Exploring Weight Functions: A Family of One-Point Methods for Multiple Roots

Abstract

In this paper we introduce a family of one-point methods for multiple roots using weight functions. We analyze the necessary conditions for the weight function to achieve third-order accuracy, and then present a sequence of rational weight functions for a complex quadratic function. The resulting members of the family exhibit Julia sets that align with the perpendicular bisector of the roots. We compare selected members with existing methods based on basin of attraction analysis, demonstrating their competitive performance. Additionally, we investigate the influence of root multiplicity on the fractal structures of basin of attraction for equations with both simple and multiple roots.