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Vincent's Theorem of 1836: Overview and Recent Developments1

Summary: Vincent's Theorem of 1836:
Overview and Recent Developments1
Alkiviadis G. Akritas
Department of Computer and Communication Engineering
University of Thessaly
To the Students of the University of Thessaly
Abstract In this paper 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. Subsequently we concentrate on the Continued Fractions method and
show how -- using a recently developed quadratic complexity bound on the values of
the positive roots of polynomials -- its performance has been improved by an average
of 40%, over its initial implementation.
Key Words: Vincent's theorem, isolation of the real roots, real root isolation meth-
ods, bisection methods, continued fractions method, positive root bounds.
1 Introduction
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


Source: Akritas, Alkiviadis G. - Department of Computer and Communication Engineering, University of Thessaly


Collections: Computer Technologies and Information Sciences