the Czech Academy of Sciences, Žitná 25, 115 67 Prague, Czech Republic'
- NVIDIA Corporation
- 'Center for Nanophase Materials Sciences, Oak Ridge National Laboratory, Oak Ridge, TN 37830, USA'
author:
- 'Victor Wen-zhe Yu'
- Jonathan Moussa
- Pavel Ks
- Andreas Marek
- Peter Messmer
- Mina Yoon
- Hermann Lederer
- Volker Blum
bibliography:
- 'elpa2gpu.bib'
title: 'GPU-Acceleration of the ELPA2 Distributed Eigensolver for Dense Symmetric and Hermitian Eigenproblems'
---

Eigensolver ,dense linear algebra ,parallel computing ,high-performance computing ,GPU ,CUDA

Introduction {#sec:intro}
============

Finding the eigenvalues and eigenvectors of large dense matrices is a frequent problem in computational science and engineering. For example, in Kohn-Sham density-functional theory (KS-DFT) [@dft_hohenberg_1964; @dft_kohn_1965], the many-electron problem for the Born-Oppenheimer electronic ground state is reduced to a system of single particle equations that can be discretized into a generalized eigenproblem in the following matrix form $$\label{eq:gevp}
\boldsymbol{H} \boldsymbol{C} = \boldsymbol{S} \boldsymbol{C} \boldsymbol{\Sigma} .$$

Here the Hamiltonian matrix $\boldsymbol{H}$ and the overlap matrix $\boldsymbol{S}$ are real symmetric or complex Hermitian. The matrix $\boldsymbol{C}$ and the diagonal matrix $\boldsymbol{\Sigma}$ are the eigenvectors and eigenvalues, respectively, of this eigensystem. In the framework of KS-DFT, $\boldsymbol{C}$ and $\boldsymbol{\Sigma}$ (or the information they carry, at least) 