Toward a high performance tile divide and conquer algorithm for the dense symmetric eigenvalue problem

SIAM Journal on Scientific Computing

Volumn 34, Issue 6, 2012, Pages

  Haidar, Azzam ^a   Ltaief, Hatem ^b   Dongarra, Jack ^a  

^a University of Tennessee   (United States)

^b KING ABDULLAH UNIVERSITY OF SCIENCE AND TECHNOLOGY   (Saudi Arabia)

Author keywords

Divide and conquer; Dynamic scheduling; Symmetric eigenvalue solver; Tile algorithms

Indexed keywords

SOFTWARE ARCHITECTURE;

% REDUCTIONS; DIVIDE-AND-CONQUER; DIVIDE-AND-CONQUER ALGORITHM; DYNAMIC SCHEDULING; EIGENVALUE SOLVERS; PERFORMANCE; SYMMETRIC EIGENVALUE PROBLEMS; SYMMETRIC EIGENVALUE SOLV; SYMMETRICS; TILE ALGORITHM;

EIGENVALUES AND EIGENFUNCTIONS;

EID: 84871558201     PISSN: 10648275     EISSN: 10957200     Source Type: Journal    
DOI: 10.1137/110823699     Document Type: Article

References (45)

1. E. Agullo, B. Hadri, H. Ltaief, and J. Dongarra, Comparative study of one-sided factorizations with multiple software packages on multi-core hardware, in Proceedings of the Conference on High Performance Computing Networking, Storage and Analysis (SC '09), 2009, pp. 1-12
2. 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, 3rd ed., SIAM, Philadelphia, PA, 1999
3. P. Arbenz, Divide and conquer algorithms for the band symmetric eigenvalue problem, Parallel Comput., 18 (1992), pp. 1105-1128
4. P. Arbenz, W. Gander, and G. H. Golub, Restricted rank modification of the symmetric eigenvalue problem: Theoretical considerations, Linear Algebra Appl., 104 (1988), pp. 75- 95
5. 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
6. Y. Bai and R. C. Ward, A parallel symmetric block-tridiagonal divide-and-conquer algorithm, ACM Trans. Math. Softw., 33 (2007), 25
7. C. H. Bischof, B. Lang, and X. Sun, Algorithm 807: The SBR toolbox - software for successive band reduction, ACM Trans. Math. Softw., 26 (2000), pp. 602-616
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. 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
10. A. Buttari, J. Langou, J. Kurzak, and J. J. Dongarra, A class of parallel tiled linear algebra algorithms for multicore architectures, Parallel Comput. Syst. Appl., 35 (2009), pp. 38-53
11. E. Chan, F. G. Van Zee, P. Bientinesi, E. S. Quintana-Orti, G. Quintana-Orti, and R. van de Geijn, Supermatrix: A multithreaded runtime scheduling system for algorithms-byblocks, in Proceedings of the 23rd ACM SIGPLAN Symposium on Principles and Practice of Parallel Programming (PPOPP'2008), Salt Lake City, UT, 2008, pp. 123-132
12. J. J. M. Cuppen, A divide and conquer method for the symmetric eigenproblem, Numer. Math., 36 (1981), pp. 177-195
13. J. W. Demmel, Applied Numerical Linear Algebra. SIAM, Philadelphia, PA, 1997
14. J. W. Demmel, O. A. Marques, B. N. Parlett, and C. Vömel, Performance and Accuracy of LAPACK's Symmetric Tridiagonal Eigensolvers, LAPACK Working Note 183, 2007
15. I. S. Dhillon. A New O(N(2)) Algorithm for the Symmetric Tridiagonal Eigenvalue/Eigenvector Problem, Ph.D. thesis, Technical Report UCB/CSD-97-971, University of California, Berkeley, CA, 1998
16. I. S. Dhillon and B. N. Parlett, Multiple representations to compute orthogonal eigenvectors of symmetric tridiagonal matrices, Linear Algebra Appl., 387 (2004), pp. 1-28
17. I. S. Dhillon, B. N. Parlett, and C. Vömel, The design and implementation of the MRRR algorithm, ACM Trans. Math. Softw., 32 (2006), pp. 533-560
18. J. J. Dongarra and D. C. Sorensen, A fully parallel algorithm for the symmetric eigenvalue problem, SIAM J. Sci. Statist. Comput., 8 (1987), pp. s139-s154
19. W. N. Gansterer, J. Schneid, and C. Ueberhuber, A low-complexity divide-and-conquer method for computing eigenvalues and eigenvectors of symmetric band matrices, BIT, 41 (2001), pp. 967-976
20. 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. Softw., 28 (2002), pp. 45-58
