On the semilocal convergence of damped Newton's method

We establish new semilocal convergence results for the damped Newton's method. Two approaches are used: the first one uses recurrent relations Ezquerro et al. (2010) and Hernández (2000) [13,19] and the second concerns recurrent functions introduced by Argyros (2011) [3]. A comparison between these two methods is provided. Some values of the iteration parameters are given which are almost optimal choices of a certain accuracy and with respect to a certain polynomial. Numerical examples illustrating the theoretical results are also presented in this study.