A unified approach for the convergence of certain numerical algorithms, using recurrent functions;

摘要

The Kantorovich analysis (Argyros in Convergence and applications of Newton-type iterations, Springer, New York, 2008; Argyros and Hilout in Efficient methods for solving equations and variational inequalities, Polimetrica Publisher, Milano, 2009; Kantorovich and Akilov in Functional analysis, Pergamon Press, Oxford, 1982), and recurrent relation's approach (Gutierrez et al. in J Comput Appl Math 115:181-192, 2000) are the most popular ways for generating sufficient conditions for the convergence of numerical algorithms to a solution of a nonlinear equations as well as providing the corresponding error estimates on the distances involved. We introduce the new approach of recurrent functions to show that a finer convergence analysis can be provided under the same hypotheses, and computational cost. Numerical examples are provided where our results apply, but not earlier ones.