  Many interesting functions arising in applications map into Riemannian
manifolds. We present an algorithm, using the manifold exponential and
logarithm, for approximating such functions. Our approach extends approximation
techniques for functions into linear spaces so that we can upper bound the
forward error in terms of a lower bound on the manifold's sectional curvature.
Furthermore, when the sectional curvature of a manifold is nonnegative, such as
for compact Lie groups, the error is guaranteed to not be worse than in the
linear case. We implement the algorithm in a Julia package and apply it to two
example problems from Krylov subspaces and dynamic low-rank approximation,
respectively. For these examples, the maps are confirmed to be well
approximated by our algorithm.
