The DQR algorithm, basic theory, convergence, and conditional stability

Numerische Mathematik

Volumn 76, Issue 4, 1997, Pages 515-553

  Uhlig, Frank ^a  

^a AUBURN UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0031487023     PISSN: 0029599X     EISSN: None     Source Type: Journal    
DOI: 10.1007/s002110050275     Document Type: Article

References (57)

1. Batterson, S. (1994): Convergence of the Francis shifted QR algorithm on normal matrices, Lin. Alg. Appl. 207, 181-1
2. Bauer, F.L. (1959): Sequential reduction to tridiagonal form, J. SIAM Appl. Math. 7, 107-113
3. Bauer, F.L., Fike, C.T. (1960): Norms ard exclusion theorems, Num. Math. 2, 137-141
4. den Boer, H., Thijsse, G.P. (1980): Semi-stability of partial multiplicities of matrices, Int. Equat. and Op. Thy. 3, 23-42
5. Bognar, J. (1974): Indefinite Inner Product Spaces, Springer
6. Boley, D.L., Elhay, S., Golub, G.H., Gutknecht, M.H. (1991): Nonsymmetric Lanczos and finding orthogonal polynomials associated with indefinite weights, Num. Alg. 1, 21-44
7. Brebner, M.A., Grad, J. (1982): Eigenvalues of Ax = λBx for real symmetric matrices A and B computed by reduction to a pseudosymmetric form and the HR process, Lin. Alg. Appl. 43, 99-118.
8. Bunse, W., Bunse-Gerstner, A. (1985): Numerische Lineare Algebra, Teubner, 314
9. Bunse-Gerstner, A. (1978): Der HR-Algorithmus zur Bestimmung der Eigenwerte einer Matrix, Ph.D. Thesis, Bielefeld, 148
10. Bunse-Gerstner, A. (1981): An analysis of the HR algorithm for computing the eigenvalues of a matrix, Lin. Alg. Appl. 35, 155-173
11. Bunse-Gerstner, A. (1982): On the similarity transformation to tridiagonal form, Lin. Multilin. Alg. 12, 51-56
12. Bunse-Gerstner, A. (1984): An algorithm for the symmetric generalized eigenvalue problem, Lin. Alg. Appl., 58, 43-68
13. Bunse-Gerstner, A., Mehrmann, V. (1989): The HHDR algorithm and its applications to optimal control, Commande Alg. des Systèmes 23, 305-329.
14. Chu, M.T., Morris, L.K. (1988): Isospectral flows and abstract matrix factorizations, SIAM J. Numer. Anal. 25, 1383-1391
15. Cullum, J., Kerner, W., Willoughby, R. (1989): A generalized nonsymmetric Lanczos procedure, Computer Phys. Comm. 53, 19-48
16. Cullum, J., Willoughby, R.A. (1985): Lanczos algorithm for large symmetric eigenvalue computations, Birkhäuser Verlag, Boston, 257
17. Cuppen, J.J.M. (1981): A divide and conquer method for the tridiagonal symmetric cigenproblem, Num. Math. 36, 177-195
18. Dax, A., Kaniel, S. (1981): The ELR method for computing the eigenvalues of a general matrix, SIAM J. Num. Anal. 18, 597-605
19. Day, J., Peterson, B. (1980): Growth in Gaussian elimination, Amer. Math. Monthly 95, 489-513
20. Delia-Dora, J. (1975): Numerical linear algorithms and group theory, Lin. Alg. Appl. 10, 267-283
21. Dieudonné, J. (1963): La Géometrie des Groupes Classiques, second ed., Springer
22. Dongarra, J.J., Geist, G.A., Romine, C.H. (1990): Computing the eigenvalues and eigenvectors of a general matrix by reduction to general tridiagonal form, Tech. Rep. ORNL/TM-11669, Oak Ridge Nat. Lab.
23. Elsner, L. (1979): On some algebraic problems in connection with general eigenvalue algorithms, Lin. Alg. Appl. 26, 123-138
24. Francis, T.G.F. (1961): The QR algorithm, a unitary analogue to the LR transformation, I and II, Comp. J. 4, 265-272 and 332-345.
25. Freund, R.W., Gutknecht, M.H., Nachtigal, N.M. (1993): An implementation of the look-ahead Lanczos algorithm for non-hermitian matrices, SIAM J. Sci. Comput. 14, 137-158
26. Geist, G.A. (1991): Reduction of a general matrix to tridiagonal form, SIAM J. Matrix Anal. Appl. 12, 362-373
27. Gohberg, I., Lancaster, P., Rodman, L. (1983): Matrices and Indefinite Scalar Products, Operator Theory, Advances and Applications, vol.8, Birkhäuser, 374
28. Golub, G.H., van Loan, C.F. (1983): Matrix Computations, Johns Hopkins University Press, 476
