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inria-00116985, version 5

A note on the complexity of univariate root isolation

Ioannis Emiris () a1, Elias P. P. Tsigaridas () 2

N° RR-6043 (2006)

Abstract: This paper presents the average-case bit complexity of subdivision-based univariate solvers, namely those named after Sturm, Descartes, and Bernstein. By real solving we mean real root isolation. We prove bounds of $\sOB(N^5)$ for all methods, where $N$ bounds the polynomial degree and the coefficient bitsize, whereas their worst-case complexity is in $\sOB(N^6)$. In the case of the Sturm solver, our bound depends on the number of real roots. Our work is a step towards understanding the practical complexity of real root isolation. This enables a better juxtaposition against numerical solvers, the latter having worst-case complexity in $\sOB(N^4)$. % Our approach extends to complex root isolation, where we offer a simple proof leading to bounds % for the number of steps that the subdivision algorithm performs on the worst and average-case complexities of $\sOB(N^7 )$ and $\sOB(N^6)$ respectively, where the latter is output-sensitive.

  • a –  National Kapodistrian University of Athens
  • 1:  Department of Informatics and Telecomunications (DI)
  • (ERGA) Laboratory of algebraic and geometric algorithms and applications – National Kapodistrian University of Athens
  • 2:  GEOMETRICA (INRIA Sophia Antipolis)
  • INRIA
 
  • inria-00116985, version 5
  • oai:hal.inria.fr:inria-00116985
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  • Submitted on: Sunday, 10 December 2006 18:05:28
  • Updated on: Wednesday, 1 October 2008 11:50:34