An efficient hybrid tridiagonal divide-and-conquer algorithm on distributed memory architectures

TL;DR

This paper introduces a hybrid divide-and-conquer algorithm for symmetric tridiagonal matrices that leverages HSS matrix techniques to accelerate eigenvector computations in distributed memory systems.

Contribution

It presents a novel parallel hybrid DC algorithm that integrates HSS matrix techniques with existing methods to improve efficiency on distributed architectures.

Findings

PHDC can outperform MKL for certain matrices with fewer deflations.

The speedup diminishes as the number of processes increases.

PHDC is comparable to ELPA for some matrices but slower at high process counts.

Abstract

In this paper, an efficient divide-and-conquer (DC) algorithm is proposed for the symmetric tridiagonal matrices based on ScaLAPACK and the hierarchically semiseparable (HSS) matrices. HSS is an important type of rank-structured matrices.Most time of the DC algorithm is cost by computing the eigenvectors via the matrix-matrix multiplications (MMM). In our parallel hybrid DC (PHDC) algorithm, MMM is accelerated by using the HSS matrix techniques when the intermediate matrix is large. All the HSS algorithms are done via the package STRUMPACK. PHDC has been tested by using many different matrices. Compared with the DC implementation in MKL, PHDC can be faster for some matrices with few deflations when using hundreds of processes. However, the gains decrease as the number of processes increases. The comparisons of PHDC with ELPA (the Eigenvalue soLvers for Petascale Applications library) are similar. PHDC is usually slower than MKL and ELPA when using 300 or more processes on Tianhe-2 supercomputer.