Convergence Analysis Of Several Inexact Iterative Methods For Solving Nonlinear Equations

The problem of solving nonlinear equation F(x)=0has been an important problem of numerical mathematic. One of the most efficient algorithms to solve this equation is iterative method. In this thesis, we study semilocal convergence of inexact Newton method under weak conditions, and local convergence of inexact Newton-like methods and inexact Gauss-Newton like methods under majorant conditions. Our work weakens some relevant convergent conditions and improves some results. The contents are as follows:In Chapter1, it presents the background and current situation for solving nonlin-ear equations. Various variants of Lipschitz condition and the associated results are summarized. Also, it introduces relevant definitions and preliminary knowledge, such as convergence order, condition of convergence and relevant knowledge in Banach space. Some of the concepts used in the thesis are also presented.In Chapter2, under some kinds of weak Lipschitz conditions, a new semilocal conver-gence analysis for inexact Newton method is obtained by majorizing sequence analysis. Unified convergence criteria ensuring the convergence are established. This approach also reaches a new error estimation and the unique ball of solution, which generalize the ex-isting relevant ones. It is worth pointing out that applications to some special cases such as the Kantorovich type conditions and7-conditions are provided and some well-known convergence theorems for Newton’s method are obtained as corollaries.In Chapter3, we define the center majorant conditions. And under the assumptions that F satisfies relevant majorant conditions, we establish a new local convergence analy-sis for inexact Newton-like method and inexact Gauss-Newton like method respectively. This analysis simplifies the conditions of convergence in relevant references and provides a clear relationship between the majorizing function and the nonlinear operator. Partic-ularly, the relevant results under center Lipschitz conditions and center γ-conditions as some special cases are also established.