On an efficient simultaneous method for finding polynomial zeros

A new iterative method for the simultaneous determination of simple zeros of algebraic polynomials is stated. This method is more efficient compared to the all existing simultaneous methods based on fixed point relations. A very high computational efficiency is obtained using suitable corrections resulting from the Kung–Traub three-step method of low computational complexity. The presented convergence analysis shows that the convergence rate of the basic third order method is increased from 3 to 10 using this special type of corrections and applying 2n2n additional polynomial evaluations per iteration. Some computational aspects and numerical examples are given to demonstrate a very fast convergence and high computational efficiency of the proposed zero-finding method.