{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:34:46Z","timestamp":1787330086490,"version":"build-2736575974"},"reference-count":27,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"5","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Numer. Anal."],"published-print":{"date-parts":[[2023,10,31]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>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\u201350] (but not named there as such) to find geodesics joining two given points [Formula: see text] and [Formula: see text] on a path-connected complete Riemannian manifold. The basic idea is to choose some junctions between [Formula: see text] and [Formula: see text] that can be joined by geodesics locally and then adjust these junctions. It was proved that the sequence of piecewise geodesics [Formula: see text] generated by this algorithm converges to a geodesic joining [Formula: see text] and [Formula: see text]. The present paper investigates leapfrog\u2019s convergence rate [Formula: see text] of ith junction depending on the manifold M. A relationship is found with the maximal root [Formula: see text] of a polynomial of degree [Formula: see text], where n [Formula: see text] is the number of geodesic segments. That is, the minimal [Formula: see text] is upper bounded by [Formula: see text], where [Formula: see text] is a sufficiently small positive constant depending on the curvature of the manifold M. Moreover, we show that [Formula: see text] increases as n increases. These results are illustrated by implementing leapfrog on two Riemannian manifolds: the unit 2-sphere and the manifold of all [Formula: see text] symmetric positive definite matrices.<\/jats:p>","DOI":"10.1137\/22m1515173","type":"journal-article","created":{"date-parts":[[2023,10,6]],"date-time":"2023-10-06T04:53:02Z","timestamp":1696567982000},"page":"2261-2284","source":"Crossref","is-referenced-by-count":0,"title":["Convergence Analysis of Leapfrog for Geodesics"],"prefix":"10.1137","volume":"61","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4005-5431","authenticated-orcid":true,"given":"Erchuan","family":"Zhang","sequence":"first","affiliation":[{"name":"School of Science, Edith Cowan University, 270 Joondalup Drive, Joondalup, WA, 6027, Australia."},{"name":"Department of Mathematics and Statistics, The University of Western Australia, 35 Stirling Highway, Crawley, WA, 6009, Australia."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Lyle","family":"Noakes","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, The University of Western Australia, 35 Stirling Highway, Crawley, WA, 6009, Australia."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2023,10,6]]},"reference":[{"key":"ref1","doi-asserted-by":"publisher","DOI":"10.1137\/16M1103099"},{"key":"ref2","volume-title":"Comparison Theorems in Riemannian Geometry","author":"Cheeger J.","year":"1975"},{"key":"ref3","series-title":"Appl. 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