Newton Method on Riemannian Manifolds: Covariant Alpha-Theory

Nf (z) = expz(−Df(z)f(z)) where expz : TzMn → Mn denotes the exponential map. When, instead of a mapping we consider a vector field X, in order to define Newton method, we resort to an object studied in differential geometry; namely, the covariant derivative of vector fields denoted here by DX. We let NX(z) = expz(−DX(z)X(z)). These definitions coincide with the usual one when Mn = Rn because expz(u) = z+ u and also because, in this context, the covariant derivative is just the usual derivative. The first to consider Newton method on a manifold is Rayleigh 1899 [7] who defined what we call today ”Rayleigh Quotient Iteration” which is in fact a Newton iteration for a vector field on the sphere. Then Shub 1986 [8] defined Newton’s method for the problem of finding the zeros of a vector field on a manifold and used retractions to send a neighborhood of the origin in the tangent space onto the manifold itself. In our paper we do not use general retractions but exponential maps. Udriste 1994 [19] studied Newton’s method to find the zeros of a gradient vector field defined on a Riemannian manifold; Owren and Welfert 1996 [6] defined Newton iteration for solving the equation F (x) = 0 where F is a map from a Lie group to its corresponding Lie algebra; Smith 1994 [18] and Edelman-Arias-Smith 1998 [3] developed Newton and conjugate gradient algorithms on the Grassmann and Stiefel manifolds. These authors define Newton’s method via the exponential map like we do here. Shub 1993 [9], Shub and Smale 1993-1996 [10], [11], [12], [13], [14], Malajovich 1994 [5], Dedieu and Shub 2000 [2] introduce and study Newton’s method on projective spaces and their products. Another paper about this subject is Adler-Dedieu-Margulies-Martens-Shub 2001 [1] where qualitative aspects of Newton method on Riemannian manifolds are investigated for both mappings and vector fields with an application to a geometric model for the human spine represented as a 18−tuple of 3×3 orthogonal matrices. Recently Ferreira-Svaiter [4] gave a Kantorovich like theorem for Newton method for vector fields defined on Riemannian manifolds.