On an improved convergence analysis of Newton's method

We present a local as well as a semi-local convergence analysis of Newton's method for solving nonlinear equations in a Banach space setting. The new approach leads in the local case to larger convergence radius than before (Argyros and Hilout, 2013, 2012 [5,6] and Rheinboldt, 1977 [17]). In the semi-local case, we obtain weaker sufficient convergence conditions; tighter error bounds distances involved and a more precise information on the location of the solution than in earlier studies such as Argyros (2007) [2], Argyros and Hilout (2013, 2012) [5,6] and Ortega and Rheinboldt (1970) [13]. Upper and lower bounds on the limit points of the majorizing sequences are also provided in this study. These advantages are obtained under the same computational cost as in the earlier stated studies. Finally, the numerical examples illustrate the theoretical results.