A globally convergent inexact Newton method with a new choice for the forcing term

In inexact Newton methods for solving nonlinear systems of equations, an approximation to the step s _k of the Newton’s system J(x _k )s=−F(x _k ) is found. This means that s _k must satisfy a condition like ‖F(x _k )+J(x _k )s _k ‖≤η _k ‖F(x _k )‖ for a forcing term η _k ∈0,1). Possible choices for η _k have already been presented. In this work, a new choice for η _k is proposed. The method is globalized using a robust backtracking strategy proposed by Birgin et al. (Numerical Algorithms 32:249–260, 2003), and its convergence properties are proved. Several numerical experiments with boundary value problems are presented. The numerical performance of the proposed algorithm is analyzed by the performance profile tool proposed by Dolan and Moré (Mathematical Programming Series A 91:201–213, 2002). The results obtained show a competitive inexact Newton method for solving academic and applied problems in several areas. Supported by FAPESP, CNPq, PRONEX-Optimization.