  This paper is concerned with the problem of finding a zero of a tangent
vector field on a Riemannian manifold. We first reformulate the problem as an
equivalent Riemannian optimization problem. Then we propose a Riemannian
derivative-free Polak-Ribi\'ere-Polyak method for solving the Riemannian
optimization problem, where a non-monotone line search is employed. The global
convergence of the proposed method is established under some mild assumptions.
To further improve the efficiency, we also provide a hybrid method, which
combines the proposed geometric method with the Riemannian Newton method.
Finally, some numerical experiments are reported to illustrate the efficiency
of the proposed method.
