On the dynamics of a triparametric family of optimal fourth-order multiple-zero finders with a weight function of the principal mth root of a function-to function ratio

Under the assumption of known root multiplicity m∈N, a triparametric family of two-point optimal quartic-order methods locating multiple zeros are investigated in this paper by introducing a weight function dependent on a function-to-function ratio. Special cases of weight functions with selected free parameters are considered and studied through various test equations and numerical experiments to support the theory developed in this paper. In addition, we explore the relevant dynamics of proposed methods via Möbius conjugacy map when applied to a prototype polynomial (z−a)m(z−b)m. The results of such dynamics are visually illustrated through a variety of parameter spaces as well as dynamical planes.