Some Researches On Iterative Methods For Solving Nonlinear Equations

Numerical solutions of systems of nonlinear equations are becoming one of most important research topics in the field of computational mathematics, and it also has a wide application in many practical problems. In recent years, this field has got the rapid development, Scholars have proposed a variety of numerical methods of solving nonlinear equations. The classic method is iterative method, in this thesis, we disscuss the iterative methods to solve systems of nonlinear equations.In the Introduction, we start from the current situation and background about nonlinear equations, giving a simple introduction of some existing classic numerical methods for solving the nonlinear equations. Finally, the organization of this thesis is offered briefly.In Chapter 1, firstly several well known iterative methods are introduced for solving the system of nonlinear equations, including Newton iterative method, Os-trowski iterative method and so on. Then based on weight function methods, a family of seventh-order iterative method for solving nonlinear equation is develope-d, and the convergence proof is given. Per iteration of this method require three evaluations of the function and one evaluation of the first derivative, so the efficien-cy index of the developed methods is 1.627. Some numerical comparisons are made with several other existing methods to illustrate the efficiency and the performance of the newly developed method confirms the theoretical results.In Chapter 2, a new family of eighth-order iterative method for solving simple roots of nonlinear equation is developed by using weight function methods, and the convergence proof is given. Per iteration of this method require three evaluations of the function and one evaluation of the first derivative, so the efficiency index of the developed methods is 1.682. Some numerical comparisons are made with several other existing methods to illustrate the efficiency and the performance of the newly developed method confirms the theoretical results.In Chapter 3, two variants of iterative methods for solving systems of nonlinear equations with fifth-order convergence are developed. The proposed methods are of the convergence of fifth order. Some numerical comparisons are made with several other existing methods to illustrate the efficiency and the performance of the newly developed method confirms the theoretical results.In Chapter 4, two variants of iterative methods are developed in this work in order to solve a system of nonlinear equations with higher order convergence. We prove that these new methods have cubic convergence. Some numerical examples are given to show the efficiency and the performance of the new iterative methods, which confirm the good theoretical properties of the approach.In Chapter 5, we give the summary of the thesis and point out how to further our research work, such as ideas, suggestions and problems.