  This paper shows how to use the shooting method, a classical numerical
algorithm for solving boundary value problems, to compute the Riemannian
distance on the Stiefel manifold $\mathrm{St}(n,p)$, the set of $ n \times p $
matrices with orthonormal columns. The main feature is that we provide neat,
explicit expressions for the Jacobians. To the author's knowledge, this is the
first time some explicit formulas are given for the Jacobians involved in the
shooting methods to find the distance between two given points on the Stiefel
manifold. This allows us to perform a preliminary analysis for the single
shooting method. Numerical experiments demonstrate the algorithms in terms of
accuracy and performance. Finally, we showcase three example applications in
summary statistics, shape analysis, and model order reduction.
