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Local convergence for the curve tracing of the homotopy method
Palabras clave:
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 (en)Descargas
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 established. A numerical example is also provided where our results compare favourably with the corresponding ones in [6].
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 más 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].
Referencias
E. A. Allgower, A survey of homotopy methods for smooth mappings, in Numerical Solution of Nonlinear Equations, Lecture Notes in Math, vol. 878, Springer-Verlag, Berlin-New York 1980, 1-29.
I. K. Argyros, On the Newton-Kantorovich hypothesis for solving equations, J. Comput. Appl. Math., 16 9 (2004), 313-332.
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.
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.
I. K. Argyros, Newton Methods, Nova Science Publ. Inc., New York, 2005.
M. T. Chu, On a numerical treatment for the curve tracing of the homotopy method, Numer. Math. 42 (1983), 323-329.
L. V. Kantorovich, & G. P. Akilov, Functional Analysis in Normed Spaces, Pergamon Press, Oxford, 1982.
W. C. Rhelnboldt, Solution fields of nonlinear equations and continuation methods, SIAM J. Numer. Anal 17 (1980), 221-237.
