Iterative Methods And Further Research For Solving Nonlinear Equations

Solving nonlinear equation (systems) is a classical problem and an important subject.It is widely used in many practical problems, such as nonlinear mechanics problems, circuitissues, economics and nonlinear programming problem. All of them can be abstracted as amathematical model f(x)=0,x(?)R~n.We make a further research on iterative methods fornonlinear equations in this paper.This paper is divided into five parts. The first part introduces backgrounds and thepresent achievements about nonlinear equations. The second part is a brief summary aboutthe existing iterative methods for nonlinear equations and their convergence rates. Part3focuses on a new variant of Newton’s method based on the complex trapezoidal formulaafter a profound study of different Newton’s method: the midpoint Newton method,arithmetical Newton method and harmonic average Newton’s method. Although ourmethod is slightly complex, it is much faster than the others. Its convergence properties arethoroughly discussed and a detailed comparison with the other Newton’s methods is alsomade to prove the efficiency of our method. In the fourth part, we construct the basesfunctions based on the geometry and a new class of fifth-order Newton iteration is given bychanging the coefficients of bases functions. Numerical experiments are given to show theadvantages of the method proposed in this paper. The last part is the summary and prospect.