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This paper shows how to use the shooting method, a classical numericalalgorithm for solving boundary value problems, to compute the Riemanniandistance 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 thefirst time some explicit formulas are given for the Jacobians involved in theshooting methods to find the distance between two given points on the Stiefelmanifold. This allows us to perform a preliminary analysis for the singleshooting method. Numerical experiments demonstrate the algorithms in terms ofaccuracy and performance. Finally, we showcase three example applications insummary statistics, shape analysis, and model order reduction.

Shooting methods for computing geodesics on the Stiefel manifold