Fast and accurate randomized algorithms for linear systems and eigenvalue problems

Journal article

This paper develops a class of algorithms for general linear systems and eigenvalue problems. These algorithms apply fast randomized dimension reduction (``sketching"") to accelerate standard subspace projection methods, such as GMRES and Rayleigh--Ritz. This modification makes it possible to incorporate nontraditional bases for the approximation subspace that are easier to construct. When the basis is numerically full rank, the new algorithms have accuracy similar to classic methods but run faster and may use less storage. For model problems, numerical experiments show large advantages over the optimized MATLAB routines, including a 70\times speedup over gmres and a 10\times speedup over eigs.

Authors: Nakatsukasa, Y; Tropp, JA

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Society for Industrial and Applied Mathematics
• Journal: SIAM Journal on Matrix Analysis and Applications
• Volume: 45
• Issue: 2
• Pages: 1183-1214
• Publication date: 2024-06-20
• Acceptance date: 2024-01-31
• DOI: 10.1137/23M1565413
• EISSN: 1095-7162
• ISSN: 0895-4798
• Language: English
• Keywords: Eigenvalue problem; projection method; sketching; Petrov–Galerkin 13 method; numerical linear algebra; Rayleigh–Ritz; subspace embedding; linear system; randomized algorithm