The stability of block variants of classical Gram-Schmidt
The result's identifiers
Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985840%3A_____%2F21%3A00545475" target="_blank" >RIV/67985840:_____/21:00545475 - isvavai.cz</a>
Alternative codes found
RIV/00216208:11320/21:10436234
Result on the web
<a href="https://doi.org/10.1137/21M1394424" target="_blank" >https://doi.org/10.1137/21M1394424</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1137/21M1394424" target="_blank" >10.1137/21M1394424</a>
Alternative languages
Result language
angličtina
Original language name
The stability of block variants of classical Gram-Schmidt
Original language description
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.
Czech name
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Czech description
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Classification
Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
—
OECD FORD branch
10101 - Pure mathematics
Result continuities
Project
<a href="/en/project/GA20-01074S" target="_blank" >GA20-01074S: Adaptive methods for the numerical solution of partial differential equations: analysis, error estimates and iterative solvers</a><br>
Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace
Others
Publication year
2021
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Name of the periodical
SIAM Journal on Matrix Analysis and Applications
ISSN
0895-4798
e-ISSN
1095-7162
Volume of the periodical
42
Issue of the periodical within the volume
3
Country of publishing house
US - UNITED STATES
Number of pages
16
Pages from-to
1365-1380
UT code for WoS article
000704163900013
EID of the result in the Scopus database
2-s2.0-85115015787