Computing row and column counts for sparse QR and LU Factorization

BIT Numerical Mathematics

Volumn 41, Issue 4, 2001, Pages 693-710

  Gilbert, J R ^a   Li, X S ^b   Ng, E G ^b   Peyton, B W ^c  

^a PALO ALTO RESEARCH CENTER   (United States)

^b LAWRENCE BERKELEY NATIONAL LABORATORY   (United States)

^c OAK RIDGE NATIONAL LABORATORY   (United States)

Author keywords

Column elimination tree; Disjoint set union; Row and column counts; Sparse QR and LU factorizations

Indexed keywords

EID: 0042318563     PISSN: 00063835     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1021943902025     Document Type: Article

References (27)

1. C. C. Ashcraft, R. G. Grimes, J. G. Lewis, B. W. Peyton, and H. D. Simon. Progress in sparse matrix methods for large linear systems on vector supercomputers. Internat. J. Supercomputer Appl., 1:4 (1987), pp. 10-30.
2. T. F. Coleman, A. Edenbrandt, and J. R. Gilbert. Predicting fill for sparse orthogonal factorization. J. ACM, 33 (1986), pp. 517-532.
3. J. W. Demmel, S. C. Eisenstat, J. R. Gilbert, X. S. Li, and J. W. H. Liu. A supernodal approach to sparse partial pivoting. SIAM J. Matrix Anal. Appl., 20:3 (1999), pp. 720-755.
4. J. W. Demmel, J. R. Gilbert, and X. S. Li. An asynchronous parallel supernodal algorithm for sparse Gaussian elimination. SIAM J. Matrix Anal. Appl., 20:4 (1999), pp. 915-952.
5. I. S. Duff. Algorithm 575. Permutations for a zero-free diagonal. ACM Trans. Math. Software, 7 (1981), pp. 387-390.
6. I. S. Duff. On algorithms for obtaining a maximum transversal. ACM Trans. Math. Software, 7 (1981), pp. 315-330.
7. I. S. Duff, A. M. Erisman, and J. K. Reid. Direct Methods for Sparse Matrices. Oxford University Press, Oxford, England, 1987.
8. 1/2τ) assignment algorithms. ACM Trans. Math. Software, 14 (1988), pp. 267-287.
9. A. George, J. R. Gilbert, and J. W. H. Liu. Graph Theory and Sparse Matrix Computation. Springer-Verlag, Berlin, 1993.
10. A. George and M. T. Heath. Solution of sparse linear least squares problems using Givens rotations. Linear Algebra Appl., 34 (1980), pp. 69-83.
11. A. George and J. W. H. Liu. Computer Solution of Large Sparse Positive Definite Systems. Prentice-Hall, Englewood Cliffs, NJ, 1981.
12. A. George, J. W. H. Liu, and E. G. Ng. A data structure for sparse QR and LU factors. SIAM J. Sci. Statist. Comput., 9 (1988), pp. 100-121.
13. A. George and E. G. Ng. Symbolic factorization for sparse Gaussian elimination with partial pivoting. SIAM J. Sci. Statist. Comput., 8 (1987), pp. 877-898.
14. J. R. Gilbert, C. Moler, and R. Schreiber. Sparse matrices in Matlab: Design and implementation. SIAM J. Matrix Anal. Appl., 13 (1992), pp. 333-356.
15. J. R. Gilbert and E. G. Ng. Predicting structure in nonsymmetric sparse matrix factorizations. In A. George et al. [9].
16. J. R. Gilbert, E. G. Ng, and B. W. Peyton. An efficient algorithm to compute row and column counts for sparse Cholesky factorization. SIAM J. Matrix Anal. Appl., 15 (1994), pp. 1075-1091.
17. D. Hare, C. Johnson, D. Olesky, and P. van den Driessche. Sparsity analysis of the QR factorization. SIAM J. Matrix Anal. Appl., 14 (1993), pp. 665-669.
18. X. S. Li. Sparse Gaussian elimination on high performance computers, Ph.D. Thesis, Technical Report UCB//CSD-96-919, Computer Science Division, U. C. Berkeley, CA, September 1996.
19. J. W. Liu. A compact row storage scheme for Cholesky factors using elimination trees. ACM Trans. Math. Software, 12 (1986), pp. 127-148.
20. J. W. H. Liu. The role of elimination trees in sparse factorization. Technical Report CS-87-12, Dept. of Computer Science, York University, Canada, 1987. (This is a longer version of Liu [21].)
21. J. W. H. Liu. The role of elimination trees in sparse factorization. SIAM J. Matrix Anal. Appl., 11 (1990), pp. 134-172.
22. J. W. H. Liu, E. G. Ng, and B. W. Peyton. On finding supernodes for sparse matrix computations. SIAM J. Matrix Anal. Appl., 14 (1993), pp. 242-252.
23. E. G. Ng and B. W. Peyton. Block sparse Cholesky algorithms on advanced uniprocessor computers. SIAM J. Sci. Comput., 14 (1993), pp. 1034-1056.
24. E. G. Ng and B. W. Peyton. Some results on structure prediction in sparse QR factorization. SIAM J. Matrix Anal. Appl., 17 (1996), pp. 443-459.
25. A. Pothen and C. J. Fan. Computing the block triangular form of a sparse matrix. ACM Trans. Math. Software, 16 (1990), pp. 303-324.
26. R. Schreiber. A new implementation of sparse Gaussian elimination. ACM Trans. Math. Software, 8 (1982), pp. 256-276.
27. R. E. Tarjan. Efficiency of a good but not linear set union algorithm. J. ACM, 22 (1975), pp. 215-225.