The efficient initialization strategy used in the proposed single shooting method for computing geodesics on the Stiefel manifold can be applied to other numerical methods for computing geodesics on Riemannian manifolds beyond the Stiefel manifold. By utilizing a first-order approximation of the matrix exponential and projecting the initial guess onto the tangent space, the initialization strategy can help improve the convergence properties of various numerical methods for solving geodesic problems on different Riemannian manifolds.
The key idea behind the initialization strategy is to provide a good starting point for the iterative optimization process, which can help accelerate convergence and improve the overall efficiency of the numerical method. This strategy can be adapted and implemented in a similar manner for other numerical algorithms that involve solving geodesic equations on different Riemannian manifolds.