The stability of block variants of classical Gram-Schmidt

0545475 - MÚ 2022 RIV US eng J - Článek v odborném periodiku
Carson, E. - Lund, K. - Rozložník, Miroslav
The stability of block variants of classical Gram-Schmidt.
SIAM Journal on Matrix Analysis and Applications. Roč. 42, č. 3 (2021), s. 1365-1380. ISSN 0895-4798. E-ISSN 1095-7162
Grant CEP: GA ČR(CZ) GA20-01074S
Institucionální podpora: RVO:67985840
Klíčová slova: Gram-Schmidt * block vectors * block Krylov subspace methods * loss of orthogonality
Obor OECD: Pure mathematics
Impakt faktor: 1.908, rok: 2021
Způsob publikování: Omezený přístup
https://doi.org/10.1137/21M1394424

The block version of the classical Gram--Schmidt (tt BCGS) method is often employed to efficiently compute orthogonal bases for Krylov subspace methods and eigenvalue solvers, but a rigorous proof of its stability behavior has not yet been established. It is shown that the usual implementation of tt BCGS can lose orthogonality at a rate worse than $O(varepsilon) kappa^{2}({$mathcalX$})$, where $mathcal{X}$ is the input matrix and $varepsilon$ is the unit roundoff. A useful intermediate quantity denoted as the Cholesky residual is given special attention and, along with a block generalization of the Pythagorean theorem, this quantity is used to develop more stable variants of tt BCGS. These variants are proven to have $O(varepsilon) kappa^2({$mathcalX$})$ loss of orthogonality with relatively relaxed conditions on the intrablock orthogonalization routine satisfied by the most commonly used algorithms. A variety of numerical examples illustrate the theoretical bounds.
Trvalý link: http://hdl.handle.net/11104/0322163