Eigenvalue computation for unitary rank structured matrices

Journal of Computational and Applied Mathematics

Volumn 213, Issue 1, 2008, Pages 268-287

  Delvaux, Steven ^a   Van Barel, Marc ^a  

^a UNIVERSITY OF LEUVEN   (Belgium)

Author keywords

Eigenvalue computation; Pull through operation; Rank structured matrix; Unitary matrix

Indexed keywords

ALGORITHMS; COMPUTATION THEORY; EIGENVALUES AND EIGENFUNCTIONS;

EIGENVALUE COMPUTATION; PULL-THROUGH OPERATION; RANK STRUCTURED MATRIX; UNITARY MATRIX;

MATRIX ALGEBRA;

EID: 37549005595     PISSN: 03770427     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cam.2007.01.006     Document Type: Article

References (31)

1. G.S. Ammar, W.B. Gragg, L. Reichel, On the eigenproblem for orthogonal matrices, in: 25th IEEE Conference on Decision and Control, 1986, Athens, Greece, pp. 1963-1966.
2. G.S. Ammar, W.B. Gragg, L. Reichel, Determination of Pisarenko frequency estimates as eigenvalues of an orthogonal matrix, in: F.T. Luk (Ed.), Proceedings of SPIE-The International Society for Optical Engineers, Advanced Algorithms and Architectures for Signal Processing II, California, vol. 826, Bellingham, Washington, USA, SPIE, 1987, pp. 143-145.
3. G.S. Ammar, W.B. Gragg, L. Reichel, Constructing a unitary Hessenberg matrix from spectral data, in: G.H. Golub, P. Van Dooren (Eds.), Numerical Linear Algebra, Digital Signal Processing and Parallel Algorithms, Computer and Systems Sciences, vol. 70, Springer, Berlin, 1991, pp. 385-395.
4. Ammar G.S., Gragg W.B., and Reichel L. Downdating of Szego{combining double acute accent} polynomials and data-fitting applications. Linear Algebra Appl. 172 (1992) 315-336
5. Ammar G.S., Reichel L., and Sorensen D.C. An implementation of a divide and conquer algorithm for the unitary eigenproblem. ACM Trans. Math. Software 18 3 (1992) 292-307
6. Arbenz P., and Golub G.H. On the spectral decomposition of hermitian matrices modified by low rank perturbations with applications. SIAM J. Matrix Anal. Appl. 9 1 (1988) 40-58
7. Bultheel A., and Van Barel M. Vector orthogonal polynomials and least squares approximation. SIAM J. Matrix Anal. Appl. 16 3 (1995) 863-885
8. Bunse-Gerstner A., and Elsner L. Schur parameter pencils for the solution of the unitary eigenproblem. Linear Algebra Appl. 154-156 (1991) 741-778
9. Bunse-Gerstner A., and He C. On a Sturm sequence of polynomials for unitary Hessenberg matrices. SIAM J. Matrix Anal. Appl. 16 4 (1995) 1043-1055
10. R.J.A. David, D.S. Watkins, Efficient implementation of the multishift QR algorithm for the unitary eigenvalue problem, SIAM J. Matrix Anal. Appl. 2006, to appear.
11. S. Delvaux, M. Van Barel, A Givens-weight representation for rank structured matrices, Technical Report TW453, Department of Computer Science, Katholieke Universiteit Leuven, Celestijnenlaan 200A, 3000 Leuven, Heverlee, Belgium, March 2006.
12. S. Delvaux, M. Van Barel, A Hessenberg reduction algorithm for rank structured matrices, Technical Report TW460, Department of Computer Science, Katholieke Universiteit Leuven, Celestijnenlaan 200A, 3000 Leuven, Heverlee, Belgium, May 2006.
13. Delvaux S., and Van Barel M. Rank structures preserved by the QR-algorithm: the singular case. J. Comput. Appl. Math. 189 (2006) 157-178
14. Delvaux S., and Van Barel M. Structures preserved by matrix inversion. SIAM J. Matrix Anal. Appl. 28 1 (2006) 213-228
15. S. Delvaux, M. Van Barel, Unitary rank structured matrices, Technical Report TW464, Department of Computer Science, Katholieke Universiteit Leuven, Celestijnenlaan 200A, 3000 Leuven, Heverlee, Belgium, July 2006.
16. Dewilde P., and van der Veen A.-J. Time-varying Systems and Computations (June 1998), Kluwer Academic Publishers, Boston
17. Golub G.H., and Van Loan C.F. Matrix Computations. third ed. (1996), Johns Hopkins University Press, Baltimore, MD, New York
18. Gragg W.B. The QR algorithm for unitary Hessenberg matrices. J. Comput. Appl. Math. 16 (1986) 1-8
19. Gragg W.B., and Reichel L. A divide and conquer method for unitary and orthogonal eigenproblems. Numer. Math. 57 (1990) 695-718
20. Gu M., Guzzo R., Chi X.-B., and Cao X.-Q. A stable divide and conquer algorithm for the unitary eigenproblem. SIAM J. Matrix Anal. Appl. 25 (2003) 385-404
21. Parlett B.N. The Symmetric Eigenvalue Problem, Classics in Applied Mathematics vol. 20 (1998), SIAM, Philadelphia
22. Reichel L., Ammar G.S., and Gragg W.B. Discrete least squares approximation by trigonometric polynomials. Math. Comp. 57 (1991) 273-289
23. Rutishauser H. Bestimmung der eigenwerte orthogonaler matrizen. Numer. Math. 9 (1966) 104-108
24. Stewart M. An error analysis of a unitary Hessenberg QR algorithm. SIAM J. Matrix Anal. Appl. 28 1 (2006) 40-67
25. Van Barel M., and Bultheel A. A parallel algorithm for discrete least squares rational approximation. Numer. Math. 63 (1992) 99-121
26. Van Barel M., and Bultheel A. Discrete linearized least squares approximation on the unit circle. J. Comput. Appl. Math. 50 (1994) 545-563
27. Van Barel M., and Bultheel A. Orthonormal polynomial vectors and least squares approximation for a discrete inner product. Electron. Trans. Numer. Anal. 3 (1995) 1-23
28. Vandebril R., Van Camp E., Van Barel M., and Mastronardi N. Orthogonal similarity transformation of a symmetric matrix into a diagonal-plus-semiseparable one with free choice of the diagonal. Numer. Math. 102 (2006) 709-726
29. Wang T.L., and Gragg W.B. Convergence of the shifted QR algorithm for unitary Hessenberg matrices. Math. Comp. 71 240 (2002) 1473-1496
30. Wang T.L., and Gragg W.B. Convergence of the unitary QR algorithm with unitary Wilkinson shift. Math. Comp. 72 241 (2003) 375-385
31. Watkins D.S. The transmission of shifts and shift blurring in the QR algorithm. Linear Algebra Appl. 241-243 (1996) 877-896