Local Convergence Analysis Of Multi-point Iterative Method In Banach Space

This paper studies and proposes a new class of multi-point iterative methods,including an improved sixth-order convergence iterative method for solving nonlinear equations and another third-order iterative method extended to solving nonlinear equations.Chapter 1: A brief introduction to the background and significance of the research problem,the theoretical research results in recent years,and the basic definition of mathematical concepts in the field of solving nonlinear equations.The development process of quasi-Newtonian method and the construction idea of fourth-order quasiNewtonian method are introduced.Chapter 2: Theoretical proof of local convergence analysis for a class of uncertain three-step iterative methods for solving nonlinear equations,mainly studies its local convergence theory under certain assumptions,and applies the theorem to Newton’s method and a fifth-order iterative method.The theoretical analysis of this chapter lays the foundation for the work of Chapter 3.Chapter 3: A new class of sixth-order iterative methods for solving nonlinear equations(groups)is constructed.Its local convergence is studied,its convergence domain is given,and the convergence order of the new iterative method is proved.Compared with the original iterative method before the improvement,the new iterative method improves the convergence order by one order without changing the computational cost.At the same time,the effectiveness of the new iterative method is verified by physical experiments and practical problems in chemical engineering,and the numerical results show that compared with other sixth-order methods,the new iterative method has a larger convergence domain radius and faster convergence speed.Further study of the real dynamic behavior and complex dynamic behavior of the new sixth-order iterative method,the fractal plot shows that the new sixth-order iterative method has better stability.Chapter 4: A class of iterative methods for solving nonlinear equations is proposed,the convergence theory is proved,and finally numerical experiments are carried out,and the numerical results and theoretical analysis verify the stability and efficient computational performance of the new method.At the same time,suggestions for parameter selection of the new iterative method in actual calculation are given.Chapter 5: summarizes the work of the full paper and looks forward to possible future research directions and content.This article has a total of 28 figures,23 tables,and references 59.