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Local convergence for some third-order iterative methods under weak conditions
(Korean Mathematical Society kms@kms.or.kr, 2016) Argyros, I.K.; Cho, Y.J.; George, S.
The solutions of equations are usually found using iterative methods whose convergence order is determined by Taylor expansions. In particular, the local convergence of the method we study in this paper is shown under hypotheses reaching the third derivative of the operator involved. These hypotheses limit the applicability of the method. In our study we show convergence of the method using only the first derivative. This way we expand the applicability of the method. Numerical examples show the applicability of our results in cases earlier results cannot. © 2016 Korean Mathematical Society.
Improved local convergence analysis for a three point method of convergence order 1.839…
(Korean Mathematical Society kms@kms.or.kr, 2019) Argyros, I.K.; Cho, Y.J.; George, S.
In this paper, we present a local convergence analysis of a three point method with convergence order 1.839… for approximating a locally unique solution of a nonlinear operator equation in setting of Banach spaces. Using weaker hypotheses than in earlier studies, we obtain: larger radius of convergence and more precise error estimates on the distances involved. Finally, numerical examples are used to show the advantages of the main results over earlier results. © 2019 Korean Mathematical Society.