REORTHOGONALIZED PYTHAGOREAN VARIANTS OF BLOCK CLASSICAL GRAM--SCHMIDT

Kód výsledku v 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>

Výsledek na webu

<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>

Jazyk výsledku

angličtina

Název v původním jazyce

REORTHOGONALIZED PYTHAGOREAN VARIANTS OF BLOCK CLASSICAL GRAM--SCHMIDT

Popis výsledku v původním jazyce

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 &quot;Pythagorean&quot; variant of BCGS, introduced in [E. Carson, K. Lund, and M. Rozlo &amp; zcaron;n &amp; 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)&lt;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.

Název v anglickém jazyce

REORTHOGONALIZED PYTHAGOREAN VARIANTS OF BLOCK CLASSICAL GRAM--SCHMIDT

Popis výsledku anglicky

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 &quot;Pythagorean&quot; variant of BCGS, introduced in [E. Carson, K. Lund, and M. Rozlo &amp; zcaron;n &amp; 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)&lt;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.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10102 - Applied mathematics

Projekt

—

Návaznosti

S - Specificky vyzkum na vysokych skolach<br>I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2025

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

SIAM Journal on Matrix Analysis and Applications

ISSN

0895-4798

e-ISSN

1095-7162

Svazek periodika

46

Číslo periodika v rámci svazku

1

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

31

Strana od-do

310-340

Kód UT WoS článku

001450484100003

EID výsledku v databázi Scopus

2-s2.0-85217929124