Complexity analysis of root clustering for a complex polynomial

Ruben Becker, Michael Sagraloff, Vikram Sharma, Juan Xu, Chee Yap

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

Abstract

Let F(z) be an arbitrary complex polynomial. We introduce the local root clustering problem, to compute a set of natural 7epsi;-clusters of roots of F(z) in some box region B0 in the complex plane. This may be viewed as an extension of the classical root isolation problem. Our contribution is twofold: we provide an efficient certified subdivision algorithm for this problem, and we provide a bit-complexity analysis based on the local geometry of the root clusters. Our computational model assumes that arbitrarily good approximations of the coefficients of F are provided by means of an oracle at the cost of reading the coefficients. Our algorithmic techniques come from a companion paper [3] and are based on the Pellet test, Graeffe and Newton iterations, and are independent of Schonhage's splitting circle method. Our algorithm is relatively simple and promises to be efficient in practice.

Original languageEnglish (US)
Title of host publicationISSAC 2016 - Proceedings of the 2016 ACM International Symposium on Symbolic and Algebraic Computation
PublisherAssociation for Computing Machinery
Pages71-78
Number of pages8
Volume20-22-July-2016
ISBN (Electronic)9781450343800
DOIs
StatePublished - Jul 20 2016
Event41st ACM International Symposium on Symbolic and Algebraic Computation, ISSAC 2016 - Waterloo, Canada
Duration: Jul 20 2016Jul 22 2016

Other

Other41st ACM International Symposium on Symbolic and Algebraic Computation, ISSAC 2016
CountryCanada
CityWaterloo
Period7/20/167/22/16

Fingerprint

Complex Polynomials
Complexity Analysis
Roots
Clustering
Circle Method
Subdivision Algorithm
Newton Iteration
Splitting Method
Coefficient
Computational Model
Argand diagram
Isolation
Efficient Algorithms
Arbitrary
Approximation

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Becker, R., Sagraloff, M., Sharma, V., Xu, J., & Yap, C. (2016). Complexity analysis of root clustering for a complex polynomial. In ISSAC 2016 - Proceedings of the 2016 ACM International Symposium on Symbolic and Algebraic Computation (Vol. 20-22-July-2016, pp. 71-78). Association for Computing Machinery. https://doi.org/10.1145/2930889.2930939

Complexity analysis of root clustering for a complex polynomial. / Becker, Ruben; Sagraloff, Michael; Sharma, Vikram; Xu, Juan; Yap, Chee.

ISSAC 2016 - Proceedings of the 2016 ACM International Symposium on Symbolic and Algebraic Computation. Vol. 20-22-July-2016 Association for Computing Machinery, 2016. p. 71-78.

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

Becker, R, Sagraloff, M, Sharma, V, Xu, J & Yap, C 2016, Complexity analysis of root clustering for a complex polynomial. in ISSAC 2016 - Proceedings of the 2016 ACM International Symposium on Symbolic and Algebraic Computation. vol. 20-22-July-2016, Association for Computing Machinery, pp. 71-78, 41st ACM International Symposium on Symbolic and Algebraic Computation, ISSAC 2016, Waterloo, Canada, 7/20/16. https://doi.org/10.1145/2930889.2930939
Becker R, Sagraloff M, Sharma V, Xu J, Yap C. Complexity analysis of root clustering for a complex polynomial. In ISSAC 2016 - Proceedings of the 2016 ACM International Symposium on Symbolic and Algebraic Computation. Vol. 20-22-July-2016. Association for Computing Machinery. 2016. p. 71-78 https://doi.org/10.1145/2930889.2930939
Becker, Ruben ; Sagraloff, Michael ; Sharma, Vikram ; Xu, Juan ; Yap, Chee. / Complexity analysis of root clustering for a complex polynomial. ISSAC 2016 - Proceedings of the 2016 ACM International Symposium on Symbolic and Algebraic Computation. Vol. 20-22-July-2016 Association for Computing Machinery, 2016. pp. 71-78
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