A new fourth-order family of simultaneous methods for finding polynomial zeros

Using a suitable fixed point relation with a complex parameter, a new one parameter family of simultaneous methods of the fourth order for finding complex zeros of a polynomial is derived in ordinary complex arithmetic. Convergence analysis of the presented family is performed under computationally verifiable initial conditions which depend only on polynomial coefficients and initial approximations. Further improvements of the proposed family of methods are discussed and a modification for finding multiple zeros is presented. Some numerical results for various values of the parameter are given.