1Cameron University, Lawton, USA. Email: iargyros@cameron.edu
]]> Abstract
We provide a semilocal convergence analysis for a cubically convergent two-step Newton method (2) recently introduced by H. Homeier [8], [9], and also studied by A. Özban [13]. In contrast to the above works we examine the semilocal convergence of the method in a Banach space setting, instead of the local in the real or complex number case. A comparison is given with a two step Newton--like method using the same information.
Key words: Two-step Newton method, Newton method, Banach space, majorizing sequence, Newton--Kantorovich hypothesis, semilocal convergence, Fréchet-derivative.
Proporcionamos un análisis de convergencia semilocal para un método de Newton de dos pasos, cúbicamente convergente, recientemente introducido por H. Homeier [8], [9], también estudiado por A. Özban [13]. En contraste con esto, examinamos la convergencia local del método en espacios de Banach en lugar del local, en el caso real y complejo. Damos una comparación con el método de Newton de dos pasos usando la misma información.
Palabras clave: Método de Newton de dos pasos, método de Newton, espacio de Banach, secuencia mayorante, hipótesis de Newton--Kantorovich, convergencia semilocal, derivada de Fréchet.
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References
[1] Amat, S., Busquier, S. & Salanova, M., `A fast Chebyshev's method for quadratic equations´, Appl. Math. Comput. 148, (2004), 461-474. [ Links ]
[2] Appel, J., DePascale, E., Lysenko, J. & Zabrejko, P., `New result on Newton-Kantorovich approximations with applications to nonlinear integral equations´, Numer. Funct. Anal. and Optimiz. 18, 1--2 (1997), 1-17. [ Links ]
[3] Argyros, I., `A unifying local-semilocal convergence analysis and applications for two-point Newton-like methods in Banach space´, J. Math. Anal. Applic. 298, (2004), 374-397. [ Links ]
[4] Argyros, I., Convergence and applications of Newton-type iterations, Springer Verlag, New York, 2008. [ Links ]
[5] Argyros, I. & Chen, D., `On the midpoint method for solving nonlinear operator equations and applications to the solution of integral equations´, Revue d'Analyse Numérique et de Théorie de l'Approximation 23, (1994), 139-152. [ Links ]
[6] Gutierrez, J. & Hernandez, M., `An acceleration of Newton's method: super-Halley method´, Appl. Math. Comp. 117, (2001), 223-239. [ Links ]
[7] Hernandez, M. & Salanova, M., `Modification of the Kantorovich assumptions for semilocal convergence of the Chebyshev methods´, J. Comput. Appl. Math. 126, (2000), 131-143. [ Links ]
[8] Homeir, H., `A modified method for root finding with cubic convergence´, J. Comput. Appl. Math. 157, (2003), 227-230. [ Links ]
[9] Homeir, H., `A modified Newton method with cubic convergence´, J. Comput. Appl. Math. 169, (2004), 161-169. [ Links ]
[10] I. K.Argyros, & Chen, D., `The midpoint method for solving equations in Banach spaces´, Appl. Math. Letters 5, (1992), 7-9. [ Links ]
[11] Kantorovich, L. & Akilov, G., Functional Analysis in Normed Spaces, Pergamon Press, Oxford, 1982. [ Links ]
[12] Ortega, J. & Rheinboldt, W., Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970. [ Links ]
[13] Ozban, A., `Some new variants of Newton's method´, Appl. Math. Letters 17, (2004), 677-682. [ Links ]
[14] Yamamoto, T., `A convergence theorem for Newton-like methods in Banach spaces´, Numer. Math. 51, (1987), 545-557. [ Links ]
Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:
@ARTICLE{RCMv42n1a02,
AUTHOR = {Argyros, Ioannis K.},
TITLE = {{On the semilocal convergence of a fast two-step Newton method}},
JOURNAL = {Revista Colombiana de Matemáticas},
YEAR = {2008},
volume = {42}, ]]>
number = {1},
pages = {15-24}
}