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Centre Scientific Computing: Applied Linear Algebra


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2016  Spring semester February 25, 12:00, Room #60 This lecture aims to give an overview of the study concerning singular values of lowerdimension normmatrices, as suggested by H.B.Li, T.Z.Huang, X.P.Liu and H.Li (Journal of Computational and Applied Mathematics 234 (2010) 2943–2952). It is wellknown that the problem of dealing with singular values of a general complex matrix can be observed from two different perspectives and the choice of a direction depends on whether it is necessary to obtain inclusion sets (intervals) for all the singular values, or if the approximation of the two extreme values occupies the focal point. As far as applications are concerned, the latter approach has attracted much more attention, owing to the fact that obtaining an improved lower (upper) bound for the smallest (largest) singular value, can indeed enhance the estimate of the spectral condition number. Additionally, some relationships between the singular values of a matrix and its lower dimension block normmatrix are established. Based on these relationships, one may obtain effective estimates for the extreme singular values of large matrices by using the lower dimension normmatrices with the block approach taken into account, making the computation much more convenient from the practical point of view. Finally, some key numerical experiments are provided to exemplify the benefits of using the block approach. 