Family of simultaneous methods of Hansen–Patrickʹs type Original Research Article

A one parameter family of iterative methods for the simultaneous determination of all (simple or multiple) complex zeros of a polynomial, based on a cubically convergent Hansen–Patrickʹs family, is considered. The proposed class of methods is adapted for the determination of simple zeros of a class of analytic functions which have only simple zeros inside a simple smooth closed contour in the complex plane. The presented convergence analysis shows that the order of convergence of the considered family is four. Numerical examples and a short description of the implementation on parallel computers are also given.