Development and convergence analysis of some new iterative methods for non linear equations

Abstract

Solving nonlinear equations stands as a crucial and challenging pursuit in scientific computing, resonating across various fields of science and engineering. Analytical methods for solving such equations are rarely available, making it essential to obtain approximate solutions through numerical methods based on iterative procedures. While the classical Newton s method serves as a well-known iterative approach, recent years have witnessed the development and analysis of various modifications, demonstrating comparable or enhanced performance in terms of convergence speed and efficiency. The present thesis mainly focuses on developing new and improved iterative methods for solving both single and systems of nonlinear equations. The work is structured into seven chapters. Chapter 1 serves as an introductory overview, highlighting the importance of iterative methods in scientific and engineering problems. We introduce fundamental terminologies, concepts and some classical methods, along with their merits and limitations. Essential definitions are provided for various aspects, including simple roots, multiple roots, convergence rate and computational efficiency. The chapter explores the classification of iterative methods into with- and without-memory methods, incorporating a detailed analysis of one-point and multipoint methods. Additionally, some basic concepts and definitions related to multiple roots and systems of nonlinear equations are introduced.