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Nikola Obreschkoff's contribution to the problem of isolating real roots of polynomials with continued fractions
 

Summary: Nikola Obreschkoff's contribution to the problem of isolating real
roots of polynomials with continued fractions
Alkiviadis G. Akritas and Panagiotis S. Vigklas
{akritas, pviglas}@uth.gr
Department of Computer and Communication Engineering
University of Thessaly
Volos, Greece
Presented by Alkiviadis G. Akritas in Varna, Bulgaria, on the occasion of
Obreschkoff's 110 anniversary
Abstract: In this talk we first mention some key facts of Obreschkoff's life and work and then delve into the
influence of Obreschkoff's book Verteilung und Berechnung der Nullstellen reeller Polynome, VEB Deutscher
Verlag der Wissenschaften, Berlin, 1963, on the real root isolation problem.
Obreschkoff is one of only two authors in the literature (Uspensky being the other one) who mention Vincent's
theorem of 1836. This theorem is the core of the continued fractions (CF) real root isolation method, which turns out
to be the fastest method known today. A little known "secret" about this method is that it can be implemented only if
we are able to compute "efficient" bounds on the values of the positive roots of polynomials -- as opposed to bounds
on the absolute values of the roots. Surprisingly, very little is still known about such positive root bounds. There-
fore, it is remarkable that Obreschkoff is the only author we know of to have included Cauchy's theorem -- for
computing positive root bounds -- in the above mentioned book of his that came out in 1963, the year he died.
Without Cauchy's theorem, the implementation of the CF algorithm would have been impossible and without

  

Source: Akritas, Alkiviadis G. - Department of Computer and Communication Engineering, University of Thessaly

 

Collections: Computer Technologies and Information Sciences