Solution of Systems of Polynomial Equations by Using Bernstein Expansion

  • Jürgen Garloff
  • Andrew P. Smith


Systems of polynomial equations appear in a great variety of applications, e.g., in geometrie interseetion eomputations (Hu et al. 1996), ehemieal equilibrium problems, combustion, and kinematies, to name only a few. Examples ean be found in the monograph Morgan (1987). Following Sherbrooke and Patrikalakis (1993), most of the methods for the solution of sueh a system ean be dassitied as techniques based on elimination theory, continuation, and subdivision. Elimination theory-based methods for eonstructing Gröbner bases rely on symbolic manipulations, making those methods seem somewhat unsuitable for larger problems. This class and also the second of the methods based on continuation frequently give us more information than we need since they determine all complex solutions of the system, whereas in applications often only the solutions in a given area of interest - typically a box - are sought. In the last category we collect all methods which apply a domain-splitting approach: Starting with the box of interest, such an algorithrn sequentially splits it into subboxes, eliminating infeasible boxes by using bounds for the range of the polynomials under consideration over each of them, and ending up with a union of boxes that contains all solutions to the system which lie within the given box. Methods utilising this approach indude interval computation techniques as well as methods which apply the expansion of a multivariate polynomial into Bernstein polynomials


Geometric Design Polynomial System Bernstein Polynomial Multivariate Polynomial Bernstein Form 
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. Cargo, G.T., Shisha, O. (1966): The Bernstein form of a polynomial. J. Res. Nat. Bur. Standards Sect. B, vol. 70B (Math. Sci.), 1: 79–81MathSciNetCrossRefGoogle Scholar
  2. Farouki, R.T., Rajan, V.T. (1988): Algorithms for polynomials in Bernstein form. Computer Aided Geometrie Design 5: 1–26MathSciNetMATHCrossRefGoogle Scholar
  3. Fischer, H.C. (1990): Range computations and applications. In: Ullrich, C. (ed.): Contributions to computer arithmetic and self-validating numerical methods. J.C.Baltzer, Amsterdam, pp. 197–211Google Scholar
  4. Garloff, J. (1986): Convergent bounds for the range of multivariate polynomials. In: Nickel, K. (ed.): Interval mathematics 1985. Springer, Berlin Heidelberg New York, pp. 37–56 (Lecture notes in computer science, vol. 212)Google Scholar
  5. Garloff, J. (1993): The Bernstein algorithm. Interval Comp. 2: 154–168MathSciNetGoogle Scholar
  6. Garloff, J. (2000): Applications of Bernstein expansion to the solution of control problems. Reliable Comp. 6: 303–320MathSciNetMATHCrossRefGoogle Scholar
  7. Garloff, J., Graf, B. (1999): Solving strict polynomial inequalities by Bernstein expansion. In: Munro, N. (ed.): The use of symbolic methods in control system analysis and design. IEE, London, pp. 329–352Google Scholar
  8. Garloff, J., Smith, A.P. (2001): Improvements of a subdivision-based algorithm for solving systems of polynomial equations, to appear in the Proceedings of the 3rd World Congress of Nonlinear Analysts, July 19–26 2000, Catania, Italy, special series of the Journal of Nonlinear Analysis, Elsevier Sci. Publ.Google Scholar
  9. Granvilliers, L. (2000): Towards unnperative interval narrowing. In: Proceedings 3rd Intern. Workshop on Frontiers of Combining Systems, FroCoS’2000, Nancy, France, Springer, Berlin Heidelberg New York (Lecture notes in artificial intelligence, vol. 1794)Google Scholar
  10. Hu, Chun-Yi, Maekawa, T., Sherbrooke, E.C., Patrikalakis, N.M. (1996): Robust interval algorithm for curve intersections. Computer-Aided Design 28:495–506CrossRefGoogle Scholar
  11. Jüger, C., Ratz, D. (1995): A combined method for enclosing all solutions ofnonlinear systems of polynomial equations, Reliable Comp. 1: 41–64CrossRefGoogle Scholar
  12. Kearfott, R.B. (1996): Rigorous global search: continuous problems, Kluwer Acad. Publ., Dordrecht Boston LondonMATHGoogle Scholar
  13. Kioustelidis, J.B. (1978): Algorithmic error estimation for approximate solutions of nonlinear systems of equations. Computing 19: 313–320MathSciNetMATHCrossRefGoogle Scholar
  14. Miranda, C. (1941): Un’ osservazione su un teorema di Brouwer. Boll. Un. Mat. Ital. Ser.2, 3: 5–7MathSciNetGoogle Scholar
  15. Moore, R.E., Kioustelidis, J.B. (1980): A simple test for accuracy of approximate solutions to nonlinear (or linear) systems. SIAM J. Numer. Anal. 17:521–529MathSciNetMATHCrossRefGoogle Scholar
  16. Morgan, A.P.(1987): Solving polynomial systems using continuation for engineering and scientific problems. Prentice Hall, Englewood-Cliffs, N.J.MATHGoogle Scholar
  17. Neumaier, A. (1990): Interval methods for systems of equations. Cambridge University Press, Cambridge (Encyclopedia of mathematics and its applications)MATHGoogle Scholar
  18. Sherbrooke, E.C., Patrikalakis, N.M. (1993): Computation of the solutions ofnonlinear polynomial systems. Computer Aided Geometric Design 10: 379–405MathSciNetMATHCrossRefGoogle Scholar
  19. Van Hentenryck, P., McAllester, D., Kapur, D. (1997): Solving polynomial systems using a branch and prune approach. SIAM J. Numer. Anal. 34:797–827MathSciNetMATHCrossRefGoogle Scholar
  20. Vrahatis, M.N. (1989): A short proof and a generalization of Miranda’s existence theorem. Proc. Amer. Math. Soc. 107: 701–703MathSciNetMATHGoogle Scholar
  21. Zettler, M., Garloff, J. (1998): Robustness analysis of polynomials with polynomial parameter dependency using Bernstein expansion. IEEE Trans. Automat. Contr. 43:425–431MathSciNetMATHCrossRefGoogle Scholar
  22. Zuhe, S., Neumaier, A. (1988.): A note on Moore’s interval test for zeros of nonlinear systems. Computing 40: 85–90MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Wien 2001

Authors and Affiliations

  • Jürgen Garloff
  • Andrew P. Smith

There are no affiliations available

Personalised recommendations