Enlarging the convergence domain of secant-like methods for equations

Abstract

We present two new semilocal convergence analyses for secant-like methods in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. These methods include the secant, Newton's method and other popular methods as special cases. The convergence analysis is based on our idea of recurrent functions. Using more precise majorizing sequences than before we obtain weaker convergence criteria. These advantages are obtained because we use more precise estimates for the upper bounds on the norm of the inverse of the linear operators involved than in earlier studies. Numerical examples are given to illustrate the advantages of the new approaches.