A generalized rayleigh quotient iteration for computing simple eigenvalues of nonnormal matrices

ZAMM Zeitschrift fur Angewandte Mathematik und Mechanik

Volumn 80, Issue 1, 2000, Pages 9-25

  Schwetlick, H ^a   Losche R ^a  

^a DRESDEN UNIVERSITY OF TECHNOLOGY   (Germany)

Author keywords

Invariant subspace; Inverse iteration; Newton iteration, simple bifurcation point; Nonnormal eigenvalue problem; Rayleigh quotient iteration

Indexed keywords

EID: 22844455049     PISSN: 00442267     EISSN: 15214001     Source Type: Journal    
DOI: 10.1002/(sici)1521-4001(200001)80:1<9::aid-zamm9>3.3.co;2-%23     Document Type: Article

References (24)

1. ALLGOWER, E. L.; GEORG, K.: Numerical continuation methods: An introduction. Springer, Berlin, 1990.
2. ALLGOWER, E. L.; SCHWETLICK, H.: A general view of minimally extended systems for simple bifurcation points. ZAMM 77 (1997), 83-97.
3. CHATELIN, F.: Simultaneous Newton's iteration for the eigenproblem. Computing Suppl. 5 (1984), 67-74.
4. DEMMEL, J. W.: Three methods for refining estimates of invarint subspaces. Computing 38 (1987), 43-57.
5. DONGARRA, J. J.; MOLER, C. B.; WILKINSON, J. H.: Improving the accuracy of computed eigenvalues and eigenvectors. SIAM J. Numer. Anal. 46 (1983), 46-58.
6. EIERMANN, M.: Semiiterative Verfahren für nichtsymmetrische lineare Gleichungssysteme. Habilitationsschrift, Universität Karlsruhe, Germany, 1989.
7. GOLUB, G. H.; VAN LOAN, C. F.: Matrix computations. 3rd ed. Johns Hopkins University Press, Baltimore 1996.
8. GRIEWANK, A.; REDDIEN, G. W.: Characterization and computation of generalized turning points. SIAM J. Numer. Anal. 21 (1984), 176-185.
9. KIEŁBASIŃSKI, A.; SCHWETLICK, H.: Numerische lineare Algebra. Eine computerorientierte Einführung. Deutscher Verlag der Wissenschaften, Berlin 1988. Auch: Harri Deutsch Verlag, Thun-Frankfurt 1988.
10. LÖSCHE, R.; SCHWETLICK, H.; TIMMERMANN, G.: A modified block Newton iteration for approximating an invariant subspace of a symmetric matrix. Linear Algebra Appl. 275/276 (1998), 381-400.
11. LUDYK, G.: CAE von Dynamischen Systemen. Analyse, Simulation, Entwurf von Regelungssystemen. Springer, Berlin 1990.
12. O'LEARY, D. P.; STEWART, G. W.: On the convergence of a new Rayleigh quotient method with applications to large eigenproblems. Techn. Rep. TR-97-74, TR-3839, Institute for Advanced Computer Studies, Department of Computer Science, University of Maryland, College Park 1997;
13. appeared in: Electronic. Trans. Numer. Anal. 7 (1998), 182-189.
14. OSTROWSKI, A.: On the convergence of the Rayleigh quotient iteration for the computation of the characteristic roots and vectors. I-VI. Arch. Rat. Mech. Anal. 1-4 (1958/59), 1: 233-241, 2: 423-428, 3: 325-340, 3: 341-347, 3: 472-481, 4: 153-165.
15. PARLETT, B. N.: The Rayleigh quotient iteration and some generalizations for nonnormal matrices. Math. Comp. 28 (1974), 679-693.
16. PETERS, G.; WILKINSON, J. H.: Inverse iteration, ill-conditioned equations and Newton's method. SIAM Rev. 21 (1979), 339-360.
17. PÖNISCH, G.; SCHWETLICK, H.: Computing turning points of curves implicitly defined by nonlinear equations depending on a parameter. Computing, 26 (1981), 107-121.
18. SCHWETLICK, H.; SCHNABEL, U.: Iterative computation of the smallest singular value and the corresponding singular vectors of a matrix. Preprint IOKOMO-06-97, Techn. Univ. Dresden 1997.
19. STEWART, G. W.: Error and perturbation bounds for subspaces associated with certain eigenvalue problems. SIAM Rev. 15 (1973), 727-764.
20. STEWART, G. W.; SUN, J.: Matrix perturbation theory. Academic Press, Boston 1990.
21. TREFETHEN, L. N.; BAU, D.: Numerical linear algebra. SIAM, Philadelphia 1997.
22. UNGER, H.: Nichtlineare Behandlung von Eigenwertaufgaben. ZAMM 30 (1950), 281-282.
23. WIELANDT, H.: Bestimmung höherer Eigenwerte durch gebrochene Iteration. Ber. B. 44/I/37, Aerodynamische Versuchsanstalt Göttingen 1944.
24. WILKINSON, J. H.: The algebraic eigenvalue problem. Clarendon Press, Oxford 1963.