Newton - Like Method Of Convergence And Error Analysis

When we use mathematical methods to study natural and social phenomena or to solve engineering technique problems, we often regard the algorithm of most practical problems as one of nonlinear equations in form ofF(x) = 0in Banach space. Newton iteration is the most elementary and important method for solving nonlinear equation. At present, many effective iterations are all derived from Newton method.In this paper,we introduce new "Newton-like" methods and their deformations. This dissertation consists of four chapters ,which mainly analyze the semi-convergence and error estimate of these two iterations under Ostrowski-Kanorovich conditions.In chapter l,we summarize not only Newton iteration and some deformed methods, but also the various corrections of Lipschitz condition since Kanorovich condition was put forward.In chapter 2,we discuss new "Newton-Like" methods which are founded on the basis of Liapunov's methods of dynamic system. These new methods preserve quadratic convergence as well as remove the monotoneity condition F'(x)≠0.In chapter 3,we establish convergence theorem under Ostrowski-Kanorovich condition for Newton-like method by using majorizing method. Moreover an error estimate is given.In chapter 4,we introduce the deformed Newton-Like methods which are cu-bically convergent. Under the same Ostrowski-Kantorovich as for " Newton-Like" method ,we establish convergence theorem and give an error estimate.Finally ,we give two numberical examples.