Ioannis K. Argyros

USA Department of Mathematical Sciences Cameron University OK 73505 Lawton, USA

e-mail: iargyros@cameron.edu

Abstract. The local convergence of a Newton-method for the tracing of an implicitly defined smooth curve is analyzed. The domain of attraction is shown to be larger than in [6]. Moreover finer error bounds on the distances involved are obtained and quadratic instead of geometrical order of convergence is es- tablished. A numerical example is also provided where our results compare favourably with the corresponding ones in [6].

Keywords and phrases. Curve tracing, homotopy method, domain of attraction, radius of convergence, Newton-Kantorovich theorem/hypothesis, smooth curve, Moore-Penrose generalized inverse.

2000 Mathematics Subject Classification. Primary: 65K05, 65G99. Secondary: 47H17, 49M15.

Resumen. Se analiza la convergencia local de un método de Newton para trazado de una curva suave definida implícitamente. Se muestra que el dominio de atracción es más grande que en [6]. Además se obtienen errores mas finos para las cotas de las distancias involucradas y se establece orden cuadrático en lugar de lineal para la convergencia. Se da un ejemplo numérico donde nuestro resultado se compara favorablemente con los resultados correspondientes en [6].

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[2] I. K. Argyros, On the Newton-Kantorovich hypothesis for solving equations, J. Comput. Appl. Math., 169 (2004), 313-332.         [ Links ]

[3] I. K. Argyros, A note on a new way for enlarging the convergence radius for Newton's method, Math. Sci. Res. J. 8 (2004) 5, 147-153.         [ Links ]

[4] I. K. Argyros, A convergence analysis of Newton-like methods for singular equations using outer or generalized inverses, Applicationes Mathematicae 32 (2005) 1, 37-49.         [ Links ]

[5] I. K. Argyros, Newton Methods, Nova Science Publ. Inc., New York, 2005.         [ Links ]

[6] M. T. Chu, On a numerical treatment for the curve tracing of the homotopy method, Numer. Math. 42 (1983), 323-329.         [ Links ]

[7] L. V. Kantorovich, & G. P. Akilov, Functional Analysis in Normed Spaces, Pergamon Press, Oxford, 1982.         [ Links ]

[8] W. C. Rheinboldt, Solution fields of nonlinear equations and continuation meth- ods, SIAM J. Numer. Anal. 17 (1980), 221-237.         [ Links ]

(Recibido en marzo de 2006. Aceptado en agosto de 2006)