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:R(n) right arrow R(n), in which, at the kth iteration, the step s(k) from the current approximate solution x(k) is required to satisfy a condition parallel to F(x(k))+F'(x(k))s(k) parallel to less than or equal to eta(k) parallel to F(x(k)) parallel to for a ''forcing term'' eta(k) is an element of [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.