Reliable Computing

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

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

Article

Abstract

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.

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References

  1. 1.
    Aberth O. (1997) The Solution of Linear Interval Equations by a Linear Programming Method. Linear Algebra and Its Applications 259: 271–279MATHCrossRefMathSciNetGoogle 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): 443–478MATHCrossRefMathSciNetGoogle 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): 1493–1503MATHCrossRefMathSciNetGoogle 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): 131–136MATHCrossRefMathSciNetGoogle 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): 1289–1305MATHCrossRefMathSciNetGoogle 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): 321–418MATHCrossRefMathSciNetGoogle 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): 141–155MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Shary S.P. (1999) Outer Estimation of Generalized Solution Sets to Interval Linear Systems. Reliable Computing 5(3): 323–335MATHCrossRefMathSciNetGoogle 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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