Institute of Mathematics and Informatics Bulgarian Academy of Sciences
Serdica Journal of Computing, Vol. 12, No 4, (2018), 227p-246p
In this paper we present Lagrange's Joseph-Louis Lagrange,
born Giuseppe Lodovico Lagrangia (25 January 1736 - 10 April 1813):
Italian mathematician. 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-Strzeboński Continued Fractions
(VAS-CF) real root isolation method, the latter becomes
measurably slower for some classes of polynomials.