New sufficient convergence conditions for the secant method. (English) Zbl 1081.65043

Summary: 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.


65H10 Numerical computation of solutions to systems of equations
65B05 Extrapolation to the limit, deferred corrections
65G99 Error analysis and interval analysis
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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