Computing approximate eigenpairs of symmetric block tridiagonal matrices

SIAM Journal on Scientific Computing

Volumn 25, Issue 1, 2003, Pages 65-85

  Gansterer, Wilfried N ^a   Ward, Robert C ^a   Muller, Richard P ^b   Goddard III, William A ^b  

^a University of Tennessee   (United States)

^b CALIFORNIA INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

Approximate eigenpairs; Block tridiagonal matrix; Divide and conquer method; Eigenvalue problem

Indexed keywords

APPROXIMATION THEORY; COMPUTATIONAL METHODS; EIGENVALUES AND EIGENFUNCTIONS; MATHEMATICAL MODELS; VECTOR QUANTIZATION;

DIVIDE AND CONQUER METHOD; OFF DIAGONAL BLOCKS; SYMMETRIC BLOCK TRIDIAGONAL MATRIX;

MATRIX ALGEBRA;

EID: 1842531885     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/S1064827501399432     Document Type: Article

References (34)

1. E. Anderson, Z. Bai, C. H. Bischof, S. Blackford, J. W. Demmel, J. J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. C. Sorensen, Lapack User's Guide, 3rd ed., SIAM, Philadelphia, 1999.
2. P. Arbenz, Divide and conquer algorithms for the bandsymmetric eigenvalue problem, Parallel Comput., 18 (1992), pp. 1105-1128.
3. P. Arbenz, W. Gander and G. H. Golub, Restricted runk modification of the symmetric eigenvalue problem: Theoretical considerations, Linear Algebra Appl., 104 (1988), pp. 75-95.
4. P. Arbenz and G. H. Golub, On the spectral decomposition of Hermitian matrices modified by low rank perturbations with applications, SIAM J. Matrix Anal. Appl., 9 (1988), pp. 40-58.
5. Y. Bai, W. N. Gansterer, and R. C. Ward, Block Tridiagonalization of "Effectively" Sparse Symmetric Matrices, Technical report UT-CS-02-492. Department of Computer Science, University of Tennessee, Knoxville, TN, 2002; also available online from http://www.cs.utk.edu/~library/techreportsmain.php; ACM Trans. Math. Software, submitted.
6. J. L. Barlow, Error analysis of update methods for the symmetric eigenvalue problem, SIAM J. Matrix Anal. Appl., 14 (1993), pp. 598-618.
7. C. H. Bischof, B. Lang, and X. Sun, A framework for symmetric band reduction, ACM Trans. Math. Software, 26 (2000), pp. 581-601.
8. J. R. Bunch, C. P. Nielsen, and D. C. Sorensen, Rank-one modification of the symmetric eigenproblem, Numer. Math., 31 (1978), pp. 31-48.
9. J. J. M. Cuppen, A divide and conquer method for the symmetric tridiagonal eigenproblem, Numer. Math., 36 (1981), pp. 177-195.
10. J. W. Demmel, Applied Numerical Linear Algebra, SIAM, Philadelphia, 1997.
11. J. J. Dongarra and D. C. Sorensen, A fully parallel algorithm for the symmetric eigenvalue problem. SIAM J. Sci. Stat. Comput., 8 (1987), pp. s139-s154.
12. W. N. Gansterer, D. F. Kvasnicka, and C. W. Ueberhuber, Multi-sweep algorithms for the symmetric eigenproblem, in VECPAR'98 - Third International Conference for Vector and Parallel Processing, Lecture Notes in Computer Science 1573, J. M. L. M. Palma, J. J. Dongarra, and V. Hernandez, eds., Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1998, pp. 20-28.
13. W. N. Gansterer, J. Schneid and C. W. Ueberhuber, A Divide-and-Conquer Method for Symmetric Banded Eigenproblems. Part II: Complexity Analysis, Technical report AURORA TR1999-14, Vienna University of Technology, Vienna, Austria 1999; also available online from http://www.vcpc.univie.ac.at/aurora/publications/.
