Skip to main content
SpringerLink
Log in

Accurate Enclosure of the Zero Set of Multivariate Polynomials

  • Published:
BIT Numerical Mathematics Aims and scope Submit manuscript

Abstract

The zero set of one general multivariate polynomial is enclosed by unions and intersections of funnel-shaped unbounded sets. There are sharper enclosures for the zero set of a polynomial in two complex variables with complex interval coefficients. Common zeros of a polynomial system can be located by an appropriate intersection of these enclosure sets in an appropriate space. The resulting domain is directly brought into polynomial equation solvers.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  1. F. G. Boese, On the bounded-input bounded-output stability of linear, time-invariant, time-discrete, multi-input, multi-output, multivariate dynamical system, ZAMM, 106 (1999), Supplement 2, pp. 335-336.

    Google Scholar 

  2. F. G. Boese, and W. Luther, Enclosure of the Zero Set of Polynomials in Several Complex Variables, Multidimensional Systems and Signal Processing, 12 (2001) 2, pp. 165-197.

    Google Scholar 

  3. N. K. Bose, Applied Multidimensional Systems Theory, Van Nostrand Reinhold Company, New York, 1982.

    Google Scholar 

  4. A. M. Cohen, H. Cuypers, and H. Sterk (eds.), Some Tapas of Computer Algebra, Springer-Verlag, Berlin, 1999.

    Google Scholar 

  5. D. Fausten, and W. Luther, Verified solution of systems of nonlinear polynomial equations, in W. Krämer, W. J. v. Gudenberg (eds.), Validated Numerics, Interval Methods, Kluwer Academic Publishers, Dordrecht, 2001, pp. 141-152.

    Google Scholar 

  6. G. Fruchter, U. Srebro, and E. Zeheb, Conditions on the boundary of the Zero Set and Application to Stabilization of Systems with uncertainty, JMAA 161 (1991), pp. 148-175.

    Google Scholar 

  7. R. Hammer, M. Hocks, U. Kulisch, and D. Ratz, Numerical Toolbox for Verified Computing I, Springer, Berlin, 1993.

    Google Scholar 

  8. E. Hille, Analytic Function Theory, vol. II., Chelsea Publishing Company, New York, 1962.

    Google Scholar 

  9. E. I. Jury, Stability of Multidimensional Systems and Related Problems, in S. G. Tzafestas (ed.), Multidimensional Systems, Techniques and Applications, M. Dekker, New York, 1986, pp. 89-159.

    Google Scholar 

  10. R. Klatte, U. Kulisch, A. Wiethoff, C. Lawo, and M. Rauch, C-XSC, Springer-Verlag, Berlin, 1993. http://www.math.uni-wuppertal.de/wrswt/xsc-sprachen.html

    Google Scholar 

  11. B. Sturmfels, Polynomial Equations and Convex Polytopes, Amer. Math. Monthly, 105 (1998), pp. 907-922.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Max-Planck-Institut für extraterrestrische Physik, Giessenbachstraße, D-85748, Garching, Germany

    Fritz G. Boese

  2. Institut für Informatik und Interaktive Systeme, University of Duisburg, Lotharstraße 65, D-47057, Duisburg, Germany

    Wolfram J. Luther

Authors
  1. Fritz G. Boese
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wolfram J. Luther
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Boese, F.G., Luther, W.J. Accurate Enclosure of the Zero Set of Multivariate Polynomials. BIT Numerical Mathematics 43, 245–261 (2003). https://doi.org/10.1023/A:1026093018189

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1026093018189

Search

Navigation