Residual Replacement in Mixed-Precision Iterative Refinement for Sparse Linear Systems

  • Hartwig Anzt
  • Goran Flegar
  • Vedran Novaković
  • Enrique S. Quintana-Ortí
  • Andrés E. TomásEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11203)


We investigate the solution of sparse linear systems via iterative methods based on Krylov subspaces. Concretely, we combine the use of extended precision in the outer iterative refinement with a reduced precision in the inner Conjugate Gradient solver. This method is additionally enhanced with different residual replacement strategies that aim to avoid the pitfalls due to the divergence between the actual residual and the recurrence formula for this parameter computed during the iteration. Our experiments using a significant part of the SuiteSparse Matrix Collection illustrate the potential benefits of this technique from the point of view, for example, of energy and performance.


Sparse linear systems Krylov solvers Iterative refinement Mixed precision Residual replacement Performance and energy modelling 



This research was partially sponsored by the EU H2020 project 732631 OPRECOMP and the CICYT project TIN2017-82972-R of the MINECO and FEDER.


  1. 1.
    Barrachina, S., Castillo, M., Igual, F.D., Mayo, R., Quintana-Ortí, E.S., Quintana-Ortí, G.: Exploiting the capabilities of modern GPUs for dense matrix computations. Conc. Comput. Pract. Experience 21(18), 2457–2477 (2009). Scholar
  2. 2.
    Buttari, A., Dongarra, J.J., Langou, J., Langou, J., Luszcek, P., Kurzak, J.: Mixed precision iterative refinement techniques for the solution of dense linear systems. Int. J. High Perform. Comput. Appl. 21(4), 457–486 (2007). Scholar
  3. 3.
    Davis, T.A., Hu, Y.: The University of Florida sparse matrix collection. ACM Trans. Math. Softw. 38(1), 1–25 (2011). Scholar
  4. 4.
    Golub, G.H., Loan, C.F.V.: Matrix Computations, 3rd edn. The Johns Hopkins University Press, Baltimore (1996)zbMATHGoogle Scholar
  5. 5.
    Saad, Y.: Iterative Methods for Sparse Linear Systems, 3rd edn. Society for Industrial and Applied Mathematics, Philadelphia (2003)CrossRefGoogle Scholar
  6. 6.
    Shalf, J.: The evolution of programming models in response to energy efficiency constraints. Slides Presented at Oklahoma Supercomputing Symposium 2013, October 2013.
  7. 7.
    van der Vorst, H.A., Ye, Q.: Residual replacement strategies for Krylov subspace iterative methods for the convergence of true residuals. SIAM J. Sci. Comput. 22(3), 835–852 (2000). Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Hartwig Anzt
    • 1
  • Goran Flegar
    • 2
  • Vedran Novaković
    • 2
  • Enrique S. Quintana-Ortí
    • 2
  • Andrés E. Tomás
    • 2
    Email author
  1. 1.Karlsruhe Institute of TechnologyKarlsruheGermany
  2. 2.Dpto. de Ingeniería y Ciencia de ComputadoresUniversidad Jaume ICastellónSpain

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