A total least squares method for Toeplitz systems of equations

BIT Numerical Mathematics

Volumn 38, Issue 3, 1998, Pages 560-582

  Kamm, Julie ^a   Nagy, James G ^b  

^a RAYTHEON COMPANY   (United States)

^b SOUTHERN METHODIST UNIVERSITY   (United States)

Author keywords

Bisection; Circulant preconditioner; Newton's method; Toeplitz matrix; Total least squares

Indexed keywords

EID: 0009897843     PISSN: 00063835     EISSN: None     Source Type: Journal    
DOI: 10.1007/BF02510260     Document Type: Article

References (33)

1. T. J. Abatzoglou, J. M. Mendel, and G. A. Harada, The constrained total least squares technique and its applications to harmonic superresolution, IEEE Trans. Signal Proc., 39 (1991), pp. 1070-1087.
2. Å. Björck, Numerical Methods for Least Squares Problems, SIAM, Philadelphia, PA, 1996.
3. Å. Björck, Newton and Rayleigh quotient methods for total least squares problems, in Recent Advances in Total Least Squares Techniques and Errors-in-Variables Modeling: Proceedings of the Second International Workshop on Total Least Squares and Errors-in-Variables Modeling, S. Van Huffel, ed., SIAM, Philadelphia, PA, 1997, pp. 149-160.
4. A. W. Bojanczyk, R. P. Brent, and F. R. de Hoog, QR factorization of Toeplitz matrices, Numer. Math., 49 (1986), pp. 81-94.
5. J. R. Bunch, Stability of methods for solving Toeplitz systems of equations, SIAM J. Sci. Stat. Comput., 6 (1985), pp. 349-364.
6. R. H. Chan, X.-Q. Jin, and M.-C. Yeung, The circulant operator in the Banach algebra of matrices, Linear Algebra Appl., 149 (1991), pp. 41-53.
7. R. H. Chan, J. G. Nagy, and R. J. Plemmons, Circulant preconditioned Toeplitz least squares iterations, SIAM J. Matrix Anal. Appl., 15 (1994), pp. 80-97.
8. R. H. Chan and M. K. Ng, Conjugate gradient methods for Toeplitz systems, SIAM Review, 38 (1996), pp. 427-482.
9. T. F. Chan, An optimal circulant preconditioner for Toeplitz systems, SIAM J. Sci. Stat. Comput., 9 (1988), pp. 766-771.
10. J. Chun, T. Kailath, and H. Lev-Ari, Fast parallel algorithms for QR and triangular factorization, SIAM J. Sci. Stat. Comput., 8 (1987), pp. 899-913.
11. G. Cybenko, Fast Toeplitz orthogonalization using inner products, SIAM J. Sci. Stat. Comput., 8 (1987), pp. 734-740.
12. G. Cybenko and C. Van Loan, Computing the minimum eigenvalue of a symmetric positive definite Toeplitz matrix, SIAM J. Sci. Stat. Comput., 7 (1986), pp. 123-131.
13. P. J. Davis, Circulant Matrices, Wiley, New York, 1979.
14. R. D. Fierro, G. H. Golub, P. C. Hansen, and D. P. O'Leary, Regularization by truncated total least squares, SIAM J. Sci. Comput., 18 (1997), pp. 1223-1241.
15. R. D. Fierro and P. C. Hansen, Low-rank revealing two-sided orthogonal decompositions, Numer. Algorithms, 15 (1997), pp. 37-55.
16. T factorizations, Math. Comp., 28 (1974), pp. 1067-1087.
17. P. E. Gill, G. H. Golub, W. Murray, and M. A. Saunders, Methods for modifying matrix factorizations, Math. Comp., 28 (1974), pp. 505-535.
18. U. Grenander and G. Szegö, Toeplitz Forms and their Applications, 2nd ed., Chelsea, New York, 1984.
19. M. Gu, New fast algorithms for structured linear least squares problems, Preprint, 1995.
20. S. Van Huffel, P. Lemmerling, and L. Vanhamme, Fast algorithms for signal subspace fitting with Toeplitz matrices and applications to exponential data modeling, in SVD and Signal Processing III: Algorithms, Architectures and Applications, M. Moonen and B. DeMoor, eds., Elsevier Science Publishers B.V. (North-Holland), 1995, pp. 191-198.
21. S. Van Huffel, H. Park, and J. B. Rosen, Formulation and solution of structured total least norm problems for parameter estimation, IEEE Trans. Signal Processing, 44 (1996), pp. 2464-2474.
22. S. Van Huffel and J. Vandewalle, The Total Least Squares Problem: Computational Aspects and Analysis, SIAM, Philadelphia, PA, 1991.
23. P. Lemmerling, B. De Moor, and S. Van Huffel, On the equivalence between constrained total least squares and structured total least squares, IEEE Trans. Signal Processing, 44 (1996), pp. 2908-2911.
24. J. G. Nagy, Applications of Toeplitz systems, SIAM News, Volume 28:8 (1995).
25. H. Park and L. Eldén, Stability analysis and fast algorithms for triangularization of Toeplitz matrices, Numer. Math., 76 (1997), pp. 383-402.
26. H. Park, J. B. Rosen, and S. Van Huffel, Structure preserving total least squares method and its application to parameter estimation, in SVD and Signal Processing III: Algorithms, Architectures and Applications, M. Moonen and B. De Moor, eds., Elsevier Science Publishers B.V. (North-Holland), 1995, pp. 399-408.
27. B. N. Parlett, The Symmetric Eigenvalue Problem. Prentice-Hall, Englewood Cliffs, NJ, 1980.
28. D. R. Sweet, Fast Toeplitz orthogonalization, Numer. Math., 43 (1984), pp. 1-21.
29. C. F. Tirendi and J. F. Martin, Quantitative analysis of NMR spectra by linear prediction and total least squares, J. Magn. Reson.. 85 (1989), pp. 162-169.
30. H. J. Trussell, Convergence criteria for iterative restoration methods, IEEE Trans. Acoust., Speech, Signal Processing, 31 (1983), pp. 129-136.
31. E. E. Tyrtyshnikov, Optimal and super-optimal circulant preconditioners, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 459-473.
32. D. S. Watkins, Fundamentals of Matrix Computations, Wiley, New York, 1991.
33. T. Yang, Iterative Methods for Least Squares and Total Least Squares Problems, Lic. thesis, Linköping University, Linköping, Sweden, 1996.