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On the behavior of the four order iteration in Euler’s family near a zero. (English) Zbl 1340.65110
Summary: The aim of this paper is to study the local convergence of the four order iteration of Euler’s family for solving nonlinear operator equations. We get the optimal radius of the local convergence ball of the method for operators satisfying the weak third order generalized Lipschitz condition with $$L$$-average. We also show that the local convergence of the method is determined by a period 2 orbit of the method itself applied to a real function.
##### MSC:
 65J15 Numerical solutions to equations with nonlinear operators (do not use 65Hxx) 47J25 Iterative procedures involving nonlinear operators
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