The danger of combining block red–black ordering with modified incomplete factorizations and its remedy by perturbation or relaxation

  • Akemi ShioyaEmail author
  • Yusaku Yamamoto
Original Paper Area 2


Modified incomplete LU/Cholesky factorizations without fill-ins are popular preconditioners for Krylov subspace methods, because they require no extra memory and have more potential of accelerating the convergence than simple ILU/IC preconditioners. For parallelizing preconditioners, the block red–black ordering is attractive due to its highly parallel nature and small number of synchronization points. Hence, their combination seems to produce powerful and parallelizable preconditioners. In fact, however, this combination can cause breakdown of the factorization due to the occurrence of zero pivots. We analyze this phenomenon and give necessary and sufficient conditions of zero pivots in the case of a regular grid. We also show both theoretically and experimentally that adding perturbation to the diagonal elements or relaxing the compensation of dropped fill-ins is useful to alleviate the problem. Numerical tests show that the resulting preconditioners are highly effective and are applicable for up to \(10^3\) level of parallelism.


Linear system Modified incomplete factorizations Block red–black ordering Parallel processing 

Mathematics Subject Classification




We are grateful to the anonymous reviewer, whose comments helped us to improve the quality of this paper. We thank Prof. Takeshi Iwashita of Hokkaido University for valuable comments on our work. The present study is supported in part by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Scientific Research (nos. 26286087, 15H02708, 15H02709, 16KT0016, 17H02828, 17K19966). The computational experiments in this paper were performed using the FX10 parallel computer at the Education Center on Computational Science and Engineering of Kobe University.


  1. 1.
    Birdsall, C.K.: Particle-in-cell charged-particle simulations, plus Monte Carlo collisions with neutral atoms, PIC-MCC. IEEE Trans. Plasma Sci. 19(2), 65–85 (1991)CrossRefGoogle Scholar
  2. 2.
    Hirsch, C.: Numerical Computation of Internal and External Flows: The Fundamentals of Computational Fluid Dynamics. Butterworth-Heinemann, Oxford (2007)Google Scholar
  3. 3.
    Meijerink, J.A., van der Vorst, H.A.: An iterative solution method for linear systems of which the coefficient matrix is a symmetric \(M\)-matrix. Math. Comput. 31, 148–162 (1977)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Gustafsson, I.: A class of first order factorization methods. BIT 18, 142–156 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Greenbaum, A.: Iterative Methods for Solving Linear Systems. SIAM, Philadelphia (1997)CrossRefzbMATHGoogle Scholar
  6. 6.
    Fujino, S., Mori, M., Takeuchi, T.: Performance of hyperplane ordering on vector computers. J. Comput. Appl. Math. 38, 125–136 (1991)CrossRefzbMATHGoogle Scholar
  7. 7.
    George, A., Liu, J.W.H.: Computer Solution of Large Sparse Positive Definite Systems. Prentice-Hall, Englewood Cliffs (1981)zbMATHGoogle Scholar
  8. 8.
    Davis, T.A.: Direct Methods for Sparse Linear Systems. SIAM, Philadelphia (2006)CrossRefzbMATHGoogle Scholar
  9. 9.
    Saad, Y.: Iterative Methods for Sparse Linear Systems, 2nd edn. SIAM, Philadelphia (2003)CrossRefzbMATHGoogle Scholar
  10. 10.
    Iwashita, T., Shimasaki, M.: Block red-black ordering for parallelized ICCG solver with fewer synchronization points. IPSJ J. 43, 893–904 (2002–2004)Google Scholar
  11. 11.
    Iwashita, T., Shimasaki, M.: Algebraic block red-black ordering method for parallelized ICCG solver with fast convergence and low communication costs. IEEE Trans. Magn. 39, 1713–1716 (2003)CrossRefGoogle Scholar
  12. 12.
    Semba, K., Tani, K., Yamada, T., Iwashita, T., Takahashi, Y., Nakashima, H.: Parallel performance of multi-threaded ICCG solver based on algebraic block multi-color ordering in finite element electromagnetic field analyses. IEEE Trans. Magn. 49, 1581–1584 (2013)CrossRefGoogle Scholar
  13. 13.
    Iwashita, T., Nakashima, H., Takahashi, Y.: Algebraic block multi-color ordering method for parallel multi-threaded sparse triangular solver in ICCG method. In: 2012 IEEE 26th International Parallel and Distributed Processing Symposium (IPDPS), pp. 474–483. IEEE (2012)Google Scholar
  14. 14.
    Guessous, N., Souhar, O.: The effect of block red-black ordering on block ILU preconditioner for sparse matrices. J. Appl. Math. Comput. 17, 283–296 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Eijkhout, V.: Beware of unperturbed modified incomplete factorizations. In: Belgium, B., Beauwens, R., de Groen, P. (eds.) Proc. of the IMACS International Symposium on Iterative Methods in Linear Algebra (1992)Google Scholar
  16. 16.
    Gustafsson, I.: Modified incomplete Cholesky (MIC) methods. In: Evans, D. (ed.) Preconditioning Methods Theory and Applications, pp. 265–293. Gordon and Breach, New York (1983)Google Scholar
  17. 17.
    Axelsson, O., Lindskog, G.: On the eigenvalue distribution of a class of preconditioning method. Numer. Math. 48, 479–498 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Eijkhout, V.: Analysis of parallel incomplete point factorizations. Linear Algebra Appl. 154, 723–740 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Van der Vorst, H.A.: BI-CGSTAB: a fast and smoothly converging variant of BI-CG for the solution of nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 13(2), 631–644 (1992)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The JJIAM Publishing Committee and Springer Japan KK 2017

Authors and Affiliations

  1. 1.The University of Electro-CommunicationsChofuJapan

Personalised recommendations