On the shifted QR iteration applied to companion matrices

Electronic Transactions on Numerical Analysis

Volumn 18, Issue , 2004, Pages 137-152

  Bini, Dario A ^a   Daddi, Francesco ^a   Gemignani, Luca ^a  

^a UNIVERSITY OF PISA   (Italy)

Author keywords

Companion matrices; Eigenvalues; Polynomial roots; QR factorization; QR iteration; Semiseparable matrices

Indexed keywords

EID: 22744450946     PISSN: 10689613     EISSN: 10689613     Source Type: Journal    
DOI: None     Document Type: Article

References (19)

1. G. S. AMMAR AND W. B. GRAGG, Schur flows for orthogonal Hessenberg matrices, in Hamiltonian and gradient flows, algorithms and control, Fields Inst. Commun., 3, Amer. Math. Soc., Providence, RI, 1994, pp. 27-34.
2. D. A. BINI, L. GEMIGNANI, AND V. Y. PAN, QR-like algorithms for generalized semiseparable matrices, Technical Report 1470, Dipartimento di Matematica, Università di Pisa, 2003.
3. D.A. BINI, L. GEMIGNANI, AND F. TISSEUR, The Ehrlich-Aberth method for the nonsymmetric tridiagonal eigenvalue problem, Technical Report 428, Manchester Centre for Computational Mathematics, University of Manchester, 2003.
4. D. CALVETTI, S. KIM, AND L. REICHEL, The restarted QR-algorithm for eigenvalue computation of structured matrices, J. Comput. Appl. Math., 149 (2002), no. 2, pp. 415-422.
5. S. CHANDRASEKARAN AND M. GU, A fast and stable solver for recursively semi-separable systems of linear equations, in Structured matrices in mathematics, computer science, and engineering, II, Boulder, CO, 1999, Contemp. Math., 281, Amer. Math. Soc., Providence, RI, 2001, pp. 39-53.
6. Y. EIDELMAN AND I. GOHBERG, Inversion formulas and linear complexity algorithm for diagonal plus semiseparable matrices, Comput. Math. Appl., 33 (1997), pp. 69-79.
7. Y. EIDELMAN AND I. GOHBERG, On a new class of structured matrices, Integral Equations Operator Theory, 34 (1999), pp. 293-324.
8. D. FASINO, Personal communication.
9. D. FASINO, N. MASTRONARDI, AND M. VAN BAREL, Fast and stable algorithms for reducing diagonal plus semiseparable matrices to tridiagonal and bidiagonal form, in Fast Algorithms for Structured Matrices: Theory and Applications, Vadim Olshevsky, ed., Contemp. Math., 323, Amer. Math. Soc., Providence, RI, 2003, pp. 105-118.
10. P. E. GILL, G. H. GOLUB, W. MURRAY, AND M. A. SAUNDERS, Methods for modifying matrix factorizations, Math. Comp., 28 (1974), pp. 505-535.
11. I. GOHBERG, T. KAILATH, AND I. KOLTRACHT, Linear complexity algorithms for semiseparable matrices, Integral Equations Operator Theory, 8 (1985), no. 6, pp. 780-804.
12. I. KOLTRACHT, Linear complexity algorithm for semiseparable matrices, Integral Equations Operator Theory, 29 (1997), no. 3, pp. 313-319.
13. P. FAVATI P. RÓZSA, R. BEVILACQUA AND F. ROMANI, On the inverse of block tridiagonal matrices with applications to the inverses of band matrices and block band matrices, Oper. Theory Adv. Appl., 40 (1989), pp. 447-469.
14. P. RÓZSA, R. BEVILACQUA, F. ROMANI, AND P. FAVATI, On band matrices and their inverses, in Proceedings of the First Conference of the International Linear Algebra Society, Provo, UT, 1989, Linear Algebra Appl., 150 (1991), pp. 287-295.
15. E. E. TYRTYSHNIKOV, Mosaic ranks for weakly semiseparable matrices, in Large-scale scientific computations of engineering and environmental problems, II, Sozopol, 1999, Notes Numer. Fluid Mech., 73, Vieweg, Braunschweig, 2000, pp. 36-41.
16. R. VANDEBRIL, M. VAN BAREL, AND N. MASTRONARDI, An implicit qr algorithm for semiseparable matrices, to compute the eigendecomposition of symmetric matrices, Technical Report TW367, Department of Computer Science, Katholieke Universiteit, Leuven, August 2003.
17. R. VANDEBRIL, M. VAN BAREL, AND N. MASTRONARDI, An orthogonal similarity reduction of a matrix to semiseparable form, Technical Report TW355, Department of Computer Science, Katholieke Universiteit, Leuven, February 2003.
18. T. WANG AND W. B. GRAGG, Convergence of the shifted QR algorithm for unitary Hessenberg matrices, Math. Comp., 71 (2002), no. 240, pp. 1473-1496.
19. H. XU, The relation between the QR and LR algorithms, SIAM J. Matrix Anal. Appl., 19 (1998), no. 2, pp. 551-555.