In this paper, a parallel structured divide-and-conquer (PSDC) eigensolver is proposed for symmetric tridiagonal matrices based on ScaLAPACK and a parallel structured matrix multiplication algorithm, called PSMMA. Computing the eigenvectors via matrix-matrix multiplications is the most computationally expensive part of the divide-and-conquer algorithm, and one of the matrices involved in such multiplications is a rank-structured Cauchy-like matrix. By exploiting this particular property, PSMMA constructs the local matrices by using generators of Cauchy-like matrices without any communication, and further reduces the computation costs by using a structured low-rank approximation algorithm. Thus, both the communication and computation costs are reduced. Experimental results show that both PSMMA and PSDC are highly scalable and scale to 4096 processes at least. PSDC has better scalability than PHDC that was proposed in [J. Comput. Appl. Math. 344 (2018) 512--520] and only scaled to 300 processes for the same matrices. Comparing with \texttt{PDSTEDC} in ScaLAPACK, PSDC is always faster and achieves $1.4$x--$1.6$x speedup for some matrices with few deflations. PSDC is also comparable with ELPA, with PSDC being faster than ELPA when using few processes and a little slower when using many processes.

翻译：在本文中,为基于 ScaLAPACK 的对称三对角三角矩阵和称为 PSMA 的平行结构化三对角矩阵提议了一个平行结构分解和正方格(PSDC) 。通过矩阵矩阵矩阵矩阵乘法乘数计算出导出方方程式是最计算昂贵的部分,而这种乘法所涉的矩阵之一是一个等级结构化的孔式矩阵。通过利用这一特定属性,PSMMA在没有任何通信的情况下使用像Caus一样的矩阵生成器来构建本地矩阵,并通过使用结构化的低级近似矩阵乘法进一步降低计算成本。因此,通信和计算成本都降低了。实验结果表明,PSMMA和SMC的乘法都高度可伸缩,至少达到4096个过程。 私营部门司比[J. 略调控系统算数. 344 (2018) 512-5200] 中提议的PSMMTFC 的缩算法要好得多,而在同一矩阵中仅达到300个流程。