Lanczos with compression for symmetric eigenvalue problems

TL;DR

This paper introduces a novel Lanczos with compression method that uses rational approximation to compress Krylov subspaces, reducing matrix-vector products while maintaining convergence properties.

Contribution

It proposes a new compression strategy for Lanczos that differs from polynomial filtering, with theoretical guarantees and practical efficiency improvements.

Findings

Compression introduces minimal error, preserving convergence.

The method often outperforms Krylov--Schur in matrix-vector product efficiency.

The approach remains compatible with subsequent Lanczos steps.

Abstract

The Lanczos method with implicit restarting is one of the most popular methods for finding a few exterior eigenpairs of a large symmetric matrix $A$. Usually based on polynomial filtering, restarting is crucial to limit memory and the cost of orthogonalization. In this work, we propose a novel strategy for the same purpose, called Lanczos with compression. Unlike polynomial filtering, our approach compresses the Krylov subspace using rational approximation and, in doing so, it sacrifices the structure of the associated Krylov decomposition. Nevertheless, it remains compatible with subsequent Lanczos steps and the overall algorithm is still solely based on matrix-vector products with $A$. On the theoretical side, we show that compression introduces only a small error compared to standard (unrestarted) Lanczos and therefore has only a negligible impact on convergence. Comparable guarantees are not available for commonly used implicit restarting strategies, including the Krylov--Schur method. On the practical side, our numerical experiments demonstrate that compression often outperforms the Krylov--Schur method in terms of matrix-vector products.