Numerical Solutions Of The System Of Singular Nonlinear Equations

This thesis focuses on the Levenberg-Marquardt method for solving the singular nonlinear system of equations. We consider the choice of the Levenberg-Marquardt parameter as some combination of the norm of the function and the norm of the gradient. We prove under the local error bound condition that the Levenberg-Marquardt method with this parameter convergences quadratically to a solution of the system of the equations by the technique of the singular value decomposition of the Jacobi matrix. And we also present the globally convergent Levenberg-Marquardt algorithm by using trust region approach and the inexact Levenberg-Marquardt algorithm with the conjugate gradient method for linear equations. This paper consists of five chapters.In Chapter One, we briefly introduce the background and the research significance of the system of singular nonlinear equations.In Chapter Two, under the weaker conditions than the non-singularity, we prove that the Levenberg-Marquardt method has locally quadratic convergence by the technique of the singular value decomposition of the Jacobi matrix. Some new Levenberg-Marquardt parameters and the corresponding Levenberg-Marquardt algorithms are given and the numerical experiments are carried out.In Chapter Three, using the trust-region techniques, we present the globally convergent Levenberg-Marquardt method. We prove that the algorithm has the global convergence and the locally quadratic convergence and give the corresponding numerical experiments.In Chapter Four, we present the inexact Levenberg-Marquardt method with the conjugate gradient method for linear equations per iteration. We prove that the algorithm has the locally suplinear convergence and the locally quadratic convergence and give the corresponding numerical experiments. At last, some conclusions are given.