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:\mathbb{R}^n \to \mathbb{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 $\|F(x_k ) + F'(x_k )s_k \| \leqslant \eta _k \|F(x_k )\|$ for a “forcing term” $\eta _k \in [0,1)$. 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.