Solution of the $k$-th eigenvalue problem in large-scale electronic structure calculations

TL;DR

This paper introduces a three-stage algorithm for efficiently computing the k-th eigenpair of large sparse Hermitian matrices, crucial for electronic structure calculations, with validation and demonstrated effectiveness on problems up to 1.5 million size.

Contribution

A novel three-stage method combining Lanczos, spectral bisection, and shift-invert Lanczos for accurate large-scale eigenpair computation with index validation.

Findings

Effective for matrices up to 1.5 million size

Accurate eigenpair computation with validated index

Reduced iterations through combined methods

Abstract

We consider computing the $k$-th eigenvalue and its corresponding eigenvector of a generalized Hermitian eigenvalue problem of $n\times n$ large sparse matrices. In electronic structure calculations, several properties of materials, such as those of optoelectronic device materials, are governed by the eigenpair with a material-specific index $k.$ We present a three-stage algorithm for computing the $k$-th eigenpair with validation of its index. In the first stage of the algorithm, we propose an efficient way of finding an interval containing the $k$-th eigenvalue $(1 \ll k \ll n)$ with a non-standard application of the Lanczos method. In the second stage, spectral bisection for large-scale problems is realized using a sparse direct linear solver to narrow down the interval of the $k$-th eigenvalue. In the third stage, we switch to a modified shift-and-invert Lanczos method to reduce bisection iterations and compute the $k$-th eigenpair with validation. Numerical results with problem sizes up to 1.5 million are reported, and the results demonstrate the accuracy and efficiency of the three-stage algorithm.