Vincent's theorem of 1836: Overview and future research

Abstract

In this paper, we present two different versions of Vincent's theorem of 1836 and discuss various real root isolation methods derived from them: one using continued fractions and two using bisections, the former being the fastest real root isolation method. Regarding the continued fractions method, we first 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, and then we indicate directions for future research. Bibliography: 45 titles. © 2010 Springer Science+Business Media, Inc.