基于两步对角化的对称稠密矩阵特征值问题快速求解算法

A New Algorithm For Rapid Symmetric Eigenvalue Problems Based on Two-step Tridiagonalization

计算对称矩阵中的某些特定的特征值和特征向量问题是很多科学计算领域中都存在的重要课题。特别在电子结构的计算中,特征值计算成为计算瓶颈。以往在需要求解大部分特征值和特征向量的应用场合,一般使用直接求解的方式。为了更好地利用存储器性能优势,我们设计了对角化算法,对规约与逆变换过程进行拆分处理,通过对整个过程的重新设计,充分利用存储器结构上的优势,提升单核计算速度,同时改进并行效率。本文中我们重点讨论三对角矩阵到带状矩阵逆变换过程。本文中所提及到的算法应用于MESIA电子结构计算软件包之中,取得了一定的性能提升。

In many different applications, to compute the eigenpairs that includes the eigenvalue and the according eigenvector of a symmetric matrix is a meaningful and challenging problem. Especially in the calculation of electronic structure the eigenproblem solving procedure can be a significant portion of the whole calculation time. The common way of the solution is to calculate the eigenpairs directly. To get better use of the processor and the memory hierarchy, we re-designed the entire diagonalization algorithm. We split the reduction phase and back-transforming phase into two stages separately, consequently we can use the memory hierarchy and improve the parallel efficiency simultaneously to gain a better performance and to reduce the computation cost. This method has been improvemented on the electronic structure procedure of the MESIA software package.