SuperDC：稳定的超快速分治eigenvalue分解
SuperDC: Stable superfast divide-and-conquer eigenvalue decomposition
摘要：
对于具有小的非对角（数值）秩和分层半可分形式的稠密埃尔米特矩阵，我们给出了一种具有近似线性复杂度的稳定分治特征分解方法（称为SuperDC），该方法显着改进了[Vogel，Xia，et al.，Siam J. Sci.计算：38（2016）]。我们采用了一系列的关键稳定性技术，并提供了许多改进的算法设计。分析了原算法的各种稳定性风险，包括潜在的指数范数增长，取消，与集群特征值或中间特征值的准确性损失等。在划分阶段，我们给出了一个新的结构化低秩更新策略与平衡，消除指数范数增长，也最大限度地减少低秩更新的秩。在低秩更新特征值解的征服阶段，原算法直接使用正则快速多极子方法（FMM）加速函数求值，存在取消、被零除和收敛速度慢的风险。在这里，我们设计了一个三角形FMM，以避免取消。此外，当有集群的中间特征值或现有的特征值的更新是非常小的，我们设计了一种新的本地移位策略集成FMM加速度到移位长期方程的解决方案，以实现效率和可靠性。我们还提供了一些改进或澄清的一些结构和技术，在以前的工作中缺失或不清楚。由此产生的SuperDC特征值求解器具有显着更好的稳定性，同时保持接近线性的复杂性，找到整个特征值分解。在一组综合测试中，SuperDC显示出比Matlab $\textsf{eig}$函数显著更低的运行时间和存储，以及更多的稳定性优势。
英文摘要：
For dense Hermitian matrices with small off-diagonal (numerical) ranks and in a hierarchically semiseparable form, we give a stable divide-and-conquer eigendecomposition method with nearly linear complexity (called SuperDC) that significantly improves an earlier basic algorithm in [Vogel, Xia, et al., SIAM J. Sci. Comput., 38 (2016)]. We incorporate a sequence of key stability techniques and provide many improvements in the algorithm design. Various stability risks in the original algorithm are analyzed, including potential exponential norm growth, cancellations, loss of accuracy with clustered eigenvalues or intermediate eigenvalues, etc. In the dividing stage, we give a new structured low-rank update strategy with balancing that eliminates the exponential norm growth and also minimizes the ranks of low-rank updates. In the conquering stage with low-rank updated eigenvalue solution, the original algorithm directly uses the regular fast multipole method (FMM) to accelerate function evaluations, which has the risks of cancellation, division by zero, and slow convergence. Here, we design a triangular FMM to avoid cancellation. Furthermore, when there are clustered intermediate eigenvalues or when updates to existing eigenvalues are very small, we design a novel local shifting strategy to integrate FMM accelerations into the solution of shifted secular equations so as to achieve both the efficiency and the reliability. We also provide several improvements or clarifications on some structures and techniques that are missing or unclear in the previous work. The resulting SuperDC eigensolver has significantly better stability while keeping the nearly linear complexity for finding the entire eigenvalue decomposition. In a set of comprehensive tests, SuperDC shows dramatically lower runtime and storage, and more stability advantages than the Matlab $\textsf{eig}$ function.
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