Fast computation of the principal singular vectors of Toeplitz matrices arising in exponential data modelling

Signal Processing

Volumn 50, Issue 1-2 SPEC. ISS., 1996, Pages 151-164

  Eldén, Lars ^a   Sjöström, Eva ^a  

^a LINKÖPING UNIVERSITY   (Sweden)

Author keywords

Algorithms; Fast fourier transform (FFT); Lanczos procedure; Signal subspace; Singular vector; Toeplitz matrix

Indexed keywords

ALGORITHMS; COMPUTATIONAL METHODS; COMPUTER SIMULATION; DIGITAL ARITHMETIC; FAST FOURIER TRANSFORMS; MATRIX ALGEBRA; VECTORS;

EXPONENTIAL DATA MODELLING; LANCZOS METHOD; MATRIX VECTOR MULTIPLICATION; PRINCIPAL SINGULAR VECTORS; SIGNAL SUBSPACE; TOEPLITZ MATRIX;

SIGNAL THEORY;

EID: 0030123520     PISSN: 01651684     EISSN: None     Source Type: Journal    
DOI: 10.1016/0165-1684(96)00009-6     Document Type: Article

References (29)

1. E. Anderson, Z. Bai and C. Bischof et al., LAPACK User's Guide, SIAM, Philadelphia, 1992.
2. H. Barkhuijsen, R. De Beer and D. Van Ormondt, "Improved algorithm for non-iterative time-domain model fitting to exponentially damped magnetic resonance signals", J. Magnetic Resonance, Vol. 73, 1987, pp. 553-557.
3. D. Calvetti, L. Reichel and D.C. Sorensen, "An implicitly restarted Lanczos method for large symmetric eigenvalue problems", Electronic Trans. Numer. Anal., Vol. 2, 1994, pp. 1-21.
4. P. Comon and G.H. Golub, "Tracking a few extreme singular values and vectors in signal processing", Proc. IEEE, Vol 78, No. 8, August 1990, pp. 1327-1343.
5. J. Cullum and R.A. Willoughby, "Lanczos and the computation in specified intervals of the spectrum of large, sparse real symmetric matrices", in: I. Duff and G.W. Stewart, eds., Sparse Matrix Proc. 1978, SIAM, Philadelphia, 1979.
6. J. Cullum and R.A. Willoughby, "Computing eigenvectors (and eigenvalues) of large, sparse symmetric matrices using Lanczos tridiagonalization", in: G.A. Watson, ed., Proc. Biennial Conf. on Num. Anal., University of Dundee, Scotland, 1980, Springer, New York.
7. P.J. Davis, Circulant Matrices, Wiley, New York, 1979.
8. R. de Beer and D. van Ormondt, Analysis of NMR Data using Time Domain Fitting Procedures, NMR Basic Principles and Progress, Vol. 26, Springer, Berlin, 1992, pp. 201-248.
9. J.J. Dongarra, J. Du Croz, S. Hammarling and R.J. Hanson, "An extended set of Fortran basic linear algebra subprograms", ACM Trans. Math. Soft., Vol. 14, 1988, pp. 1-17.
10. R.D. Fierro and P.C. Hansen, Low-rank revealing two-sided orthogonal decompositions, Technical Report UNIC-95-06, UNI-C, Copenhagen, 1995.
11. H. Gesmar, J.J. Led and F. Abildgaard, "Improved methods for quantitative spectral analysis of NMR data", Progress NMR Spectroscopy, Vol. 22, 1990, pp. 255-288.
12. I. Gohberg and V. Olshevsky, "Complexity of multiplication with vectors for structured matrices", Linear Algebra Appl., Vol. 202, 1994, pp. 163-192.
13. G.H. Golub and C.F. Van Loan, Matrix Computations, 2nd Edition, Johns Hopkins Press, Baltimore, MD, 1989.
14. W. Kahan and B.N. Parlett, "How far should you go with the Lanczos process"?, Sparse Matrix Comput., 1976, pp. 131-144.
15. S.Y. Kung, K.S. Arun and D.V. Baskar Rao, "State-space and singular value decomposition-based approximation methods for the harmonic retrieval problem", J. Opt. Soc. Amer., Vol. 73, 1983, pp. 1799-1811.
16. C. Lanczos, "An iterative method for the solution of the eigenvalue problem of linear differential and integral operators", J. Res. Nat. Bureau Standards, B, Vol. 45, 1950, pp. 225-280.
17. G.L. Millhauser, A.A. Carter, D.J. Schneider, J.H. Freed and R.E Oswald, "Rapid singular value decomposition for time-domain analysis of magnetic resonance signals by use of the Lanczos algorithm", J. Magnetic Resonance, Vol. 82, 1989, pp. 150-155.
18. J.G. Nagy, Toeplitz least squares computations, Ph.D. Thesis, Raleigh, 1991.
19. D.P. O'Leary and J.A. Simmons, "A bidiagonalization-regularization procedure for large scale discretizations of ill-posed problems", SIAM J. Sci. Stat. Comput., Vol. 2, No. 4, 1981, pp. 474-489.
20. C.C. Paige, 'Error analysis of the Lanczos algorithm for tridiagonalizing a symmetric matrix", J. Inst. Maths. Applics, Vol. 18, 1976, pp. 341-349.
21. B.N. Parlett, The symmetric Eigenvalue Problem, Prentice-Hall, Englewood Cliffs, NJ, 1980.
22. B.N. Parlett and D.S. Scott, "The Lanczos algorithm with selective orthogonalization", Math. Computation, Vol. 33, No. 145, 1979, pp. 217-238.
23. W.W.F. Pijnappel, A. van den Boogaart, R. de Beer and D. van Ormondt, "Svd-based quantification of magnetic resonance signals", J. Magnetic Resonance, Vol. 97, 1992, pp. 122-134.
24. H.D. Simon, "Analysis of the symmetric Lanczos algorithm with reorthogonalization methods", Linear Algebra Appl., Vol. 61, 1984, pp. 101-131.
25. D.C. Sorensen, "Implicit application of polynomial filters in a k-step Arnoldi method", SIAM J. Matrix Anal. Appl., Vol. 13, 1992, pp. 357-385.
26. P.N. Swarztrauber, R.A. Sweet, W.L. Briggs, Van Emden Hensen and J. Otto, "Bluestein's FFT for arbitrary N on the hypercube", Parallell Comput., Vol. 17, 1991, pp. 607-617.
27. S. Van Huffel, L. Aerts, J. Bervoets, J. Vandewalle, C. Decanniere and P. Van Hecke, Improved Quantitative Time-Domain Analysis of NMR Data by Total Least Squares, Signal Processing VI: Theories and Applications, Vol III, Elsevier, Amsterdam, 1992, pp. 1721-1724.
28. C.F. Van Loan, Computational Framework for the Fast Fourier Transform, SIAM, Philadelphia, 1992.
29. G. Xu and T. Kailath, "Fast estimation of principal eigenspace using Lanczos algorithm", SIAM J. Matrix Anal. Appl., Vol. 15, No. 3, July 1994, pp. 974-994.