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New efficient iterative method of Halley’s type for the simultaneous inclusion of all simple complex zeros of a polynomial is proposed. The presented convergence analysis, which uses the concept of the R -order of convergence of mutually dependent sequences, shows that the convergence rate of the basic fourth order method is increased from 4 to 9 using a two-point correction. The proposed inclusion method possesses high computational efficiency since the increase of convergence is attained with only one additional function evaluation per sought zero. Further acceleration of the proposed method is carried out using the Gauss–Seidel procedure. Some computational aspects and three numerical examples are given in order to demonstrate high computational efficiency and the convergence properties of the proposed methods. [ABSTRACT FROM AUTHOR]

Higher-order methods for the simultaneous inclusion of complex zeros of algebraic polynomials are presented in parallel (total-step) and serial (single-step) versions. If the multiplicities of each zeros are given in advance, the proposed methods can be extended for multiple zeros using appropriate corrections. These methods are constructed on the basis of the zero-relation of Gargantini's type, the inclusion isotonicity property and suitable corrections that appear in two-point methods of the fourth order for solving nonlinear equations. It is proved that the order of convergence of the proposed methods is at least six. The computational efficiency of the new methods is very high since the acceleration of convergence order from 3 (basic methods) to 6 (new methods) is attained using only n polynomial evaluations per iteration. Computational efficiency of the considered methods is studied in detail and two numerical examples are given to demonstrate the convergence behavior of the proposed methods. [ABSTRACT FROM AUTHOR]

The aim of this paper is to present some modifications of Newton's type method for the simultaneous inclusion of all simple complex zeros of a polynomial. Using the concept of the R-order of convergence of mutually dependent sequences, the convergence analysis shows that the convergence rate of the basic method is increased from 3 to 6 using Jarratt's corrections. The proposed method possesses a great computational efficiency since the acceleration of convergence is attained with only few additional calculations. Numerical results are given to demonstrate convergence properties of the considered methods. [ABSTRACT FROM AUTHOR]