Cholesky-like Factorization of Symmetric Indefinite Matrices and Orthogonalization with Respect to Bilinear Forms

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985807%3A_____%2F15%3A00399416" target="_blank" >RIV/67985807:_____/15:00399416 - isvavai.cz</a>

Výsledek na webu

<a href="http://dx.doi.org/10.1137/130947003" target="_blank" >http://dx.doi.org/10.1137/130947003</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1137/130947003" target="_blank" >10.1137/130947003</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Cholesky-like Factorization of Symmetric Indefinite Matrices and Orthogonalization with Respect to Bilinear Forms

Popis výsledku v původním jazyce

It is well known that orthogonalization of column vectors in a rectangular matrix $B$ with respect to the bilinear form induced by a nonsingular symmetric indefinite matrix $A$ can be eventually seen as its factorization $B=QR$ that is equivalent to theCholesky-like factorization in the form $B^TAB=R^T Omega R$, where $R$ is upper triangular and $Omega$ is a signature matrix. Under the assumption of nonzero principal minors of the matrix $M=B^T A B$ we give bounds for the conditioning of the triangular factor $R$ in terms of extremal singular values of $M$ and of only those principal submatrices of $M$ where there is a change of sign in $Omega$. Using these results we study the numerical behavior of two types of orthogonalization schemes and we give the worst-case bounds for quantities computed in finite precision arithmetic. In particular, we analyze the implementation based on the Cholesky-like factorization of $M$ and the Gram--Schmidt process with respect to the bilinear form i

Název v anglickém jazyce

Cholesky-like Factorization of Symmetric Indefinite Matrices and Orthogonalization with Respect to Bilinear Forms

Popis výsledku anglicky

It is well known that orthogonalization of column vectors in a rectangular matrix $B$ with respect to the bilinear form induced by a nonsingular symmetric indefinite matrix $A$ can be eventually seen as its factorization $B=QR$ that is equivalent to theCholesky-like factorization in the form $B^TAB=R^T Omega R$, where $R$ is upper triangular and $Omega$ is a signature matrix. Under the assumption of nonzero principal minors of the matrix $M=B^T A B$ we give bounds for the conditioning of the triangular factor $R$ in terms of extremal singular values of $M$ and of only those principal submatrices of $M$ where there is a change of sign in $Omega$. Using these results we study the numerical behavior of two types of orthogonalization schemes and we give the worst-case bounds for quantities computed in finite precision arithmetic. In particular, we analyze the implementation based on the Cholesky-like factorization of $M$ and the Gram--Schmidt process with respect to the bilinear form i

Druh

J_x - Nezařazeno - Článek v odborném periodiku (Jimp, Jsc a Jost)

CEP obor

BA - Obecná matematika

OECD FORD obor

—

Projekt

<a href="/cs/project/GAP108%2F11%2F0853" target="_blank" >GAP108/11/0853: Nanostruktury obsahující tranzitivní kovy: Směrem k ab-initio materiálovému designu</a><br>

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2015

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

—

Svazek periodika

36

Číslo periodika v rámci svazku

2

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

25

Strana od-do

727-751

Kód UT WoS článku

000357407800019

EID výsledku v databázi Scopus

2-s2.0-84936754616