About Half Of The Newton Method, Local Convergence Of The Basic Assumptions

When we use the mathematical thought to solve the engineering technical prob-lems, quite a lot of practical problems are often regarded as the nonlinear equations in form of F(x)=0 in Banach space. Newton method is a kind of basic and very important iterative method for solving the nonlinear function, firstly, L.V.Kantorovich gave a theorem about whether the Newton method is convergent in Banach space. Later, there are lots of articles about weakening the Kantorovich condition for researching the convergence of the Newton method, Professor X.H.Wang made many outstanding contributions in this field. Among them, in 1999, he gave a Kantorovich-type theorem under the center Lipschitz condition in the inscribed sphere with the L average, and made a unified hypothesis of convergence including the Kantorovich-type hypothesis, Smale-type a theory and other convergence theorems. In 2009, O.P.Ferreira and B.F.Svaiter estab-lished a corresponding convergence theorem for Newton method supposing that the majorant function's derived function is convex and increasing. Both of the above two basic hypothesis conditions weaken the Kantorovich-type condition, and the main con-tent of this paper is to prove the equivalence of the two basic semi-local convergence conditions. The paper is organized as follows:Chapter 1:Mainly summarize the historical development and research current situation of Newton method and review the modified conditions for the Kantorovich condition by the researchers and the corresponding convergence theorems since the Newton-Kantorovich theorem was established.Chapter 2:Prove the equivalence of the two basic conditions.Chapter 3:List the corresponding two basic equivalent forms of number of con-vergence theorems, such as Newton-Kantorovich theorem, Smale's a theory and so on.