A unified convergence analysis for secant-type methods (Q2929755)

A unified local and semilocal convergence analysis for secant-type methods for the approximation of a locally unique solution of a nonlinear equation in a Banach space setting is presented. The analysis performed includes the computation of the bounds on the limit points of the majorizing sequences involved. Under the same computational cost the semilocal convergence criteria can be weaker, the error bounds more precise and in the local case the convergence balls can be larger and the error bounds tighter than in existing studies, at least for the cases of Newton's method and the secant method. Three representative numerical examples are used to illustrate the theoretical results.