A Riemannian nonmonotone spectral method for self-adjoint tangent vector field

Based on the requirement of specific problems, for instance unconstrained and equality-constrained Rayleigh quotient problems, we consider the problem of finding zeros of a tangent vector field on Riemannian manifold. More precisely, we focus on the study of self-adjoint tangent vector field in this paper. By making full use of the self-adjointness property of the tangent vector field, we propose an effective Riemannian spectral method to solve the problem, which is derivative free with nonmonotone line search employed. Through analysis, we find that the algorithm can achieve global convergence under certain conditions, which is a good result. At the end of the paper, numerical test results of the algorithm are given. We find that the proposed algorithm not only has an improvement in speed and time, but also is applicable to large-scale problems.