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Solvability of the core problem with multiple right-hand sides in the TLS sense. (English) Zbl 1343.15002

MSC:
15A06 Linear equations (linear algebraic aspects)
15A18 Eigenvalues, singular values, and eigenvectors
15A21 Canonical forms, reductions, classification
15A24 Matrix equations and identities
65F20 Numerical solutions to overdetermined systems, pseudoinverses
65F25 Orthogonalization in numerical linear algebra
Software:
VanHuffel
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References:
[1] \AA. Björck, A band-Lanczos algorithm for least squares and total least squares problems, in Book of Abstracts of the 4th Total Least Squares and Errors-in-Variables Modeling Workshop, Leuven, Katholieke Universiteit Leuven, Leuven, Belgium, 2006, pp. 22–23; http://homepages.vub.ac.be/ imarkovs/workshop/program.pdf.
[2] \AA. Björck, Block Bidiagonal Decomposition and Least Squares Problems with Multiple Right-Hand Sides, unpublished manuscript, Department of Mathematics, University of Linköping, 2008.
[3] G. H. Golub and C. F. Van Loan, An analysis of the total least squares problem, SIAM J. Numer. Anal., 17 (1980), pp. 883–893, http://dx.doi.org/10.1137/0717073 doi:10.1137/0717073. · Zbl 0468.65011
[4] I. Hnětynková, M. Plešinger, D. M. Sima, Z. Strakoš, and S. Van Huffel, The total least squares problem in \(AX≈ B\): A new classification with the relationship to the classical works, SIAM J. Matrix Anal. Appl., 32 (2011), pp. 748–770, http://dx.doi.org/10.1137/100813348 doi:10.1137/100813348.
[5] I. Hnětynková, M. Plešinger, and Z. Strakoš, The core problem within linear approximation problem \(AX≈ B\) with multiple right-hand sides, SIAM J. Matrix Anal. Appl., 34 (2013), pp. 917–931, http://dx.doi.org/10.1137/120884237 doi:10.1137/120884237.
[6] I. Hnětynková, M. Plešinger, and Z. Strakoš, Band generalization of the Golub–Kahan bidiagonalization, generalized Jacobi matrices, and the core problem, SIAM J. Matrix Anal. Appl., 36 (2015), pp. 417–434, http://dx.doi.org/10.1137/140968914 doi:10.1137/140968914.
[7] C. C. Paige and Z. Strakoš, Core problem in linear algebraic systems, SIAM J. Matrix Anal. Appl., 27 (2006), pp. 861–875, http://dx.doi.org/10.1137/040616991 doi:10.1137/040616991. · Zbl 1097.15003
[8] M. Plešinger, The Total Least Squares Problem and Reduction of Data in \(AX≈ B\), Ph.D. thesis, Faculty of Mechatronics, Technical University of Liberec, Liberec, Czech Republic, 2008.
[9] D. M. Sima, Regularization Techniques in Model Fitting and Parameter Estimation, Ph.D. thesis, ESAT, Katholieke Universiteit Leuven, Leuven, Belgium, 2006.
[10] S. Van Huffel, Documented Fortran 77 Programs of the Extended Classical Total Least Squares Algorithm, the Partial Singular Value Decomposition Algorithm and the Partial Total Least Squares Algorithm, Internal report ESAT-KUL 88/1, Katholieke Universiteit Leuven, Leuven, Belgium, 1988.
[11] S. Van Huffel, The extended classical total least squares algorithm, J. Comput. Appl. Math., 25 (1989), pp. 111–119, http://dx.doi.org/10.1016/0377-0427(89)90080-0 doi:10.1016/0377-0427(89)90080-0. · Zbl 0664.65035
[12] S. Van Huffel and J. Vandewalle, The Total Least Squares Problem: Computational Aspects and Analysis, Frontiers in Appl. Math. 9, SIAM, Philadelphia, 1991, http://dx.doi.org/10.1137/1.9781611971002 doi:10.1137/1.9781611971002. · Zbl 0789.62054
[13] M. Wei, The analysis for the total least squares problem with more than one solution, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 746–763, http://dx.doi.org/10.1137/0613047 doi:10.1137/0613047. · Zbl 0758.65039
[14] M. Wei, Algebraic relations between the total least squares and least squares problems with more than one solution, Numer. Math., 62 (1992), pp. 123–148, http://dx.doi.org/10.1007/BF01396223 doi:10.1007/BF01396223. · Zbl 0761.65030
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