Several Numerical Methods For Solving Nonlinear Equations And Their Convergence Analyses

The problems of solving nonlinear equations have a widely application background in Pattern Recognition, Circuit Problems, Electric Power Systems Calculation and so on. Recently, the algorithm problem of solving nonlinear equation F(x)=0has been studied by many numerical scientists, some of the most efficient methods to solve this kind of equations are iterative methods. In this work, we focus on constructing united structural approach and higher order convergence numerical methods. The thesis consists of four chapters as follows:Firstly, in Chapter1, the research background and current situation of iterative methods for nonlinear equations are introduced. Some of the concepts and preliminaries used in this thesis are also presented in the chapter.Secondly, in Chapter2, we modify the Chebyshcv-Halley type methods and obtain a family of at least third order convergence methods free from second derivatives for nonlinear equations, which includes a fourth order convergence method free from sec-ond derivatives, In Subsection2.1.4, we present some numerical tests to illustrate the performance of our methods by comparing with some other methods.Thirdly, in Chapter3, we present almost all methods for finding multiple roots of nonlinear equations in recent years. Motivated by their work, we develop three new fam-ilies of methods for multiple roots which include almost all methods for finding multiple roots of nonlinear equations we can find. From these three families, we could obtain many third order and fourth order convergence methods for solving multiple roots of nonlin-ear equations. In Section3.3, numerical illustrations are given to compare the proposed methods with some other methods of the same kind.Finally, in Chapter4, the contents of this thesis are summarized, and then, some further questions in the field of solving nonlinear equations with iterative methods are proposed.