A Study On Iterative Algorithm For Solving Nonlinear Equations

When we make use of mathematical tools to research social phenomena and natural phenomena, or to resolve the engineering and other problems, a lot of problems can be ended up with solving the equation f ( x ) = 0, the nonlinear equations have a very important role in research of both theory and practical application. The iterative method is an important method to slove the equation f ( x ) = 0, whether the nonlinear equations will be solved well or not is directly affected by the choice of iterative method. Therefore, the research on iterative method with high efficiency means a lot in terms of both scientific research and practical application.This paper discusses the research of iterative method for solving nonlinear equations, here refers to the iterative method is based on the improved algorithm of Newton's method. This paper discusses the method based on Newton's iteration function, by increasing the iteration, similar to replace or increase the parameters, thus put forward some new variants of Newton's method, and give the iterative method for solving simple root in R, and through numerical examples are given to illustrate the effectiveness of the new method. This paper consists of six chapters.Chapter 1 we summarize mainly the associated basic theory, mainly introduces the research background of nonlinear equations and the common methods for solving nonlinear equations - iterative methods, a detailed review Newton's method and the study status.Chapter 2 discusses through a combination of classical Newton's method and the geometric mean Newton's method, and proposed a new sixth order convergence of Newon's method. Which only requires two evaluations of the function and two evaluations of the derivative per iteration, but does not require the second derivative. For a group of widely used testing questions, numerical conclusions show that the efficiency of the method is better than Classical Newton's method and Geometric mean Newton's method for most questions.Chapter 3 presents a common framework of new algorithm for solving nonlinear equations, which based on the existing algorithm. That is, comprehensive utilization of the advantages of different interpolation methods, let two methods of the same order approximately equate, the approximation of a particular statement holds on behalf of the other into the same order of the iterative, we can export the same order convergence and has its own characteristics of new or existing methods. Some common numerical examples show that the new method can compete with the classical Newton's method. Moreover, many algorithms for solving nonlinear equations such as the famous fourth-order convergence Ostrowski's algorithm can also be obtained within this framework.Chapter 4 discusses the existence of the existing algorithms form deformation, can be summarized as a unified form, by adding parameters to get a more general algorithm, convergence analysis shows that in the relations between the parameters satisfy certain conditions, and will be obtain different convergence order of the new algorithms or pre-existing algorithms.Chapter 5 from the theory illustrated the convergence of two added parameters iterative.Chapter 6 summarize the main conclusions of this paper and propose the prospects for research on Newton's method and the movement of future research.