Three-point Iterative Methods For Solving Nonlinear Equations

The problem of solving nonlinear equations not only plays an important role in the theory and practice of mathematics,but also has a wide range of applications in the fields of physics,engineering,and computer science.The study of nonlinear equations promotes the fusion and development of mathematics and other disciplines.Therefore,it is of great theoretical and practical significance to study the iterative solution method of nonlinear equations.This paper mainly studies the three-point iterative method of nonlinear equations.The main objective is to propose an iterative method with the highest convergence order and computational efficiency.The main contents of this article are as follows:Firstly,based on the two-point iterative method without memory,two kinds of three-point iteration methods without memory are obtained by two improved methods:the first improved method is direct interpolation method,and the direct interpolation method is used to obtain three points without memory.The convergence order of the method is 8th order.The second improvement method is the weight function approximation method.The convergence order of three-point method without memory obtained by the weight function approximation method is 7th order,and it can reach 8th order under some special conditions.Secondly,in order to obtain an iterative method with higher convergence order,the acceleration parameters in the both three-point iteration method without memory are approximated by Newton interpolation.According to the difference of the number of interpolation nodes,four different degrees of interpolation can be obtained.Approximation,so that both methods without memory can obtain methods with memory with four different convergence orders.The convergence order of the first method with memory can be increased from the 8th order of the original method without memory to 8.69,9.42,11 and 14,and the convergence order of the second method with memory can be increased from the original 7th order to 7.89,8.80,10.66 and 14,under certain special conditions,can be increased from the original 8th order to 8.47,9,10 and 12.Finally,all the iterative methods and related theorems proposed in this paper are verified by numerical experiments.