Relative perturbation bounds for eigenvalues of symmetric positive definite diagonally dominant matrices

SIAM Journal on Matrix Analysis and Applications

Volumn 31, Issue 1, 2009, Pages 11-17

  Ye, Qiang ^a  

^a UNIVERSITY OF KENTUCKY   (United States)

Author keywords

Diagonal dominant matrix; Eigenvalues; Relative perturbation

Indexed keywords

ABSOLUTE VALUES; DIAGONALLY DOMINANT; DIAGONALLY DOMINANT MATRIX; DOMINANT MATRIX; EIGENVALUES; GENERALIZED EIGENVALUE PROBLEMS; MATRIX; RELATIVE ACCURACY; RELATIVE ERRORS; RELATIVE PERTURBATION; SYMMETRIC POSITIVE DEFINITE;

ERRORS; MATRIX ALGEBRA; SWITCHING SYSTEMS;

EIGENVALUES AND EIGENFUNCTIONS;

EID: 73649111754     PISSN: 08954798     EISSN: 10957162     Source Type: Journal    
DOI: 10.1137/060676349     Document Type: Conference Paper

References (22)

1. A. S. ALFA, J. XUE, AND Q. YE, Entrywise perturbation theory for diagonally dominant Mmatrices with applications, Numer. Math., 90 (2002), pp. 401-414.
2. A. S. ALFA, J. XUE, AND Q. YE, Accurate computation of the smallest eigenvalue of a diagonally dominant M-matrix, Math. Comp., 71 (2002), pp. 217-236.
3. J. BARLOW AND J. DEMMEL, Computing accurate eigensystems of scaled diagonally dominant matrices, SIAM J. Numer. Anal., 27 (1990), pp. 762-791.
4. J. W. DEMMEL, Applied Numerical Linear Algebra, SIAM, Philadelphia, 1997.
5. J. DEMMEL ANDW. GRAGG, On computing accurate singular values and eigenvalues of matrices with acyclic graphs, Linear Algebra Appl., 185 (1993), pp. 203-217.
6. J. DEMMEL AND W. KAHAN, Accurate singular values of bidiagonal matrices, SIAM J. Sci. Statist. Comput., 11 (1990), pp. 873-912.
7. J. DEMMEL, M. GU, S. EISENSTAT, I. SLAPNIČAR, K. VESELIĆ, AND Z. DRMAČ, Computing the singular value decomposition with high relative accuracy, Linear Algebra Appl., 299 (1999), pp. 21-80.
8. J. W. DEMMEL AND P. KOEV, Accurate SVDs of weakly diagonally dominant M-matrices, Numer. Math., 98 (2004), pp. 99-104.
9. F. M. DOPICO AND P. KOEV, Accurate symmetric rank revealing and eigendecompositions of symmetric structured matrices, SIAM J. Matrix Anal. Appl., 28 (2006), pp. 1126-1156.
10. S. C. EISENSTAT AND I. C. F. IPSEN, Relative perturbation techniques for singular value problems, SIAM J. Numer. Anal., 32 (1995), pp. 1972-1988.
11. L. ELSNER, Bounds for determinants of perturbed M-matrices, Linear Algebra Appl., 257 (1997), pp. 283-288.
12. N. J. HIGHAM, A survey of componentwise perturbation theory in numerical linear algebra, in Mathematics of Computation 1943-1993, Proceedings of Symposia in Applied Mathematics 48, W. Gautschi, ed., AMS, Providence, RI, 1994, pp. 49-77.
13. I. C. F. IPSEN, Relative perturbation bounds for matrix eigenvalues and singular values, in Acta Numerica 1998, Acta Numer. 7, Cambridge University Press, Cambridge, UK, 1998, pp. 151-201.
14. P. KOEV, Accurate eigenvalues and SVDs of totally nonnegative matrices, SIAM J. Matrix Anal. Appl., 27 (2005), pp. 1-23.
15. R.-C. LI, Relative perturbation theory: I. Eigenvalue and singular value variations, SIAM J. Matrix Anal. Appl., 19 (1998), pp. 956-982.
16. R. C. LI, Relative perturbation theory: (III) More bounds on eigenvalue variation, Linear Algebra Appl., 266 (1997), pp. 337-345.
17. J. M. PẼNA, LDU decompositions with L and U well conditioned, Electron. Trans. Numer. Anal., 18 (2004), pp. 198-208.
18. D. A. SPIELMAN AND S. TENG, Solving sparse, symmetric, diagonally-dominant linear systems in time O(m1.31), in Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science, IEEE Computer Society, Los Alamitos, CA, 2003, pp. 416-427.
19. K. VESELIC AND I. SLAPNICAR, Floating point perturbations of Hermitian matrices, Linear Algebra Appl., 195 (1993), pp. 81-116.
20. J. XUE AND E. JIANG, Entrywise relative perturbation theory for nonsingular M-matrices and applications, BIT, 35 (1995), pp. 417-427.
21. J. XUE, Computing the smallest eigenvalue of an M-matrix, SIAM J. Matrix Anal. Appl., 17 (1996), pp. 748-762.
22. Q. YE, Computing singular values of diagonally dominant matrices to high relative accuracy, Math. Comp., to appear.