Summary: Digital Object Identifier (DOI) 10.1007/s00607-006-0186-y
Computing 78, 355367 (2006)
Implementations of a New Theorem for Computing Bounds
for Positive Roots of Polynomials
A. Akritas, Volos, A. StrzeboŽnski, Champaign, and P. Vigklas, Volos
Received August 24, 2006; revised October 16, 2006
Published online: December 5, 2006
© Springer-Verlag 2006
Finding an upper bound for the positive roots of univariate polynomials is an important step of the
continued fractions real root isolation algorithm. The revived interest in this algorithm has highlighted
the need for better estimations of upper bounds of positive roots. In this paper we present a new theorem,
based on a generalization of a theorem by D. Stefanescu, and describe several implementations of it
including Cauchy's and Kioustelidis' rules as well as two new rules recently developed by us. From
the empirical results presented here we see that applying various implementations of our theorem and
taking the minimum of the computed values greatly improves the estimation of the upper bound and
hopefully that will affect the performance of the continued fractions real root isolation method.
AMS Subject Classifications: 65H05, 68W30, 26C10.
Keywords: Upper bounds, positive roots, polynomial real root isolation.