The Convergence Analysis Of Some High-Order Iterative Methods For Solving Nonlinear Equations

When we use mathematical methods to study natural and social phenomena or to solve engineering technique problems, we often regard the solutions of most practical problems as the ones of nonlinear equations in form ofF(x) = 0in Banach space. While iterative methods are the most efficient algorithms for solving nonlinear equations. So it is very important and meaningful to do the research of iterative methods.This thesis consists of five chapters. In Chapter 1, we give some basic definitions and notations which will be used throughout the whole thesis; summarize the techniques in proving iterative methods' convergence theorems and the convergence conditions of several famous iterative methods.In Chapter 2, under the (K,p)-Ho|¨lder continuous condition of the first Fréchet derivative, we discuss the convergence property of a third order composite Newton-Steffensen method, which was introduced by Sharma , and obtain a semilocal convergence theorem for it. Meanwhile, a local convergence theorem is also given under the similar conditions.In Chapter 3, we study the semilocal convergence for the method appeared in Chapter 2 under Lipschitz continuous condition of the second Frechet derivative, and get the corresponding error estimation by majoriz-ing function technique. And also, an application of the theorem to a non- linear equation is given.In Chapter 4, we present a(?)united structural approach for a special type of high-order iterative methods which were proposed in recent years. By this means, we deduce a new high-order Halley type method based on the famous Halley's method, and proves that its local convergence order is five. Finally, we give some numerical results with a number of function tests to show that the new method performs better in efficiency comparing to some classical methods.In Chapter 5, by using the majorizing function technique, we establish a semilocal convergence theorem for the method obtained in Chapter 4 under generalized Lipschitz continuous condition of the second Fréchet derivative.