering applications is to find the solution of the nonlinear equations. Analytical methods for such equations rarely exist and therefore we can only hope to obtain approximate solutions by relying on numerical methods based on iteration procedures. Newton method is probably the best known iterative method for solving nonlinear equations. In recent years, several modifications of Newton method have been proposed and analyzed, which have either equal or better performance than Newton method. The present thesis deals mainly with the development of iterative methods with improved order and efficiency to solve nonlinear equations and systems of nonlinear equations. Work of the thesis is divided into 6 chapters. The main contents of each chapter are furnished in the following text. In chapter 1, we give brief explanation about the need of iterative methods in scientific and engineering problems. Some classical methods are introduced, their merits and demerits are discussed. The basic definitions and classification of iterative methods are presented. Some basic concepts and definitions regarding multiple roots and systems of nonlinear equations are introduced. The important features of one-point and multipoint methods are stated. The various techniques, which are used by researchers to generate higher order iterative methods such as functional approximation, sampling, composition, geometric and Adomian approaches, are presented. Lastly, the chapter ends with the summery of the main results embodied in the thesis. In chapter 2, an attempt is made to develop a unified scheme to obtain one-point methods without memory with at least cubic convergence. Our approach is based on a simple modification of Newton method. The scheme is powerful and interesting since it generates almost all available one-point third order methods in literature. Moreover, many new methods can be generated.