Convergence Analysis of Leapfrog for Geodesics

Erchuan Zhang, Lyle Noakes

Abstract: Geodesics are of fundamental interest in mathematics, physics, computer science, and many other subjects. The so-called leapfrog algorithm was proposed in [L. Noakes, J. Aust. Math. Soc., 65 (1998), pp. 37–50] (but not named there as such) to find geodesics joining two given points and on a path-connected complete Riemannian manifold. The basic idea is to choose some junctions between and that can be joined by geodesics locally and then adjust these junctions. It was proved that the sequence of piecewise geodesics generated by this algorithm converges to a geodesic joining and . The present paper investigates leapfrog’s convergence rate of ith junction depending on the manifold M. A relationship is found with the maximal root of a polynomial of degree , where n is the number of geodesic segments. That is, the minimal is upper bounded by , where is a sufficiently small positive constant depending on the curvature of the manifold M. Moreover, we show that increases as n increases. These results are illustrated by implementing leapfrog on two Riemannian manifolds: the unit 2-sphere and the manifold of all symmetric positive definite matrices.