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Univariate Polynomial Real Root Isolation: Continued Fractions Revisited

Summary: Univariate Polynomial Real Root Isolation:
Continued Fractions Revisited
Elias P. Tsigaridas and Ioannis Z. Emiris
Department of Informatics and Telecommunications,
National Kapodistrian University of Athens, HELLAS
{et, emiris}@di.uoa.gr
Abstract. We present algorithmic, complexity and implementation re-
sults concerning real root isolation of integer univariate polynomials us-
ing the continued fraction expansion of real numbers. We improve the
previously known bound by a factor of d , where d is the polynomial
degree and bounds the coefficient bitsize, thus matching the current
record complexity for real root isolation by exact methods. Namely, the
complexity bound is OB(d4
) using a standard bound on the expected
bitsize of the integers in the continued fraction expansion. We show how
to compute the multiplicities within the same complexity and extend the
algorithm to non square-free polynomials. Finally, we present an efficient
open-source C++ implementation in the algebraic library synaps, and il-
lustrate its efficiency as compared to other available software. We use


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


Collections: Computer Technologies and Information Sciences