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 - 2012
Event37th International Symposium on Symbolic and Algebraic Computation, ISSAC 2012 - Grenoble, France
Duration: Jul 22 2012Jul 25 2012

Other

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

Fingerprint

Real Roots
Isolation
Interval
Bisect
Ray Shooting
Roots
Square free
G-function
Exact Algorithms
Absolute value
Recursion
Numerical Algorithms
Predicate
Range of data
Polynomial
Integer
Coefficient

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) https://doi.org/10.1145/2442829.2442875

Near optimal tree size bounds on a simple real root isolation algorithm. / Sharma, Vikram; Yap, Chee K.

ISSAC 2012 - Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation. 2012. p. 319-326.

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

Sharma, V & Yap, CK 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, 37th International Symposium on Symbolic and Algebraic Computation, ISSAC 2012, Grenoble, France, 7/22/12. https://doi.org/10.1145/2442829.2442875
Sharma V, Yap CK. 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. 2012. p. 319-326 https://doi.org/10.1145/2442829.2442875
Sharma, Vikram ; Yap, Chee K. / Near optimal tree size bounds on a simple real root isolation algorithm. ISSAC 2012 - Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation. 2012. pp. 319-326
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