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SUMMARYIterative methods for the simultaneous determination of simple or multiple complex zeros of a polynomial, based on a cubically convergent Chebyshev method, are considered. Using Newton's and Halley's corrections the convergence of the basic method of the fourth order is increased to five and six, respectively. The improved convergence is achieved with negligible number of additional calculations, which significantly increases the computational efficiency of the accelerated methods. One of the most important problems in solving polynomial equations, the construction of initial conditions that enable both guaranteed and fast convergence, is also studied for the proposed methods. These conditions are computationally verifiable since they depend only on initial approximations, the polynomial coefficients and the polynomial degree, which is of practical importance. Finally, modified methods of Chebyshev's type for finding multiple zeros and single-step methods based on the Gauss-Seidel approach are constructed.

a b s t r a c tA 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 Schröder 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.