CALU: A communication optimal LU factorization algorithm

SIAM Journal on Matrix Analysis and Applications

Volumn 32, Issue 4, 2011, Pages 1317-1350

  Grigori, Laura ^a   Demmel, James W ^b   Xiang, Hua ^c  

^a INRIA SACLAY   (France)

^b UNIVERSITY OF CALIFORNIA   (United States)

^c WUHAN UNIVERSITY   (China)

Author keywords

Communication optimal algorithm; LU factorization; Numerical stability

Indexed keywords

COMPUTING PLATFORM; EXPONENTIAL GROWTH; GAUSSIAN ELIMINATION; GROWTH FACTOR; LU FACTORIZATION; MATRIX; NUMERICAL STABILITIES; ON CURRENTS; OPTIMAL ALGORITHM; RANDOM MATRICES; SCHUR COMPLEMENT; SPARSE MATRICES; UPPER BOUND;

ALGORITHMS; COMMUNICATION; FACTORIZATION; OPTIMIZATION; PEPTIDES;

MATRIX ALGEBRA;

EID: 84856277432     PISSN: 08954798     EISSN: 10957162     Source Type: Journal    
DOI: 10.1137/100788926     Document Type: Article

References (28)

1. E. ANDERSON, Z. BAI, C. BISCHOF, S. BLACKFORD, J. DEMMEL, J. DONGARRA, J. DU CROZ, A. GREENBAUM, S. HAMMARLING, A. MCKENNEY, AND D. SORENSEN, LAPACK Users' Guide, SIAM, Philadelphia, 1999.
2. G. BALLARD, J. DEMMEL, O. HOLTZ, AND O. SCHWARTZ, Minimizing communication in numerical linear algebra, SIAM J. Matrix Anal. Appl., 32 (2011), pp. 866-901.
3. D. W. BARRON AND H. P. F. SWINNERTON-DYER, Solution of simultaneous linear equations using a magnetic-tape store, Comput. J., 3 (1960), pp. 28-33.
4. A. BUTTARI, J. LANGOU, J. KURZAK, AND J. DONGARRA, A class of parallel tiled linear algebra algorithms for multicore architectures, Parallel Comput., 35 (2009), pp. 38-53.
5. J. CHOI, J. DONGARRA, L. S. OSTROUCHOV, A. P. PETITET, D. W. WALKER, AND R. C. WHALEY, The design and implementation of the ScaLAPACK LU, QR and Cholesky factorization routines, Scientific Programming, 5 (1996), pp. 173-184.
6. J. DEMMEL, L. GRIGORI, M. HOEMMEN, AND J. LANGOU, Communication-Optimal Parallel and Sequential QR and LU Factorizations, Technical report UCB/EECS-2008-89, UC Berkeley, Berkeley, CA, 2008.
7. J. W. DEMMEL, Applied Numerical Linear Algebra, SIAM, Philadelphia, 1997.
8. S. DONFACK, L. GRIGORI, AND A. KUMAR GUPTA, Adapting communication- avoiding LU and QR factorizations to multicore architectures, Proceedings of IPDPS, 2010.
9. J. DONGARRA, P. LUSZCZEK, AND A. PETITET, The LINPACK benchmark: Past, present and future, Concurrency: Practice and Experience, 15 (2003), pp. 803-820.
10. T. ENDO AND K. TAURA, Highly Latency Tolerant Gaussian Elimination, in Proceedings of 6th IEEE/ACM International Workshop on Grid Computing, 2005, pp. 91-98. (Pubitemid 44549207)
11. L. GRIGORI, J. W. DEMMEL, AND H. XIANG, Communication avoiding Gaussian elimination, Proceedings of the ACM/IEEE SC08 Conference, 2008.
12. L. GRIGORI, J. W. DEMMEL, AND H. XIANG, CALU: A Communication Optimal LU Factorization Algorithm, Technical report UCB/EECS-2010-29, EECS Department, University of California, Berkeley, Berkeley, CA, 2010.
13. F. GUSTAVSON, Recursion leads to automatic variable blocking for dense linear-algebra algorithms, IBM J. Res. Development, 41 (1997), pp. 737-755. (Pubitemid 127557046)
14. N. J. HIGHAM, The Matrix Function Toolbox; available online at http://www.ma.man.ac.uk/~higham/mftoolbox.
15. N. J. HIGHAM AND D. J. HIGHAM, Large growth factors in Gaussian elimination with pivoting, SIAM J. Matrix Anal. Appl., 10 (1989), pp. 155-164.
16. N. J. HIGHAM, Accuracy and Stability of Numerical Algorithms, 2nd ed., SIAM, Philadelphia 2002.
17. J.-W. HONG AND H. T. KUNG, I/O complexity: The red-blue pebble game, in STOC '81: Proceedings of the Thirteenth Annual ACM Symposium on Theory of Computing, New York, 1981, ACM, pp. 326-333.
18. D. IRONY, S. TOLEDO, AND A. TISKIN, Communication lower bounds for distributed-memory matrix multiplication, J. Parallel Distrib. Comput., 64 (2004), pp. 1017-1026. (Pubitemid 40000755)
19. T. JOFFRAIN, E. S. QUINTANA-ORTÍ, AND R. A. VAN DE GEIJN, Rapid development of highperformance out-of-core solvers, in Proceedings of PARA 2004, Lecture Notes in Comput. Sci. 3732, Springer-Verlag, Berlin, Heidelberg, 2005, pp. 413-422.
20. E. QUINTANA-ORTÍ AND R. A. VAN DE GEIJN, Updating an LU factorization with pivoting, ACM Trans. Math. Software, 35 (2008), pp. 11:1-11:16.
21. G. QUINTANA-ORTI, E. S. QUINTANA-ORTI, E. CHAN, F. G. VAN ZEE, AND R. VAN DE GEIJN, Programming Algorithms-By-Blocks for Matrix Computations on Multithreaded Architectures, Technical report TR-08-04, University of Texas at Austin, Austin, TX, 2008.
22. R. D. SKEEL, Iterative refinement implies numerical stability for Gaussian elimination, Math. Comp., 35 (1980), pp. 817-832.
23. D. C. SORENSEN, Analysis of pairwise pivoting in Gaussian elimination, IEEE Trans. Comput., 34 (1985), pp. 274-278.
24. S. TOLEDO, Locality of reference in LU Decomposition with partial pivoting, SIAM J. Matrix Anal. Appl., 18 (1997), pp. 1065-1081.
25. S. TOMOV, J. DONGARRA, AND M. BABOULIN, Towards dense linear algebra for hybrid gpu accelerated manycore systems, Parallel Computing, 36 (2010), pp. 232-240.
26. L. N. TREFETHEN AND R. S. SCHREIBER, Average-case stability of Gaussian elimination, SIAM J. Matrix Anal. Appl., 11 (1990), pp. 335-360.
27. V. VOLKOV, Private communication.
28. E. L. YIP, Subroutines for Out of Core Solutions of Large Complex Linear Systems, Technical report NASA-CR-159142, NASA, Washington, DC, 1979.