×

Univariate polynomial real root isolation: Continued fractions revisited. (English) Zbl 1131.68596

Azar, Yossi (ed.) et al., Algorithms – ESA 2006. 14th annual European symposium, Zurich, Switzerland, September 11–13, 2006. Proceedings. Berlin: Springer (ISBN 978-3-540-38875-3/pbk). Lecture Notes in Computer Science 4168, 817-828 (2006).
Summary: We present algorithmic, complexity and implementation results concerning real root isolation of integer univariate polynomials using the continued fraction expansion of real numbers. We improve the previously known bound by a factor of \(d \tau \), where \(d\) is the polynomial degree and \(\tau \) bounds the coefficient bitsize, thus matching the current record complexity for real root isolation by exact methods. Namely, the complexity bound is \({{\widetilde{\mathcal{O}}_B}(d^4 \tau^2)}\) 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 illustrate its efficiency as compared to other available software. We use polynomials with coefficient bitsize up to 8000 and degree up to 1000.
For the entire collection see [Zbl 1130.68002].

MSC:

68W30 Symbolic computation and algebraic computation
68Q25 Analysis of algorithms and problem complexity

Software:

ISOLATE; na20; SYNAPS
PDFBibTeX XMLCite
Full Text: DOI arXiv