Two Family Of High Order Iterative Method For Solving Nonlinear Equation F(x)=0

With the development of scientific technology and electric computer,one of the most important and challenging problems in scientific and engineering applications is to find the solutions of the nonlinear equations.The nonlinear finite element problem,the boundary value problems appearing in Kinetic theory of gases,elasticity and other applied areas are reduced to solving these equations.Many optimization problems also lead to such equations.This paper is concerned with the high-order iterative methods for finding a simple root r,i.e.,f(r)=0 and f′(r)≠0 of f(x)=0,where f:R→R,be the continuously differentiable real function.This paper is divided into four chapters:In first chapter,the actual back-ground of this problem is mainly introduced,and the development situation of this problem is recollected, then some elementary knowledge is outlined.In second chapter,a family of seventh-order iterative methods for solving nonlinear equation f(x)=0 is presented and analyzed.This family of seventh-order methods contains the Bi's seventh-order methods [10]and many other seventh-order iterative methods as special cases.In terms of computational cost,per iteration the new methods require three evaluations of the function and one evaluation of its first derivative,therefore their efficiency index is 1.627.The convergence of this family of methods is analyzed to establish its seventh-order convergence. Numerical comparisons are made with several other existing methods to show the performance of the presented methods.In third chapter,a family of new eighth-order iterative methods for solving nonlinear equation f(x)=0 are presented and analyzed.The new methods are based on Kou's seventh-order methods[5].In terms of computational cost, per iteration the new methods require three evaluations of the function and one evaluation of its first derivative,therefore their efficiency index is 1.682.The convergence of the new iterative methods is analyzed to establish their eighth-order convergence.Several examples are given to illustrate the efficiency and the performance of the new methods.Finally, we extend the present methods to the general multi-step iterative method and obtain a new family of multi-step iterative methods.In the fourth chapter,we make a simple summary to this paper.