Local and Semilocal Convergence of a Family of Multi-point Weierstrass-Type Root-Finding Methods

Abstract

Weierstrass (Sitzungsber Konigl Preuss Akad Wiss Berlin II:1085–1101, 1891) introduced his famous iterative method for numerical finding all zeros of a polynomial simultaneously. Kyurkchiev and Ivanov (Ann Univ Sofia Fac Math Mech 78:132–136, 1984) constructed a family of multi-point root-finding methods which are based on the Weierstrass method. The purpose of this research is threefold: (1) to develop a new simple approach for the study of the local convergence of the multi-point simultaneous iterative methods; (2) to present a new local convergence result for this family which improves in several directions the result of Kyurkchiev and Ivanov; (3) to provide semilocal convergence results for Kyurkchiev–Ivanov’s family of iterative methods.