Convergence Analysis Of Iteration Methods On Nondifferentiable Or Derivative Irreversible Equations

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 convergence of generalized Newton method and Gauss-Newton method exploring the approximate so-lution of 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 traditional iterations. Also, it presents relevant preliminary knowledge, such as pattern of iteration, condition of convergence, B-subdifferential and relevant knowledge in Banach space. Some of the concepts used in the thesis are also presented.In Chapter2, when solving nonlinear equation F(x)=0, if the derivative of F(x) doesn’t exist, we divide F into differentiable part and non-differentiable part. By using any matrix from the B-subdifferential of the non-differentiable term, we present a new generalized Newton method and obtain its local convergence and semilocal convergence under ω-condition. The main theorems obtained in this section extend some results of [10,34].In Chapter3, we use Gauss-Newton method to explore the approximate solution of F(x)=0when the derivative of F is not invertible. By letting F satisfy a majorant condition and a center majorant condition at the same time, we obtain its local conver-gence. The main theorems obtained in this section expand the range of convergence and improve the results of [33].