Numerical Algorithms

, Volume 71, Issue 1, pp 41–57 | Cite as

A block algorithm for computing antitriangular factorizations of symmetric matrices

  • Zvonimir BujanovićEmail author
  • Daniel Kressner
Original Paper


Any symmetric matrix can be reduced to antitriangular form in finitely many steps by orthogonal similarity transformations. This form reveals the inertia of the matrix and has found applications in, e.g., model predictive control and constraint preconditioning. Originally proposed by Mastronardi and Van Dooren, the existing algorithm for performing the reduction to antitriangular form is primarily based on Householder reflectors and Givens rotations. The poor memory access pattern of these operations implies that the performance of the algorithm is bound by the memory bandwidth. In this work, we develop a block algorithm that performs all operations almost entirely in terms of level 3 BLAS operations, which feature a more favorable memory access pattern and lead to better performance. These performance gains are confirmed by numerical experiments that cover a wide range of different inertia.


Antitriangular factorization Matrix inertia Block algorithm Symmetric matrix 

Mathematics Subject Classification (2000)

15A18 15A23 15A57 65F15 65Y20 


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  1. 1.
    Auckenthaler, T., Blum, V., Bungartz, H.J., Huckle, T., Johanni, R., Krämer, L., Lang, B., Lederer, H., Willems, P.: Parallel solution of partial symmetric eigenvalue problems from electronic structure calculations. Parallel Comput. 37(12), 783–794 (2011)CrossRefGoogle Scholar
  2. 2.
    Bientinesi, P., Igual, F.D., Kressner, D., Petschow, M., Quintana-Ortí, E.S.: Condensed forms for the symmetric eigenvalue problem on multi-threaded architectures. Concurr. Comput.: Pract. Experience 23(7), 694–707 (2011)CrossRefGoogle Scholar
  3. 3.
    Bischof, C.H., Quintana-Ortí, G.: Computing rank-revealing QR factorizations of dense matrices. ACM Trans. Math. Softw. 24(2), 226–253 (1998)CrossRefzbMATHGoogle Scholar
  4. 4.
    Buttari, A., Langou, J., Kurzak, J., Dongarra, J.: A class of parallel tiled linear algebra algorithms for multicore architectures. Parallel Comput. 35(1), 38–53 (2009)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Davis, T.A.: Algorithm 915, SuiteSparseQR: multifrontal multithreaded rank-revealing sparse QR factorization. ACM Trans. Math. Softw. 38(1) (2011). Art. 8, 22Google Scholar
  6. 6.
    Drmaċ, Z., Bujanović, Z.: On the failure of rank-revealing QR factorization software—a case study. ACM Trans. Math. Softw. 35(2) (2009). Art. 12, 28Google Scholar
  7. 7.
    Golub, G.H., Van Loan, C.F.: Matrix Computations, 3rd. Johns Hopkins University Press, Baltimore (1996)zbMATHGoogle Scholar
  8. 8.
    Haidar, A., Ltaief, H., Dongarra, J.: Parallel reduction to condensed forms for symmetric eigenvalue problems using aggregated fine-grained and memory-aware kernels. In: Proceedings of 2011 International Conference for High Performance Computing, Networking, Storage and Analysis, SC ’11, pp 8:1–8:11. ACM, NY, USA (2011)Google Scholar
  9. 9.
    Haidar, A., Solcà, R., Gates, M., Tomov, S., Schulthess, T., Dongarra, J.: Leading edge hybrid multi-GPU algorithms for generalized eigenproblems in electronic structure calculations. In: Kunkel, J., Ludwig, T., Meuer, H. (eds.) Supercomputing, Lecture Notes in Computer Science, vol. 7905, pp 67–80. Springer Berlin Heidelberg (2013)Google Scholar
  10. 10.
    Mastronardi, N., Van Dooren, P.: Recursive approximation of the dominant eigenspace of an indefinite matrix. J. Comput. Appl. Math. 236(16), 4090–4104 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Mastronardi, N., Van Dooren, P.: The antitriangular factorization of symmetric matrices. SIAM J. Matrix Anal. Appl. 34(1), 173–196 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Mastronardi, N., Van Dooren, P.: On solving KKT linear systems with antitriangular matrices (2013). TUM-IAS Workshop on Novel Numerical Methods, Munich Germany. Presentation available from
  13. 13.
    Mastronardi, N., Van Dooren, P.: An algorithm for solving the indefinite least squares problem with equality constraints. BIT 54(1), 201–218 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Mastronardi, N., Van Dooren, P., Vandebril, R.: On solving KKT linear systems arising in model predictive control via recursive antitriangular factorization (2014). Presentation at Householder Symposium XIX, 8–13, Spa, BelgiumGoogle Scholar
  15. 15.
    Parlett, B.N.: The Symmetric Eigenvalue Problem. Society for Industrial and Applied Mathematics (1998)Google Scholar
  16. 16.
    Pestana, J., Wathen, A.J.: The antitriangular factorization of saddle point matrices. SIAM J. Matrix Anal. Appl. 35(2), 339–353 (2014)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ZagrebZagrebCroatia
  2. 2.École polytechnique fédérale de Lausanne, SB MATHICSE ANCHPLausanneSwitzerland

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