Mathematical Programming

, Volume 104, Issue 2, pp 263–292

Variational analysis of functions of the roots of polynomials

  • James V. Burke
  • Adrian S. Lewis
  • Michael L. Overton

DOI: 10.1007/s10107-005-0616-1

Cite this article as:
Burke, J., Lewis, A. & Overton, M. Math. Program. (2005) 104: 263. doi:10.1007/s10107-005-0616-1


The Gauss-Lucas Theorem on the roots of polynomials nicely simplifies the computation of the subderivative and regular subdifferential of the abscissa mapping on polynomials (the maximum of the real parts of the roots). This paper extends this approach to more general functions of the roots. By combining the Gauss-Lucas methodology with an analysis of the splitting behavior of the roots, we obtain characterizations of the subderivative and regular subdifferential for these functions as well. In particular, we completely characterize the subderivative and regular subdifferential of the radius mapping (the maximum of the moduli of the roots). The abscissa and radius mappings are important for the study of continuous and discrete time linear dynamical systems.

Mathematics Subject Classification (1991)


Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • James V. Burke
    • 1
  • Adrian S. Lewis
    • 2
  • Michael L. Overton
    • 3
  1. 1.Department of MathematicsUniversity of WashingtonSeattleUSA
  2. 2.School of Operations Research and Industrial EngineeringCornell UniversityIthacaUSA
  3. 3.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA