Mixed Precision Bisection



We discuss the implementation of the bisection algorithm for the computation of the eigenvalues of symmetric tridiagonal matrices in a context of mixed precision arithmetic. This approach is motivated by the emergence of processors which carry out floating-point operations much faster in single precision than they do in double precision. Perturbation theory results are used to decide when to switch from single to double precision. Numerical examples are presented.


Eigenvalues Bisection algorithm Mixed precision 

Mathematics Subject Classification



Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Anderson, E., et al.: LAPACK Users’ Guide. SIAM, Philadelphia (1999)CrossRefMATHGoogle Scholar
  2. 2.
    Blackford, L., et al.: ScaLAPACK Users’ Guide. SIAM, Philadelphia (1997)CrossRefMATHGoogle Scholar
  3. 3.
    Buttari, A., et al.: Mixed precision iterative refinement techniques for the solution of dense linear systems. Int. J. High Perform. Comput. Appl. 21, 457–466 (2007)CrossRefGoogle Scholar
  4. 4.
    Buttari, A., et al.: Using mixed precision for sparse matrix computations to enhance the performance while achieving 64-bit accuracy. ACM Trans. Math. Softw. TOMS 34(4), 17:1–17:22 (2008)MathSciNetMATHGoogle Scholar
  5. 5.
    Demmel, J.W.: The inherent inaccuracy of implicit tridiagonal QR. LAPACK working note 15 (1992)Google Scholar
  6. 6.
    Demmel, J.W., Li, X.: Faster numerical algorithms via exception handling. IEEE Trans. Comput. 43, 983–992 (1992)CrossRefMATHGoogle Scholar
  7. 7.
    Demmel, J.W., Dhillon, I., Ren, H.: On the correctness of some bisection-like parallel eigenvalue algorithms in floating point arithmetic. Electr. Trans. Numer. Anal. 3, 116–149 (1995)MathSciNetMATHGoogle Scholar
  8. 8.
    Demmel, J.W.: Applied Numerical Linear Algebra. SIAM, Philadelphia (1997)CrossRefMATHGoogle Scholar
  9. 9.
    Giraud, l, Haidar, A., Watson, L.: Mixed-precision preconditioners in parallel domain decomposition solvers. Lect. Notes Comput. Sci. Eng 60, 357–364 (2008)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Golub, G., Van Loan, C.: Matrix Computations, 2nd edn. The Johns Hopkins University Press, Baltimore (1989)MATHGoogle Scholar
  11. 11.
    Higham, N.: The rise of mixed precision arithmetic. https://nickhigham.wordpress.com/2015/10/20/the-rise-of-mixed-precision-arithmetic/. Accessed Nov 2015
  12. 12.
    Kurzak, J., Dongarra, J.: Implementation of mixed precision in solving systems of linear equations on the CELL processor. Concurr. Comput. Pract. Exp. 19, 1371–1385 (2007)CrossRefGoogle Scholar
  13. 13.
    Langou, J., et al.: Exploiting the performance of 32 bit floating point arithmetic in obtaining 64 bit accuracy (revisiting iterative refinement for linear systems. In: Proceedings of the ACM/IEEE 2006 Conference. IEEE Computer Society Press (2006)Google Scholar
  14. 14.
    Parlett, B.N.: The Symmetric Eigenvalue Problem. Prentice-Hall, Englewood Cliffs (1998)CrossRefMATHGoogle Scholar
  15. 15.
    Petschow, M., Quintana-Orti, E.S., Bientinesi, P.: Improved accuracy and parallelism for MRRR-based eigensolvers—a mixed precision approach. SIAM J. Sci. Comput. 36(2), C240–C263 (2014)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Ralha, R.: Perturbation splitting for more accurate eigenvalues. SIAM J. Matrix Anal. Appl. 31, 75–91 (2009)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Volkov, V., Demmel, J.W.: Using GPUs to accelerate the bisection algorithm for finding eigenvalues of symmetric tridiagonal matrices. LAPACK working note 197 (2008)Google Scholar
  18. 18.
    Wilkinson, J.H.: The Algebraic Eigenvalue Problem. Oxford University Press, London (1965)MATHGoogle Scholar
  19. 19.
    Yamazaki, I., Tomov, S., Dongarra, J.: Mixed-precision Choleski QR factorization and its case studies on multicore CPU with multiple GPUs. SIAM J. Sci. Comput. 37(3), C307–C330 (2015)CrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Center of Mathematics, School of SciencesUniversity of MinhoBragaPortugal

Personalised recommendations