Root-Finding with Implicit Deflation

Rémi Imbach, Victor Y. Pan, Chee Yap, Ilias S. Kotsireas, Vitaly Zaderman

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

Abstract

Functional iterations such as Newton’s are a popular tool for polynomial root-finding. We consider realistic situation where some (e.g., better-conditioned) roots have already been approximated and where further computations is directed to the approximation of the remaining roots. Such a situation is also realistic for root by means of subdivision iterations. A natural approach of applying explicit deflation has been much studied and recently advanced by one of the authors of this paper, but presently we consider the alternative of implicit deflation combined with the mapping of the variable and reversion of an input polynomial. We also show another unexplored direction for substantial further progress in this long and extensively studied area. Namely we dramatically increase the local efficiency of root-finding by means of the incorporation of fast algorithms for multipoint polynomial evaluation and Fast Multipole Method.

Original languageEnglish (US)
Title of host publicationComputer Algebra in Scientific Computing - 21st International Workshop, CASC 2019, Proceedings
EditorsEvgenii V. Vorozhtsov, Timur M. Sadykov, Werner M. Seiler, Wolfram Koepf, Matthew England
PublisherSpringer-Verlag
Pages236-245
Number of pages10
ISBN (Print)9783030268305
DOIs
StatePublished - Jan 1 2019
Event21st International Workshop on Computer Algebra in Scientific Computing, CASC 2019 - Moscow, Russian Federation
Duration: Aug 26 2019Aug 30 2019

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume11661 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

Conference21st International Workshop on Computer Algebra in Scientific Computing, CASC 2019
CountryRussian Federation
CityMoscow
Period8/26/198/30/19

Fingerprint

Deflation
Root-finding
Polynomials
Roots
Polynomial Evaluation
Polynomial Roots
Iteration
Fast multipole Method
Subdivision
Fast Algorithm
Polynomial
Alternatives
Approximation

Keywords

  • Deflation
  • Efficiency
  • Ehrlich’s iterations
  • Functional iterations
  • Maps of the variable
  • Newton’s iterations
  • Polynomial roots
  • Taming wild roots
  • Weierstrass’s iterations

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)

Cite this

Imbach, R., Pan, V. Y., Yap, C., Kotsireas, I. S., & Zaderman, V. (2019). Root-Finding with Implicit Deflation. In E. V. Vorozhtsov, T. M. Sadykov, W. M. Seiler, W. Koepf, & M. England (Eds.), Computer Algebra in Scientific Computing - 21st International Workshop, CASC 2019, Proceedings (pp. 236-245). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 11661 LNCS). Springer-Verlag. https://doi.org/10.1007/978-3-030-26831-2_16

Root-Finding with Implicit Deflation. / Imbach, Rémi; Pan, Victor Y.; Yap, Chee; Kotsireas, Ilias S.; Zaderman, Vitaly.

Computer Algebra in Scientific Computing - 21st International Workshop, CASC 2019, Proceedings. ed. / Evgenii V. Vorozhtsov; Timur M. Sadykov; Werner M. Seiler; Wolfram Koepf; Matthew England. Springer-Verlag, 2019. p. 236-245 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 11661 LNCS).

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

Imbach, R, Pan, VY, Yap, C, Kotsireas, IS & Zaderman, V 2019, Root-Finding with Implicit Deflation. in EV Vorozhtsov, TM Sadykov, WM Seiler, W Koepf & M England (eds), Computer Algebra in Scientific Computing - 21st International Workshop, CASC 2019, Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 11661 LNCS, Springer-Verlag, pp. 236-245, 21st International Workshop on Computer Algebra in Scientific Computing, CASC 2019, Moscow, Russian Federation, 8/26/19. https://doi.org/10.1007/978-3-030-26831-2_16
Imbach R, Pan VY, Yap C, Kotsireas IS, Zaderman V. Root-Finding with Implicit Deflation. In Vorozhtsov EV, Sadykov TM, Seiler WM, Koepf W, England M, editors, Computer Algebra in Scientific Computing - 21st International Workshop, CASC 2019, Proceedings. Springer-Verlag. 2019. p. 236-245. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). https://doi.org/10.1007/978-3-030-26831-2_16
Imbach, Rémi ; Pan, Victor Y. ; Yap, Chee ; Kotsireas, Ilias S. ; Zaderman, Vitaly. / Root-Finding with Implicit Deflation. Computer Algebra in Scientific Computing - 21st International Workshop, CASC 2019, Proceedings. editor / Evgenii V. Vorozhtsov ; Timur M. Sadykov ; Werner M. Seiler ; Wolfram Koepf ; Matthew England. Springer-Verlag, 2019. pp. 236-245 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).
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