On the Gauss-Newton Method for Solving Equations
DOI:
https://doi.org/10.4067/S0716-09172012000100002Keywords:
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.Abstract
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.References
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[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).
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[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).
[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).
Published
2012-01-29
How to Cite
[1]
I. K. Argyros and S. Hilout, “On the Gauss-Newton Method for Solving Equations”, Proyecciones (Antofagasta, On line), vol. 31, no. 1, pp. 11-24, Jan. 2012.
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