On the new fourth-order methods for the simultaneous approximation of polynomial zeros

A new iterative method of the fourth-order for the simultaneous determination of polynomial zeros is proposed. This method is based on a suitable zero-relation derived from the fourth-order method for a single zero belonging to the Schrِder basic sequence. One of the most important problems in solving polynomial equations, the construction of initial conditions that enable both guaranteed and fast convergence, is studied in detail for the proposed method. These conditions are computationally verifiable since they depend only on initial approximations, the polynomial coefficients and the polynomial degree, which is of practical importance. The construction of improved methods in ordinary complex arithmetic and complex circular arithmetic is discussed. Finally, numerical examples and the comparison with existing fourth-order methods are given.