  In this paper, the spectral algorithm for nonlinear equations (SANE) is
adapted to the problem of finding a zero of a given tangent vector field on a
Riemannian manifold. The generalized version of SANE uses, in a systematic way,
the tangent vector field as a search direction and a continuous real-valued
function that adapts this direction and ensures that it verifies a descent
condition for an associated merit function. In order to speed up the
convergence of the proposed method, we incorporate a Riemannian adaptive
spectral parameter in combination with a non-monotone globalization technique.
The global convergence of the proposed procedure is established under some
standard assumptions. Numerical results indicate that our algorithm is very
effective and efficient solving tangent vector field on different Riemannian
manifolds and competes favorably with a Polak-Ribi\'ere-Polyak Method recently
published and other methods existing in the literature.
