  Parallel transport is an important step in many discrete algorithms for
statistical computing on manifolds. Numerical methods based on Jacobi fields or
geodesics parallelograms are currently used in geometric data processing. In
this last class, pole ladder is a simplification of Schild's ladder for the
parallel transport along geodesics that was shown to be particularly simple and
numerically stable in Lie groups. So far, these methods were shown to be first
order approximations of the Riemannian parallel transport, but higher order
error terms are difficult to establish. In this paper, we build on a BCH-type
formula on affine connection spaces to establish the behavior of one pole
ladder step up to order 5. It is remarkable that the scheme is of order three
in general affine connection spaces with a symmetric connection, much higher
than expected. Moreover, the fourth-order term involves the covariant
derivative of the curvature only, which is vanishing in locally symmetric
space. We show that pole ladder is actually locally exact in these spaces, and
even almost surely globally exact in Riemannian symmetric manifolds. These
properties make pole ladder a very attractive alternative to other methods in
general affine manifolds with a symmetric connection.
