Advertisement

Solving the Symmetric Tridiagonal Eigenproblem Using MPI/OpenMP Hybrid Parallelization

  • Yonghua Zhao
  • Jiang Chen
  • Xuebin Chi
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3756)

Abstract

We present a hybrid MPI/OpenMP parallel implementation for the eigenvalues of symmetric tridiagonal matrices on cluster of SMP’s environments. The algorithm is based on a divide-and-conquer method which uses the split-merge technique and Laguerre’s iteration. We study two different implementations of the algorithm: one based on MPI and the other based on a hybrid parallel paradigm with MPI/OpenMP. We take a coarse grain OpenMP approach to parallel implementation for solving the eigenvalues of symmetric tridiagonal submatrices within a SMP node. And dynamic work sharing is used in Laguerre’s iterations. This has two effects: first, the amount of synchronization has been reduced; secondly, this could have an effect on the load balance. In addition, we analyze the communication overhead on two different implementations. An experimental analysis on the DeepComp 6800 shows the hybrid algorithm performs good scalability.

Keywords

Tridiagonal Matrix Hybrid Program Tridiagonal Matrice Parallel Speedup Master Processor 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Sameh, A., Kuck, D.: A parallel QR algorithm for symmetric tridiagonal matrices. IEEE tans. Comput. C-26, 81–91 (1977)Google Scholar
  2. 2.
    Arbenz, P., Gates, D., Sprenger, C.: A parallel implementation of the symmetric tridiagonal QR algorithm. In: Proc. Fourth Symp. On the Frontiers of Massively Parallel Computation. IEEE CS Press, Los Alamitos (1992)Google Scholar
  3. 3.
    Cuppen, J.J.M.: A divide and conquer method for symmetric tridiagonal eigenproblem. Numer. Mathematik 2(36), 177–195 (1981)MathSciNetGoogle Scholar
  4. 4.
    Wilkinson, J.H.: The Algebraic Eigenvalue Problem. Clarendon Press, Oxford (1965)Google Scholar
  5. 5.
    Li, T.Y., Zhang, H., Sun, X.H.: Parallel homotopy algorithm for the symmetric tridiagonal eigenvalue problem. SIAM J. Scientific and Statistical Comput. 12, 469–487 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Dongarra, J.J., Soorenson, D.C.: A Fully Parallel Algorithm for the symmetric Eigenvalue Problem. SIAM J.Sci.Stat. Comput. 8(2), s139–s154 (1987)CrossRefGoogle Scholar
  7. 7.
    Dhillon, I.S., Fannm, G., Parlett, B.N.: Application of new algorithm for the symmetric eigenproblem to computational quantum chemistry. In: processings of the Eigen SIAM Conference on Parallel processing for Scientific Computing, Minneapolis, MN, March 1997. SIAM, Philadelphia (1997)Google Scholar
  8. 8.
    Dhillon, I.S., Parlett, B.N.: Multiple representations compute orthogonal eigenvertors of symmetric tridiagonal matrices. Lin. Alg. Appl. 387, 1–28 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Luong, P., Breshears, C.P., Ly, L.N.: Coastal ocean modeling of the U.S. west coast with multiblock grid and dual-level parallelism. In: Supercomputing 2001: High Performance Networking and Computing, SC 2001 (2001)Google Scholar
  10. 10.
    Pavani, R., De Ros, U.: Solving the tridiagonal symmetric eigenvalue problem on a transputer network. n. 146/p, Dipartimento di mathematica, Politecino di Milano (1994)Google Scholar
  11. 11.
    LI, T.Y., Zeng, Z.: Lagurre’s iteration in solving the symmetric tridiagonal eigenproblem. SIAM J. Scientific Comput. 15, 1145–1173 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Pavani, R., De Ros, U.: A Distributed divide-and-conquer approach to the parallel symmetric eigenvalue problem. In: Hertzberger, B., Serazzi, G. (eds.) HPCN-Europe 1995. LNCS, vol. 919. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  13. 13.
    Bova, S.W., Breshears, C., Cuicchi, C., Demirbilek, Z., Gabb, H.A.: Dual-level parallel analysis of harbor wave response using MPI and OpenMP. Int. J. High Perform Comput. Appl. 14, 49–64 (2000)CrossRefGoogle Scholar
  14. 14.
    Trefftz, C., Huang, C.C., Li, T.Y., Zeng, Z.: A scalable eigenvalue solver for symmetric tridiagonal matrices. Parallel Computing 21, 1213–1240 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Cappello, F., Etiemble, D.: MPI versus MPI+OpenMP on the IBM SP for the NAS Benchmarks. In:Supercomputing 2000: High Performance Networking and Computing (SC 2000) (2000)Google Scholar
  16. 16.
    Loft, R.D., Thomas, S.J., Dennis, J.M.: Terascale spectral element dynamical core for atmospheric general circulation models. In: Supercomputing 2001: High Performance Networking and Computing, SC2001 (2001)Google Scholar
  17. 17.
    Crawford, C.H., Evangelinos, C., Newman, D., Karniadakis, G.E.: Parallel benchmarks of turbulence in complex geometries. Comput. Fluids 25, 677–698 (1996)zbMATHCrossRefGoogle Scholar
  18. 18.
    Henty, D.S.: Performance of hybrid message-passing and shared-memory parallelism for discrete element modeling. Supercomputing 2000: High Performance Networking and Computing, SC2000 (2000)Google Scholar
  19. 19.
    Dong, S.H., Em Karniadakis, G.: Dual-level parallelism for high-order CFD methods. Parallel Computing 30, 1–20 (2004)CrossRefGoogle Scholar
  20. 20.
    Liu, W., Zheng, W.M., Zheng, X.W.: The concept of node-oriented speedup on SMP cluster. Computer engineering and design 21(5) (October 2000)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Yonghua Zhao
    • 1
    • 2
    • 3
  • Jiang Chen
    • 1
  • Xuebin Chi
    • 1
  1. 1.Supercomputing Center, Computer Network Information CenterChinese Academy of SciencesBeijingChina
  2. 2.Graduate SchoolChinese Academy of SciencesBeijingChina
  3. 3.Department of Computer ScienceDezhou UniversityShandongChina

Personalised recommendations