Algorithms For Solving Interval Nonlinear Equation(s)

In 1966,American mathematician Moore pioneered the discipline of interval anal-ysis which is an important branch of numerical analysis and widely used in many disci-plines.Interval iteration method is an important application of interval analysis,which has great advantages in controlling errors and determining the existence and unique-ness of solutions.In this paper,the interval nonlinear equation,interval non-smooth equation and interval nonlinear equations are studied and some important theoretical results are obtained.The content of this paper is as follows.In chapter 1,the research background and significance,present situation of home and abroad of this paper are introduced.Some important definitions of interval analysis and some important theoretical results are introduced in the preliminary knowledge,interval Newton method and interval Krawczyk method and their important properties are also introduced.In chapter 2,the problem of solving the interval nonlinear equation is studied.The extended interval Newton method proposed by Nikas[25]has been improved based on the monotonic segment technique.The non-monotonic interval is divided into sever-al monotone subintervals,and then the interval zero of interval nonlinear equation is solved in the monotonic subinterval.The high order convergence numerical algorithm for solving interval zero of interval nonlinear equations is proposed,and the conver-gence rate of the method is proved.The numerical example shows that the new method improves the computational efficiency.In chapter 3,the problem of solving interval non-smooth equation is studied and interval slope method proposed by Lin[40]is promoted.Using interval derivative to calculate the smooth part of interval non-smooth equation and interval slope to calculate the non-smooth part,and then interval non-smooth equation is solved by the monotone segment technique.The convergence and convergence rate of the method are proved and the numerical example shows that the new method improves the computational efficiency.In chapter 4,the problem of solving the interval nonlinear equations is studied.The interval Krawczyk operator is improved to determine the exact region zero of the interval nonlinear equations.Converting interval nonlinear equations into general non-linear equations to determine the vertices and edges of the region zero.The relevant theoretical results are obtained and the feasibility and validity of this method are veri-fied by numerical examples.