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Cubo (Temuco)

versión On-line ISSN 0719-0646

Cubo vol.13 no.3 Temuco oct. 2011

http://dx.doi.org/10.4067/S0719-06462011000300001 

CUBO A Mathematical Journal Vol.13, Nº03, (1-15). October 2011

 

On the semilocal convergence of Newton-type methods, when the derivative is not continuously invertible

 

Ioannis K. Argyros and Saïd Hilout

Department of Mathematics Sciences, Cameron University, Lawton, OK 73505, USA. email: iargyros@cameron.edu

Laboratoire de Mathematiques et Applications, Poitiers University, Bd. Pierre et Marie Curie, Teleport 2, B.P. 30179 86962 Futuroscope Chasseneuil Cedex, France email: said.hilout@math.univ—poitiers.fr


ABSTRACT

We provide a semilocal convergence analysis for Newton-type methods to approximate a locally unique solution of a nonlinear equation in a Banach space setting. The Frechet-derivative of the operator involved is not necessarily continuous invertible. This way we extend the applicability of Newton-type methods [1]—[12].

We also provide weaker sufficient convergence conditions, and finer error bound on the distances involved (under the same computational cost) than [1]-[12], in some intersting cases. Numerical examples are also provided in this study

Keywords: Newton-type methods, Banach space, small divisors, non-invertible operators, semilocal convergence, Newton-Kantorovich-type hypothesis.

Mathematics Subject Classification: 65H10, 65G99, 65J15, 47H17, 49M15.


RESUMEN

Ofrecemos un análisis de convergencia semilocal de los metodos de Newton type para aproximar una solución local unica de una ecuación no lineal en un entorno de un espacio de Banach. L derivada de Frechet del operador en cuestion no es necesariamente invertible continua. De esta manera ampliamos la aplicabilidad de los metodos del tipo Newton [1]-[12].

Tambien proporcionamos condiciones suficientes mas debiles de convergencia, y una cota de error más fina de las distancias involucradas que [1]-[12] (en el mismo coste computacional), en algunos casos interesantes. tambien presentamos ejemplos numericos.


References

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[2] Argyros, I.K., On the Newton-Kantorovich hypothesis for solving equations, J. Comput. Appl. Math., 169 (2004), 315-332.         [ Links ]

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

[4] Argyros, I.K., Computational theory of iterative methods, Series: Studies in Computational Mathematics, 15, Editors: C.K. Chui and L. Wuytack, Elsevier Publ. Co., New York, USA, 2007.         [ Links ]

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[7] Craven, B.D., Nashed, M Z., Generalized implicit function theorems when the derivative has no bounded inverse, Nonlinear Anal., 6 (1982), 375-387.         [ Links ]

[8] Hald, O.H., On a Newton-Moser type method, Numer. Math., 23 (1975), 411-426.         [ Links ]

[9] Jerome, J.W., An adaptive Newton algorithm based on numerical inversion: regularization as postconditioner., Numer. Math., 47 (1985), 123-138.         [ Links ]

[10] Moret, I., On a general iterative scheme for Newton-type methods, Numer. Funct. Anal. Optim., 9 (1987-1988), 1115-1137.         [ Links ]

[11] Potra, F.A., Sharp error bounds for a class of Newton-like methods, Libertas Mathematica, 5 (1985), 71-84.         [ Links ]

[12] Yamamoto, T., A convergence theorem for Newton-like methods in Banach spaces, Numer. Math., 51 (1987), 545-557.         [ Links ]


Received: September 2009.

Revised: October 2009.