Orthogonal eigenvectors and relative gaps

SIAM Journal on Matrix Analysis and Applications

Volumn 25, Issue 3, 2004, Pages 858-899

  Dhillon, Inderjit S ^a   Parlett, Beresford N ^b  

^a University of Texas at Austin   (United States)

^b UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Bidiagonal factorization; Eigenvectors; High relative accuracy; Orthogonality; Symmetric tridiagonal matrix

Indexed keywords

EID: 2942514399     PISSN: 08954798     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0895479800370111     Document Type: Article

References (39)

1. ANSI/IEEE, IEEE Standard for Binary Floating Point Arithmetic, Std 754-1985 ed., New York, 1985.
2. J. BARLOW, B. PARLETT, AND K. VESELIĆ, EDS., Linear Algebra Appl., 309, no. 1-3 (2000), pp. 1-361.
3. A. W. BOJANCZYK, R. ONN, AND A. O. STEINHARDT, Existence of the hyperbolic singular value decomposition, Linear Algebra Appl., 185 (1993), pp. 21-30.
4. J. R. BUNCH, The weak and strong stability of algorithms in numerical linear algebra, Linear Algebra Appl., 88/89 (1987), pp. 49-66.
5. B. W. CHAR, K. O. GEDDES, G. H. GONNET, B. L. LEONG, M. B. MONAGAN, AND S. M. WATT, Maple V Library Reference Manual, Springer-Verlag, Berlin, 1991.
6. L. S. DEJONG, Towards a formal definition of numerical stability, Numer. Math., 28 (1977), pp. 211-220.
7. J. DEMMEL AND W. KAHAN, Accurate singular values of bidiagonal matrices, SIAM J. Sci. Statist. Comput., 11 (1990), pp. 873-912.
8. J. W. DEMMEL, Applied Numerical Linear Algebra, SIAM, Philadelphia, PA, 1997.
9. 2) Algorithm for the Symmetric Tridiagonal Eigenvalue/Eigenvector Problem, Ph.D. thesis, Computer Science Division, University of California, Berkeley, California, 1997. Available as UC Berkeley Technical Report UCB//CSD-97-971.
10. I. S. DHILLON AND B. N. PARLETT, Multiple representations to compute orthogonal eigenvectors of symmetric tridiagonal matrices, Linear Algebra Appl., to appear.
11. S. EISENSTAT AND I. IPSEN, Relative perturbation techniques for singular value problems, SIAM J. Numer. Anal., 32 (1995), pp. 1972-1988.
12. S. C. EISENSTAT AND I. C. F. IPSEN, Relative perturbation bounds for eigenspaces and singular vector subspaces, in Proceedings of the Fifth SIAM Conference on Applied Linear Algebra, J. G. Lewis, ed., SIAM, 1994, pp. 62-66.
13. K. FERNANDO AND B. PARLETT, Accurate singular values and differential qd algorithms, Numer. Math., 67 (1994), pp. 191-229.
14. K. V. FERNANDO, On computing an eigenvector of a tridiagonal matrix. Part I: Basic results, SIAM J. Matrix Anal. Appl., 18 (1997), pp. 1013-1034.
15. J. G. F. FRANCIS, The QR transformation, part I, Comput. J., 4 (1961-62), pp. 265-271.
16. J. G. F. FRANCIS, The QR transformation, part II, Comput. J., 4 (1961-62), pp. 332-345.
17. S. K. GODUNOV, A. G. ANTONOV, O. P. KIRILJUK, AND V. I. KOSTIN, Guaranteed Accuracy in Numerical Linear Algebra, Kluwer Academic Publishers, Dordrecht, the Netherlands, 1993.
18. S. K. GODUNOV, V. I. KOSTIN, AND A. D. MITCHENKO, Computation of an eigenvector of symmetric tridiagonal matrices, Siberian Math. J., 26 (1985), pp. 71-85.
19. A. GREENBAUM AND J. DONGARRA, Experiments with QL/QR Methods for the Symmetric Tridiagonal Eigenproblem, Computer Science Dept. Technical Report CS-89-92, University of Tennessee, Knoxville, 1989. (LAPACK Working Note 17, available electronically on netlib.)
20. A. GREENBAUM AND J. DONGARRA, Experiments with QL/QR Methods for the Symmetric Tridiagonal Eigenproblem, Computer Science Dept. Technical Report CS-89-92, University of Tennessee, Knoxville, 1989. (LAPACK Working Note 17, available electronically on netlib.)
21. 2) algorithm for the bidiagonal SVD, Linear Algebra Appl., 358 (2002), pp. 45-70.
22. M. GU AND S. C. EISENSTAT, A divide-and-conquer algorithm for the symmetric tridiagonal eigenproblem, SIAM J. Matrix Anal. Appl., 16 (1995), pp. 172-191.
23. P. HENRICI, The quotient-difference algorithm, Nat. Bur. Standards Appl. Math. Series, 19 (1958), pp. 23-46.
24. P. HENRICI, Applied and Computational Complex Analysis, Vol. I, John Wiley & Sons, New York, 1974.
25. N. J. HIGHAM, Accuracy and Stability of Numerical Algorithms, 2nd ed., SIAM, Philadelphia, 2002.
26. E. R. JESSUP AND I. IPSEN, Improving the accuracy of inverse iteration, SIAM J. Sci. Statist. Comput., 13 (1992), pp. 550-572.
27. V. N. KUBLANOVSKAYA, On some algorithms for the solution of the complete eigenvalue problem, Zh. Vychisl. Mat., 1 (1961), pp. 555-570.
28. R.-C. LI, Relative perturbation theory: II. Eigenspace and singular subspace variations, SIAM J. Matrix Anal. Appl., 20 (1998), pp. 471-492.
29. B. PARLETT, The Symmetric Eigenvalue Problem, Classics Appl. Math. 20, SIAM, Philadelphia, 1997.
30. B. N. PARLETT, Invariant subspaces for tightly clustered eigenvalues of tridiagonals, BIT, 36 (1996), pp. 542-562.
31. B. N. PARLETT, Spectral sensitivity of products of bidiagonals, Linear Algebra Appl., 275/276 (1998), pp. 417-431.
32. B. N. PARLETT AND I. S. DHILLON, Fernando's solution to Wilkinson's problem: An application of double factorization, Linear Algebra Appl., 267 (1997), pp. 247-279.
33. B. N. PARLETT AND I. S. DHILLON, Relatively robust representations of symmetric tridiagonals, Linear Algebra Appl., 309 (2000), pp. 121-151.
34. B. N. PARLETT, A bidiagonal matrix determines its hyperbolic SVD to (varied) relative accuracy, SIAM J. Matrix Anal. Appl., submitted.
35. H. RUTISHAUSER, Der Quotienten-Differenzen-Algorithmus. Z. Angew. Math. Phys., 5 (1954), pp. 223-251.
36. H. RUTISHAUSER, Vorlesungen über numerische Mathematik, Birkhäuser, Basel, 1976.
37. H. RUTISHAUSER, Lectures on Numerical Mathematics, Birkhäuser, Boston, 1990.
38. S. WOLFRAM, Mathematica: A System for Doing Mathematics by Computer, 2nd ed., Addison-Wesley, Reading, MA, 1991.
39. Y. YANG, Error Analysis of the dqds Algorithm, Ph.D. thesis, University of California, Berkeley, California, 1994.