29. Golub, G.H., van Loan, C.F. (1989): Matrix Computations, second ed., Johns Hopkins University Press, 642
30. Golub, G.H., Wilkinson, J.H. (1976): Ill-conditioned eigensystems and computation of the Jordan canonical form, SIAM Rev. 18, 578-619
31. Gregory, R.T., Karney, A.L. (1978): A Collection of Matrices for Testing Computational Algorithms, Wiley, 154
32. Gutknecht, M.H. (1992): A completed theory of the unsymmetric Lanczos process and related algorithms, part I, SIAM J. Matr. Anal. Appl. 13, 594-639
33. Gutknecht, M.H. (1994): A completed theory of the unsymmetric Lanczos process and related algorithms, part II, SIAM J. Matr. Anal. Appl. 15, 15-58
34. Jessup, E. (1993): A case against a divide and conquer approach to the nonsymmetric eigenproblem, Appl. Num. Math. 12, 403-420
35. Klein, F. (1884): Ueber die Transformation der allgemeinen Gleichung des zweiten Grades zwischen Linien-Coordinaten auf eine canonische Form, Dissertation, Bonn, 1968; Math. Ann. 23, 539-578
36. Kublanoskaja, V.N. (1961): On some algorithms for the solution of the complete eigenvalue problem (Russian), Zh. vych. mat. 1, 555-570
37. LaBudde, C.D. (1963): The reduction of an arbitrary real square matrix to tri-diagonal form using similarity transformations, Math. Comp. 17, 433-437
38. Lanczos, C. (1950): An iteration method for the solution of the eigenvalue problem of linear differential and integral operators, J. Res. Nat. Bur. Stand. 45, 255-282
39. Markus, A.S., Parilis, E.E. (1983): The change of the Jordan structure of a matrix under small perturbations, Mat. Issled., 54 (1980), 98-109; English translation in Lin. Alg. Appl. 54, 139-152
40. Markus, A.S., Parilis, E.E. (1983): The change of the Jordan structure of a matrix under small perturbations, Mat. Issled., 54 (1980), 98-109; English translation in Lin. Alg. Appl. 54, 139-152
41. Martin, R.S., Peters, G., Wilkinson, J.H. (1970): The QR algorithm for real Hessenberg matrices, Num. Math. 14, 219-231
42. Nagy, T.G. (1993): Fast inverse QR factorization of Toeplitz matrices, SIAM J. Sci. Comput. 14, 1174-1193
43. Kreyszig, E., Todd, J. (1976): QR in two dimensions, Elemente der Mathematik 31, 109-114
44. Parlett, B.N. (1980): The Symmetric Eigenvalue Problem, Prentice Hall, 348
45. Parlett, B.N. (1992): Reduction to tridiagonal form and minimal realizations, SIAM J. Matrix Anal. Appl. 13, 567-593
46. Rath, W. (1982): Fast Givens rotations for orthogonal similarity transformations, Num. Math. 40, 47-56
47. Ruhe, A. (1994): The rational Krylov algorithm for generalized eigenvalue problems, Numerical Algebra, Proceedings of the '92 Shanghai International Num. Alg. Conference, China Science and Techn. Press, 105-112
48. Rutishauser, H. (1958): Solution of the eigenvalue problem with the LR-transformation, Nat. Bur. Stand. Appl. Math. Ser. 49, 47-81.
49. Stewart, G.W. (1973): Introduction to Matrix Computations, Academic Press, 441
50. Stoer, J., Bulirsch, R. (1980): Introduction to Numerical Analysis, Springer, 609
51. Strachey, C., Francis, J. (1961): The reduction of a matrix to co-diagonal form by elimination, Comput. J. 4, 168
52. Taussky, O. (1972): The role of symmetric matrices in the study of general matrices. Lin. Alg. Appl. 5, 147-154
53. Uhlig, F. (1981): Inertia and eigenvalue relations between symmetrized and symmetrizing matrices for the real and general field case, Lin. Alg. Appl. 35, 203-226
54. Watkins, D., Elsner, L. (1989): Self-equivalent flows associated with generalized eigenvalue problems. Lin. Alg. Appl. 118, 107-127
55. Wiedemann, T. (1989): Das HDR Verfahren zur Lösung des allgemeinen Eigenwertproblems Ax = λBx, Diplomarbeit, Universität Bielefeld, 128
56. Wilkinson, J.H. (1963): Rounding Errors in Algebraic Processes, Prentice-Hall
57. Wilkinson, J.H. (1965): The Algebraic Eigenvalue Problem, Clarendon Press, 662