Abstract
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.
Chapter PDF
Similar content being viewed by others
Keywords
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.
References
Barton, P.I.: The equation oriented strategy for process flowsheeting. Dept. of Chemical Eng. MIT, Cambridge (2000)
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)
Duff, I.S.: On algorithms for obtaining a maximum transversal. ACM Trans. Math. Software 7(3), 315–330 (1981)
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)
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)
Hansen, E.R., Sengupta, S.: Bounding solutions of systems of equations using interval analysis. BIT 21, 203–211 (1981)
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)
Hickey, T.J., Ju, Q., Van Emden, M.H.: Interval arithmetic: from principles to implementation. J. ACM 48(5), 1038–1068 (2001)
INRIA project COPRIN: Contraintes, OPtimisation, Résolution par INtervalles. The COPRIN examples page. Web page at http://www-sop.inria.fr/coprin/logiciels/ALIAS/Benches/benches.html
Moore, R.E.: Interval Analysis. Prentice-Hall, Englewood Cliffs (1966)
Neumaier, A.: Interval methods for systems of equations. Encyclopedia of Mathematics and its Applications, vol. 37. Cambridge University Press, Cambridge (1990)
Ortega, J.M., Rheinboldt, W.C.: Iterative solutions of nonlinear equations in several variables. Academic Press Inc., London (1970)
Ratschek, H., Rokne, J.: Interval methods. In: Handbook of Global Optimization, pp. 751–828. Kluwer Academic, Dordrecht (1995)
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)
Sutton, R.S., Barto, A.G.: Reinforcement Learning: An Introduction. The MIT Press, Cambridge (1998)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Goualard, F., Jermann, C. (2006). On the Selection of a Transversal to Solve Nonlinear Systems with Interval Arithmetic. In: Alexandrov, V.N., van Albada, G.D., Sloot, P.M.A., Dongarra, J. (eds) Computational Science – ICCS 2006. ICCS 2006. Lecture Notes in Computer Science, vol 3991. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11758501_47
Download citation
DOI: https://doi.org/10.1007/11758501_47
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-34379-0
Online ISBN: 978-3-540-34380-6
eBook Packages: Computer ScienceComputer Science (R0)