This paper explores variants of the subspace iteration algorithm for computing approximate invariant subspaces. The standard subspace iteration approach is revisited and new variants that exploit gradient-type techniques combined with a Grassmann manifold viewpoint are developed. A gradient method as well as a nonlinear conjugate gradient technique are described. Convergence of the gradient-based algorithm is analyzed and a few numerical experiments are reported, indicating that the proposed algorithms are sometimes superior to standard algorithms. This includes the Chebyshev-based subspace iteration and the locally optimal block conjugate gradient method, when compared in terms of number of matrix vector products and computational time, resp. The new methods, on the other hand, do not require estimating optimal parameters. An important contribution of this paper to achieve this good performance is the accurate and efficient implementation of an exact line search. In addition, new convergence proofs are presented for the non-accelerated gradient method that includes a locally exponential convergence if started in a $\mathcal{O(\sqrt{\delta})}$ neighbourhood of the dominant subspace with spectral gap $\delta$.

翻译：本文探讨了用于计算近似不变子空间的子空间迭代算法的变体。本文重新审视了标准的子空间迭代方法，并开发了结合Grassmann流形视角与梯度技巧的新变体。文中描述了梯度方法以及非线性共轭梯度技术。分析了基于梯度的算法的收敛性，并报告了一些数值实验，表明所提出的算法有时优于标准算法。这包括基于切比雪夫的子空间迭代和局部最优块共轭梯度方法，在矩阵向量乘积次数和计算时间方面进行比较时，分别显示出优势。另一方面，新方法不需要估计最优参数。实现这种良好性能的一个重要贡献是精确高效的精确线搜索实现。此外，为非加速梯度方法提出了新的收敛性证明，包括如果从主不变子空间的$\mathcal{O(\sqrt{\delta})}$邻域内开始（其中谱间隙为$\delta$），则可实现局部指数收敛。