Geodesic
New sufficient convergence conditions for the secant method
Czechoslovak Mathematical Journal, Tome 55 (2005) no. 1, pp. 175-187
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We provide new sufficient 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 “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 the earlier ones and under our convergence hypotheses we can cover cases where the earlier conditions are violated.
We provide new sufficient 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 “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 the earlier ones and under our convergence hypotheses we can cover cases where the earlier conditions are violated.
Classification : 47H17, 47J25, 49M15, 65B05, 65G99, 65H10, 65J15, 65N30
Keywords: secant method; Banach space; majorizing sequence; divided difference; Fréchet-derivative 
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Argyros, Ioannis K. New sufficient convergence conditions for the secant method. Czechoslovak Mathematical Journal, Tome 55 (2005) no. 1, pp. 175-187. http://geodesic.mathdoc.fr/item/CMJ_2005_55_1_a12/

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