A new choice for the forcing term and a global convergent inexact Newton method 1

Inexact Newton methods for solving F (x) = 0, F: D ∈ IRn → IRn with F ∈ CI1(D), where D is an open and convex set, are characterized by finding approximately the step sk of the Newton’s systems J(xk)s = −F (xk), instead of solving it exactly as done by Newton’s method. This means that sk must satisfy a condition like ||F (xk) + J(xk)(xk+1 − xk)| | ≤ ηk||F (xk)|| for a forcing term ηk ∈ [0, 1] ([3]). Many authors have presented possible choices for ηk (see [6], for example). In this work, it is presented a new way of choosing the forcing term ηk with its geometrical motivation. A backtracking strategy (see [4],[1]) is incorporated, with which a global convergence result is obtained for the final algorithm. Convergence properties of the new method are also proved. To solve the linear systems of the inexact Newton method, the direction s̄k is found by the GMRES with restart [7]. In order to avoid stagnation in the process of solving the linear systems, a strategy based on the hybrid ideas, proposed by Brezinski and Redivo-Zaglia in [2] is included. This hybrid strategy is used to find an initial approximation for each new cycle of GMRES. Besides this, some other strategies to prevent an extremely large number os inner iterations, or linear searches in directions which do not give good decrease in ||F ||, are introduced. The analysis of the numerical performance of this new inexact Newto