Sturm's Theorem - History Section and Other Related Methods

History Section and Other Related Methods

For counting and isolating the real roots, other methods are usually preferred, because they are computationally more efficient; these methods all use Descartes' rule of signs (just like Fourier did back in 1820) and Vincent's theorem. Interestingly, the very first one of those methods was initially called "modified Uspensky's algorithm" by its inventors, but it was later shown that there is no Uspensky's method; afterwards, people started calling it either "Collins-Akritas method" or "Descartes' method" only to be shown that there is no Descartes' method either. Finally, François Boulier, of the University of Lille, p. 24, gave it the name "Vincent-Collins-Akritas" (VCA for short) to also give credit to Vincent. VCA is a bisection method; there exists also a continued fractions method based on Vincent's theorem namely, the Vincent-Akritas-Strzeboński (VAS) method.

VAS is based on Budan's theorem whereas Sturm's method has been inspired by Fourier's theorem. In fact Sturm himself, p. 108, acknowledges the great influence Fourier's theorem had on him: « C'est en m'appuyant sur les principes qu'il a posés, et en imitant ses démonstrations, que j'ai trouvé les nouveaux théorèmes que je vais énoncer. » which translates to "It is by relying upon the principles he has laid out and by imitating his proofs that I have found the new theorems which I am about to announce."