A Generalization of the Orlando Formula — Symbolic Manipulation Approach

  • Henryk Górecki
  • Maciej Szymkat
  • Mieczyslaw Zaczyk
Conference paper
Part of the ISNM International Series of Numerical Mathematics book series (ISNM, volume 121)


In this paper some new results on polynomials whose roots are linear combinations (with fixed integer coefficients) of the roots of other polynomial are presented. This research is motivated by a new method for calculation of a generalized integral criterion with the integrand function being an arbitrary polynomial of the transient error.

The well known Orlando formula establishes the useful relation between the Hurwitz determinants and the polynomial whose roots are sums of the roots of a given polynomial. However, this is only a special case of the formulae derived in this paper.

In the paper two approaches are presented. The first one uses a recursive procedure based on resultants and Hurwitz determinants, and the second exploits the theory of symmetric functions. Both methods are implemented as symbolic manipulation algorithms in Maple V. The results obtained are illustrated with an example of application.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    H. Górecki, A new method for calculation of the generalized integral criterion, Bull. Pol. Acad. Sci. Tech. Sci., vol. 42, no. 4, pp. 595–603, 1994.Google Scholar
  2. [2]
    H.Górecki, M.Szymkat, Application of an elimination method to the study of the geometry of zeros of real polynomials, International Journal of Control, vol. 38, no. 1, pp. 1–26, 1983.MathSciNetCrossRefGoogle Scholar
  3. [3]
    L. Orlando, Sul problema di Hurwitz relativo alle parti reali delle radici di un’equatione algebrica, Math. Ann., vol. 71, pp. 233–245, 1911.MathSciNetCrossRefGoogle Scholar
  4. [4]
    E.I. Jury, From J.J. Sylvester to Adolf Hurwitz: A historical review, these proceedings.Google Scholar
  5. [5]
    P.C. Parks, A new proof of Hermite’s stability criterion and a generalization of Orlando’s formula, Int. J. Cont., vol. 26, no. 2, pp. 197–206, 1977.zbMATHCrossRefGoogle Scholar
  6. [6]
    E.I. Jury, Inners and Stability of Dynamic Systems, Second Edition, Malabar, FL: Krieger, 1982.zbMATHGoogle Scholar
  7. [7]
    H. Górecki and A.Korytowski (eds.), Advances in optimization and stability analysis of dynamical systems, Cracow, Academy of Mining and Metallurgy Publishers, 1993.Google Scholar
  8. [8]
    D. Redfern, Maple Handbook, New York, Springer Verlag, 1993.zbMATHGoogle Scholar
  9. [9]
    I.G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford, Clarendon Press, 1979.zbMATHGoogle Scholar
  10. [10]
    J.R. Stembridge, A Maple package for symmetric functions, Department of Mathematics, Ann Arbor, MI, June 1989.Google Scholar

Copyright information

© Birkhäuser Verlag Basel 1996

Authors and Affiliations

  • Henryk Górecki
    • 1
  • Maciej Szymkat
    • 1
  • Mieczyslaw Zaczyk
    • 1
  1. 1.Institute of AutomaticsSt. Staszic Technical UniversityKrakówPoland

Personalised recommendations