Performance Evaluation of Parallel Gram-Schmidt Re-orthogonalization Methods

  • Takahiro Katagiri
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2565)


In this paper, the performance of the five kinds of parallel reorthogonalization methods by using the Gram-Schmidt (G-S) method is reported. Parallelization of the re-orthogonalization process depends on the implementation of G-S orthogonalization process, i.e. Classical G-S (CG-S) and Modified G-S (MG-S). To relax the parallelization problem, we propose a new hybrid method by using both the CG-S and MG-S. The HITACHI SR8000/MPP of 128 PEs, which is a distributed memory super-computer, is used in this performance evaluation.




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  1. [1]
    S. Balay, W. Gropp, L. C. McInnes, and B. Smith. Petsc 2.0 users manual, 1995. ANL-95/11-Revision 2.0.24, 309
  2. [2]
    C. Bischof and C. van Loan. The wy representation for products of householder matrices. SIAM J. Sci. Stat. Comput., 8(1):s2–s13, 1987. 302CrossRefGoogle Scholar
  3. [3]
    J.W. Demmel. Applied Numerical Linear Algebra. SIAM, 1997. 302Google Scholar
  4. [4]
    J. J. Dongarra, I. S. Du., D.C. Sorensen, and H.A. van der Vorst. Numerical Linear Algebra for High-Performance Computers. SIAM, 1998. 302Google Scholar
  5. [5]
    J. J. Dongarra and R.A. van de Geijn. Reduction to condensed form for the eigenvalue problem on distributed memory architectures. Parallel Computing, 18:973–982, 1992. 302MATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    B.A. Hendrickson and D.E. Womble. The tours-wrap mapping for dense matrix calculation on massively parallel computers. SIAM Sci. Comput., 15(5):1201–1226, 1994. 302MATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    T. Katagiri. A study on parallel implementation of large scale eigenproblem solver for distributed memory architecture parallel machines. Master’s Degree Thesis, the Department of Information Science, the University of Tokyo, 1998. 302Google Scholar
  8. [8]
    T. Katagiri. A study on large scale eigensolvers for distributed memory parallel machines. Ph.D Thesis, the Department of Information Science, the University of Tokyo, 2000. 302, 303, 304Google Scholar
  9. [9]
    T. Katagiri and Y. Kanada. An efficient implementation of parallel eigenvalue computation for massively parallel processing. Parallel Computing, 27:1831–1845, 2001. 309MATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    B.N. Parlett. The Symmetric Eigenvalue Problem. SIAM, 1997. 302, 308, 309Google Scholar
  11. [11]
    G.W. Stewart. Matrix Algorithms Volume II:Eigensystems. SIAM, 2001. 302Google Scholar
  12. [12]
    D. Vanderstraeten. A parallel block gram-schmidt algorithm with controlled loss of orthogonality. Proceedings of the Ninth SIAM Conference on Parallel Processing for Scientific Computing, 1999. 302Google Scholar
  13. [13]
    Y. Yamamoto, M. Igai, and K. Naono. A new algorithm for accurate computation of eigenvectors on shared-memory parallel processors. Proceedings of Joint Symposium on Parallel Processing (JSPP)’2000, pages 19–26, 2000. in Japanese. 302Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Takahiro Katagiri
    • 1
    • 2
  1. 1.PRESTO, Japan Science and Technology Corporation (JST) JST Katagiri Laboratory, Computer Centre Division, Information Technology CenterThe University of TokyoBunkyo-ku, TokyoJAPAN
  2. 2.Department of Information Network ScienceGraduate School of Information Systems, The University of Electro-CommunicationsChoufu-shi, TokyoJAPAN

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