Support theory for preconditioning

SIAM Journal on Matrix Analysis and Applications

Volumn 25, Issue 3, 2004, Pages 694-717

  Boman, Erik G ^a   Hendrickson, Bruce ^a  

^a SANDIA NATIONAL LABORATORIES   (United States)

Author keywords

Condition number; Eigenvalue bounds; Preconditioning; Support theory

Indexed keywords

EID: 3142718480     PISSN: 08954798     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0895479801390637     Document Type: Article

References (32)

1. O. AXELSSON, Bounds of eigenvalues of preconditioned matrices, SIAM J. Matrix Anal., 13 (1992), pp. 847-862.
2. O. AXELSSON, Iterative Solution Methods, Cambridge University Press, Cambridge, UK, 1994.
3. O. AXELSSON AND V. A. BARKER, Finite Element Solution of Boundary Value Problems. Theory and Computation, Academic Press, Orlando, FL, 1984.
4. R. BEAUWENS, Upper eigenvalue bounds for pencils of matrices, Linear Algebra Appl., 62 (1984), pp. 87-104.
5. R. BEAUWENS, Lower eigenvalue bounds for pencils of matrices, Linear Algebra Appl., 85 (1987), pp. 101-119.
6. M. BERN, J. R. GILBERT, B. HENDRICKSON, N. NGUYEN, AND S. TOLEDO, Support-Graph Pre-conditioners, tech. report, School of Computer Science, Tel-Aviv University, 2001; SIAM J. Matrix Anal. Appl., submitted.
7. E. G. BOMAN, D. CHEN, B. HENDRICKSON, AND S. TOLEDO, Maximum-weight-basis preconditioners, Numer. Linear Algebra Appl., to appear.
8. M. Y. CHANG AND M. H. SCHULTZ, Bounds on block diagonal preconditioning, Parallel Algorithms Appl., 1 (1993), pp. 141-164.
9. M. FIEDLER AND T. L. MARKHAM, An observation on the Hadamard product of Hermitian matrices, Linear Algebra Appl., 215 (1995), pp. 179-182.
10. G. H. GOLUB AND C. F. VAN LOAN, Matrix Computations, 2nd ed., The Johns Hopkins University Press, Baltimore, MD, 1989.
11. K. GREMBAN, Combinatorial Preconditioners for Sparse, Symmetric, Diagonally Dominant Linear Systems, Ph.D. thesis, School of Computer Science, Carnegie Mellon University, Pittsburgh, PA, 1996; available as Tech. Report CMU-CS-96-123.
12. K. GREMBAN, G.L. MILLER, AND M. ZAGHA, Performance evaluation of a new parallel pre-conditioner, in Proceedings of the 9th International Parallel Processing Symposium, IEEE Computer Society Press, Los Alamitos, CA, 1995, pp. 65-69; a longer version is available as Tech. Report CMU-CS-94-205, Carnegie Mellon University, Pittsburgh, PA, 1994.
13. S. GUATTERY, Graph Embedding Techniques for Bounding Condition Numbers of Incomplete Factor Preconditioners, Tech. Report 97-47, ICASE, NASA Langley Research Center, Hampton, VA, 1997.
14. S. GUATTERY, F. T. LEIGHTON, AND G. MILLER, The path resistance method for bounding λ2 of a Laplacian, in Proceedings of the 8th Annual ACM-SIAM Symposium on Discrete Algorithms, ACM, New York, 1997, pp. 201-210.
15. R. A. HORN, Personal communication, 2001.
16. V. HOWLE AND S. VAVASIS, An Iterative Method for Solving Complex-Symmetric Systems Arising in Electric Power Modeling, tech. report, Cornell University, Ithaca, NY, 2000; SIAM J. Matrix Anal. Appl., submitted.
17. B. M. IRONS AND G. TREHARNE, A bound theorem in eigenvalues and its practical applications, in Proceedings of the Third Conference on Matrix Methods in Structural Mechanics, Wright-Patterson AFB, Dayton, Ohio, 1971.
18. H.-C. LEE AND A. J. WATHEN, On element-by-element preconditioning for general elliptic problems, Comput. Methods Appl. Mech. Engrg., 92 (1991), pp. 215-229.
19. M. M. MAGOLU, Analytical bounds for block approximate factorizations, Linear Algebra Appl., 179 (1993), pp. 33-57.
20. M. M. MAGOLU AND Y. NOTAY, On the conditioning analysis of block approximate factorization methods, Linear Algebra Appl., 154/156 (1991), pp. 583-599.
21. Y. NOTAY, Conditioning analysis of modified block incomplete factorizations, Linear Algebra Appl., 154/156 (1991), pp. 711-722.
22. Y. NOTAY, Upper eigenvalue bounds and related modified incomplete factorizations, in Iterative Methods in Linear Algebra, North-Holland, Amsterdam, 1992, pp. 551-562.
23. R. REAMS, Hadamard inverses, square roots and products of almost semidefinite matrices, Linear Algebra Appl., 288 (1999), pp. 35-43.
24. J. H. REIF, Efficient approximate solution of sparse linear systems, Comput. Math. Appl., 36 (1998) pp. 37-58; see also errata in Comput. Math. Appl., 38 (1999), p. 141.
25. J. H. REIF, Efficient approximate solution of sparse linear systems, Comput. Math. Appl., 36 (1998) pp. 37-58; see also errata in Comput. Math. Appl., 38 (1999), p. 141.
26. I. SCHUR, Bemerkungen zur Theorie der beschränkten Bilinearformen mit unendlich vielen Veränderlichen, J. Reine Angew. Math., 140 (1911), pp. 1-28.
27. B. SMITH, P. E. BJØRSTAD, AND W. GROPP, Domain Decomposition. Parallel Multilevel Methods for Elliptic Partial Differential Equations, Cambridge University Press, Cambridge, UK, 1996.
28. G. STARKE, Field-of-values analysis of preconditioned iterative methods for nonsymmetric elliptic problems, Numer. Math., 78 (1997), pp. 103-117.
29. P. M. VAIDYA, Solving Linear Equations with Symmetric Diagonally Dominant Matrices by Constructing Good Preconditioners, manuscript.
30. R. S. VARGA, Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1962.
31. A. J. WATHEN, An analysis of some element-by-element preconditioning techniques, Comput. Methods Appl. Mech. Engrg., 74 (1989), pp. 271-287.
32. D. M. YOUNG, Iterative Solution of Large Linear Systems, Academic Press, New York, 1971.