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Band generalization of the Golub-Kahan bidiagonalization, generalized Jacobi matrices, and the core problem. (English) Zbl 1320.65057

##### MSC:
 65F20 Numerical solutions to overdetermined systems, pseudoinverses 15A21 Canonical forms, reductions, classification 15A06 Linear equations (linear algebraic aspects) 15A18 Eigenvalues, singular values, and eigenvectors 15A24 Matrix equations and identities 65F25 Orthogonalization in numerical linear algebra
##### Software:
VanHuffel; symrcm; OPQ
Full Text:
##### References:
 [1] J. L. Barlow, Reorthogonalization for the Golub–Kahan–Lanczos bidiagonal reduction, Numer. Math., 124 (2013), pp. 237–278. · Zbl 1272.65034 [2] \rA. Björck, Bidiagonal Decomposition and Least Squares, Presentation, Canberra, Australia, 2005. [3] \rA. Björck, A Band-Lanczos Generalization of Bidiagonal Decomposition, Presentation, Conference in Honor of G. Dahlquist, Stockholm, Sweden, 2006. [4] \rA. Björck, A band-Lanczos algorithm for least squares and total least squares problems, in Book of Abstracts of 4th Total Least Squares and Errors-in-Variables Modeling Workshop, Leuven, Katholieke Universiteit Leuven, Leuven, Belgium, 2006, pp, 22–23. [5] \rA. Björck, Block Bidiagonal Decomposition and Least Squares Problems with Multiple Right-Hand Sides, manuscript, 2008. [6] B. Bohnhorst, Beiträge zur Numerischen Behandlung des Unitären Eigenwertproblems, Ph.D. thesis, Universität Bielefeld, Bielefeld, Germany, 1993. [7] W. Gautschi, Orthogonal Polynomials, Computation and Approximation, Oxford University Press, New York, 2004. · Zbl 1130.42300 [8] A. George and J. W. H. Liu, Computer Solution of Large Sparse Positive Definite Systems, Prentice-Hall, Englewood Cliffs, NJ, 1981. · Zbl 0516.65010 [9] G. Golub and W. Kahan, Calculating the singular values and pseudo-inverse of a matrix, J. Soc. Ind. Appl. Math. Ser. B Numer. Anal., 2 (1965), pp. 205–224. · Zbl 0194.18201 [10] 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. · Zbl 0468.65011 [11] 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. [12] I. Hnětynková, M. Plešinger, and Z. Strakoš, Lanczos tridiagonalization, Golub–Kahan bidiagonalization and core problem, Proc. Appl. Math. Mech., 6 (2006), pp, 717–718. [13] I. Hnětynková, M. Plešinger, and Z. Strakoš, The core problem within a linear approximation problem $$AX≈ B$$ with multiple right-hand sides, SIAM J. Matrix Anal. Appl., 34 (2013), pp, 917–931. [14] I. Hnětynková and Z. Strakoš, Lanczos tridiagonalization and core problems, Linear Algebra Appl., 421 (2007), pp, 243–251. [15] J. Liesen and Z. Strakoš, Krylov Subspace Methods, Principles and Analysis, Oxford University Press, Oxford, 2013. · Zbl 1263.65034 [16] C. C. Paige and Z. Strakoš, Core problem in linear algebraic systems, SIAM J. Matrix Anal. Appl., 27 (2005), pp, 861–875. [17] B. N. Parlett, The Symmetric Eigenvalue Problem, Classics Appl. Math. 20, SIAM, Philadelphia, 1998. · Zbl 0885.65039 [18] M. Plešinger, The Total Least Squares Problem and Reduction of Data in $$AX≈ B$$, Ph.D. thesis, Technical University of Liberec, Liberec, Czech Republic, 2008. [19] D. M. Sima, Regularization Techniques in Model Fitting and Parameter Estimation, Ph.D. thesis, Katholieke Universiteit Leuven, Leuven, Belgium, 2006. [20] D. M. Sima and S. Van Huffel, Core Problems in $$AX≈ B$$, Technical report, Department of Electrical Engineering, Katholieke Universiteit Leuven, Leuven, Belgium, 2006. [21] S. Van Huffel and J. Vandewalle, The Total Least Squares Problem: Computational Aspects and Analysis, Front. Appl. Math. 9, SIAM, Philadelphia, 1991. · Zbl 0789.62054 [22] 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. · Zbl 0758.65039 [23] 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. · Zbl 0761.65030 [24] J. H. Wilkinson, The Algebraic Eigenvalue Problem, Clarendon Press, Oxford, England, 1965. · Zbl 0258.65037
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