Variational analysis of functions of the roots of polynomials

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

Research output: Contribution to journalArticle

Abstract

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.

Original languageEnglish (US)
Pages (from-to)263-292
Number of pages30
JournalMathematical Programming
Volume104
Issue number2-3
DOIs
StatePublished - Nov 2005

Fingerprint

Variational Analysis
Polynomials
Roots
Subdifferential
Polynomial
Abscissa
Gauss
Radius
Dynamical systems
Discrete-time Dynamical Systems
Linear Dynamical Systems
Modulus
Simplify
Methodology
Theorem

ASJC Scopus subject areas

  • Applied Mathematics
  • Mathematics(all)
  • Safety, Risk, Reliability and Quality
  • Management Science and Operations Research
  • Software
  • Computer Graphics and Computer-Aided Design
  • Computer Science(all)

Cite this

Variational analysis of functions of the roots of polynomials. / Burke, James V.; Lewis, Adrian S.; Overton, Michael L.

In: Mathematical Programming, Vol. 104, No. 2-3, 11.2005, p. 263-292.

Research output: Contribution to journalArticle

Burke, James V. ; Lewis, Adrian S. ; Overton, Michael L. / Variational analysis of functions of the roots of polynomials. In: Mathematical Programming. 2005 ; Vol. 104, No. 2-3. pp. 263-292.
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