Look-ahead in the two-sided reduction to compact band forms for symmetric eigenvalue problems and the SVD
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We address the reduction to compact band forms, via unitary similarity transformations, for the solution of symmetric eigenvalue problems and the computation of the singular value decomposition (SVD). Concretely, in the first case, we revisit the reduction to symmetric band form, while, for the second case, we propose a similar alternative, which transforms the original matrix to (unsymmetric) band form, replacing the conventional reduction method that produces a triangular–band output. In both cases, we describe algorithmic variants of the standard Level 3 Basic Linear Algebra Subroutines (BLAS)-based procedures, enhanced with look-ahead, to overcome the performance bottleneck imposed by the panel factorization. Furthermore, our solutions employ an algorithmic block size that differs from the target bandwidth, illustrating the important performance benefits of this decision. Finally, we show that our alternative compact band form for the SVD is key to introduce an effective look-ahead strategy into the corresponding reduction procedure.
KeywordsTwo-sided reduction to compact band form Look-ahead Symmetric eigenvalue problems Singular value decomposition
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This research was partially sponsored by projects TIN2014-53495-R and TIN2015-65316-P of the Spanish Ministerio de Economía y Competitividad, project 2014-SGR-1051 from the Generalitat de Catalunya, and the EU H2020 project 732631 OPRECOMP.
- 1.Aliaga, J.I., Alonso, P., Badía, J.M., Chacn, P., Davidović, D., López-Blanco, J.R., Quintana-Ortí, E.S.: A fast band–krylov eigensolver for macromolecular functional motion simulation on multicore architectures and graphics processors. J. Comput. Phys. 309(Supplement C), 314–323 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
- 2.Anderson, E., Bai, Z., Blackford, L.S., Demmel, J., Dongarra, J.J., Du Croz, J., Hammarling, S., Greenbaum, A., McKenney, A., Sorensen, D.C.: LAPACK Users’ Guide, 3rd edn. SIAM (1999)Google Scholar
- 8.Catalȧn, S., Herrero, J.R., Quintana-ortí, E.S., Rodríguez-Sȧnchez, R., van de Geijn, R.A.: A case for malleable thread-level linear algebra libraries: The LU factorization with partial pivoting. CoRR, arXiv:1611.06365 (2016)
- 9.Davidović, D., Quintana-Ortí, E.S.: Applying OOC techniques in the reduction to condensed form for very large symmetric eigenproblems on GPUs. In: Proceedings of the 20th Euromicro Conference on Parallel, Distributed and Network Based Processing – PDP 2012, pp. 442–449 (2012)Google Scholar
- 10.Davis, T.A., Rajamanickam, S.: Algorithm 8xx: PIRO BAND, pipelined plane rotations for band reduction. ACM Trans. Math. Soft. SubmittedGoogle Scholar
- 15.Golub, G.H., Van Loan, C.F.: Matrix Computations, 3rd edn. The Johns Hopkins University Press, Baltimore (1996)Google Scholar
- 18.Haidar, A., Ltaief, H., Dongarra, J.: Parallel reduction to condensed forms for symmetric eigenvalue problems using aggregated fine-grained and memory-aware kernels. In: 2011 International Conference for High Performance Computing, Networking, Storage and Analysis (SC), pp. 1–11 (2011)Google Scholar
- 19.Haidar, A., Kurzak, J., Luszczek, P.: An improved parallel singular value algorithm and its implementation for multicore hardware. In: Proceedings of the International Conference on High Performance Computing, Networking, Storage and Analysis, SC ’13, pp. 90:1–90:12. ACM, New York (2013)Google Scholar
- 23.Strazdins, P.: A comparison of lookahead and algorithmic blocking techniques for parallel matrix factorization. Technical Report TR-CS-98-07, Department of Computer Science, The Australian National University Canberra 0200 ACT, Australia (1998)Google Scholar