Inexact Broyden Methods For Solving Nonlinear Equations

This paper studies numerical algorithms for solving nonlinear equations.Numerical methods for nonlinear equations is an important research branch in the field of computational mathematics.There are some efficient algorithms such as Newton methods,quasi-Newton methods and Gauss-Newton methods et.al..These methods possess high precision and fast convergence.Some methods have become their inexact versions,for example,the inexact Newton method can reduce calculated amount,which only requires computing linear equations subproblem each iteration.Based on the exact Broyden methods in the papers[25,26],we propose an inexact Broyden method for smooth nonlinear equations and a smoothing inexact Broyden method for nonsmooth nonlinear equations and discuss their convergence properties.The paper is organized as follows:In Chapter 1,we simply introduce the background and related preparatory knowledge.In Chapter 2,we propose an inexact Broyden method for solving smooth nonlinear equations and establish its global convergence and superlinear convergence.Moreover,we do some numerical experiments,numerical results show that the method performs well.In Chapter 3,we introduce a smoothing inexact Broyden method for solving nonsmooth nonlinear equations and prove its global convergence and superlinear convergence.In our numerical computing,we convert an nonlinear complementarity problem into an equivalent nonlinear system by the use of Fischer-Burmeister function.Numerical results show that the method is efficient.