On the row merge tree for sparse LU factorization with partial pivoting

BIT Numerical Mathematics

Volumn 47, Issue 1, 2007, Pages 45-76

  Grigori, L ^a   Cosnard, M ^b   Ng, E G ^c  

^a INRIA   (France)

^b INRIA   (France)

^c LAWRENCE BERKELEY NATIONAL LABORATORY   (United States)

Author keywords

Column elimination tree; Partial pivoting; Row merge tree; Sparse LU factorization; Structure prediction

Indexed keywords

EID: 33947267971     PISSN: 00063835     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10543-007-0116-1     Document Type: Article

References (27)

1. H. Avron, G. Shklarski, and S. Toledo, Parallel unsymmetric-pattern multifrontal sparse LU with column preordering, Submitted for publication in ACM Trans. Math. Softw., (2004).
2. R. A. Brualdi and B. A. Shader, Strong Hall matrices, SIAM J. Matrix Anal. Appl., 15 (1994), pp. 359-365.
3. T. F. Coleman, A. Edenbrandt, and J. R. Gilbert, Predicting fill for sparse orthogonal factorization, J. ACM, 33 (1986), pp. 517-532.
4. T. A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method, ACM Trans. Math. Softw., 30 (2004), pp. 165-195.
5. J. W. Demmel, S. C. Eisenstat, J. R. Gilbert, X. S. Li, and J. W. Liu, A supernodal approach to sparse partial pivoting, SIAM J. Matrix Anal. Appl., 20 (1999), pp. 720-755.
6. J. W. Demmel, J. R. Gilbert, and X. S. Li, An asynchronous parallel supernodal algorithm for sparse Gaussian elimination, SIAM J. Matrix Anal. Appl., 20 (1999), pp. 915-952.
7. I. S. Duff, Algorithm 575. Permutations for a zero-free diagonal, ACM Trans. Math. Softw., 7 (1981), pp. 387-390.
8. I. S. Duff, On algorithms for obtaining a maximum transversal, ACM Trans. Math. Softw., 7 (1981), pp. 315-330.
9. I. S. Duff, A. M. Erisman, and J. K. Reid, Direct Methods for Sparse Matrices, Clarendon Press, Oxford, 1986.
10. I. S. Duff and J. Koster, On algorithms for permuting large entries to the diagonal of a sparse matrix, SIAM J. Matrix Anal. Appl., 22 (2001), pp. 973-996.
11. I. S. Duff and J. K. Reid, An implementation of Tarjan's algorithm for the block triangularization of a matrix, ACM Trans. Math. Softw., 4 (1978), pp. 137-147.
12. A. George and E. G. Ng, Symbolic factorization for sparse Gaussian elimination with partial pivoting, SIAM J. Sci. Stat. Comput., 8 (1987), pp. 877-898.
13. A. George and E. G. Ng, Parallel sparse Gaussian elimination with partial pivoting, Ann. Oper. Res., 22 (1990), pp. 219-240.
14. J. R. Gilbert, An efficient parallel sparse partial pivoting algorithm, Tech. Report 88/45052-1, Christian Michelsen Institute, 1988.
15. J. R. Gilbert, X. S. Li, E. G. Ng, and B. W. Peyton, Computing row and column counts for sparse QR and LU factorization, BIT, 41 (2001), pp. 693-711.
16. J. R. Gilbert and E. G. Ng, Predicting Structure in Nonsymmetric Sparse Matrix Factorizations, in Graph Theory and Sparse Matrix Computation, A. George, J. R. Gilbert, and J. W. H. Liu, eds., pp. 107-139, Springer, 1994.
17. L. Grigori, Structure Prediction and Parallel Algorithms for LU Factorization of Sparse Matrices, PhD thesis, University Henri Poincare, France, 2001.
18. L Grigori, J. R. Gilbert, and M. Cosnard, Structure prediction for sparse Gaussian elimination with partial pivoting, Submitted for publication to SIAM J. Matrix Anal. Appl., (2005).
19. J. W. H. Liu, Modification of the minimum degree algorithm by multiple elimination, ACM Trans. Math. Softw., 11 (1985), pp. 141-153.
20. J. W. H. Liu, A compact row storage scheme for Cholesky factors using elimination trees, ACM Trans. Math. Softw., 12 (1986), pp. 127-148.
21. J. W. H. Liu, On general row merging schemes for sparse Givens transformations, SIAM J. Sci. Stat. Comput., 7 (1986), pp. 1190-1211.
22. J. W. H. Liu, The role of elimination trees in sparse factorization, SIAM J. Matrix Anal. Appl., 11 (1990), pp. 134-172.
23. S. Oliveira, Exact prediction of QR fill-in by row-merge trees, SIAM J. Sci. Comput., 22 (2001), pp. 1962-1973.
24. A. Pothen and C.-J. Fan, Computing the block triangular form of a sparse matrix, ACM Trans. Math. Softw., 16 (1990), pp. 303-324.
25. K. Shen, X. Jiao, and T. Yang, Elimination forest guided 2D sparse LU factorization, ACM Symp. on Parallel Architectures and Algorithms (SPAA'98), (1998), pp. 5-15.
26. K. Shen, T. Yang, and X. Jiao, S+: Efficient 2D sparse LU factorization on parallel machines, SIAM J. Matrix Anal. Appl., 22 (2000), pp. 282-305.
27. R. E. Tarjan, Efficiency of a good but not linear set union algorithm, J. ACM, 22 (1975), pp. 215-225.