In this paper, we propose, analyze and demonstrate a dynamic momentum method to accelerate power and inverse power iterations with minimal computational overhead. The method is appropriate for real, diagonalizable matrices, and does not require a priori spectral knowledge. We review and extend background results on previously developed static momentum accelerations for the power iteration through the connection between the momentum accelerated iteration and the standard power iteration applied to an augmented matrix. We show that the augmented matrix is defective for the optimal parameter choice. We then present our dynamic method which updates the momentum parameter at each iteration based on the Rayleigh quotient and two previous residuals. We present convergence and stability theory for the method by considering a power-like method consisting of multiplying an initial vector by a sequence of augmented matrices. We demonstrate the developed method on a number of benchmark problems, and see that it outperforms both the power iteration and often the static momentum acceleration with optimal parameter choice. Finally, we present and demonstrate an explicit extension of the algorithm to inverse power iterations.

翻译：本文提出、分析并展示了一种动态动量方法，用于以极小计算开销加速幂迭代和反幂迭代。该方法适用于实可对角化矩阵，且无需先验谱知识。我们通过动量加速迭代与应用于增广矩阵的标准幂迭代之间的关联，回顾并扩展了先前发展的静态动量加速幂迭代的背景结果，并指出最优参数选择下增广矩阵是亏损的。随后提出动态方法：每次迭代基于瑞利商和两个先前残差更新动量参数。通过考虑由初始向量与一系列增广矩阵相乘构成的类幂迭代方法，建立了该方法的收敛性与稳定性理论。我们在多个基准问题上验证所提方法，表明其不仅优于幂迭代，且通常超越采用最优参数选择的静态动量加速。最后，我们展示并验证了该算法向反幂迭代的显式扩展。