Development and analysis of some new iterative methods for numerical solutions of nonlinear equations

Abstract

One of the most important and challenging problems in scientific and engineering applications is to find solutions of nonlinear equations, unconstrained optimization problems and systems of nonlinear equations. In order to find out the approximate solutions of these equations, one has to adopt numerical techniques based on iteration procedures. Newton?s method is probably the well-known iterative method for solving these equations. In recent years, many modifications of Newton?s method has been proposed in literature, which have either equal or better performance than Newton?s method. In the present thesis an attempt has been made to unify the classical existing methods and to remove their (Newton?s method and its variants) defects. The thesis consists of six chapters and main contents of each chapter are furnished as follows : newlineChapter 1 is an introductory chapter and gives a brief survey of literature. Several real world problems have been considered for which the numerical solutions are required for solving scalar nonlinear equations, unconstrained optimization problems and systems of nonlinear equations. Further, some classical methods have been introduced by discussing their merits and demerits. The fundamental concepts and classification of iterative methods and their striking features are also stated. The research work on the iterative methods carried out in solving nonlinear equations (for simple as well as multiple roots), unconstrained optimization problems and systems of nonlinear equations is also reviewed briefly. Chapter 2, eliminates the defects of the classical Newton?s method by simple modifi- cations of iterative processes. The proposed iterative methods are derived by implementing approximations through a straight line, parabolic and elliptical curves in the vicinity of the required root. Further, the extensions of these iterative methods to the unconstrained optimization problems and to systems of nonlinear equations have been proposed.