Influence of the Condition Number on Interval Computations: Illustration on Some Examples

  • Nathalie RevolEmail author
Part of the Studies in Computational Intelligence book series (SCI, volume 835)


The condition number is a quantity that is well-known in “classical” numerical analysis, that is, where numerical computations are performed using floating-point numbers. This quantity appears much less frequently in interval numerical analysis, that is, where the computations are performed on intervals. The goal of this paper is twofold. On the one hand, it is stressed that the notion of condition number already appears in the literature on interval analysis, even if it does not bear that name. On the other hand, three small examples are used to illustrate experimentally the impact of the condition number on interval computations. As expected, problems with a larger condition number are more difficult to solve: this means either that the solution is not very accurate (for moderate condition numbers) or that the method fails to solve the problem, even inaccurately (for larger condition numbers). Different strategies to counteract the impact of the condition number are discussed and experimented: use of a higher precision, iterative refinement, bisection of the input. More strategies are discussed as a conclusion.



This work has been partially supported by the ANR project MetaLibm ANR-13-INSE-0007-04.

The author thanks H. D. Nguyen for his kind permission to reproduce the material of Sect. 4. She also thanks Olga Kosheleva for her kind invitation. Last but not least, she wishes to thank Vladik Kreinovich for his kindness and friendship during many years, and for his vitality that is contagious and beneficial to the community.


  1. 1.
    E. Carson, N.J. Higham, A new analysis of iterative refinement and its application to accurate solution of ill-conditioned sparse linear systems. Technical Report MIMS Eprint 2017.12, Manchester Institute for Mathematical Sciences, (The University of Manchester, UK, 2017). SIAM J. Sci. Comput.,39(6), A2834–A2856, (2017)Google Scholar
  2. 2.
    E. Carson, N.J. Higham, Accelerating the solution of linear systems by iterative refinement in three precisions. Technical Report MIMS Eprint 2017.24, Manchester Institute for Mathematical Sciences, (The University of Manchester, UK, 2017). SIAM J. Sci. Comput.,40(2), A817–A847, (2018)Google Scholar
  3. 3.
    J. Demmel, Y. Hida, W. Kahan, X.S. Li, S. Mukherjee, E.J. Riedy, Error bounds from extra-precise iterative refinement. ACM Trans. Math. Softw. 32(2), 325–351 (2006)MathSciNetCrossRefGoogle Scholar
  4. 4.
    O. Heimlich, Interval arithmetic in GNU Octave, In SWIM 2016: Summer Workshop on Interval Methods (2016)Google Scholar
  5. 5.
    N. Higham, Accuracy and Stability of Numerical Algorithms, 2nd edn. (SIAM Press, 2002)Google Scholar
  6. 6.
    V. Kreinovich, S. Rump, Towards optimal use of multi-precision arithmetic: a remark. Reliab. Comput. 12(5), 365–369 (2006)MathSciNetCrossRefGoogle Scholar
  7. 7.
    J.-M. Muller, N. Brunie, de Dinechin, F. Jeannerod, C.-P. Joldes, M. Lefèvre, V. Melquiond, G. Revol, S. Torres, Handbook of Floating-Point Arithmetic (2nd edition, 2018)Google Scholar
  8. 8.
    A. Neumaier, Interval Methods for Systems of Equations (Cambridge University Press, 1990)Google Scholar
  9. 9.
    A. Neumaier, A simple derivation of the Hansen-Bliek-Rohn-Ning-Kearfott enclosure for linear interval equations. Reliab. Comput. 5, 131–136 (1999) (+ Erratum: Reliable Computing 6, p. 227, 2000)Google Scholar
  10. 10.
    H.D. Nguyen, Efficient algorithms for verified scientific computing: numerical linear algebra using interval arithmetic. Ph.D. Thesis, École Normale Supérieure de Lyon - ENS LYON, (2011).
  11. 11.
    N. Revol, Y. Denneulin, J.-F. Méhaut, B. Planquelle, A methodology of parallelization for continuous verified global optimization. LNCS 2328, 803–810 (2002)Google Scholar
  12. 12.
    S. Rump, Developments in Reliable Computing, T. Csendes ed., chapter INTLAB - Interval Laboratory (Kluwer, 1999), pp. 77–104Google Scholar
  13. 13.
    J.H. Wilkinson, Rounding Errors in Algebraic Processes (Prentice-Hall, Englewood Cliffs, NJ, 1963)zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.InriaUniversity of Lyon—LIPLyon Cedex 07France

Personalised recommendations