Amortized Bound for Root Isolation via Sturm Sequences

  • Zilin Du
  • Vikram Sharma
  • Chee K. Yap
Conference paper

DOI: 10.1007/978-3-7643-7984-1_8

Part of the Trends in Mathematics book series (TM)
Cite this paper as:
Du Z., Sharma V., Yap C.K. (2007) Amortized Bound for Root Isolation via Sturm Sequences. In: Wang D., Zhi L. (eds) Symbolic-Numeric Computation. Trends in Mathematics. Birkhäuser Basel

Abstract

This paper presents two results on the complexity of root isolation via Sturm sequences. Both results exploit amortization arguments.

For a square-free polynomial A (X) of degree d with L-bit integer coefficients, we use an amortization argument to show that all the roots, real or complex, can be isolated using at most O(dL + dlgd) Sturm probes. This extends Davenport’s result for the case of isolating all real roots.

We also show that a relatively straightforward algorithm, based on the classical subresultant PQS, allows us to evaluate the Sturm sequence of A(X) at rational Õ(dL)-bit values in time Õ(d3L); here the Õ-notation means we ignore logarithmic factors. Again, an amortization argument is used. We provide a family of examples to show that such amortization is necessary.

Keywords

Sturm sequence Davenport-Mahler bound subresultant complexity root isolation separation bound 

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Copyright information

© Birkhäuser Verlag Basel/Switzerland 2007

Authors and Affiliations

  • Zilin Du
    • 1
  • Vikram Sharma
    • 1
  • Chee K. Yap
    • 1
  1. 1.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA

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