Newton Dogleg Methof Of The Under-Determined System

In this thesis, we consider the nonlinear equations of the under-determined sys-tem This system has been widely applied in the field of science and technology, such as the nonlinear eigenvalue problems, the problems of periodic orbits in dynamic systems, the nonlinear finite element problems, the nonlinear programming problems and so on. How to solve the under-determined system is one of the basic problems in scientific computation. So the research about the under-determined system has very important actual significance.Newton method is also the most classical method for solving the nonlinear equa-tions of the under-determined system. Newton method has the perfect theory basis, and is easy to operate. But, the choice of the initial value is relatively strict. In addi-tion, it needs the accurate solution of the linear equations giving by every iteration. So, Newton method will cost a lot of time for large scale problems. Thus, we derive various modified Newton Method and the Newton like method(such as the Inexact Newton Method, the Globalized Newton Method)[16]. Then we can solve the nonlin-ear equations in effective way.In this thesis, firstly, we give more sufficient theoretical basis about the Ex-act Newton Single Dogleg Method of the under-determined system given by J. P. Simonis[35]. Secondly, we apply the double Dogleg Method to the exact Newton Method of the under-determined system and show the monotonicity analysis of the algorithm. Thirdly, We apply the single Dogleg Method and double Dogleg Method to the Inexact Newton method of the under-determined system and save a lot of time when solving the nonlinear equations. Then we prove the the convergence of the two algorithms. Finally, we show the feasibility and effectiveness of the algorithms given in this thesis by some numerical examples (the Bratu problem and the Chan problem).