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On a quadratic eigenproblem occurring in regularized total least squares. (English) Zbl 1393.15013
Summary: A computational approach for solving regularized total least squares problems via a sequence of quadratic eigenvalue problems has recently been proposed. Taking advantage of a variational characterization of real eigenvalues of nonlinear eigenproblems the existence of a real right-most eigenvalue for each quadratic eigenvalue problem in the sequence is proven. For large problems the approach is improved considerably utilizing information from the previous quadratic problems and early updates in a nonlinear Arnoldi method.

15A18 Eigenvalues, singular values, and eigenvectors
65F15 Numerical computation of eigenvalues and eigenvectors of matrices
65F20 Numerical solutions to overdetermined systems, pseudoinverses
65F22 Ill-posedness and regularization problems in numerical linear algebra
Full Text: DOI
[1] Bai, Z.; Su, Y., SOAR: a second order Arnoldi method for the solution of the quadratic eigenvalue problem, SIAM J. matrix anal. appl., 26, 540-659, (2005) · Zbl 1080.65024
[2] Beck, A.; Ben-Tal, A., On the solution of the Tikhonov regularization of the total least squares problem, SIAM J. optim., 17, 98-118, (2006) · Zbl 1112.65034
[3] Beck, A.; Ben-Tal, A.; Teboulle, M., Finding a global optimal solution for a quadratically constrained fractional quadratic problem with applications to the regularized total least squares problem, SIAM J. matrix anal. appl., 28, 425-445, (2006) · Zbl 1115.65065
[4] Engl, H.; Hanke, M.; Neubauer, A., Regularization of inverse problems, (1996), Kluwer Academic Publishers Dodrecht · Zbl 0859.65054
[5] Gander, W.; Golub, G.; von Matt, U., A constrained eigenvalue problem, Linear algebra appl., 114-115, 815-839, (1989) · Zbl 0666.15006
[6] Golub, G.; Hansen, P.; O’Leary, D., Tikhonov regularization and total least squares, SIAM J. matrix anal. appl., 21, 185-194, (1999) · Zbl 0945.65042
[7] Golub, G.; Van Loan, C., An analysis of the total least squares problem, SIAM J. numer. anal., 17, 883-893, (1980) · Zbl 0468.65011
[8] Golub, G.; Van Loan, C., Matrix computations, (1996), The John Hopkins University Press Baltimore and London · Zbl 0865.65009
[9] Groetsch, C., Inverse problems in the mathematical sciences, (1993), Vieweg Wiesbaden, Germany · Zbl 0779.45001
[10] Guo, H.; Renaut, R., A regularized total least squares algorithm, () · Zbl 0995.65042
[11] Hansen, P., Regularization tools, a Matlab package for analysis of discrete regularization problems, Numer. algorithm, 6, 1-35, (1994) · Zbl 0789.65029
[12] Hansen, P., Rank-deficient and discrete ill-posed problems: numerical aspects of linear inversion, (1998), SIAM Philadelphia
[13] Lehoucq, R.; Sorensen, D.; Yang, C., ARPACK users’ guide. solution of large-scale eigenvalue problems with implicitly restarted Arnoldi methods, (1998), SIAM Philadelphia · Zbl 0901.65021
[14] Li, R.-C.; Ye, Q., A Krylov subspace method for quadratic matrix polynomials with application to constrained least squares problems, SIAM J. matrix anal. appl., 25, 405-428, (2003) · Zbl 1050.65038
[15] Renaut, R.; Guo, H., Efficient algorithms for solution of regularized total least squares, SIAM J. matrix anal. appl., 26, 457-476, (2005) · Zbl 1082.65045
[16] Sima, D., 2006. Regularization techniques in model fitting and parameter estimation. Ph.D. Thesis, Katolieke Universiteit Leuven, Leuven, Belgium.
[17] Sima, D.; Huffel, S.V.; Golub, G., Regularized total least squares based on quadratic eigenvalue problem solvers, BIT numer. math., 44, 793-812, (2004) · Zbl 1079.65048
[18] Van Huffel, S., Vandevalle, J., 1991. The total least squares problems: computational aspects and analysis. In: Frontiers in Applied Mathematics, vol. 9, SIAM, Philadelphia.
[19] Voss, H., An Arnoldi method for nonlinear eigenvalue problems, BIT numer. math., 44, 387-401, (2004) · Zbl 1066.65059
[20] Voss, H.; Werner, B., A minimax principle for nonlinear eigenvalue problems with applications to nonoverdamped systems, Math. meth. appl. sci., 4, 415-424, (1982) · Zbl 0489.49029
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