REORTHOGONALIZED PYTHAGOREAN VARIANTS OF BLOCK CLASSICAL GRAM--SCHMIDT
The result's identifiers
Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F25%3A10509854" target="_blank" >RIV/00216208:11320/25:10509854 - isvavai.cz</a>
Result on the web
<a href="https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=oMRyUQ.DOn" target="_blank" >https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=oMRyUQ.DOn</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1137/24M1658723" target="_blank" >10.1137/24M1658723</a>
Alternative languages
Result language
angličtina
Original language name
REORTHOGONALIZED PYTHAGOREAN VARIANTS OF BLOCK CLASSICAL GRAM--SCHMIDT
Original language description
Block classical Gram-Schmidt (BCGS) is commonly used for orthogonalizing a set of vectors X in distributed computing environments due to its favorable communication properties relative to other orthogonalization approaches, such as modified Gram-Schmidt or Householder. However, it is known that BCGS (as well as recently developed low-synchronization variants of BCGS) can suffer from a significant loss of orthogonality in finite-precision arithmetic, which can contribute to instability and inaccurate solutions in downstream applications such as s-step Krylov subspace methods. A common solution to improve the orthogonality among the vectors is reorthogonalization. Focusing on the "Pythagorean" variant of BCGS, introduced in [E. Carson, K. Lund, and M. Rozlo & zcaron;n & iacute;k, SIAM J. Matrix Anal. Appl., 42 (2021), pp. 1365-1380], which guarantees an O(epsilon)kappa(2)(X) bound on the loss of orthogonality as long as O(epsilon)kappa(2)(X)<1, where epsilon denotes the unit roundoff, we introduce and analyze two reorthogonalized Pythagorean BCGS variants. These variants feature favorable communication properties, with asymptotically two synchronization points per block column, as well as an improved O(epsilon) bound on the loss of orthogonality. Our bounds are derived in a general fashion to additionally allow for the analysis of mixed-precision variants. We verify our theoretical results with a panel of test matrices and experiments from a new version of the BlockStab toolbox.
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
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OECD FORD branch
10102 - Applied mathematics
Result continuities
Project
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Continuities
S - Specificky vyzkum na vysokych skolach<br>I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace
Others
Publication year
2025
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
46
Issue of the periodical within the volume
1
Country of publishing house
US - UNITED STATES
Number of pages
31
Pages from-to
310-340
UT code for WoS article
001450484100003
EID of the result in the Scopus database
2-s2.0-85217929124