Saadatmandi-Bisheh-Niasar Root Finding Method via the S-iteration with Periodic Parameters and Its Polynomiography

Abstract

In recent years many researchers have focused their attention on the use of different iteration process - known from fixed point theory - in the generation of different kinds of patterns. In this paper, we propose modifications of the Saadatmandi and Bisheh-Niasar root finding method. In the first modification we modify the formula of the method and in the second modification we use the S-iteration with periodic parameters. Moreover, we numerically investigate some properties of the proposed methods and modification using three measures, i.e., the generation time, mean number of iterations and convergence area index. The obtained polynomiographs show that the proposed methods have a potential artistic applications, and the numerical results show that there is no obvious dependency of the considered measures on the sequences of the parameters used in the S-iteration.