Traub-Gander’s family for the simultaneous determination of multiple zeros of polynomials

Abstract

By combining Traub-Gander’s family of third order for finding a multiple zero and suitable corrective approximations od Schröder’s and Halley’s type, a new family of iterative methods for the simultaneous approximation of multiple zeros of algebraic polynomials is proposed. Taking various forms of a function involved in the iterative formula, a number of different simultaneous methods can be obtained. It is proved that the order of convergence is 4, 5 or 6, depending of the type of employed corrective approximations. Two numerical examples are given to demonstrate the convergence properties of the proposed family of simultaneous methods. Displayed trajectories of the sequences of approximations point to global characteristics of the proposed family of iterative methods.