Interative Solutions Of Non-linear Equations With Parameters

Nonlinear problems is one of the main fields in modern mathematical research, the numerical solution of the nonlinear equations with parameters is a very important direction in the researching of the nonlinear problems. Because of its extensive practical background and important theoretical value, the numerical solution of the nonlinear equations with parameters is always focused by many numerical analysis researchers. In this paper, the numerical solution of the nonlinear equations with parameters F ( x , s ) = 0 is studied, under the point estimates and weak conditions the major sequence is constructed, at the same time, the existence and convergence theorem is obtained.The main results of the thesis are summarized as following:Firstly, some methods for solving nonlinear equations under point estimates and weak conditions are introduced. Some results of solving nonlinear equations with parameters under the Kantorovich condition are presented. Simultaneously the main content of the article is given.Secondly, the major function and major sequence are constructed. The existence and convergence theorem of Newton method and quasi Newton method solving nonlinear equation with parameters F ( x , s ) = 0 is obtained under point estimates and weak conditions. The error estimates are also given.Thirdly, theγcondition is given, and theγcondition is compared with point estimates condition. The relation of some criterions is considered and the conditions in Smale's results are weakened.Fourthly, Broyden iterative method is studied to solve nonlinear equation with parameters and the existence and convergence theorem is also given.