In this paper, a Parallel Direct Eigensolver for Sequences of Hermitian Eigenvalue Problems with no tridiagonalization is proposed, denoted by \texttt{PDESHEP}, and it combines direct methods with iterative methods. \texttt{PDESHEP} first reduces a Hermitian matrix to its banded form, then applies a spectrum slicing algorithm to the banded matrix, and finally computes the eigenvectors of the original matrix via backtransform. Therefore, compared with conventional direct eigensolvers, \texttt{PDESHEP} avoids tridiagonalization, which consists of many memory-bounded operations. In this work, the iterative method in \texttt{PDESHEP} is based on the contour integral method implemented in FEAST. The combination of direct methods with iterative methods for banded matrices requires some efficient data redistribution algorithms both from 2D to 1D and from 1D to 2D data structures. Hence, some two-step data redistribution algorithms are proposed, which can be $10\times$ faster than ScaLAPACK routine \texttt{PXGEMR2D}. For the symmetric self-consistent field (SCF) eigenvalue problems, \texttt{PDESHEP} can be on average $1.25\times$ faster than the state-of-the-art direct solver in ELPA when using $4096$ processes. Numerical results are obtained for dense Hermitian matrices from real applications and large real sparse matrices from the SuiteSparse collection.

中文翻译：

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无三对角线化的Hermitian特征值问题序列的并行直接特征求解器

本文提出了一种并行的直接特征求解器，用于求解无三对角化的埃尔米特特征值问题序列，用\ texttt {PDESHEP}表示，它将直接方法与迭代方法结合起来。\ texttt {PDESHEP}首先将Hermitian矩阵简化为带状形式，然后将频谱切片算法应用于带状矩阵，最后通过逆变换计算原始矩阵的特征向量。因此，与传统的直接本征求解器相比，\ texttt {PDESHEP}避免了由许多内存受限操作组成的三对角化。在本文中，\ texttt {PDESHEP}中的迭代方法基于FEAST中实现的轮廓积分方法。带状矩阵的直接方法与迭代方法的结合要求从2D到1D以及从1D到2D数据结构的一些有效数据重新分配算法。因此，提出了一些两步数据重新分配算法，该算法比ScaLAPACK例程\ texttt {PXGEMR2D}快10倍。对于对称自洽字段（SCF）特征值问题，当使用$ 4096 $进程时，\ texttt {PDESHEP}的平均速度比ELPA中最先进的直接求解器快$ 1.25 \ times $。从实际应用中获得稠密Hermitian矩阵的数值结果，从SuiteSparse集合中获得大型实际稀疏矩阵的数值结果。对于对称自洽字段（SCF）特征值问题，当使用$ 4096 $进程时，\ texttt {PDESHEP}的平均速度比ELPA中最先进的直接求解器快$ 1.25 \ times $。从实际应用中获得稠密Hermitian矩阵的数值结果，从SuiteSparse集合中获得大型实际稀疏矩阵的数值结果。对于对称自洽字段（SCF）特征值问题，当使用$ 4096 $进程时，\ texttt {PDESHEP}的平均速度比ELPA中最先进的直接求解器快$ 1.25 \ times $。从实际应用中获得稠密Hermitian矩阵的数值结果，从SuiteSparse集合中获得大型实际稀疏矩阵的数值结果。