Reliable computation of the condition number of a tridiagonal matrix in O(n) time

SIAM Journal on Matrix Analysis and Applications

Volumn 19, Issue 3, 1998, Pages 776-796

  Dhillon, Inderjit S ^a  

^a IBM ALMADEN RESEARCH CENTER   (United States)

Author keywords

Condition number; Inverse; Norm; Overflow; Tridiagonal matrix; Underflow

Indexed keywords

EID: 0032395502     PISSN: 08954798     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0895479897314747     Document Type: Article

References (26)

1. E. ANDERSON, Z. BAI, C. BISCHOF, J. DEMMEL, J. DONGARRA, J. DU CROZ, A. GREENBAUM, S. HAMMARLING, A. MCKENNEY, S. OSTROUCHOV, AND D. SORENSEN, LAPACK Users' Guide, 2nd ed., SIAM, Philadelphia, PA, 1995.
2. ANSI/IEEE, NEW YORK, IEEE Standard for Binary Floating Point Arithmetic, Std 754-1985 ed., 1985.
3. ij, = 0 for j > i + p, Math. Scand., 7 (1959), pp. 57-60.
4. S. O. ASPLUND, Finite boundary value problems solved by Green's matrix, Math. Scand., 7 (1959), pp. 49-56.
5. J. BARLOW, Private communication, Pennsylvania State University, University Park, PA, 1996.
6. B. BUKHBERGER AND G. A. EMEL'YANENKO, Methods of inverting tridiagonal matrices, Comput. Math. Math. Phys., 13 (1973), pp. 10-20.
7. L. S. DEJONG, Towards a formal definition of numerical stability, Numer. Math., 28 (1977), pp. 211-220.
8. J. DEMMEL AND A. MCKENNEY, A Test Matrix Generation Suite, LAPACK Working Note #9, Technical report, Courant Institute, Computer Science Department, New York, 1989.
9. J. W. DEMMEL, The Complexity of Condition Estimation, manuscript, University of California, Berkeley, CA, 1997.
10. JAMES W. DEMMEL, Open Problems in Numerical Linear Algebra, LAPACK Working Note 47, IMA Preprint Series #961, Institute for Mathematics and Its Applications, University of Minnesota, Minneapolis, MN, April 1992.
11. JAMES W. DEMMEL AND XIAOYE LI, Faster numerical algorithms via exception handling, IEEE Trans. Comput., 43 (1994), pp. 983-992.
12. 2) Algorithm for the Symmetric Tridiagonal Eigenvalue/Eigenvector Problem, Ph.D. thesis, Computer Science Division, EECS Dept., University of California, Berkeley, CA, May 1997; also available as Computer Science Division Technical report UCB//CSD-97-971.
13. I. S. DHILLON AND B. N. PARLETT, Orthogonal Eigenvectors Without Gram-Schmidt, 1997, manuscript.
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. W. W. HAGER, Condition estimators, SIAM J. Sci. Stat. Comput., 5 (1984), pp. 311-316.
16. N. J. HIGHAM, Matrix condition numbers, M.Sc. thesis, Dept. of Mathematics, University of Manchester, Manchester, England, 1983.
17. N. J. HIGHAM, Efficient algorithms for computing the condition number of a tridiagonal matrix, SIAM J. Sci. Statist. Comput., 7 (1986), pp. 150-165.
18. N. J. HIGHAM, A survey of condition number estimation for triangular matrices, SIAM Rev., 29 (1987), pp. 575-596.
19. N. J. HIGHAM, FORTRAN codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation, ACM Trans. Math. Software, 14 (1988), pp. 381-396.
20. N. J. HIGHAM, Accuracy and stability of numerical algorithms, SIAM, Philadelphia, PA, 1996.
21. I. IKEBE, On inverses of Hessenberg matrices, Linear Algebra Appl., 24 (1979), pp. 93-97.
22. G. MEURANT, A review on the inverse of symmetric tridiagonal and block tridiagonal matrices, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 707-728.
23. 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.
24. G. W. STEWART, Introduction to Matrix Computations, Academic Press, New York, 1973.
25. J. H. WILKINSON, The calculation of the eigenvectors of codiagonal matrices, Computer J., 1 (1958), pp. 90-96.
26. J. H. WILKINSON, The Algebraic Eigenvalue Problem, Oxford University Press, Oxford, 1965.