Semi-local Convergence Analysis Of Inexact Newton Methods For Solving Nonlinear Equations

The problem of solving nonlinear equation f(x)=0has always been a very impor-tant project in mathematical theory and applications. A large number of problems in applied mathematics and also in engineering are solved by finding the solutions of cer-tain equations, for example, dynamic systems are mathematically modeled by differential equation. Now, more effective algorithms to solve this problem are iterative method. This thesis mainly studies the semi-local convergence of inexact Newton methods in solving equation f(x)=0. Our work weakens some relevant convergence conditions and improves some results. The contents are as follows:Chapter1mainly presents the background and current situation of various itera-tions. Also, it introduces relevant definitions and preliminary knowledge, such as iterative scheme, conditions of convergence, convergence order and relevant knowledge in Banach space. Some of the concepts used in the thesis are also presented.Chapter2introduces the center γ0-condition. And combine the γ-condition, we es-tablish the Kantorovich semilocal convergence for Inexact Newton methods by using the technique of majorant functions. Also, we get a finer error estimate and uniqueness of the solution is proved in the thesis.Chapter3under the assumption that f has the first Frechet-derivatives, we establish the semilocal convergence by relaxing the convergence conditions and choosing appropri-ate control on the basis of the known convergence analysis results of inexact Newton methods. Results are more general than before. Also, by modifying the convergence con-ditions, the corresponding local convergence analysis is presented in the paper.