The stability of block variants of classical Gram-Schmidt

Kód výsledku v 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>

Nalezeny alternativní kódy

RIV/00216208:11320/21:10436234

Výsledek na webu

<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>

Jazyk výsledku

angličtina

Název v původním jazyce

The stability of block variants of classical Gram-Schmidt

Popis výsledku v původním jazyce

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.

Název v anglickém jazyce

The stability of block variants of classical Gram-Schmidt

Popis výsledku anglicky

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.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

<a href="/cs/project/GA20-01074S" target="_blank" >GA20-01074S: Adaptivní metody pro numerické řešení parciálních diferenciálních rovnic: analýza, odhady chyb a iterativní řešiče</a><br>

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2021

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

42

Číslo periodika v rámci svazku

3

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

16

Strana od-do

1365-1380

Kód UT WoS článku

000704163900013

EID výsledku v databázi Scopus

2-s2.0-85115015787