Geodesic
Convergence conditions for Secant-type methods
Czechoslovak Mathematical Journal, Tome 60 (2010) no. 1, pp. 253-272

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We provide new sufficient convergence conditions for the convergence of the secant-type methods to a locally unique solution of a nonlinear equation in a Banach space. Our new idea uses recurrent functions, and 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.
Classification : 49M15, 65B05, 65G99, 65H10, 65N30
Keywords: secant method; Banach space; majorizing sequence; divided difference; Fréchet-derivative 
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     author = {Argyros, Ioannis K. and Hilout, Said},
     title = {Convergence conditions for {Secant-type} methods},
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Argyros, Ioannis K.; Hilout, Said. Convergence conditions for Secant-type methods. Czechoslovak Mathematical Journal, Tome 60 (2010) no. 1, pp. 253-272. http://geodesic.mathdoc.fr/item/CMJ_2010__60_1_a20/