Interval Multiple Iteration Method For Nonlinear Equations

A large number of nonlinear equation(groups)problems encountered in practical problems are mainly solved by numerical methods.The numerical solution of nonlinear equations(groups)is also an important basis for solving many nonlinear problems.Because the nonlinearity of nonlinear equations(groups)and various nonlinear problems brings more difficulties to the solution of the problem,its numerical solution methods need to be explored and perfected constantly.In this thesis,several interval multiple iteration methods for solving nonlinear equations are discussed by combining interval analysis method with multiple iteration method.First of all,based on Newton method,two kinds of interval multiple iteration algorithms are constructed by constructing different Newton interval multiple iteration operators,namely P-Newton interval multiple iteration method and Newton-S interval multiple iteration method.The correlation properties of iteration operators and the convergence and convergence order of the algorithm are proved.The convergence order of the algorithm is relatively high,and the efficiency index is improved.Both are higher than the interval Newton method(?)=1.4142….Among them,two specific P-Newton interval multiple iteration method and Newton-S interval multiple iteration method are given,and the performance indexes are (?)=1.4309…and (?)=1.5874….Secondly,based on interval slope method,a multi-iteration interval slope scheme for solving non-differentiable nonlinear equations is established,and the properties of the operators concerned and the convergence of the algorithm are proved.Finally,by using Hansen’s generalized interval operation and the introduction of auxiliary equations,the corresponding interval multiple iteration method for finding the multiple roots of nonlinear equations is established,and the correlation properties and convergence are proved.In this thesis,the corresponding software is compiled for the established interval multiple iteration algorithms.The numerical calculation experiment is carried out.The numerical example results are given to verify the convergence theory and the actual calculation effect of the algorithm.The results show that the speed of the algorithm is improved.These algorithms are feasible,reliable and effective.