  Several applications in optimization, image, and signal processing deal with
data that belong to the Stiefel manifold St(n,p), that is, the set of n-by-p
matrices with orthonormal columns. Some applications, like the Riemannian
center of mass, require evaluating the Riemannian distance between two
arbitrary points on St(n,p). This can be done by explicitly constructing the
geodesic connecting these two points. An existing method for finding geodesics
is the leapfrog algorithm of J. L. Noakes. This algorithm is related to the
Gauss-Seidel method, a classical iterative method for solving a linear system
of equations that can be extended to nonlinear systems. We propose a
convergence proof of leapfrog as a nonlinear Gauss-Seidel method. Our
discussion is limited to the case of the Stiefel manifold, however, it may be
generalized to other embedded submanifolds. We discuss other aspects of
leapfrog and present some numerical experiments.
