The Semilocal Convergence Properties Of Super-Halley Method And Newton Method Under Weak Conditions

The algorithm problem of solving nonlinear operator equation F(x)=0 has been one of the most interesting problems to many numerical scientists. One of the most efficient algorithms to solve this equation is iterative method. For an iterative method, one mainly studies convergence, convergence rate, error estimation and so on. In this thesis, we study semilocal convergence of super-Halley method under affine covariant conditions and Newton method under the two types of affine contravariant conditions, and obtain some results, which generalize or improve the existing relevant ones. The thesis consists of three chapters as follows:In Chapter 1, the research background and current situation of iterative methods for solving nonlinear equations are given, which include three types of convergence analyses and several commonly used iterative methods. Various variants of Lipschitz condition related to Kantorovich condition and the associated results are summarized. Some of the concepts used in the thesis are presented.In Chapter 2, under some majorant conditions which are weaker than L-average Lipschitz conditions, a new semilocal convergence analysis for super-Halley method is obtained. This analysis provides a clear relationship between the majorizing function and the nonlinear operator. This approach also guarantees Q-cubic convergence rate, reach a new error estimation in terms of a directional derivative of the second derivative of the majorizing function and the unique ball of solution. The obtained results generalize the related existing ones.In Chapter 3, Two new affine Contravariants, w-condition andγ-condition, are in-troduced and the semilocal convergence of Newton method under these two conditions are obtained. The results generalize the ones given by Deuflhard and Hohmann.