Abstract
Many scientific applications require the computation of about 10–30 % of the eigenvalues and eigenvectors of large dense symmetric or complex hermitian matrices. In this paper we will present performance evaluation results of the eigensolvers of the three libraries Elemental, ELPA, and ScaLAPACK on the BlueGene/Q architecture. All libraries include solvers for the computation of only a part of the spectrum. The most time-consuming part of the eigensolver is the reduction of the full eigenproblem to a tridiagonal one. Whereas Elemental and ScaLAPACK only offer routines to directly reduce the full matrix to a tridiagonal one, which only allows the use of BLAS 2 matrix-vector operations and needs a lot of communication, ELPA also offers a two-step reduction routine, first transforming the full matrix to banded form and thereafter to tridiagonal form. This two-step reduction shortens the reduction time significantly but at the cost of a higher complexity of the back transformation step. We will show up to which part of the eigenspectrum the use of the two-step reduction pays off.
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References
FLEUR: The Jülich FLAPW code family. Website (May 2013). http://www.flapw.de
Choi, J., Demmel, J., Dhillon, I., Dongarra, J., Ostrouchov, S., Petitet, A., Stanley, K., Walker, D., Whaley, R.: Scalapack: a portable linear algebra library for distributed memory computers-design issues and performance. Comput. Phys. Commun. 97(1–2), 1–15 (1996)
Poulson, J., Marker, B., van de Geijn, R.A., Hammond, J.R., Romero, N.A.: Elemental: a new framework for distributed memory dense matrix computations. ACM Trans. Math. Softw. 39(2), 13:1–13:24 (2013)
ELPA: Eigenvalue Solvers for Petaflop Applications home page. Website (May 2013). http://elpa.rzg.mpg.de
FZJ-JSC: IBM Blue Gene/Q - JUQUEEN home page. Website (May 2013). http://www.fz-juelich.de/ias/jsc/juqueen
Gutheil, I.: Performance evaluation of scalapack eigensolver routines on two hpc systems. In: 6th International Workshop on Parallel Matrix Algorithms and Applications (PMAA’10) (2010). http://juser.fz-juelich.de/record/10376
Dhillon, I., Parlett, B., Vömel, C.: The design and implementation of the MRRR algorithm. ACM Trans. Math. Softw. (TOMS) 32(4), 533–560 (2006)
Dhillon, I.: A new O(\(n^2\)) algorithm for the symmetric tridiagonal eigenvalue eigenvector problem. Ph.D. thesis, University of California, Berkeley (1997)
Auckenthaler, T., Blum, V., Bungartz, H.J., Huckle, T., Johanni, R., Krämer, L., Lang, B., Lederer, H., Willems, P.: Parallel solution of partial symmetric eigenvalue problems from electronic structure calculations. Parallel Comput. 37(12), 783–794 (2011)
Auckenthaler, T., Bungartz, H.J., Huckle, T., Krämer, L., Lang, B., Willems, P.: Developing algorithms and software for the parallel solution of the symmetric eigenvalue problem. J. Comput. Sci. 2(3), 272–278 (2011)
Gutheil, I., Berg, T., Grotendorst, J.: Performance analysis of parallel eigensolvers of two libraries on BlueGene/p. J. Math. Syst. Sci. 2(4), 231–236 (2012)
Petschow, M., Peise, E., Bientinesi, P.: High-performance solvers for dense hermitian eigenproblems. SIAM J. Sci. Comput. (SISC) 35(1), C1–C22 (2013). arXiv:1205.2107v2[cs.MS]
Münchhalfen, J.: Performance analysis and comparison of parallel eigensolvers on blue gene architectures. Berichte des Forschungszentrums Jülich (4359) 65 p. (2013). http://juser.fz-juelich.de/record/128657
Acknowledgements
The authors thank Jack Poulson, the author of the Elemental library and the ELPA team, especially Thomas Auckenthaler, for their immediate responses to problem reports.
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Gutheil, I., Münchhalfen, J.F., Grotendorst, J. (2014). Performance of Dense Eigensolvers on BlueGene/Q. In: Wyrzykowski, R., Dongarra, J., Karczewski, K., Waśniewski, J. (eds) Parallel Processing and Applied Mathematics. PPAM 2013. Lecture Notes in Computer Science(), vol 8384. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-55224-3_3
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DOI: https://doi.org/10.1007/978-3-642-55224-3_3
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