ABSTRACT
This chapter uses the iterative methods to find solutions of nonlinear equations. Concerning the semilocal convergence of Newton’s method, one of the most important results is the famous Kantorovich theorem for solving nonlinear equations. This theorem provides a simple and transparent convergence criterion for operators with bounded second derivatives or the Lipschitz continuous first derivatives. The chapter presents standard results on convex functions as well as main semilocal convergence result. It describes the iterations performed for nonlinear equations to obtain the asymptotic bounds by the inexact Newton method. The chapter also provides special cases and applications of the inexact Newton method to some interesting cases such as Kantorovich’s theory and even Smale’s alpha-theory or Wang’s gamma-theory.
