Newton's method, zeroes of vector fields, and the Riemannian center of mass
| dc.creator | Groisser, David | |
| dc.date | 2003-11-02 | |
| dc.date.accessioned | 2026-07-07T05:02:26Z | |
| dc.date.available | 2026-07-07T05:02:26Z | |
| dc.description | We present an iterative technique for finding zeroes of vector fields on Riemannian manifolds. As a special case we obtain a ``nonlinear averaging algorithm'' that computes the centroid of a mass distribution supported in a set of small enough diameter D in a Riemannian manifold M. We estimate the convergence rate of our general algorithm and the more special Riemannian averaging algorithm. The algorithm is also used to provide a constructive proof of Karcher's theorem on the existence and local uniqueness of the center of mass, under a somewhat stronger requirement than Karcher's on D. Another corollary of our results is a proof of convergence, for a fairly large open set of initial conditions, of the ``GPA algorithm'' used in statistics to average points in a shape-space, and a quantitative explanation of why the GPA algorithm converges rapidly in practice. We also show that a mass distribution in M with support Q has a unique center of mass in a (suitably defined) convex hull of Q. | |
| dc.description | 43 pages, 1 figure | |
| dc.identifier | https://arxiv.org/abs/math/0311011 | |
| dc.identifier | http://arxiv.org/abs/math/0311011 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/69049 | |
| dc.subject | Differential Geometry | |
| dc.subject | 53B21 (primary), 60D05 (primary), 53C99 (secondary) | |
| dc.title | Newton's method, zeroes of vector fields, and the Riemannian center of mass | |
| dc.type | text |