  The conformal barycenter of a point cloud on the sphere at infinity of the
Poincar\'e ball model of hyperbolic space is a hyperbolic analogue of the
geometric median of a point cloud in Euclidean space. It was defined by Douady
and Earle as part of a construction of a conformally natural way to extend
homeomorphisms of the circle to homeomorphisms of the disk, and it plays a
central role in Millson and Kapovich's model of the configuration space of
cyclic linkages with fixed edgelengths.
  In this paper we consider the problem of computing the conformal barycenter.
Abikoff and Ye have given an iterative algorithm for measures on $\mathbb{S}^1$
which is guaranteed to converge. We analyze Riemannian versions of Newton's
method computed in the intrinsic geometry of the Poincare ball model. We give
Newton-Kantorovich (NK) conditions under which we show that Newton's method
with fixed step size is guaranteed to converge quadratically to the conformal
barycenter for measures on any $\mathbb{S}^d$ (including infinite-dimensional
spheres). For measures given by $n$ atoms on a finite dimensional sphere which
obey the NK conditions, we give an explicit linear bound on the computation
time required to approximate the conformal barycenter to fixed error. We prove
that our NK conditions hold for all but exponentially few $n$ atom measures.
For all measures with a unique conformal barycenter we show that a regularized
Newton's method with line search will always converge (eventually
superlinearly) to the conformal barycenter. Though we do not have hard time
bounds for this algorithm, experiments show that it is extremely efficient in
practice and in particular much faster than the Abikoff-Ye iteration.
