  The circular Sitnikov problem, where the two primary bodies are prolate or
oblate spheroids, is numerically investigated. In particular, the basins of
convergence on the complex plane are revealed by using a large collection of
numerical methods of several order. We consider four cases, regarding the value
of the oblateness coefficient which determines the nature of the roots
(attractors) of the system. For all cases we use the iterative schemes for
performing a thorough and systematic classification of the nodes on the complex
plane. The distribution of the iterations as well as the probability and their
correlations with the corresponding basins of convergence are also discussed.
Our numerical computations indicate that most of the iterative schemes provide
relatively similar convergence structures on the complex plane. However, there
are some numerical methods for which the corresponding basins of attraction are
extremely complicated with highly fractal basin boundaries. Moreover, it is
proved that the efficiency strongly varies between the numerical methods.
