Convergence Analysis Of Inexact Newton-type Iteration Methods

Using Iterative algorithms to solve nonlinear equation F(x)=0is not only an im-portant mathematical problem, but also having a wide range of practical applications in engineering, economics and other disciplines. This thesis mainly concerns the semilocal and local convergence of inexact Newton-type Methods solving F(x)=0. Our work weakens some relevant convergence 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, condi-tion of convergence and relevant knowledge in Banach space. Some of the concepts used in the thesis are also presented.In Chapter2, when the derivative of F(x) doesn’t exist, we divide F into differen-tiable part and non-differentiable part. By using some majorant functions, its semilocal convergence and local convergence are proved. The main theorems obtained in this section extend some known results.Chapter3uses the inexact Newton-type method to solve nonlinear equation F(x)=0. By controling the residual sequence, its semilocal convergence and local convergence are proved when F has the first and second Frechet-derivatives. At last, this chapter uses a numerical example to illustrate the advantages of the obtained results.