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Proyecciones (Antofagasta)

versión impresa ISSN 0716-0917

Proyecciones (Antofagasta) vol.31 no.1 Antofagasta mar. 2012 

Proyecciones Journal of Mathematics Vol. 31, No 1, pp. 11-24, March 2012. Universidad Catolica del Norte Antofagasta - Chile

On the Gauss—Newton method for solving equations

Ioaniss K. Argyros

Cameron University, U.S.A.

Saïd Hitlout

Poitiers University, France



We use a combination of the center—Lipschitz condition with the Lipschitz condition condition on the Frechet—derivative of the operator involved to provide a semilocal convergence analysis of the Gauss-Newton method to a solution of an equation. Using more precise estimates on the distances involved, under weaker hypotheses, and under the same computational cost, we provide an analysis of the Gauss— Newton method with the following advantages over the corresponding results in [8]: larger convergence domain; finer error estimates on the distances involved, and an at least as precise information on the location ofthe solution

AMS Subject Classification. 65F20, 65G99, 65H10, 49M15.

Key Words. Gauss—Newton method, semilocal convergence, Frechet— derivative, Lipschitz/center—Lipschitz condition, convergence domain.



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[3] I. K. Argyros, Convergence and applications of Newton-type iterations, Springer Verlag Publ., New York, (2008).

[4] A. Ben-Israel, A Newton-Raphson method for the solution of systems ofequations, J. Math. Anal. Appl., 15, pp. 243-252, (1966).

[5] J.M.Gutierrez, A new semilocal convergence theorem for Newton's method, 79, pp. 131-145, (1997).

[6] Z. Huang, The convergence ball of Newton's method and the uniqueness ball of equations under Holder continuous derivatives, Comput. Appl. Math., 47, pp. 247-251, (2004).

[7] L. V. Kantorovich, G.P. Akilov, Functional analysis in normed spaces, Pergamon Press, New York, (1982).

[8] C. Li, W. Zhang, Convergence ofGauss-Newton's method, J. ofSouth-east University, (Don Nan Da Xue Xue Bao), (Natural Science Edition in Chinese), Vol. 31, 5, sept., pp. 135-138, (2001).

[9] P. A. Wedin, Perturbation theory for pseudo-inverse, BIT, 13, pp. 217-232, (1973).

[10] Y. Yuan, W. Sun, Optimization theory and methods. Nonlinear Programming. Springer Optimization and Its Applications, Springer, New York, (2006).

Ioannis K. Argyros

Department of Mathematics Sciences Cameron university

Lawton, OK 73505, U.S.A.

e-mail :

Said Hilout

Laboratoire de Mathematiques et Applications

Poitiers university

Bd. Pierre et Marie Curie,

Teleport 2, B.P. 30179

86962 Futuroscope Chasseneuil Cedex,


e-mail :

Received : January 2011. Accepted : October 2011

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