SuperDC:提高秩结构矩阵稳定性的超快速分治特征值分解

Xiaofeng Ou, J. Xia
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引用次数: 3

摘要

. 对于具有小非对角线(数值)秩和5层次半可分形式的密集对称矩阵,我们给出了具有6近线性复杂度的分治特征分解方法(称为SuperDC),该方法显着改进了先前的基本算法[Vogel, Xia, et al., SIAM J. Sci]。第一版。, 38(2016)]。分析了原算法存在的稳定性风险,包括指数范数增长、消去、聚类特征值或中间特征值导致的精度损失等。在划分阶段,我们给出了一种新的具有平衡的结构化低秩更新策略,该策略消除了指数范数增长,并且最小化了低秩更新的秩。在低秩更新特征值解的征服阶段,原算法直接使用标准快速多极法(FMM)加速函数13的求值,存在消去、除零、收敛慢的风险。在这里,我们设计了一个三角形FMM来避免抵消。此外,当存在聚类中间特征值时,我们设计了一种新颖的局部移位策略,将FMM加速度集成到移位长期方程的解中。这有助于实现效率和可靠性。我们还提供了一种用户提供公差的压缩策略,并给出了结果特征向量矩阵结构的精确描述。SuperDC特征求解器在保持整个特征值分解的近似线性复杂度的同时,显著提高了稳定性。通过大量的20个数值试验,验证了SuperDC的效率和准确性。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
SuperDC: Superfast Divide-And-Conquer Eigenvalue Decomposition With Improved Stability for Rank-Structured Matrices
. For dense symmetric matrices with small off-diagonal (numerical) ranks and in a 5 hierarchically semiseparable form, we give a divide-and-conquer eigendecomposition method with 6 nearly linear complexity (called SuperDC) that significantly improves an earlier basic algorithm in 7 [Vogel, Xia, et al., SIAM J. Sci. Comput., 38 (2016)]. Some stability risks in the original algorithm are 8 analyzed, including potential exponential norm growth, cancellations, loss of accuracy with clustered 9 eigenvalues or intermediate eigenvalues, etc. In the dividing stage, we give a new structured low-rank 10 updating strategy with balancing that eliminates the exponential norm growth and also minimizes 11 the ranks of low-rank updates. In the conquering stage with low-rank updated eigenvalue solution, 12 the original algorithm directly uses the standard fast multipole method (FMM) to accelerate function 13 evaluations, which has the risks of cancellation, division by zero, and slow convergence. Here, we 14 design a triangular FMM to avoid cancellation. Furthermore, when there are clustered intermediate 15 eigenvalues, we design a novel local shifting strategy to integrate FMM accelerations into the solution 16 of shifted secular equations. This helps achieve both the efficiency and the reliability. We also provide 17 a deflation strategy with a user-supplied tolerance and give a precise description of the structure of 18 the resulting eigenvector matrix. The SuperDC eigensolver has significantly improved stability while 19 keeping the nearly linear complexity for finding the entire eigenvalue decomposition. Extensive 20 numerical tests are used to show the efficiency and accuracy of SuperDC.
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