Convergence Analysis Of A Newton-like Method

Solving non-linear operator equations is a very important research field in computa-tional mathematics.It has a wide range of applications in many scientific fields such as engineering and physics.Generally,we approximate the solution of an operator equation by using an iterative method to obtain a convergent sequence.Newton's method is one of the most widely used methods.Based on Newton's method,many scholars have proposed a variety of iterative methods such as Quasi-Newton's method,Newton-like method and etc.In this thesis,we study the convergence of a Newton-like method to solve operator equations with the form F(x)+G(x)=0.The layout of this thesis is as follows:The first chapter introduces the development process and current research status of studying nonlinear equation operators by using Newton's method under Kantorovich-type conditions.At the same time,it provides the background knowledge involved in this thesis.It mainly includes Newton-like iterative methods,central affine Lipschitz condi-tion,H(?)lder condition,convergence conditions,basic definitions and related theorems.The main results of this thesis are given at the end.The second chapter studies the local convergence of a Newton-like method under the central affine Lipschitz condition,and gives error estimates and uniqueness proof.In addition,some specific cases are discussed and a numerical example is given to explain the effectiveness of the method.The third chapter studies the semi-local convergence of a Newton-like method under the H(?)lder condition,and proves the existence and uniqueness of the solution.In addition,specific numerical examples are also given to compare the speed of convergence for some cases of A(x).