Lanczos with compression for symmetric eigenvalue problems

The Lanczos method with implicit restarting is one of the most successful algorithms for computing a few eigenpairs of large-scale symmetric matrices.Despite its widespread use, the core idea of employing polynomial filtering for restarting has remained essentially unchanged for over two decades. In this talk, we introduce a novel compression strategy, termed Lanczos with compression, as an alternative to restarting. Unlike traditional restarting, Lanczos with compression sacrifices the Krylov subspace structure but preserves the subsequent Lanczos sequence. Our theoretical analysis shows that the compression introduces only a small error compared to the standard Lanczos method. This talk is based on joint work with Angelo A. Casulli (GSSI) and Daniel Kressner (EPFL).