Abstract 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.

中文翻译：

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自伴随切向量场的黎曼非单调谱方法

摘要 根据具体问题的要求，例如无约束和等式约束的瑞利商问题，我们考虑在黎曼流形上求切向量场的零点问题。更准确地说，我们在本文中专注于自伴随切向量场的研究。充分利用切向量场的自伴随性，我们提出了一种有效的黎曼谱方法来解决该问题，该方法是采用非单调线搜索的无导数方法。通过分析，我们发现该算法在一定条件下可以实现全局收敛，是一个很好的结果。文末给出了算法的数值试验结果。我们发现所提出的算法不仅在速度和时间上都有了提升，