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.

翻译：本文比较了斯梯弗流形$\textbf{St}_{n,k}$上黎曼三次多项式积分的两种数值方法。第一种是修正的德卡斯特里奥算法，第二种是通过离散化映射构造的辛积分器。特别地，我们选取了$n=3$且$k=1$与$k=2$的情形。第一种情形与球面微分同胚，修正的德卡斯特里奥算法中出现的拟测地线实际上是测地线。第二种情形则展示了不同于测地线的纯拟测地线实例。我们对两种方法进行了数值比较，并讨论了所得结果，以突出每种方法的优势。