On the Selection of a Transversal to Solve Nonlinear Systems with Interval Arithmetic

  • Frédéric Goualard
  • Christophe Jermann
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3991)


This paper investigates the impact of the selection of a transversal on the speed of convergence of interval methods based on the nonlinear Gauss-Seidel scheme to solve nonlinear systems of equations. It is shown that, in a marked contrast with the linear case, such a selection does not speed up the computation in the general case; directions for researches on more flexible methods to select projections are then discussed.


Static Transversal Interval Arithmetic Interval Method Good Projection Interval Constraint 
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.


  1. 1.
    Barton, P.I.: The equation oriented strategy for process flowsheeting. Dept. of Chemical Eng. MIT, Cambridge (2000)Google Scholar
  2. 2.
    Benhamou, F., McAllester, D., Van Hentenryck, P.: CLP(Intervals) revisited. In: Procs. Intl. Symp. on Logic Prog., Ithaca, NY, pp. 124–138. The MIT Press, Cambridge (1994)Google Scholar
  3. 3.
    Duff, I.S.: On algorithms for obtaining a maximum transversal. ACM Trans. Math. Software 7(3), 315–330 (1981)CrossRefGoogle Scholar
  4. 4.
    Goualard, F.: On considering an interval constraint solving algorithm as a free-steering nonlinear gauss-seidel procedure. In: Procs. 20th Annual ACM Symp. on Applied Computing (Reliable Comp. and Applications track), March 2005, vol. 2, pp. 1434–1438. Ass. for Comp. Machinery, Inc. (2005)Google Scholar
  5. 5.
    Hughes Hallett, A.J., Piscitelli, L.: Simple reordering techniques for expanding the convergence radius of first-order iterative techniques. J. Econom. Dynam. Control 22, 1319–1333 (1998)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Hansen, E.R., Sengupta, S.: Bounding solutions of systems of equations using interval analysis. BIT 21, 203–211 (1981)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Herbort, S., Ratz, D.: Improving the efficiency of a nonlinear-system-solver using a componentwise newton method. Research report 2/1997, Institut für Angewandte Mathematik, Universität Karslruhe, TH (1997)Google Scholar
  8. 8.
    Hickey, T.J., Ju, Q., Van Emden, M.H.: Interval arithmetic: from principles to implementation. J. ACM 48(5), 1038–1068 (2001)CrossRefMathSciNetGoogle Scholar
  9. 9.
    INRIA project COPRIN: Contraintes, OPtimisation, Résolution par INtervalles. The COPRIN examples page. Web page at
  10. 10.
    Moore, R.E.: Interval Analysis. Prentice-Hall, Englewood Cliffs (1966)MATHGoogle Scholar
  11. 11.
    Neumaier, A.: Interval methods for systems of equations. Encyclopedia of Mathematics and its Applications, vol. 37. Cambridge University Press, Cambridge (1990)MATHGoogle Scholar
  12. 12.
    Ortega, J.M., Rheinboldt, W.C.: Iterative solutions of nonlinear equations in several variables. Academic Press Inc., London (1970)Google Scholar
  13. 13.
    Ratschek, H., Rokne, J.: Interval methods. In: Handbook of Global Optimization, pp. 751–828. Kluwer Academic, Dordrecht (1995)Google Scholar
  14. 14.
    Sotiropoulos, D.G., Nikas, J.A., Grapsa, T.N.: Improving the efficiency of a polynomial system solver via a reordering technique. In: Procs. 4th GRACM Congress on Computational Mechanics, vol. III, pp. 970–976 (2002)Google Scholar
  15. 15.
    Sutton, R.S., Barto, A.G.: Reinforcement Learning: An Introduction. The MIT Press, Cambridge (1998)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Frédéric Goualard
    • 1
  • Christophe Jermann
    • 1
  1. 1.LINA, FRE CNRS 2729University of Nantes – FranceNantes cedex 3

Personalised recommendations