Choosing the Forcing Terms in an Inexact Newton Method

. An inexact Newton method is a generalization of Newton's method for solving F (x) = 0, F : IR n ! IR n , in which, at the kth iteration, the step sk from the current approximate solution xk is required to satisfy a condition kF (xk) + F 0 (xk ) skk jkkF (xk )k for a "forcing term" jk 2 [0; 1). In typical applications, the choice of the forcing terms is critical to the efficiency of the method and can affect robustness as well. Promising choices of the forcing terms are given, their local convergence properties are analyzed, and their practical performance is shown on a representative set of test problems. Key words. forcing terms, inexact Newton methods, Newton iterative methods, truncated Newton methods, Newton's method, iterative linear algebra methods, GMRES AMS(MOS) subject classifications. 65H10, 65F10 1. Introduction. Suppose that F : IR n ! IR n is continuously differentiable in a neighborhood of x 2 IR n for which F (x ) = 0 and F 0 (x ) is nonsingular....