Inexact Newton-like Method And Its Application

The numerical method for solving the nonlinear equation systems is widely used in practice. Its importance is especially shown in the scientific computation of the diversified nonlinear problems. Moreover, while computers are widely used in various fields, nonlinear equation systems are dealt with in more and more fields, such as the dynamic systems, nonlinear finite elements, the nonlinear dynamics and the nonlinear optimization and the nonlinear programming. As a result, it is of practical significance to probe into the solutions for the nonlinear equation systems. Due to the complexity of the nonlinear equation systems, except some extremely special nonlinear equation systems, iteration method instead of direct method, which can hardly be used for such equation systems, should be adopted.Although Newton's iteration is a classical method for solving the nonlinear equation systems, each step of iteration in the Newton method involves computation of the Jacobian matrix or its inversion as well as the solution of the linear equation systems. When there are comparatively many variables, huge amounts of calculation are needed. Besides, when the F'(x_k) in the Newton's iteration is singular or ill-conditioned, the iteration process will not work or hardly work. Especially, when x_k is far from the solution of x~* of the linear equation systems, the iterationpoints accurately found with the direct elimination method tend to be helpless for getting the solution of the Newton equations.Based on the Newton method, the thesis mainly probes into the Newton-like method and inexact Newton-like method employed for solving the nonlinear equation systems and the two methods' convergence. Theoretically, the thesis studies their local convergence and semilocal convergence and gets some new conclusions under the reasonable assumptions . Meanwhile, under the appropriate condition, we prove Kantorvich-type theorem on the semilocal convergence of the inexact Newton method in the paper. In terms of application, the two methods are not only directly applied for solving the nonlinear equation systems but also for the numerical solutions of the unconstrained optimization and nonlinear partial differential equations. The results of the numerical experiments show the necessity and feasibility of these two methods. In addition, we discuss a variant of the Newton method in terms of its local and semilocal convergence, such variant is proved to be third-order convergent and numerical examples are given.