Geodesic
Estimates for the bilinear form $x^T A^{-1} y$ with applications to linear algebra problems
Electronic transactions on numerical analysis, Tome 43 (2015).

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: Let $A \in \mathbb{R}^{p\times p}$ be a nonsingular matrix and $x,y$ vectors in $\mathbb{R}^p$. The task of this paper is to develop efficient estimation methods for the bilinear form $x^TA^{-1}y$ based on the extrapolation of moments of the matrix $A$ at the point $-1.$ The extrapolation method and estimates for the trace of $A^{-1}$ presented in Brezinski et al. [Numer. Linear Algebra Appl., 19 (2012), pp. 937 -- 953] are extended, and families of estimates efficiently approximating the bilinear form requiring only few matrix vector products are derived. Numerical approximations of the entries and the trace of the inverse of any real nonsingular matrix are presented and several numerical results, discussions, and comparisons are given.
Classification : 65F15, 65F30, 65B05, 65C05, 65J10, 15A18, 15A45
Keywords: extrapolation, matrix moments, matrix inverse, trace 
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     author = {Fika, Paraskevi and Mitrouli, Marilena and Roupa, Paraskevi},
     title = {Estimates for the bilinear form $x^T A^{-1} y$ with applications to linear algebra problems},
     journal = {Electronic transactions on numerical analysis},
     publisher = {mathdoc},
     volume = {43},
     year = {2015},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ETNA_2015__43__a7/}
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Fika, Paraskevi; Mitrouli, Marilena; Roupa, Paraskevi. Estimates for the bilinear form $x^T A^{-1} y$ with applications to linear algebra problems. Electronic transactions on numerical analysis, Tome 43 (2015). http://geodesic.mathdoc.fr/item/ETNA_2015__43__a7/