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

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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.

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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.

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Received: October, 2008.

Revised: January, 2009.