Convergence Analysis Of Two Types Of Modified Newton Methods

The problem of solving nonlinear equation F(x)=0has been an important problem of numerical mathematic. This thesis mainly concerns the semilocal and local convergence of two types of modified Newton Methods for solving F(x)=0. Our work weakens some relevant convergent conditions and improves some results. The contents are as follows:Chapter1introduces the background and current situation of typical iterations. Also, it presents relevant preliminary knowledge, such as convergence of iteration, con-vergence order, condition of convergence and relevant knowledge in Banach space. Some of the concepts used in the thesis are also presented.Chapter2studies solving nonlinear equation F(x)=O when the derivative does not exist. By using the modified Newton method and some majorant functions, its semilocal convergence and local convergence are proved. We also give the convergence radius of the iterative method when the operator have a unique solution on it. The main theorems obtained in this section extend some known results.Chapter3uses generalized inexact Newton method to solve nonsmooth equations. On one hand, we prove the iterative method is locally and convergent under the residual control and semismooth conditions. On the other hand, combined with related control condition, we further obtain a type of generalized inexact Newton iterations and it con-vergence.