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Title: Numerical methods for SVD and its generalizations with applications in computational statistics Authors:  Zlatko Drmac - University of Zagreb (Croatia) [presenting]
Abstract: The singular value decomposition (SVD) and its generalization, the GSVD (including the QSVD, PSVD and the cosine-sine decomposition CSD of partitioned orthonormal matrices) 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, numerical algorithms are not that simple. We will review the recent advances in this important part of numerical linear algebra and propose improvements, with particular attention to \emph{(i)} numerical robustness, where we show how the new generation of numerical algorithms returns accurate decomposition even in the cases that are considered ill-conditioned in the classical sense; \emph{(ii)} development of reliable mathematical software that performs as predicted by error analysis and perturbation theory. Then, we illustrate the numerical performances on selected applications from computational statistics.