A New Choice for the Forcing Term and a Global Convergent Inexact Newton Method

Inexact Newton methods for solving $F(x) = 0,\,F:D \subset \R^{n} \rightarrow \R^{n}$ with $F \in C{^1}(D)$,where $D$ is an open and convex set, find approximation to the step $s_{k}$ of the Newton's systems $J(x_{k})s = - F(x_{k})$, instead of solving this system exactly as done by Newton's method. This means that $s_{k}$ must satisfy a condition like $||F(x_{k}) + J(x_{k})s_{k}|| \leq \eta_{k}||F(x_{k})||$ for a forcing term $\eta_{k} \in [0,1]$ (\cite{des}). Many authors have presented possible choices for $\eta_{k}$ (see \cite{ew}). In this work, a new choice for $\eta_{k}$ is introduced, the new method obtained is globalized by the introduction of a robust backtracking strategy (see \cite{natasha}, \cite{dglm}), and its convergence properties are proved. The numerical performance of the new method is presented by plotting the performance profile of the method as proposed in \cite{dm}. The results obtained show a competitive new inexact Newton method.