Computational efficiency of some combined methods for polynomial equations

The iterative methods for the simultaneous determination of all simple complex zeros of algebraic polynomials, based on the fixed point relation of Ehrlich’s type, are considered. Using the iterative correction appearing in the Jarratt method of the fourth order, it is proved that the convergence rate of the modified Ehrlich method is increased from 3 to 6. This acceleration of the convergence is obtained with few additional numerical operations which means that the proposed combined method possesses very high computational efficiency. Moreover, the convergence rate can be further accelerated using the Gauss–Siedel approach (single-step or serial mode). A great part of the paper is devoted to the computational aspects of the discussed methods, including numerical examples. A comparison procedure shows that the new iterative method is more efficient than existing methods in the considered class.