Some Convergence Results for the Newton - Gmres Algorithm Rémi Choquet , Jocelyne Erhel
نویسنده
چکیده
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 suucient 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 nite diierence 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. (URA 227) Université de Rennes 1 – Insa de Rennes et en Automatique – unité de recherche de Rennes Quelques r esultats de convergence pour l'algorithme de Newton-GMRES R esum e : Dans cet article, nous etudions la convergence de l'algorithme de Newton appliqu e a un probl eme non lin eaire, lorsqu'' a chaque it eration GMRES r esout le sys-t eme lin eaire de mani ere approch ee. Une condition suusante sur la solution calcul ee par GMRES est donn ee pour qu'elle soit une direction de descente aan d'utiliser une technique de hh backtracking ii. Puis, nous etendons ce r esultat au cas o u chaque produit matrice vecteur est approch ee par un sch ema aux dii erences nies, en consi-d erant aussi l'utilisation d'un pr econditionnement. Finalement la convergence locale de Newton-GMRES est d emontr ee dans ce contexte en soulignant l'importance d'un bon pr econditionnement.
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