In this paper we study the convergence, stability and data independence of the \({S_n}\) iteration procedure in the W-hyperbolic setting, in connection to hyperbolic contractive mappings. Examples of hyperbolic contractions are provided, based on the convexity map. In particular, we associate the Poincaré disk with the appropriate convexity map, for which we determine the specific formula. Finally, we numerically test the convergence of the orbits under the \({S_n}\) iteration procedure. This study brings a new contribution to a problem of recent interest: the qualitative analysis of iterative procedures in nonlinear framework.
