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 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 a standard Chebyshev-based subspace iteration when compared in terms of number of matrix vector products, but 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：本文探讨了子空间迭代算法的变体，用于计算近似不变子空间。我们重新审视了标准子空间迭代方法，并开发了结合梯度型技术与Grassmann流形视角的新变体。描述了一种梯度方法以及一种共轭梯度技术。分析了基于梯度算法的收敛性，并报告了一些数值实验，表明所提出的算法在矩阵向量乘积数量方面有时优于标准的Chebyshev基子空间迭代，且无需估计最优参数。实现这一良好性能的重要贡献在于精确且高效地实现了精确线搜索。此外，针对非加速梯度方法给出了新的收敛性证明，包括当从主子空间邻域$\mathcal{O(\sqrt{\delta})}$（谱间隙为$\delta$）出发时，局部指数收敛的结果。