Fast computation of spectral projectors of banded matrices

TL;DR

This paper introduces a fast, robust method using hierarchical matrices and the QDWH algorithm for approximating spectral projectors of symmetric banded matrices, effective even with small spectral gaps.

Contribution

It presents a novel hierarchical matrix-based approach combined with QDWH, providing a priori error bounds and demonstrating improved performance over existing methods.

Findings

Algorithm is robust to small spectral gaps.

Preliminary Matlab implementation outperforms eig for matrices of a few thousand size.

Provides a priori bounds on approximation error.

Abstract

We consider the approximate computation of spectral projectors for symmetric banded matrices. While this problem has received considerable attention, especially in the context of linear scaling electronic structure methods, the presence of small relative spectral gaps challenges existing methods based on approximate sparsity. In this work, we show how a data-sparse approximation based on hierarchical matrices can be used to overcome this problem. We prove a priori bounds on the approximation error and propose a fast algo- rithm based on the QDWH algorithm, along the works by Nakatsukasa et al. Numerical experiments demonstrate that the performance of our algorithm is robust with respect to the spectral gap. A preliminary Matlab implementation becomes faster than eig already for matrix sizes of a few thousand.