The numerical solution of eigenvalue problems is essential in various application areas of scientific and engineering domains. In many problem classes, the practical interest is only a small subset of eigenvalues so it is unnecessary to compute all of the eigenvalues. Notable examples are the electronic structure problems where the $k$-th smallest eigenvalue is closely related to the electronic properties of materials. In this paper, we consider the $k$-th eigenvalue problems of symmetric dense matrices with low-rank off-diagonal blocks. We present a linear time generalized LDL decomposition of $\mathcal{H}^2$ matrices and combine it with the bisection eigenvalue algorithm to compute the $k$-th eigenvalue with controllable accuracy. In addition, if more than one eigenvalue is required, some of the previous computations can be reused to compute the other eigenvalues in parallel. Numerical experiments show that our method is more efficient than the state-of-the-art dense eigenvalue solver in LAPACK/ScaLAPACK and ELPA. Furthermore, tests on electronic state calculations of carbon nanomaterials demonstrate that our method outperforms the existing HSS-based bisection eigenvalue algorithm on 3D problems.

翻译：特征值问题的数值求解在科学与工程领域的诸多应用场景中至关重要。在许多问题类别中，实际关注的仅是特征值的一个小子集，因此无需计算全部特征值。典型例子是电子结构问题，其中第$k$小特征值与材料的电子特性密切相关。本文研究具有低秩非对角块的对称稠密矩阵的第$k$特征值问题。我们提出一种$\mathcal{H}^2$矩阵的线性时间广义LDL分解方法，并将其与二分特征值算法相结合，以可控精度计算第$k$特征值。此外，若需计算多个特征值，可复用部分先前计算结果并行求解其余特征值。数值实验表明，本方法较LAPACK/ScaLAPACK及ELPA中最先进的稠密特征值求解器更为高效。同时，碳纳米材料电子态计算测试显示，针对三维问题，本方法性能优于现有基于HSS的二分特征值算法。