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Local Convergence Results for an Optimal Iterative Method for Multiple Roots

Abstract:

In this paper our aim is to perform a local convergence study of a fourth order iterative method in the case of multiple roots. As far as we know, these kind of studies have only been performed for iterative methods of second and third order of convergence in the case of multiple roots. So it is our purpose to analyze the radius of local convergence for higher-order methods. Usually the local convergence radius decreases when the order of the method increases, so it is necessary to study its behavior when we propose a new iterative method. In this sense, we introduce in this paper a new idea for establishing local convergence results of iterative methods for locating multiple zeros, under the assumption of a bounding condition for the (m + 1)-th derivative of the function f(x) in its existence domain. We apply this technique to the modification of the Maheshwari fourth order method for the case of multiple roots. Finally, we perform some numerical examples that confirm the theoretical results established in this paper.