Newton's method, zeroes of vector fields, and the Riemannian center of mass

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.