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On a variational formulation of the QSVD and the RSVD. (English) Zbl 0970.65037
Generalized singular value decompositions (SVDs) are studied. The authors give an alternative proof of the variational formula for the quotient singular value decomposition (QSVD) of two matrices $$A\in\mathbb{R}^{n\times m}$$ and $$C\in\mathbb{R}^{p\times m}$$ which is a generalization to two matrices of the ordinary SVD and establish an analogous variational formulation for the restricted singular value decomposition (RSVD) of a matrix triplet $$A\in\mathbb{R}^{n\times m}$$, $$B\in \mathbb{R}^{n\times\ell}$$ and $$C\in \mathbb{R}^{p\times m}$$ which provides a new understanding of the orthogonal matrices occurring in this decomposition.
In order to prove their main results (Theorem 4 and 7), they establish two condensed forms based on orthogonal matrix transformations. The proofs of Theorem 4 and 7 provide deflation procedures for the numerical computation of two orthogonal matrices in the QSVD and RSVD.

##### MSC:
 65F20 Numerical solutions to overdetermined systems, pseudoinverses 65F15 Numerical computation of eigenvalues and eigenvectors of matrices 15A18 Eigenvalues, singular values, and eigenvectors
LAPACK
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##### References:
 [1] Chu, M.T.; Funderlic, R.E.; Golub, G.H., On a variational formulation of the generalized singular value decomposition, SIAM J. matrix anal. appl., 18, 1082-1092, (1997) · Zbl 0891.65038 [2] E. Anderson, Z. Bai, C. Bischof, J.W. Demmel, J. Dongarra, J.D. Croz, A. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov, D. Sorensen, LAPACK Users’ Guide. Society for Industrial and Applied Mathematics, Philadelphia, 1992 [3] Bai, Z.; Demmel, J.W., Computing the generalized singular value decomposition, SIAM J. sci. comput., 14, 1464-1486, (1993) · Zbl 0789.65024 [4] G.H. Golub, C.F. Van Loan, Matrix Computations, third ed., John Hopkins University Press, Baltimore, MD, 1996 · Zbl 0865.65009 [5] Paige, C.C.; Saunders, M.A., Towards a generalized singular value decomposition, SIAM J. numer. anal., 18, 398-405, (1981) · Zbl 0471.65018 [6] Paige, C.C., Computing the generalized singular value decomposition, SIAM J. sci. stat. comput., 7, 1126-1146, (1986) · Zbl 0621.65030 [7] De Moor, B., On the structure and geometry of the product singular value decomposition, Linear algebra appl., 168, 95-136, (1992) · Zbl 0754.15005 [8] De Moor, B.; Golub, G.H., The restricted singular value decomposition: properties and applications, SIAM J. matrix anal. appl., 12, 401-425, (1991) · Zbl 0738.15006 [9] De Moor, B., On the structure of generalized singular value and QR decompositions, SIAM J. matrix anal. appl., 15, 347-358, (1994) · Zbl 0792.15007 [10] De Moor, B.; Van Dooren, P., Generalizations of the QR and singular value decomposition, SIAM J. matrix anal. appl., 13, 993-1014, (1992) · Zbl 0764.65014 [11] B. De Moor, G.H. Golub, Generalized singular value decompositions: a proposal for a standardized nomenclature. Internal Report 89-10, ESAT-SISTA, Leuven, Belgium, 1989 [12] Stewart, G.W., Computing the CS-decomposition of a partitioned orthogonal matrix, Numer. math., 40, 297-306, (1982) · Zbl 0516.65016 [13] Van Loan, C.F., Generalizing the singular value decomposition, SIAM J. numer. anal., 13, 76-83, (1976) · Zbl 0338.65022 [14] Van Loan, C.F., Computing the CS and the generalized singular value decomposition, Numer. math., 46, 479-491, (1985) · Zbl 0548.65020 [15] Zha, H., A numerical algorithm for computing the restricted singular value decomposition of matrix triplets, Linear algebra appl., 168, 1-25, (1992) · Zbl 0748.65038 [16] Zha, H., The restricted singular value decomposition of matrix triplets, SIAM J. matrix anal. appl., 12, 172-194, (1991) · Zbl 0722.15011
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