Families of Simultaneous Methods of Higher Order: Part II

In this chapter, we derive a fixed point relation of the square root type, which is the base for the construction of new one-parameter families of iterative methods for the simultaneous determination of simple complex zeros of a polynomial in ordinary complex arithmetic (Sect. 5.1) and circular complex arithmetic (Sect. 5.3). A slight modification of the derived fixed point relation can provide the simultaneous approximation of multiple zeros. Under computationally verifiable initial conditions, we prove that the basic method has the convergence order equal to 4. Using an approach with corrections, proposed by Carstensen and M. Petković in [17] and [111], we construct modified methods with very fast convergence on the account of only a few additional numerical operations (Sect. 5.2). In this way, we obtain a high computational efficiency of the proposed methods. Numerical results are given in Sect. 5.2 for the methods realized in ordinary complex arithmetic and in Sect. 5.3 for the methods implemented in circular complex arithmetic.