Improved Halley-like methods for the inclusion of polynomial zeros

Improved iterative methods of Halley's type for the simultaneous inclusion of all simple complex zeros of a polynomial are proposed. The presented convergence analysis, which uses the concept of the R-order of convergence of mutually dependent sequences, shows that the convergence rate of the basic fourth order method is increased to 5 and 6 using Newton's and Halley's corrections, respectively. The proposed inclusion methods possess a high computational efficiency since the increase of the convergence is attained without additional calculations. This advantage, together with very fast convergence, make that the presented methods are ranking among the most powerful inclusion methods for polynomial zeros. In order to demonstrate convergence properties of the proposed methods, two numerical examples are given.