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Some convergence results for the Newton-GMRES algorithm

In this paper, we consider both local and global convergence of the Newton algorithm to solve nonlinear problems when GMRES is used to invert the Jacobian at each Newton iteration. Under weak assumptions, we give a sufficient condition for an inexact solution of GMRES to be a descent direction in order to apply a backtracking technique. Moreover, we extend this result to a finite difference scheme considering also the use of preconditioners. Then we show the impact of the condition number of the Jacobian on the local convergence of the Newton-GMRES algorithm.