Reliable Computing

, Volume 13, Issue 4, pp 325–349 | Cite as

Extension of the Hansen-Bliek Method to Right-Quantified Linear Systems

  • Gilles Chabert
  • Alexandre Goldsztejn


The problem of finding the smallest box enclosing the united solution set of a linear interval system, also known as the “interval hull” problem, was proven to be NP-hard. However, Hansen, Bliek, and others subsequently, have provided a polynomial-time solution in the case of systems preconditioned by the midpoint inverse matrix.

Based upon a similar approach, this paper deals with the interval hull problem in the context of AE-solution sets, where parameters may be given different quantifiers. A polynomial-time algorithm is proposed for computing the hull of AE-solution sets where parameters involved in the matrix are constrained to be existentially quantified. Such AE-solution sets are called right-quantified solution sets. They have recently been shown to be of practical interest.


Candidate Point Support Line Outer Estimation Interval Vector Interval Linear 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Aberth O. (1997) The Solution of Linear Interval Equations by a Linear Programming Method. Linear Algebra and Its Applications 259: 271–279zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bliek, C.: Computer Methods for Design Automation, PhD Thesis, Massachusetts Institute of Technology, 1992.Google Scholar
  3. 3.
    Dantzig, G. B.: Linear Programming and Extensions, Princeton University Press, 1963.Google Scholar
  4. 4.
    Goldsztejn, A.: A Branch and Prune Algorithm for the Approximation of Non-Linear AE-Solution Sets, in: ACM SAC, 2006, pp. 1650–1654.Google Scholar
  5. 5.
    Goldsztejn A. (2005) A Right-Preconditioning Process for the Formal-Algebraic Approach to Inner and Outer Estimation of AE-Solution Sets. Reliable Computing 11(6): 443–478zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Goldsztejn, A.: Définition et Applications des Extensions des Fonctions Réelles aux Intervalles Généralisés, PhDThesis, Université de Nice-Sophia Antipolis, 2005.Google Scholar
  7. 7.
    Hansen E.R. (1992) Bounding the Solution of Interval Linear Equations. SIAM J. Numer. Anal. 29(5): 1493–1503zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Heindl G., Kreinovich V., Lakeyev A.V. (1998) Solving Linear Interval Systems Is NP-Hard Even If We Exclude Overflow and Underflow. Reliable Computing 4(4): 377–381CrossRefMathSciNetGoogle Scholar
  9. 9.
    Markov, S., Popova, E., and Ullrich,Ch.: On the Solution of Linear Algebraic Equations Involving Interval Coefficients, in: Margenov, S. and Vassilevski, P. (eds), Iterative Methods in Linear Algebra, II, IMACS Series in Computational and Applied Mathematics 3 (1996), pp. 216–225.Google Scholar
  10. 10.
    Moore, R.: Interval Analysis, Prentice Hall, 1966.Google Scholar
  11. 11.
    Neumaier A. (1999) A Simple Derivation of the Hansen-Bliek-Rohn-Ning-Kearfott Enclosure for Linear Interval Equations. Reliable Computing 5(2): 131–136zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Neumaier, A.: Interval Methods for Systems of Equations, Cambridge University Press, 1990.Google Scholar
  13. 13.
    Ning S., Kearfott R.B. (1997) A Comparison of Some Methods for Solving Linear Interval Equations. SIAM J. Numer. Anal. 34(1): 1289–1305zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Oettli W. (1965) On the Solution Set of a Linear System with Inaccurate Coefficients, SIAM J. Numer. Anal. 2(1): 115–118CrossRefMathSciNetGoogle Scholar
  15. 15.
    Rohn J. (1993) Cheap and Tight Bounds: The Recent Result by E.Hansen Can Be Made More Efficient. Interval Computations 1(4): 13–21MathSciNetGoogle Scholar
  16. 16.
    Shary S.P. (2002) A New Technique in Systems Analysis Under Interval Uncertainty and Ambiguity. Reliable Computing 8(5): 321–418zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Shary, S. P.: Algebraic Solutions to Interval Linear Equations and Their Application, in: IMACS— GAMM International Symposium on Numerical Methods and Error Bounds, 1996.Google Scholar
  18. 18.
    Shary S.P. (2001) Interval Gauss-Seidel Method for Generalized Solution Sets to Interval Linear Systems. Reliable Computing 7(2): 141–155zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Shary S.P. (1999) Outer Estimation of Generalized Solution Sets to Interval Linear Systems. Reliable Computing 5(3): 323–335zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science + Business Media B.V. 2007

Authors and Affiliations

  1. 1.Projet CoprinINRIASophia AntipolisFrance
  2. 2.University of Central ArkansasConwayUSA

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