Numerical Methods For System Of Singular Nonlinear Equations And Nonlinear Least Square Problems

The present dissertation is concerned with the numerical methods for nonlinear equations and nonlinear least square problems. Numerical methods for nonlinear equations and nonlinear least square problems are very heated research topics. They have wide applications in chemical engineering, aeromau-tics, mechanics, economy, product management, communications and transportations, etc..Levenberg-Marquardt method is one of the most important methods for solving systems of nonlinear equations. Recently, Yamashita and Fukushima[4] show that the sequence produced by the Levenberg-Marquardt method converges quadraticlly to the solution set of the equations, if the parameter is chosen as the quadratic norm of the function and under the weaker condition than the nonsingularity that the function provides a local error bound near the solution. However, the quadratic term has some unsatisfactory properties. Fan and Yuan [6] uses another method that has proved under the local error bound condition, if we choice the parameter as the norm of the function, the sequence produced by the Levenberg-Marquardt method converges quadraticlly to a solution of the system of the equations. Here we consider the choice of the parameter as the norm of the gratitude of the function. We prove under the local error bound condition that the Levenberg-Marquardt method with this parameter converges quadraticlly to a solution of the system of the equations. And we also present two globally convergent Levenberg-Marquardt algorithms using line search techniques and trust region approach respectively. In addtion, we introduce some new methods and theories for solving systems of nonlinear least square problems. A new Levenberg-Marquardt method is produced to solve the high dimemsion portfolio selection models, we can solve the problem quickly and get better results than before.