21. W. N. Gansterer, R. C. Ward, R. P. Muller, and W. A. Goddard III, Computing approximate eigenpairs of symmetric block tridiagonal matrices, SIAM J. Sci. Comput., 25 (2003), pp. 65-85
22. K. Gates and P. Arbenz, Parallel Divide and Conquer Algorithms for the Symmetric Tridiagonal Eigenproblem, Technical Report 222, Department of Computer Science, ETH Zurich, 1994
23. G. H. Golub, Some modified matrix eigenvalue problems, SIAM Rev., 15 (1973), pp. 318-334
24. G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed., Johns Hopkins University Press, Baltimore, MD, 1996
25. R. Grimes, H. Krakauer, J. Lewis, H. Simon, and S.-H. Wei, The solution of large dense generalized eigenvalue problems on the Cray X-MP/24 with SSD, J. Comput. Phys., 69 (1987), pp. 471-481
26. 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
27. Intel Math Kernel Library, http://www.intel.com/cd/software/products/ asmo-na/eng/ 307757.htm
28. I. C. F. Ipsen and E. R. Jessup, Solving the symmetric tridiagonal eigenvalues problem on the hypercube, SIAM J. Sci. Statist. Comput., 11 (1990), pp. 203-229
29. J. Kurzak, A. Buttari, and J. J. Dongarra, Solving systems of linear equations on the CELL processor using Cholesky factorization, IEEE Trans. Parallel Distrib. Syst., 19 (2008), pp. 1-11
30. J. Kurzak and J. Dongarra, Fully Dynamic Scheduler for Numerical Scheduling on Multicore Processors, LAPACK Working Note 220, UT-CS-09-643, Innovative Computing Lab, University of Tennessee, Knoxville, 2009
31. J. Kurzak and J. Dongarra, QR factorization for the cell broadband engine, Sci. Programming, 17 (2009), pp. 31-42
32. O. A. Marques, C. Vömel, J. W. Demmel, and B. N. Parlett, Algorithm 880: A testing infrastructure for symmetric tridiagonal eigensolvers, ACM Trans. Math. Softw., 35 (2008), pp. 8:1-8:13
33. R. M. Martin, Electronic Structure: Basic Theory and Practical Methods, Cambridge University Press, Cambridge, UK, 2008
34. PLASMA Users' Guide, Parallel Linear Algebra Software for Multicore Architectures, Version 2.4.5, University of Tennessee Knoxville, 2011
35. E. S. Quintana-Ortï and R. A. van de Geijn, Updating an LU factorization with pivoting, ACM Trans. Math. Softw., 35 (2008), 11
36. G. Quintana-Ortï, E. S. Quintana-Ortï, E. Chan, R. A. van de Geijn, and F. G. Van Zee, Scheduling of QR factorization algorithms on SMP and multi-core architectures, in PDP'08: Proceedings of the 16th Euromicro International Conference on Parallel, Distributed and Network-Based Processing, IEEE Computer Society, 2008, pp. 301-310
37. N. Rösch, S. Krüger, V. Nasluzov, and A. Matveev, ParaGauss: The density functional program ParaGauss for complex systems in chemistry, in High Performance Computing in Science and Engineering, Springer, New York, 2005, pp. 285-296
38. J. Rutter, A Serial Implementation of Cuppen's Divide and Conquer Algorithm for the Symmetric Eigenvalue Problem, LAPACK Working Note 69, UT-CS-94-225, Department of Computer Science, University of Tennessee, Knoxville, 1994
39. SMP Superscalar (SMPSs) User's Manual, Version 2.0, Barcelona Supercomputing Center, http://www.bsc.es/media/1002.pdf, 2008
40. D. C. Sorensen and P. T. P. Tang, On the orthogonality of eigenvectors computed by divideand- conquer techniques, SIAM J. Numer. Anal., 28 (1991), pp. 1752-1775
41. R. C. Thompson, The behavior of eigenvalues and singular values under perturbations of restricted rank, Linear Algebra Appl., 13 (1976), pp. 69-78
42. 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
43. L. N. Trefethen and D. Bau III, Numerical Linear Algebra, SIAM, Philadelphia, PA, 1997
44. F. G. Van Zee, E. Chan, and R. A. van de Geijn, libflame, in Encyclopedia of Parallel Computing, Springer, New York, 2011, pp. 1010-1014
45. J. H. Wilkinson, The Algebraic Eigenvalue Problem, Clarendon Press, Oxford, UK, 1965