Inexact Newton Methods Applied to Under – Determined Systems by Joseph P . Simonis A Dissertation

Consider an under-determined system of nonlinear equations F (x) = 0, F : IR → IR, where F is continuously differentiable and m > n. This system appears in a variety of applications, including parameter–dependent systems, dynamical systems with periodic solutions, and nonlinear eigenvalue problems. Robust, efficient numerical methods are often required for the solution of this system. Newton’s method is an iterative scheme for solving the nonlinear system of equations F (x) = 0, F : IR → IR. Simple to implement and theoretically sound, it is not, however, often practical in its pure form. Inexact Newton methods and globalized inexact Newton methods are computationally efficient variations of Newton’s method commonly used on large-scale problems. Frequently, these variations are more robust than Newton’s method. Trust region methods, thought of here as globalized exact Newton methods, are not as computationally efficient in the large–scale case, yet notably more robust than Newton’s method in practice. The normal flow method is a generalization of Newton’s method for solving the system F : IR → IR, m > n. Easy to implement, this method has a simple and useful local convergence theory; however, in its pure form, it is not well suited for solving large-scale problems. This dissertation presents new methods that improve the efficiency and robustness of the normal flow method in the large–scale case. These are developed in direct analogy with inexact–Newton, globalized inexact–Newton, and trust–region methods, with particular consideration of the associated convergence theory. Included are selected problems of interest simulated in MATLAB.