Over 10 TFLOPS Computation for a Huge Sparse Eigensolver on the Earth Simulator

  • Toshiyuki Imamura
  • Susumu Yamada
  • Masahiko Machida
Conference paper


To investigate a possibility of special physical properties like superfluidity, we implement a high performance exact diagonalization code for the trapped Hubbard model on the Earth Simulator. From the numerical and computational point of view, it is found that the performance of the preconditioned conjugate gradient (PCG) method is excellent in our case. It is 1.5 times faster than the conventional Lanczos one since it can conceal the communication overhead much more effectively. Consequently, the PCG method shows 16.14 TFLOPS on 512 nodes. Furthermore, we succeed in solving a 120-billion-dimensional matrix. To our knowledge, this dimension is a world-record.


Hubbard Model Communication Overhead Preconditioned Conjugate Gradient Hamiltonian Matrix Earth Simulator 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Machida M., Yamada S., Ohashi Y., Matsumoto H.: Novel Superfluidity in a Trapped Gas of Fermi Atoms with Repulsive Interaction Loaded on an Optical Lattice. Phys. Rev. Lett., 93 (2004) 200402CrossRefGoogle Scholar
  2. 2.
    Rasetti M. (ed.): The Hubbard Model: Recent Results. Series on Advances in Statistical Mechanics, Vol. 7., World Scientific, Singapore (1991)Google Scholar
  3. 3.
    Montorsi A. (ed.): The Hubbard Model: A Collection of Reprints. World Scientific, Singapore (1992)Google Scholar
  4. 4.
    Rigol M., Muramatsu A., Batrouni G.G., Scalettar R.T.: Local Quantum Criticality in Confined Fermions on Optical Lattices. Phys. Rev. Lett., 91 (2003) 130403CrossRefGoogle Scholar
  5. 5.
    Dagotto E.: Correlated Electrons in High-temperature Superconductors. Rev.Mod. Phys., 66 (1994) 763CrossRefGoogle Scholar
  6. 6.
    The Earth Simulator Center. Scholar
  7. 7.
    TOP500 Supercomputer Sites. Scholar
  8. 8.
    Shingu S. et al.: A 26.58 Tflops Global Atmospheric Simulation with the Spectral Transform Method on the Earth Simulator. Proc. of SC2002, IEEE/ACM (2002)Google Scholar
  9. 9.
    Yamada S., Imamura T., Machida M.: 10TFLOPS Eigenvalue Solver for Strongly-Correlated Fermions on the Earth Simulator. Proc. of PDCN2005, IASTED (2005)Google Scholar
  10. 10.
    Cullum J.K., Willoughby R.A.: Lanczos Algorithms for Large Symmetric Eigenvalue Computations, Vol. 1. SIAM, Philadelphia PA (2002)zbMATHGoogle Scholar
  11. 11.
    Knyazev A.V.: Preconditioned Eigensolvers — An Oxymoron? Electr. Trans. on Numer. Anal., Vol. 7 (1998) 104–123zbMATHMathSciNetGoogle Scholar
  12. 12.
    Uehara H., Tamura M., Yokokawa M.: MPI Performance Measurement on the Earth Simulator. NEC Research & Development, Vol. 44, No. 1 (2003) 75–79Google Scholar
  13. 13.
    Vorst H.A., Dekker K.: Vectorization of Linear Recurrence Relations. SIAM J. Sci. Stat. Comput., Vol. 10, No. 1 (1989) 27–35zbMATHCrossRefGoogle Scholar
  14. 14.
    Imamura T.: A Group of Retry-type Algorithms on a Vector Computer. IPSJ, Trans., Vol. 46, SIG 7 (2005) 52–62 (written in Japanese)Google Scholar
  15. 15.
    NEC Corporation, FORTRAN90/ES Programmerfs Guide, Earth imulator Userfs Manuals. NEC Corporation (2002)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Toshiyuki Imamura
    • 1
  • Susumu Yamada
    • 2
  • Masahiko Machida
    • 2
    • 3
  1. 1.Department of Computer Sciencethe University of Electro-CommunicationsChofu-shi, TokyoJapan
  2. 2.Center for Computational Science and EngineeringJapan Atomic Energy AgencyTokyoJapan
  3. 3.CREST, JSTKawaguchi-shi, SaitamaJapan

Personalised recommendations