A fast spectral divide-and-conquer method for banded matrices

TL;DR

This paper introduces a fast, scalable spectral divide-and-conquer algorithm tailored for symmetric banded matrices, leveraging HODLR matrix formats to efficiently compute eigenvalues and eigenvectors with quasilinear complexity.

Contribution

It combines spectral projectors in HODLR format with a new basis extraction technique, enabling efficient eigenvalue computations for large-scale matrices.

Findings

Algorithm exhibits quasilinear complexity.

Effective for large-scale symmetric banded matrices.

Demonstrated efficiency through numerical experiments.

Abstract

Based on the spectral divide-and-conquer algorithm by Nakatsukasa and Higham [SIAM J. Sci. Comput., 35(3): A1325-A1349, 2013], we propose a new algorithm for computing all the eigenvalues and eigenvectors of a symmetric banded matrix. For this purpose, we combine our previous work on the fast computation of spectral projectors in the so called HODLR format, with a novel technique for extracting a basis for the range of such a HODLR matrix. The numerical experiments demonstrate that our algorithm exhibits quasilinear complexity and allows for conveniently dealing with large-scale matrices.