Amortized bound for root isolation via sturm sequences

Zilin Du, Vikram Sharma, Chee K. Yap

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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 0(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.

Original languageEnglish (US)
Title of host publicationSymbolic-Numeric Computation
PublisherSpringer International Publishing
Pages113-129
Number of pages17
Volume41
ISBN (Print)9783764379834
StatePublished - 2007
EventInternational Workshop on Symbolic-Numeric Computation, SNC 2005 - Xian, China
Duration: Jul 19 2005Jul 21 2005

Publication series

NameTrends in Mathematics
Volume41
ISSN (Print)22970215
ISSN (Electronic)2297024X

Other

OtherInternational Workshop on Symbolic-Numeric Computation, SNC 2005
CountryChina
CityXian
Period7/19/057/21/05

Fingerprint

Sturm Sequence
Isolation
Real Roots
Roots
Subresultants
Square free
Notation
Logarithmic
Probe
Polynomial
Integer
Necessary
Evaluate
Coefficient

Keywords

  • Complexity
  • Davenport-Mahler bound
  • Root isolation
  • Separation bound
  • Sturm sequence
  • Subresultant

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Du, Z., Sharma, V., & Yap, C. K. (2007). Amortized bound for root isolation via sturm sequences. In Symbolic-Numeric Computation (Vol. 41, pp. 113-129). (Trends in Mathematics; Vol. 41). Springer International Publishing.

Amortized bound for root isolation via sturm sequences. / Du, Zilin; Sharma, Vikram; Yap, Chee K.

Symbolic-Numeric Computation. Vol. 41 Springer International Publishing, 2007. p. 113-129 (Trends in Mathematics; Vol. 41).

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Du, Z, Sharma, V & Yap, CK 2007, Amortized bound for root isolation via sturm sequences. in Symbolic-Numeric Computation. vol. 41, Trends in Mathematics, vol. 41, Springer International Publishing, pp. 113-129, International Workshop on Symbolic-Numeric Computation, SNC 2005, Xian, China, 7/19/05.
Du Z, Sharma V, Yap CK. Amortized bound for root isolation via sturm sequences. In Symbolic-Numeric Computation. Vol. 41. Springer International Publishing. 2007. p. 113-129. (Trends in Mathematics).
Du, Zilin ; Sharma, Vikram ; Yap, Chee K. / Amortized bound for root isolation via sturm sequences. Symbolic-Numeric Computation. Vol. 41 Springer International Publishing, 2007. pp. 113-129 (Trends in Mathematics).
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