Based on the spectral divide-and-conquer algorithm by Nakatsukasa and Higham [SIAM J. Sci. Comput., 35(3):A1325–A1349, 2013], we propose a new algorithm for computing all the eigenvalues and eigenvectors of a symmetric banded matrix with small bandwidth, with the eigenvectors given implicitly as a product of orthonormal matrices stored in the so-called hierarchically off-diagonal low-rank (HODLR) format. For this purpose, we combine our previous work on the fast computation of spectral projectors in the HODLR format, with a novel technique for extracting a basis for the range of such a HODLR matrix. Preliminary numerical experiments demonstrate that our algorithm exhibits quasi-linear complexity for matrices that can be efficiently represented in the HODLR format throughout the divide-and-conquer algorithm, and allows for conveniently dealing with such large-scale matrices.

中文翻译：

---

带状矩阵的一种快速谱分治法

基于 Nakatsukasa 和 Higham 的光谱分治算法 [SIAM J. Sci. Comput., 35(3):A1325–A1349, 2013]，我们提出了一种新算法，用于计算具有小带宽的对称带状矩阵的所有特征值和特征向量，特征向量隐式给出为存储在所谓的分层非对角低秩 (HODLR) 格式。为此，我们结合了我们之前在 HODLR 格式中快速计算光谱投影仪的工作，以及一种提取这种 HODLR 矩阵范围基础的新技术。初步数值实验表明，我们的算法表现出矩阵的准线性复杂度，可以在整个分而治之算法中以 HODLR 格式有效表示，