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This paper deals with the study of relaxed conditions for semi-local convergence for a general iterative method, k-step Newton's method, using majorizing sequences. Dynamical behavior of the mentioned method is also analyzed via Julia set and basins of attraction. Numerical examples of nonlinear systems of equations will be examined to verify the given theory. [ABSTRACT FROM AUTHOR]

We derive new iterative methods with memory for approximating a simple zero of a nonlinear single variable function. To this end, we first consider several modifications on some existing optimal classes without memory in such a way that their extensions to cases with memory could obtain the higher efficiency index 12 1 4 ≈ 1.86120 . Furthermore, we construct our main method with memory using three self-accelerators. It is demonstrated that this new scheme possesses the very high efficiency index 7.23814 1 3 ≈ 1.93438 . [ABSTRACT FROM AUTHOR]

In this paper, we present a fourth order method for computing simple roots of nonlinear equations by using suitable Taylor and weight function approximation. The method is based on Weerakoon-Fernando method [S. Weerakoon, G.I. Fernando, A variant of Newton's method with third-order convergence, Appl. Math. Lett. 17 (2000) 87-93]. The method is optimal, as it needs three evaluations per iterate, namely one evaluation of function and two evaluations of first derivative. So, Kung and Traub's conjecture is fulfilled. We also perform some numerical tests that confirm the theoretical results and allow us to compare the proposed method with some existing methods of the same type. [ABSTRACT FROM AUTHOR]