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

A Riemannian Derivative-Free Polak-Ribiere-Polyak Method for Tangent Vector Field