QR -like algorithms for eigenvalue problems

Journal of Computational and Applied Mathematics

Volumn 123, Issue 1-2, 2000, Pages 67-83

  Watkins, David S ^a  

^a WASHINGTON STATE UNIVERSITY   (United States)

Author keywords

65F15; Eigenvalue* QR algorithm* GR algorithm

Indexed keywords

BOUNDARY CONDITIONS; COMPUTER SIMULATION; CONVERGENCE OF NUMERICAL METHODS; EIGENVALUES AND EIGENFUNCTIONS; ITERATIVE METHODS; MATRIX ALGEBRA; PARALLEL PROCESSING SYSTEMS; PROBLEM SOLVING;

EIGENVALUE PROBLEMS; GR ALGORITHMS; QR ALGORITHMS; SINGULAR VALUE DECOMPOSITION;

ALGORITHMS;

EID: 0034320028     PISSN: 03770427     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0377-0427(00)00402-7     Document Type: Article

References (63)

1. G.S. Ammar, W.B. Gragg, L. Reichel, On the eigenproblem for orthogonal matrices, in: Proceedings 25th IEEE Conference on Decision and Control, Athens, New York, 1986, pp. 1063-1066.
2. Bai Z., Demmel J. On a block implementation of the Hessenberg multishift QR iteration. Int. J. High Speed Comput. 1:1989;97-112.
3. Bai Z., Demmel J., Gu M. Inverse free parallel spectral divide and conquer algorithms for nonsymmetric eigenproblems. Numer. Math. 76:1997;279-308.
4. Bai Z.et al. The spectral decomposition of nonsymmetric matrices on distributed memory parallel computers, Technical Report CS-95-273. 1995;University of Tennessee.
5. Batterson S., Smillie J. Rayleigh quotient iteration for nonsymmetric matrices. Math. Comp. 55:1990;169-178.
6. Benner P., Fassbender H., Watkins D.S. Two connections between the. SR and HR eigenvalue algorithms Linear Algebra Appl. 272:1998;17-32.
7. B. Bohnhorst, Ein Lanczos-ähnliches Verfahren zur Lösung des unitären Eigenwertproblems, Ph.D. Thesis, University of Bielefeld, 1993.
8. K. Braman, R. Byers, R. Mathias, The multi-shift QR algorithm: aggressive deflation, maintaining well focused shifts, and level 3 performance. Manuscript, 1999.
9. Brebner M.A., Grad J. Eigenvalues of. Ax=λ Bx for real symmetric matrices A and B computed by reduction to pseudosymmetric form and the HR process Linear Algebra Appl. 43:1982;99-118.
10. Bunse-Gerstner A. An analysis of the HR algorithm for computing the eigenvalues of a matrix. Linear Algebra Appl. 35:1981;155-173.
11. Bunse-Gerstner A., Elsner L. Schur parameter pencils for the solution of the unitary eigenproblem. Linear Algebra Appl. 154-156:1991;741-778.
12. Bunse Gerstner A., Mehrmann V. A symplectic. QR -like algorithm for the solution of the real algebraic Riccati equation IEEE Trans. Auto. Control. AC-31:1986;1104-1113.
13. Bunse-Gerstner A., Mehrmann V., Watkins D.S. An. SR algorithm for Hamiltonian matrices based on Gaussian elimination Methods Oper. Res. 58:1989;339-358.
14. H.J. Buurema, A geometric proof of convergence for the QR method, Ph.D. Thesis, University of Groningen, Netherlands, 1970.
15. R. Byers, Hamiltonian and symplectic algorithms for the algebraic Riccati equation, Ph.D. Thesis, Cornell University, 1983.
16. Byers R. A Hamiltonian. QR algorithm SIAM J. Sci. Stat. Comput. 7:1986;212-229.
17. D. Day, How the QR algorithm fails to converge and how to fix it. Mathematics of Numerical Analysis: Real Number Algorithms, August 1995.
18. Della-Dora J. Numerical linear algorithms and group theory. Linear Algebra Appl. 10:1975;267-283.
19. J. Demmel, B. Kågström, GUPTRI. NETLIB, 1991. http://www.netlib.org/ .
20. Dongarra J.J., Bunch J.R., Moler C.B., Stewart G.W. LINPACK Users' Guide. 1979;SIAM, Philadelphia.
21. Dongarra J.J., Du Croz J., Duff I., Hammarling S. A set of level 3 basic linear algebra subprograms. ACM Trans. Math. Software. 16:1990;1-17.
22. Dongarra J.J., Du Croz J., Hammarling S., Hanson R. An extended set of Fortran basic linear algebra subprograms. ACM Trans. Math. Software. 14:1988;1-17.
23. van Dooren P. The computation of Kronecker's canonical form of a singular pencil. Linear Algebra Appl. 27:1979;103-140.
24. A.A. Dubrulle, The multishift QR algorithm - is it worth the trouble?, Technical Report G320-3558x, IBM Corp., Palo Alto, 1991.
25. E. Anderson et al., LAPACK Users' Guide, SIAM, Philadelphia, 2nd Edition, 1995. http://www.netlib.org/lapack/lug/lapack_lug.html .
26. L.S. Blackford et al., ScaLAPACK Users' Guide, SIAM, Philadelphia, 1997. http://www.netlib.org/scalapack/slug/scalapack_slug.html .
27. Fernando K.V., Parlett B.N. Accurate singular values and differential qd algorithms. Numer. Math. 67:1994;191-229.
