A Q R-method for computing the singular values via semiseparable matrices

Numerische Mathematik

Volumn 99, Issue 1, 2004, Pages 163-195

  Vandebril, Raf ^a   Van Barel, Marc ^a   Mastronardi, Nicola ^b  

^a Department of Computer Science ^*  (Belgium)

^b CNR   (Italy)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 11144351655     PISSN: 0029599X     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00211-004-0550-9     Document Type: Article

References (29)

1. Bevilacqua, R., Del Corso, G.M.: Existence, uniqueness and algorithms for matrix unitary reduction to semiseparable form. Tech. Report TR-03-12, University of Pisa, July 2003
2. Bini, D.A., Gemignani, L., Pan, V.: Q R-like algorithms for generalized semiseparable matrices. Tech. Report 1470, Department of Mathematics, University of Pisa, 2004
3. Chan, T.F., Hansen, P.C.: Some applications of the rank revealing Q R factorization. SIAM J. Sci. Stat. Comp. 13, 727-741 (1992)
4. Chandrasekaran, S., Ipsen, I.: On rank-revealing factorizations. SIAM J. Matrix Anal. Appl. 15, 592-622 (1994)
5. Demmel, J.W., Kahan, W.: Accurate singular values of bidiagonal matrices. SIAM J. Sci. Comp. 11(5), 873-912 (1990)
6. Elsner, L.: Some observations on inverses of band matrices and low rank perturbations of triangular matrices. Acta Technica Academiae Scientiarum Hungaricae. 108(1-2), 41-48 (1997-1999)
7. Gantmacher, F.R., Krein, M.G.: Oscillation matrices and kernels and small vibrations of mechanical systems. AMS Chelsea Publishing, Providence, Rhode Island, revised edition, 2002
8. Gohberg, I., Kailath, T., Koltracht, I.: Linear complexity algorithms for semiseparable matrices. Integral Equations and Operator Theory. 8(6), 780-804 (1985)
9. Golub, G.H., Kahan, W.: Calculating the singular values and pseudo-inverse of a matrix. SIAM J. Numer. Anal. 2, 205-224 (1965)
10. Golub, G.H., Van Loan, C.F.: Matrix Computations. The Johns Hopkins university Press, third edition, 1996
11. Joliffe, I.T.: Principal component analysis. Springer verlag, 1986
12. Koltracht, I.: Linear complexity algorithm for semiseparable matrices. Integral Equations and Operator Theory. 29(3), 313-319 (1997)
13. Minchev, B.V.: Some algorithms for solving special tridiagonal block toeplitz linear systems. J. Comput. Appl. Math. 156, 179-200 (2003)
14. Stewart, G.W.: Matrix Algorithms, Vol II Eigensystems. SIAM, 1999
15. Stewart, G.W.: The QLP approximation to the singular value decomposition. SIAM J. Sci. Stat. Comput. 20(4), 1336-1348 (1999)
16. Strang, G., Nguyen, T.: The interplay of ranks of submatrices. Submitted to SIAM Review, http://www-math.mit.edu/gs/papers/papers.html, February 2004
17. Trefethen, L.N., Bau, D.: Numer. Linear Algebra. SIAM, 1997
18. Van Barel, M., Vandebril, R., Mastronardi, N.: The Lanczos-Ritz values appearing in an orthogonal similarity reduction of a matrix into semiseparable form. Report TW 360, Department of Computer Science, K.U.Leuven, Leuven, Belgium, May 2003
19. Van Camp, E., Mastronardi, N., Van Barel, M.: Two fast algorithms for solving diagonal-plus-semiseparable linear systems. J. Comput. Appl. Math. 164-165, 731-747 (2004)
20. Vandebril, R., Van Barel, M., Mastronardi, N.: An implicit Q R algorithm for semiseparable matrices to compute the eigendecomposition of symmetric matrices. Report TW 367, Department of Computer Science, K.U.Leuven, Leuven, Belgium, August 2003
21. Vandebril, R., Van Barel, N., end Mastronardi, M.: A note on the representation and definition of semiseparable matrices. Numerical Linear Algebra with Applications. May 2004. Accepted for publication (Report TW393)
22. Wall, M.E., Dyck, P.A., Brettin, T.S.: Svdman - singular value decomposition analysis of microarray data. Bioinformatics. 17(6), 566-568 (2001)
23. Wall, M.E., Rechtsteiner, A., Rocha, L.M.: Singular value decomposition and principal component analysis. A practical approach to microarray data analysis. 2003. D.P. Berrar, W. Dubitzky, M. Granzow, eds, kluwer pp. 1-20
24. Watkins, D.S.: Q R like algorithms an overview of convergence theory and practice. Lectures Appl. Math. 32, 879-893 (1996)
25. Watkins, D.S., Elsner, L.: Convergence of algorithms of decomposition type for the eigenvalue problem. Linear Algebra and Its Applications. 143, 19-41 (1991)
26. Watkins, D.S., Elsner, L.: Chasing algorithms for the eigenvalue problem. SIAM J. Matrix Anal. Appl. 12(2), 374-384 (1991)
27. Watkins, D.S., Eisner, L.: Theory of decomposition and bulge-chasing algorithms for the generalized eigenvalue problem. SIAM J. Matrix Anal. Appl. 15(3), 943-967 (1994)
28. Yeung, M.K.S., Ruzzo, W.L.: Principal component analysis for clustering gene expression data. Bioinformatics. 17(9), 763-774 (2001)
29. Yeung, M.K.S., Tegner, J., Collins, J.J.: Reverse engineering gene networks using singular value decomposition and robust regression. In Proceedings of the national academy of sciences of the united states of america. 99, 6163-6168 (2002)