  In this paper we compare two numerical methods to integrate Riemannian cubic
polynomials on the Stiefel manifold $\textbf{St}_{n,k}$. The first one is the
adjusted de Casteljau algorithm, and the second one is a symplectic integrator
constructed through discretization maps. In particular, we choose the cases of
$n=3$ together with $k=1$ and $k=2$. The first case is diffeomorphic to the
sphere and the quasi-geodesics appearing in the adjusted de Casteljau algorithm
are actually geodesics. The second case is an example where we have a pure
quasi-geodesic different from a geodesic. We provide a numerical comparison of
both methods and discuss the obtained results to highlight the benefits of each
method.
