Document Type
Journal Article
Publication Title
SIAM Journal on Numerical Analysis
Volume
61
Issue
5
First Page
2261
Last Page
2284
Publisher
Society for Industrial and Applied Mathematics
School
School of Science
RAS ID
64575
Funders
Edith Cowan University
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 x0 and x1 on a path-connected complete Riemannian manifold. The basic idea is to choose some junctions between x0 and x1 that can be joined by geodesics locally and then adjust these junctions. It was proved that the sequence of piecewise geodesics { k}k ≥ 1 generated by this algorithm converges to a geodesic joining x0 and x1. The present paper investigates leapfrog's convergence rate i,n of ith junction depending on the manifold M. A relationship is found with the maximal root n of a polynomial of degree n-3, where n (n > 3) is the number of geodesic segments. That is, the minimal i,n is upper bounded by n(1 + c+), where c+ is a sufficiently small positive constant depending on the curvature of the manifold M. Moreover, we show that n increases as n increases. These results are illustrated by implementing leapfrog on two Riemannian manifolds: the unit 2-sphere and the manifold of all 2 × 2 symmetric positive definite matrices.
DOI
10.1137/22M1515173
Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 License.
Comments
Zhang, E., & Noakes, L. (2023). Convergence analysis of leapfrog for geodesics. SIAM Journal on Numerical Analysis, 61(5), 2261-2284. https://doi.org/10.1137/22M1515173