14. W. N. Gansterer, J. Schneid, and C. W. Ueberhuber, A low-complexity divide-and-conquer method for computing eigenvalues and eigenvectors of symmetric band matrices, BIT, 41 (2001), pp. 967-976.
15. W. N. Gansterer and R. C. Ward, Computing Approximate Eigenpairs of Symmetric Block Tridiagonal Matrices, Technical report UT-CS-01-463, Department of Computer Science, University of Tennessee, Knoxville, TN, 2001; also available online from http://www.cs.utk.edu/~library/techreportsmain.php.
16. W. N. Gansterer, R. C. Ward, and R. P. Muller, An extension of the divide-and-conquer method for a class of symmetric block-tridiagonal eigenproblems, ACM Trans. Math. Software, 28 (2002), pp. 45-58.
17. K. Gates and P. Arbenz, Parallel Divide and Conquer Algorithms for the Symmetric Tridiagonal Eigenproblem, Technical report 222, Institut für Wissenschaftliches Rechnen, ETH Zürich, Zürich, Switzerland, 1994; also available online from http://www.inf.ethz.ch/research/publications/.
18. G. H. Golub, Some modified matrix eigenvalue problems, SIAM Rev., 15 (1973), pp. 318-334.
19. G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed., Johns Hopkins University Press, Baltimore, MD, 1996.
20. M. Gu and S. C. Eisenstat, A stable and efficient algorithm for the rank-one modification of the symmetric eigenproblem, SIAM J. Matrix Anal. Appl., 15 (1994), pp. 1266-1276.
21. 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.
22. L. Kaufman, Band reduction algorithms revisited, ACM Trans. Math. Software, 26 (2000), pp. 551-567.
23. B. N. Parlett, The Symmetric Eigenvalue Problem, SIAM, Philadelphia, 1997.
24. J. A. Pople and D. L. Beveridge, Approximate Molecular Orbital Theory, 1st ed., McGraw-Hill, New York, 1970.
25. J. A. Pople, D. L. Beveridge, and P. A. Dobosh, Approximate self-consistent molecular orbital theory. V. Intermediate neglect of differential overlap., J. Chem. Phys., 47 (1967), p. 2026.
26. J. A. Pople, D. P. Santry, and G. A. Segal, Approximate self-consistent molecular orbital theory. I. Invariant procedures., J. Chem. Phys., 43 (1965), p. S129.
27. J. A. Pople and G. A. Segal, Approximate self-consistent molecular orbital theory. II. Calculations with complete neglect of differential overlap., J. Chem. Phys., 43 (1965), p. S136.
28. J. A. Pople and G. A. Segal, Approximate self-consistent molecular orbital theory. III. CNDO results for ab2 and ab3 systems., J. Chem. Phys., 44 (1966), p. 3289.
29. J. Rutter, A Serial Implementation of Cuppen's Divide and Conquer Algorithm for the Symmetric Eigenvalue Problem, LAPACK Working Note 69, Computer Science Division (EECS), University of California at Berkeley, Berkeley, CA, 1994; also available online from http://www.netlib.org/lapack/lawns/downloads/.
30. H. R. Schwarz, Tridiagonalization of a symmetric band matrix, Numer. Math., 12 (1968), pp. 231-241.
31. D. C. Sorensen and P. T. P. Tang, On the orthogonality of eigenvectors computed by divide-and-conquer techniques, SIAM J. Numer. Anal., 28 (1991), pp. 1752-1775.
32. A. Szabo and N. S. Ostlund, Modern Quantum Chemistry, Dover, Mineola, NY, 1996.
33. F. Tisseur and J. Dongarra, A parallel divide and conquer algorithm for the symmetric eigenvalue problem on distributed memory architectures, SIAM J. Sci. Comput., 20 (1999), pp. 2223-2236.
34. J. H. Wilkinson, The Algebraic Eigenvalue Problem, Oxford University Press, Oxford, 1965.