Near optimal tree size bounds on a simple real root isolation algorithm

Vikram Sharma, Chee K. Yap

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

Abstract

The problem of isolating all real roots of a square-free integer polynomial f(X) inside any given interval I0 is a fundamental problem. EVAL is a simple and practical exact numerical algorithm for this problem: it recursively bisects I0, and any sub-interval I ⊆ I0, until a certain numerical predicate C0(I) ⊆ C1(I) holds on each I. We prove that the size of the recursion tree is O(d(L + r + log d)) where f has degree d, its coefficients have absolute values < 2L, and I0 contains r roots of f. In the range L ≥ d, our bound is the sharpest known, and provably optimal. Our results are closely paralleled by recent bounds on EVAL by Sagraloff-Yap (ISSAC 2011) and Burr-Krahmer (2012). In the range L ≤ d, our bound is incomparable with those of Sagraloff-Yap or Burr-Krahmer. Similar to the Burr-Krahmer proof, we exploit the technique of "continuous amortization" from Burr-Krahmer-Yap (2009), namely to bound the tree size by an integral I0 G(x)dx over a suitable "charging function" G(x). We give an application of this feature to the problem of ray-shooting (i.e., finding smallest root in a given interval).

Original languageEnglish (US)
Title of host publicationISSAC 2012 - Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation
Pages319-326
Number of pages8
DOIs
StatePublished - Dec 1 2012
Event37th International Symposium on Symbolic and Algebraic Computation, ISSAC 2012 - Grenoble, France
Duration: Jul 22 2012Jul 25 2012

Publication series

NameProceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC

Other

Other37th International Symposium on Symbolic and Algebraic Computation, ISSAC 2012
CountryFrance
CityGrenoble
Period7/22/127/25/12

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Keywords

  • Continuous amortization
  • Integral analysis
  • Real root isolation
  • Subdivision algorithm

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Sharma, V., & Yap, C. K. (2012). Near optimal tree size bounds on a simple real root isolation algorithm. In ISSAC 2012 - Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation (pp. 319-326). (Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC). https://doi.org/10.1145/2442829.2442875