PHIDE: A Parallel Hybrid Direct–Iterative Eigensolver for Hermitian Eigenvalue Problems

Abstract

In this paper, we propose a Parallel Hybrid Direct–Iterative Eigensolver for Hermitian Eigenvalue Problems without tridiagonalization, denoted by PHIDE, which combines direct and iterative methods. PHIDE first reduces a Hermitian matrix to banded form, then applies a spectrum slicing algorithm to the banded matrix, and finally computes the eigenvectors of the original matrix via backtransformation. Compared with conventional direct eigensolvers, PHIDE avoids tridiagonalization, which involves many memory-bound operations. In PHIDE, the banded eigenvalue problem is solved using the contour integral method implemented in FEAST, which may yield slightly lower accuracy than tridiagonalization-based approaches. For sequences of correlated Hermitian eigenvalue problems arising in density functional theory (DFT), PHIDE achieves an average speedup of $1.22\times$ over the state-of-the-art direct solver in ELPA when using 1024 processes. Numerical experiments are conducted on dense Hermitian matrices from real applications as well as large sparse matrices from the SuiteSparse and ELSES collections.