Abstract
A common problem in scientific computing is to find global solutions of a system of nonlinear polynomial equations efficiently. In this paper, we describe a possible strategy to compute verified solutions of such systems in a given box. First we use the algorithms of Sherbrooke and Patrikalakis, extended to interval arithmetic, to find intervals, which possibly contain solutions, then we verify, if these intervals really contain a zero of the system. These tests are based on the criterion of Miranda, which we further developed for polynomial systems. We show that only checking a matrix to be strictly diagonally dominant and estimating of the remainder is required. At last we quote a further criterion that can be used alternatively.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Boese, F. G. and Luther, W. (2001). Enclosure of the Zero Set of Polynomials in Several Complex Variables. Multidimensional Systems and Signal Processing 12, 165–197.
Cormen, T. H., Leiserson, C. E., and Rivest, R. L. (1990). Introduction to Algorithms. MIT Press, Cambridge, MA.
Dyllong, E. and Luther, W. (2000). Distance calculation between a point and a NURBS surface. Curve and Surface Design: Saint-Malo 1999, Larry L. Schumaker et al. (Eds.), Vanderbilt University Press, Nashville, TN, 55–62.
Farin, G. (1997). Curves and Surfaces for Computer Aided Geometric Design —A Practical Guide. Academic Press, New York, 4th ed.
Fausten, D. and Luther, W. (2000). Verifizierte Lösungen von nichtlinearen polynomialen Gleichungssystemen. Technical Report SM-DU-477, Universität Duisburg.
Gass, S. I. (1985). Linear Programming: Methods and Applications. McGraw—Hill, New York.
Knüppel, 0. (1993). BIAS — basic interval subroutines. Bericht 93.3, TU Hamburg—Harburg.
Kolev, L. (1999). An Improved Method for Global Solution of Non—Linear Systems. Reliable Computing 5, 103–111.
Miranda, C. (1940). Un’ osservazione su un teorema di Brouwer. Bollettino della Unione Matematica Italiana, II. Ser. 3, 5–7.
Moore, R. E. and Kioustelidis, J. B. (1980). A simple test for accuracy of approximate solutions to nonlinear (or linear) systems. SIAM J. Numer. Anal. 17, 521–529.
Neumaier, A. (1990). Interval methods for systems of equations. Cambridge University Press, Cambridge.
Sherbrooke, E. C. and Patrikalakis, N. M. (1993). Computation of the Solution of Nonlinear Polynomial Systems. J. Comput. Aided Geom. Des. 10, No. 5, 379–405.
Walach, E. and Zeheb, E. (1980). Sign Test of Multivariable Real Polynomials. Transactions on Circuits and Systems, vol. 27, No. 7, 619–625.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer Science+Business Media New York
About this chapter
Cite this chapter
Fausten, D., Luther, W. (2001). Verified Solutions of Systems of Nonlinear Polynomial Equations. In: Krämer, W., von Gudenberg, J.W. (eds) Scientific Computing, Validated Numerics, Interval Methods. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-6484-0_12
Download citation
DOI: https://doi.org/10.1007/978-1-4757-6484-0_12
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4419-3376-8
Online ISBN: 978-1-4757-6484-0
eBook Packages: Springer Book Archive