On the convergence of inexact Newton methods

The inexact Newton method is widely used to solve systems of non-linear equations. It is well-known that forcing terms should be chosen relatively large at the start of the process, and be made smaller during the iteration process. This paper explores the mechanics behind this behavior theoretically using a simplified problem, and presents theory that shows a proper choice of the forcing terms leads to a reduction in the non-linear error that is approximately equal to the forcing term in each Newton iteration. Further it is shown that under certain conditions the inexact Newton method converges linearly in the number of linear iterations performed throughout all Newton iterations. Numerical experiments are presented to illustrate the theory in practice.Electrical Engineering, Mathematics and Computer Scienc