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A0280
Title: Numerical linear algebra for computational statistics Authors:  Zlatko Drmac - University of Zagreb (Croatia) [presenting]
Abstract: Why has a covariance matrix negative eigenvalues and other questions related to numerical procedures in computational statistics are discussed, in particular on eigenvalues, singular value decomposition (SVD) and its generalization, the GSVD. These are the tools of trade in various applications, including computational statistics, least squares modeling, vibration analysis in structural engineering - just to name a few. In essence, the GSVD can be reduced to the SVD of certain products and quotients of matrices. For instance, in the canonical correlation analysis of two sets of variables $x$, $y$, with joint distribution and the covariance matrix $C= ( C_{xx}, C_{xy} ; C_{yx}, C_{yy})$, wanted is the SVD of the product $C_{xx}^{-1/2}C_{xy}C_{yy}^{-1/2}$. However, using software implementations of numerical algorithms is not that simple, despite availability of many well-known state-of-the-art software packages. We will review the recent advances in this important part of numerical linear algebra, with particular attention to \emph{(i)} understanding the sensitivity and condition numbers; \emph{(ii)} numerical robustness and limitations of numerical algorithms; \emph{(iii)} careful selection and deployment of reliable mathematical software to be able to interpret and use the computed output with confidence in concrete applications. We illustrate the theoretical numerical issues on selected tasks from computational statistics.