On the Gauss-Newton Method for Solving Equations

  • Ioannis K. Argyros Cameron University.
  • Saïd Hilout Poitiers University.
Palabras clave: Gauss—Newton method, semilocal convergence, Frechet—derivative, Lipschitz/center—Lipschitz condition, convergence domain, método de Gauss-Newton, convergencia semilocal, derivada de Frechet, condición de Lipschitz, centro de Lipschitz.


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 ofthe Gauss-Newton method to a solution ofan 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.

Biografía del autor

Ioannis K. Argyros, Cameron University.
Department of Mathematics Sciences.
Saïd Hilout, Poitiers University.
Laboratoire de Mathématiques et Applications.


[1] I. K. Argyros, A unifying local—semilocal convergence analysis and applications for two—point Newton—like methods in Banach space, J. Math. Anal. and Appl., 298, pp. 374—397, (2004)

[2] I. K. Argyros, A convergence analysis of Newton—like methods for singular equations using outer or generalized inverses, Applicationes Mathematicae, 32, pp. 37—49, (2005).

[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 of equations, J. Math. Anal. Appl., 15, pp. 243—252, (1966).

[5] J. M. Gutiérrez, 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 H¨ older 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 of Gauss—Newton’s method, J. of Southeast 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).
Cómo citar
Argyros, I., & Hilout, S. (2012). On the Gauss-Newton Method for Solving Equations. Proyecciones. Journal of Mathematics, 31(1), 11-24. https://doi.org/10.4067/S0716-09172012000100002