Vincent's theorem of 1836: overview and future research

In this paper, we 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 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. Bibl. – 45 titles.