Summary: Vincent's Theorem of 1836:
Overview and Future Research1
Alkiviadis G. Akritas
Department of Computer and Communication Engineering
University of Thessaly
Abstract We first present the two different versions of Vincent's theorem of 1836 and
discuss the various real root isolation methods derived from them: one using continued
fractions and two bisection methods -- the former being the fastest real root isolation
method. We then concentrate on the Continued Fractions method and present: (a)
a recently developed quadratic complexity bound on the values of the positive roots
of polynomials, which helped improve its performance by an average of 40%, over its
initial implementation, and (b) directions for future research.
Key Words: Vincent's theorem, isolation of the real roots, real root isolation meth-
ods, bisection methods, continued fractions method, positive root bounds.
Isolation of the real roots of a polynomial is the process of finding real disjoint
intervals such that each contains one real root and every real root is contained
in some interval.