Block Gram-Schmidt algorithms and their stability properties

0550984 - MÚ 2023 RIV US eng J - Článek v odborném periodiku
Carson, E. - Lund, K. - Rozložník, Miroslav - Thomas, S.
Block Gram-Schmidt algorithms and their stability properties.
Linear Algebra and Its Applications. Roč. 638, April 1 (2022), s. 150-195. ISSN 0024-3795. E-ISSN 1873-1856
Grant CEP: GA ČR(CZ) GA20-01074S
Institucionální podpora: RVO:67985840
Klíčová slova: Gram-Schmidt * Block Krylov subspace methods * stabilityLoss of orthogonality
Obor OECD: Pure mathematics
Impakt faktor: 1.1, rok: 2022
Způsob publikování: Open access
https://doi.org/10.1016/j.laa.2021.12.017

Block Gram-Schmidt algorithms serve as essential kernels in many scientific computing applications, but for many commonly used variants, a rigorous treatment of their stability properties remains open. This work provides a comprehensive categorization of block Gram-Schmidt algorithms, particularly those used in Krylov subspace methods to build orthonormal bases one block vector at a time. Known stability results are assembled, and new results are summarized or conjectured for important communication-reducing variants. Additionally, new block versions of low-synchronization variants are derived, and their efficacy and stability are demonstrated for a wide range of challenging examples. Numerical examples are computed with a versatile Matlab package hosted at https://github.com/katlund/BlockStab, and scripts for reproducing all results in the paper are provided. Block Gram-Schmidt implementations in popular software packages are discussed, along with a number of open problems. An appendix containing all algorithms type-set in a uniform fashion is provided.
Trvalý link: http://hdl.handle.net/11104/0326259