A note on methods for solving systems of polynomial equations with floating point coefficients

  • J. A. van Hulzen
9. Symbolic-Numeric Interface
Part of the Lecture Notes in Computer Science book series (LNCS, volume 72)


A short description is presented of our experience with the use of resultants to solve systems of polynomial equations with floating point coefficients. To motivate the relevance of these attempts we first give a short introduction in the problem which causes this temptation. We conclude with some proposals for an alternative hybrid method.


Polynomial Equation Integer Representation Reliable Hybrid Computational Rule Format Data Structure 
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. [1]
    A.V. Aho, J.E. Hopcroft and J.D. Ullman, The design and analysis of computer programs, Addison Wesley, Reading (1974), pp. 428–451.Google Scholar
  2. [2]
    E.C. Belaga, Some problems in the computation of polynomials, Dokl.Akad.Nauk. SSSR 123 (1958), pp. 775–777.Google Scholar
  3. [3]
    W.S. Brown, ALTRAN user's manual, Bell Laboratories, Murray Hill (1977).Google Scholar
  4. [4]
    W.S. Brown and A.C. Hearn, Applications of Symbolic Algebraic Computation, Utah Computational Physics Group Report no.61 (March 1978).Google Scholar
  5. [5]
    G.E. Collins, The calculation of multivariate polynomial resultants, Journal of the ACM, vol.18, no.4 (Oct.1971), pp.515–532.Google Scholar
  6. [6]
    J. Eve, The evaluation of polynomials, Numerische Mathematik 6, (1964), pp. 17–21.Google Scholar
  7. [7]
    S.I. Feldman and J. Ho, A rational expression evaluation package, Computer Science Technical Report no. 34, Bell Laboratories, Murray Hill (Sept. 1975).Google Scholar
  8. [8]
    M.L. Griss, Using an efficient sparse minor expansion algorithm to compute polynomial subresultants and GCD, Utah Computational Physics Group Report no.53 (March 1977).Google Scholar
  9. [9]
    R. Hettich, Kriterien zweiter Ordnung für local beste Approximationen, Numerische Mathematik 22 (1974), pp. 409–417.Google Scholar
  10. [10]
    R. Hettich, A Newton-method for nonlinear Chebyshew approximation, Approximation Theory Bonn 1976 (R. Schaback and K. Scherer, editors), Springer Verlag, Berlin (1976), pp. 222–236.Google Scholar
  11. [11]
    R. Hettich and J.A. van Hulzen, Approximation with a class of rational functions, Memorandum no. 165, Department of Applied Mathematics, Twente University of Technology (May 1977).Google Scholar
  12. [12]
    R. Hettich and W. Wetterling, Non-linear Chebyshev approximation by H-polynomials, Journal of Approximation Theory, vol.7, no.2 (1973), pp.198–211.Google Scholar
  13. [13]
    J. Larson and A. Sameh, Efficient Calculation of the effects of roundoff errors, ACM TOMS Vol.4, no.3 (Sept.1978), pp.228–236.Google Scholar
  14. [14]
    D.H. Lehmer, A machine method for solving polynomial equations, Journal of the ACM, Vol.8 no.2 (April 1961), pp. 151–162.Google Scholar
  15. [15]
    W. Miller, Graph transformations for roundoff analysis, SIAM J. Comput., vol.5 no.2 (June 1976), pp. 204–216.Google Scholar
  16. [16]
    J. Moses, Solution of Systems of Polynomial equations by elimination, Communications of the ACM, vol.9 no.8 (Aug. 1966), pp. 634–637.Google Scholar
  17. [17]
    T.S. Motzkin, Evaluation of Polynomials and Evaluation of rational functions, Bulletin American Mathematical Society 61 (1955), pp. 163.Google Scholar
  18. [18]
    F.W.J. Olver, A new approach to error arithmetic, SIAM Journal Numeric. Analysis, vol.15 no.2 (April 1978), pp. 368–393.Google Scholar
  19. [19]
    V.Y. Pan, Methods of computing values of polynomials, Russian Mathematical Surveys, vol.21 no.1 (1966), pp. 105–136.Google Scholar
  20. [20]
    D.R. Stoutemeyer, Analytical optimization using computer algebraic manipulation, ACM TOMS, vol.1 no.2 (June 1975), pp.147–164.Google Scholar
  21. [21]
    Bj. Svejgaard, Zeros of polynomials, Contribution no.21, ALGOL-programming, BIT 7 (1967), pp. 240–246.Google Scholar
  22. [22]
    J.A. van Hulzen and R. Hettich, Approximation with a class of rational functions, IFIP 77 (B. Gilchrist, ed.), North-Holland Publ. Company, Amsterdam (1977), pp. 487–492.Google Scholar
  23. [23]
    J.A. van Hulzen, Experience with the ALTRAN-converter, in preparation.Google Scholar
  24. [24]
    W. Wetterling, Tschebyscheff-Approximation mit nichtlinear auftretenden Parametern, Zeitschrift für Angewandte Mathematik und Mechanik 44, (1964), pp. 85–86.Google Scholar
  25. [25]
    L.H. Williams, Algebra of polynomials in several variables for a digital computer, Journal of the ACM, vol.9 no.1 (Jan, 1962), pp. 29–40.Google Scholar
  26. [26]
    D.Y.Y. Yun, On algorithms for solving systems of polynomial equations, ACM SIGSAM Bulletin no.27 (Sept. 1973), pp. 19–25.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1979

Authors and Affiliations

  • J. A. van Hulzen
    • 1
  1. 1.Department of Applied MathematicsTwente University of TechnologyEnschedeThe Netherlands

Personalised recommendations