On-line version ISSN 0719-0646

Cubo vol.12 no.1 Temuco  2010

http://dx.doi.org/10.4067/S0719-06462010000100014

CUBO A Mathematical Journal Vol.12, N° 01, (161-174). March 2010

Convergence Conditions for the Secant Method

Ioannis K. Argyros and Saïd Hilout

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

Poitiers university, Laboratoire de Mathématiques et Applications, 86962 Futuroscope Chasseneuil Cedex, France email : said.hilout@math.univ-poitiers.fr

ABSTRACT

We provide new sufficient convergence conditions for the convergence of the Secant method to a locally unique solution of a nonlinear equation in a Banach space. Our new idea uses recurrent functions, Lipschitz-type and center-Lipschitz-type instead of just Lipschitz-type conditions on the divided difference of the operator involved. It turns out that this way our error bounds are more precise than earlier ones and under our convergence hypotheses we can cover cases where earlier conditions are violated. Numerical examples are also provided in this study.

Key words and phrases: Secant method, Banach space, majorizing sequence, divided difference, Fréchet-derivative.

RESUMEN

Son dadas nuevas condiciones suficientes para la convergencia del método de la secante para una solución localmente única de una ecuación no lineal en un espacio de Banach. Estas ideas nuevas usan funciones recurrentes, tipo-Lipschitz y tipo centro-Lipschitz sobre la diferencia dividida de los operadores envolvidos. Resulta que esta manera las cotas de errores son mas precisas que las anteriores y bajo nuestras hipótesis de convergencia nosotros podemos cubrir casos donde las condiciones previas eran violadas. Ejemplos numéricos son dados en este estudio.

Math. Subj. Class.: 65H10, 65B05, 65G99, 65N30, 47H17, 49M15.

References

[1] Argyros, I.K., The theory and application of abstract polynomial equations, St.Lucie/CRC/ Lewis Publ. Mathematics series, 1998, Boca Raton, Florida, U.S.A.

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

[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.

[4] Argyros, I.K., New sufficient convergence conditions for the Secant method, Chechoslovak Math. J., 55, (2005), 175-187.

[5] Argyros, I.K., Convergence and applications of Newton-type iterations, Springer-Verlag Publ., New-York, 2008.

[6] Argyros, I.K. and Hilout, S., Efficient methods for solving equations and variational inequalities, Polimetrica Publ. Co., Milano, Italy,

[7] Bosarge, W.E. and Falb, P.L., A multipoint method of third order, J. Optimiz. Th. Appl., 4 (1969), 156-166.

[8] Chandrasekhar, S., Radiative transfer, Dover Publ., New-York, 1960.

[9] Dennis, J.E., Toward a unified convergence theory for Newton-like methods, in Nonlinear Functional Analysis and Applications (L.B. Rall, ed.), Academic Press, New York, (1971), 425-472.

[10] Gutiérrez, J.M., A new semilocal convergence theorem for Newton's method, J. Comput. Appl. Math., 79 (1997), 131-145.

[11] Hernández, M.A., Rubio, M.J. and Ezquerro, J.A., Secant-like methods for solving nonlinear integral equations of the Hammerstein type, J. Comput. Appl. Math., 115 (2000), 245-254.

[12] Huang, Z., A note of Kantorovich theorem for Newton iteration, J. Comput. Appl. Math., 47 (1993), 211-217.

[13] Kantorovich and L.V., Akilov, G.P., Functional Analysis, Pergamon Press, Oxford, 1982.

[14] Laasonen, P., Ein überquadratisch konvergenter iterativer algorithmus, Ann. Acad. Sci. Fenn. Ser I, 450 (1969), 1-10.

[15] Ortega, J.M. and Rheinboldt, W.C., Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970.

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

[17] Schmidt, J.W., Untere Fehlerschranken fur Regula-Falsi Verhafren, Period. Hungar., 9 (1978), 241-247.

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

[19] Wolfe, M.A., Extended iterative methods for the solution of operator equations, Numer. Math., 31 (1978), 153-174

Revised: January, 2009.