A simple but exact and efficient algorithm for complex root isolation

Chee Yap, Michael Sagraloff

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

Abstract

We present a new exact subdivision algorithm CEVAL for isolating the complex roots of a square-free polynomial in any given box. It is a generalization of a previous real root isolation algorithm called EVAL. Under suitable conditions, our approach is applicable for general analytic functions. CEVAL is based on the simple Bolzano Principle and is easy to implement exactly. Preliminary experiments have shown its competitiveness. We further show that, for the "benchmark problem" of isolating all roots of a square-free polynomial with integer coefficients, the asymptotic complexity of both algorithms EVAL and CEVAL matches (up a logarithmic term) that of more sophisticated real root isolation methods which are based on Descartes' Rule of Signs, Continued Fraction or Sturm sequence. In particular, we show that the tree size of EVAL matches that of other algorithms. Our analysis is based on a novel technique called Δ-clusters from which we expect to see further applications.

Original languageEnglish (US)
Title of host publicationISSAC 2011 - Proceedings of the 36th International Symposium on Symbolic and Algebraic Computation
Pages353-360
Number of pages8
DOIs
StatePublished - 2011
Event36th International Symposium on Symbolic and Algebraic Computation, ISSAC 2011 - San Jose, CA, United States
Duration: Jun 8 2011Jun 11 2011

Other

Other36th International Symposium on Symbolic and Algebraic Computation, ISSAC 2011
CountryUnited States
CitySan Jose, CA
Period6/8/116/11/11

Fingerprint

Exact Algorithms
Isolation
Real Roots
Square free
Efficient Algorithms
Roots
Sturm Sequence
Subdivision Algorithm
René Descartes
Polynomial
Competitiveness
Continued fraction
Analytic function
Logarithmic
Benchmark
Integer
Coefficient
Term
Experiment

Keywords

  • bolzano methods
  • complexity of complex root isolation
  • evaluation-based root isolation
  • exact root isolation
  • subdivision algorithms

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Yap, C., & Sagraloff, M. (2011). A simple but exact and efficient algorithm for complex root isolation. In ISSAC 2011 - Proceedings of the 36th International Symposium on Symbolic and Algebraic Computation (pp. 353-360) https://doi.org/10.1145/1993886.1993938

A simple but exact and efficient algorithm for complex root isolation. / Yap, Chee; Sagraloff, Michael.

ISSAC 2011 - Proceedings of the 36th International Symposium on Symbolic and Algebraic Computation. 2011. p. 353-360.

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

Yap, C & Sagraloff, M 2011, A simple but exact and efficient algorithm for complex root isolation. in ISSAC 2011 - Proceedings of the 36th International Symposium on Symbolic and Algebraic Computation. pp. 353-360, 36th International Symposium on Symbolic and Algebraic Computation, ISSAC 2011, San Jose, CA, United States, 6/8/11. https://doi.org/10.1145/1993886.1993938
Yap C, Sagraloff M. A simple but exact and efficient algorithm for complex root isolation. In ISSAC 2011 - Proceedings of the 36th International Symposium on Symbolic and Algebraic Computation. 2011. p. 353-360 https://doi.org/10.1145/1993886.1993938
Yap, Chee ; Sagraloff, Michael. / A simple but exact and efficient algorithm for complex root isolation. ISSAC 2011 - Proceedings of the 36th International Symposium on Symbolic and Algebraic Computation. 2011. pp. 353-360
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