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

Result code in 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>
Result on the web
<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>

Result language
angličtina
Original language name
Cholesky-like Factorization of Symmetric Indefinite Matrices and Orthogonalization with Respect to Bilinear Forms
Original language description
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
Czech name
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Czech description
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Type
J<sub>x</sub> - Unclassified - Peer-reviewed scientific article (Jimp, Jsc and Jost)
CEP classification
BA - General mathematics
OECD FORD branch
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Project
<a href="/en/project/GAP108%2F11%2F0853" target="_blank" >GAP108/11/0853: Nanostructures with transition metals: Towards ab-initio material design</a><br>
Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year
2015
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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