Solving Nonlinear Equations by Abstraction, Gaussian Elimination, and Interval Methods

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2309)


The solving engines of most of constraint programming systems use interval-based consistency techniques to process nonlinear systems over the reals. However, few symbolic-interval cooperative solvers are implemented. The challenge is twofold: control of the amount of symbolic computations, and prediction of the accuracy of interval computations over transformed systems.

In this paper, we introduce a new symbolic pre-processing for interval branch-and-prune algorithms based on box consistency. The symbolic algorithm computes a linear relaxation by abstraction of the nonlinear terms. The resulting rectangular linear system is processed by Gaussian elimination. Control strategies of the densification of systems during elimination are devised. Three scalable problems known to be hard for box consistency are efficiently solved.


Symbolic Computation Gaussian Elimination Interval Arithmetic Linear Relaxation Real Interval 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 2.
    F. Benhamou, D. McAllester, and P. Van Hentenryck. CLP(Intervals) Revisited.In Procs. of ILPS'94, Intl. Logic Prog. Symp., pages 124–138, Ithaca, USA, 1994. MIT Press.Google Scholar
  2. 3.
    F. Benhamou and W. J. Older. Applying Interval Arithmetic to Real, Integer and Boolean Constraints. J. of Logic Programming, 32(1):1–24, 1997.zbMATHCrossRefMathSciNetGoogle Scholar
  3. 4.
    M. Ceberio and L. Granvilliers. Solving Nonlinear Systems by Constraint Inversion and Interval Arithmetic. In Procs. of AISC’2000, 5th Intl. Conf. on Artificial Intelligence and Symbolic Computation, volume 1930 of LNAI, Madrid, Spain, 2000. Springer-Verlag.Google Scholar
  4. 5.
    A. Colmerauer. Naive Solving of Non-linear Constraints. In F. Benhamou and A. Colmerauer, eds., Constraint Logic Programming: Selected Research, pages 89–112. MIT Press, 1993.Google Scholar
  5. 6.
    F. Goualard, F. Benhamou, and L. Granvilliers. An Extension of the WAM for Hybrid Interval Solvers. J. of Functional and Logic Programming, 5(4):1–31, 1999.Google Scholar
  6. 7.
    L. Granvilliers. A Symbolic-Numerical Branch and Prune Algorithm for Solving Non-linear Polynomial Systems.J. of Universal Comp. Sci., 4(2):125–146, 1998.zbMATHMathSciNetGoogle Scholar
  7. 8.
    L. Granvilliers, E. Monfroy, and F. Benhamou. Symbolic-Interval Cooperation in Constraint Programming. In Procs. of ISSAC’2001, 26th Intl. Symp. on Symbolic and Algebraic Computation, pages 150–166, Univ. of Western Ontario, London, Ontario, Canada, 2001. ACM Press.Google Scholar
  8. 9.
    T. J. Hickey. CLIP: a CLP(Intervals) Dialect for Metalevel Constraint Solving. In Procs. of PADL’2000, Intl. Workshop on Practical Aspects of Declarative Languages, volume 1753 of LNCS, pages 200–214, Boston, USA, 2000. Springer-Verlag.Google Scholar
  9. 10.
    H. Hong. RISC-CLP(Real): Constraint Logic Programming over Real Numbers. In F. Benhamou and A. Colmerauer, eds., Constraint Logic Programming: Selected Research. MIT Press, 1993.Google Scholar
  10. 11.
    J. Jaffar, S. Michaylov, P. Stuckey, and R. Yap. The CLP(ℜ) Language and System. ACM Trans. on Programming Languages and Systems, 14(3):339–395, 1992.CrossRefGoogle Scholar
  11. 12.
    P. Marti and M. Rueher. A Distributed Cooperating Constraints Solving System. Intl. J. on Artificial Intelligence Tools, 4(1–2):93–113, 1995.CrossRefGoogle Scholar
  12. 13.
    R. E. Moore. Interval Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1966.zbMATHGoogle Scholar
  13. 14.
    J.-F. Puget and M. Leconte. Beyond the Glass Box: Constraints as Objects. In Procs. of ILPS’95, Intl. Logic Programming Symposium, pages 513–527, Portland, USA, 1995. MIT Press.Google Scholar
  14. 15.
    A. Semenov and A. Leshchenko. Interval and Symbolic Computations in the Unicalc Solver. In Procs. of INTERVAL’94, pages 206–208, St-Petersburg, Russia, 1994.Google Scholar
  15. 16.
    M. H. Van Emden. Algorithmic Power from Declarative Use of Redundant Constraints. Constraints, 4(4):363–381, 1999.zbMATHCrossRefMathSciNetGoogle Scholar
  16. 17.
    P. Van Hentenryck, D. McAllester, and D. Kapur. Solving Polynomial Systems Using a Branch and Prune Approach. SIAM J. on Numerical Analysis, 34(2):797–827, 1997.zbMATHCrossRefGoogle Scholar
  17. 18.
    P. Van Hentenryck, L. Michel, and Y. Deville. Numerica: a Modeling Language for Global Optimization. MIT Press, 1997.Google Scholar
  18. 19.
    M. Wallace, S. Novello, and J. Schimpf. ECLiPSe: A Platform for Constraint Logic Programming. Technical report, IC-Parc, London, 1997.Google Scholar
  19. 20.
    K. Yamamura, H. Kawata, and A. Tokue. Interval Analysis using Linear Programming. BIT, 38:188–201, 1998.CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  1. 1.IRIN - University of NantesNantes Cedex 3France

Personalised recommendations