A new hybrid Newton-Krylov method is presented. The generalized minimal residual (GMRES) method is used to solve the Newton equation approximately. A global strategy restricted to a suitable Krylov subspace is performed. It consists of two parts. The first one is the backtracking procedure of the inexact Newton backtracking method and the second one is a backtracking technique along a piecewise linear curve that involves the current search direction and an additional direction selected using the information provided by GMRES. The method is an extension of Newton-GMRES backtracking techniques designed to improve performance when the search direction is a poor descent one. A convergence analysis is performed and the consistency with restarting and preconditioning procedures is also proved.
By intensive numerical experiments it is proved that the new strategy enhances the global convergence of Newton-GMRES backtracking method for cases where the last mentioned method fails. Numerical tests also suggest that the new approach and the classical backtracking Newton-GMRES method have similar cost.