The Levenberg-Marquardt Method For Two Classes Of Nonlinear Problems

In this thesis, we mainly discuss the Levenberg-Marquardt method for solving singular nonlinear equations and nonlinear inequalities. Nonlinear equations have been widely used in engineering, management, economy and so on. There exist plenty of singular nonlinear equations in practice, such as the problems arising from elastic mechanics and hydromechanics, which mostly can be summed up as singular nonlinear equations. The systems of nonlinear inequalities have been extensively studied due to their various applications in modern engineering calculations, set separation problems, computer aided design problems and image reconstructions.In the Introduction, we start from the research background and status of singu-lar nonlinear equations and nonlinear inequalities. Some fundamental conceptions are introduced.In Chapter One, based on the trust region technique, a new modified Levenberg-Marquardt method is presented for singular nonlinear equations. Some combination of the norm of the function and the norm of the gradient is chosen as the new Levenberg-Marquardt parameter. Without the condition of nonsingularity, the glob-al and locally quadratic convergence of the algorithm are proved under local error bound condition which is weaker than the nonsingularity. Numerical results indicate the algorithm is effective for the given test functions.In Chapter Two, by constructing a new smoothing approximate function, the system of nonlinear inequalities is transformed into the system of nonlinear equation-s, which is solved by Levenberg-Marquardt method with the trust region technique. The global and locally super linear convergence of the algorithm is established. Nu-merical results show the efficiency of the algorithm for the given test funetions.In Chapter Three, the research is summarized.Besides, the research contents in the future is also discussed.