Journal of Engineering Mathematics

, Volume 93, Issue 1, pp 21–39 | Cite as

Partitioning strategies for the block Cimmino algorithm

  • L. A. Drummond
  • Iain S. DuffEmail author
  • Ronan Guivarch
  • Daniel Ruiz
  • Mohamed Zenadi


In the context of the block Cimmino algorithm, we study preprocessing strategies to obtain block partitionings that can be applied to general linear systems of equations \(\mathbf{A}\mathbf{x}= \mathbf{b}\). We study strategies that transform the matrix \(\mathbf{A}\mathbf{A}^\mathrm{{T}}\) into a matrix with a block tridiagonal structure. This provides a partitioning of the linear system for row projection methods because block Cimmino is essentially equivalent to block Jacobi on the normal equations, and the resulting partition will yield a two-block partition of the original matrix. Therefore, the resulting block partitioning should improve the rate of convergence of block row projection methods such as block Cimmino. We discuss a method for obtaining a partitioning using a dropping strategy that gives more blocks at the cost of relaxing the two-block partitioning. We then use a hypergraph partitioning that works directly on the matrix \(\mathbf{A}\) to reduce directly the connections between blocks. We give numerical results showing the performance of these techniques both in their effect on the convergence of the block Cimmino algorithm and in their ability to exploit parallelism.


Cuthill McKee Hypergraph partitioning Iterative methods Sparse matrices Unsymmetric matrices 



This work was partially funded by the ANR-BARESAFE project, ANR-11-MONU-004, Programme Modèles Numériques 2011, supported by the French National Agency for Research. The authors were granted access to the HPC resources of CALMIP under allocation 2013-P0989. We thank the four referees for their detailed comments on the first version of the manuscript. The research of I.S. Duff was supported in part by the EPSRC Grant EP/I013067/1.


  1. 1.
    Amestoy PR, Duff IS, L’Excellent J-Y, Koster J (2001) A fully asynchronous multifrontal solver using distributed dynamic scheduling. SIAM J Matrix Anal Appl 23(1):15–41CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Elfving T (1980) Block-iterative methods for consistent and inconsistent linear equations. Numer Math 35(1):1–12CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Kamath C, Sameh A (1989) A projection method for solving nonsymmetric linear systems on multiprocessors. Parallel Comput 9(3):291–312CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Arioli M, Duff IS, Noailles J, Ruiz D (1992) A block projection method for sparse matrices. SIAM J Sci Stat Comput 13(1):47–70Google Scholar
  5. 5.
    Bramley R (1989) Row projection methods for linear systems. Ph.D. thesis, University of Illinois, Urbana, ILGoogle Scholar
  6. 6.
    Bramley R, Sameh A (1992) Row projection methods for large nonsymmetric linear systems. SIAM J Sci Stat Comput 13(1):168–193CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Ruiz D (1992) Solution of large sparse unsymmetric linear systems with a block iterative method in a multiprocessor environment. Ph.D. thesis, CERFACS, ToulouseGoogle Scholar
  8. 8.
    Duff IS, Guivarch R, Ruiz D, Zenadi M (2013) The augmented block Cimmino distributed method. Technical report TR/PA/13/11, CERFACS, ToulouseGoogle Scholar
  9. 9.
    Golub GH, Kahan W (1965) Calculating the singular values and pseudo-inverse of a matrix. SIAM J Numer Anal 2(2):205–224zbMATHMathSciNetGoogle Scholar
  10. 10.
    Golub GH, Van Loan CF (2013) Matrix computations, 4th edn. The Johns Hopkins University Press, Baltimore and LondonzbMATHGoogle Scholar
  11. 11.
    Björck A, Golub GH (1973) Numerical methods for computing angles between linear subspaces. Math Comput 27:579–594CrossRefGoogle Scholar
  12. 12.
    Arioli M, Drummond A, Duff IS, Ruiz D (1995) A parallel scheduler for block iterative solvers in heterogeneous computing environments. In: Proceedings of the seventh SIAM conference on parallel processing for scientific computing, Philadelphia, USA, pp 460–465Google Scholar
  13. 13.
    Cuthill E, McKee J (1969) Reducing the bandwidth of sparse symmetric matrices. In: Proceedings 24th national conference of the association for computing machinery, New Jersey, pp 157–172Google Scholar
  14. 14.
    Duff IS, Erisman AM, Reid J (1989) Direct methods for sparse matrices. Oxford University Press, New YorkzbMATHGoogle Scholar
  15. 15.
    George A (1971) Computer implementation of the finite-element method. Ph.D. thesis, Department of Computer Science, Stanford University, Stanford, CA. Report STAN CS-71-208Google Scholar
  16. 16.
    Ruiz D (2001) A scaling algorithm to equilibrate both row and column norms in matrices. Technical report RAL-TR-2001-034, Rutherford Appleton LaboratoryGoogle Scholar
  17. 17.
    Lengauer T (1990) Combinatorial algorithms for integrated circuit layout. Wiley, New YorkzbMATHGoogle Scholar
  18. 18.
    Çatalyürek ÜV, Aykanat C (1999) Hypergraph-partitioning-based decomposition for parallel sparse-matrix vector multiplication. IEEE Trans Parallel Distrib Syst 10(7):673–693Google Scholar
  19. 19.
    Çatalyürek ÜV, Aykanat C (1999) PaToH: a multilevel hypergraph partitioning tool, version 3.0. Bilkent University, Department of Computer Engineering, Ankara.
  20. 20.
    Arioli M, Duff IS, Ruiz D, Sadkane M (1995) Block Lanczos techniques for accelerating the block Cimmino method. SIAM J Sci Comput 16(6):1478–1511Google Scholar
  21. 21.
    Arioli M, Duff IS, Ruiz D (1992) Stopping criteria for iterative solvers. SIAM J Matrix Anal Appl 13(1):138–144Google Scholar
  22. 22.
    Oettli W, Prager W (1964) Compatibility of approximate solution of linear equations with given error bounds for coefficients and right-hand sides. Numer Math 6(1):405–409CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Duff IS, Grimes RG, Lewis JG (1992). Users guide for the Harwell–Boeing sparse matrix collection. Technical report RAL-92-086, Rutherford Appleton LaboratoryGoogle Scholar
  24. 24.
    Davis TA (2008) University of Florida sparse matrix collection.

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • L. A. Drummond
    • 1
  • Iain S. Duff
    • 2
    • 3
    Email author
  • Ronan Guivarch
    • 4
  • Daniel Ruiz
    • 4
  • Mohamed Zenadi
    • 4
  1. 1.Computational Research Division, Lawrence Berkeley National LaboratoryBerkeleyUSA
  2. 2.Centre Européen de Recherche et de Formation Avancée en Calcul Scientifique (CERFACS)Toulouse CedexFrance
  3. 3.R18, Rutherford Appleton LaboratoryOxonEngland
  4. 4.ENSEEIHT-IRITToulouse Cedex 7France

Personalised recommendations