On a high-order one-parameter family for the simultaneous determination of polynomial roots

A new one-parameter family of simultaneous methods for the determination of all (simple or multiple) zeros of a polynomial is derived. The order of the basic family of simultaneous methods is four. Using suitable corrective approximations, the order of convergence of this family is increased even to six without any additional evaluations of the polynomial and its derivatives, which points to a high computational efficiency of the new family of root solvers. Comparison with the existing methods in regard to computational efficiency and numerical examples are also performed.