28. J.G.F. Francis, The QR transformation, Parts I and II, Computer J. 4 1961 265-272, 332-345.
29. Geist G.A., Howell G.W., Watkins D.S. The BR eigenvalue algorithm. SIAM J. Matrix Anal. Appl. 20:1999;1083-1098.
30. G.H. Golub, C.F. Van Loan, Matrix Computations, Johns Hopkins University Press, Baltimore, 3rd Edition, 1996.
31. Haag J.B., Watkins D.S. QR -like algorithms for the nonsymmetric eigenvalue problem ACM Trans. Math. Software. 19:1993;407-418.
32. Heller D.E., Ipsen I.C.F. Systolic networks for orthogonal decompositions. SIAM J. Sci. Stat. Comput. 4:1983;261-269.
33. G. Henry, D. Watkins, J. Dongarra, A parallel implementation of the nonsymmetric QR algorithm for distributed memory architectures, Technical Report LAPACK Working Note 121, University of Tennessee, 1997.
34. Householder A.S. The Theory of Matrices in Numerical Analysis. 1964;Dover Books.
35. Kaufman L.C. The. LZ algorithm to solve the generalized eigenvalue problems SIAM J. Numer. Anal. 11:1974;997-1024.
36. Kaufman L.C. A parallel. QR algorithm for the symmetric tridiagonal eigenvalue problem J. Parallel and Distributed Computing. 23:1994;429-434.
37. Kublanovskaya V.N. On some algorithms for the solution of the complete eigenvalue problem. USSR Comput. Math. Math. Phys. 3:1961;637-657.
38. Moler C.B., Stewart G.W. An algorithm for generalized matrix eigenvalue problems. SIAM J. Numer. Anal. 10:1973;241-256.
39. Paige C., Van Loan C. A Schur decomposition for Hamiltonian matrices. Linear Algebra Appl. 41:1981;11-32.
40. B.N. Parlett, The Symmetric Eigenvalue Problem, Prentice-Hall, Englewood Cliffs, New Jersey, 1980. Reprinted by SIAM, 1997.
41. B.N. Parlett, The new qd algorithms, Acta Numerica (1995) 459-491.
42. Parlett B.N., Poole W.G. A geometric theory for the. QR, LU, and power iterations SIAM J. Numer. Anal. 10:1973;389-412.
43. Rutishauser H. Der Quotienten-Differenzen-Algorithmus. Z. Angew. Math. Physik. 5:1954;233-251.
44. Rutishauser H. Der Quotienten-Differenzen-Algorithmus, Mitt. Inst. Angew. Math. ETH, Vol. 7, Birkhäuser. 1957;Birkhäuser, Basel.
45. Rutishauser H. Solution of eigenvalue problems with the LR-transformation. Nat. Bur. Standards Appl. Math. Series. 49:1958;47-81.
46. B.T. Smith et al., Matrix Eigensystem Routines - EISPACK Guide, Springer, Berlin, 2nd Edition, 1976.
47. Stewart G.W. A parallel implementation of the. QR algorithm Parallel Comput. 5:1987;187-196.
48. R. van de Geijn, D.G. Hudson, An efficient parallel implementation of the nonsymmetric QR algorithm, in: Proceedings of the Fourth Conference on Hypercube Concurrent Computers and Applications, 1989.
49. R.A. van de Geijn, Implementing the QR algorithm on an array of processors, Ph.D. Thesis, University of Maryland, 1987. Department of Computer Science TR-1897.
50. van de Geijn R.A. Deferred shifting schemes for parallel. QR methods SIAM J. Matrix Anal. Appl. 14:1993;180-194.
51. Watkins D.S. Understanding the. QR algorithm SIAM Rev. 24:1982;427-440.
52. Watkins D.S. Fundamentals of Matrix Computations. 1991;Wiley, New York.
53. Watkins D.S. Some perspectives on the eigenvalue problem. SIAM Rev. 35:1993;430-471.
54. Watkins D.S. Shifting strategies for the parallel. QR algorithm SIAM J. Sci. Comput. 15:1994;953-958.
55. Watkins D.S. QR -like algorithms - an overview of convergence theory and practice Renegar J., Shub M., Smale S. The Mathematics of Numerical Analysis. Lectures in Applied Mathematics. Vol. 22:1996;American Mathematical Society, Providence, RI.
56. Watkins D.S. The transmission of shifts and shift blurring in the. QR algorithm Linear Algebra Appl. 241-243:1996;877-896.
57. Watkins D.S. Bulge exchanges in algorithms of. QR type SIAM J. Matrix Anal. Appl. 19:1998;1074-1096.
58. Watkins D.S., Elsner L. Chasing algorithms for the eigenvalue problem. SIAM J. Matrix Anal. Appl. 12:1991;374-384.
59. Watkins D.S., Elsner L. Convergence of algorithms of decomposition type for the eigenvalue problem. Linear Algebra Appl. 143:1991;19-47.
60. Watkins D.S., Elsner L. Theory of decomposition and bulge-chasing algorithms for the generalized eigenvalue problem. SIAM J. Matrix Anal. Appl. 15:1994;943-967.
61. Wilkinson J.H. The Algebraic Eigenvalue Problem. 1965;Clarendon Press, Oxford University.
62. Wilkinson J.H. Global convergence of tridiagonal. QR algorithm with origin shifts Linear Algebra Appl. 1:1968;409-420.
63. Wilkinson J.H. Kronecker's canonical form and the. QZ algorithm Linear Algebra Appl. 28:1979;285-303.