Lagrange's Bound on the Values of the Positive Roots of Polynomials

Abstract

In this paper we present Lagrange's theorem of 1767 for computing a bound on the values of the positive roots of polynomials, along with its interesting history and a short proof of it dating back to 1842. Since the bound obtained by Lagrange's theorem is of linear complexity, in the sequel it is called "Lagrange Linear", or LL for short. Despite its average good performance, LL is endowed with the weaknesses inherent in all bounds with linear complexity and, therefore, the values obtained by it can be much bigger than those obtained by our own bound "Local Max Quadratic", or LMQ for short. To level the playing field, we incorporate Lagrange's theorem into our LMQ and we present the new bound "Lagrange Quadratic", or LQ for short, the quadratic complexity version of LL. It turns out that LQ is one of the most efficient bounds available since, at best, the values obtained by it are half of those obtained by LMQ. Empirical results indicate that when LQ replaces LMQ in the Vincent-Akritas-Strzebonski Continued Fractions (VAS-CF) real root isolation method, the latter becomes measurably slower for some classes of polynomials.