Achieving an extended convergence analysis for the secant method under a restricted Hölder continuity condition

We are concerned with the improvement of the estimations for the attraction balls without adding conditions. The novelty of our paper is that in the case of Secant method we introduce a smaller domain than before containing the iterates. That is now we find more precise Hölder constants leading under the same conditions and computational effort as before to: a larger radius of convergence (i.e. a wider choice of initial points) as well as more precise estimates on the distances involved leading to the computation of fewer iterates to attain a predetermined error tolerance. Our local analysis includes verifying the convergence order of the Secant method but with a better estimate for the asymptotic constant. The new idea can be used to extend the applicability of other iterative methods along the same lines and in the semilocal convergence analysis too. A numerical example is given to test the convergence criteria and show the validity of the advantages claimed in this abstract.