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Root-Finding with Implicit Deflation

  • Rémi Imbach
  • Victor Y. PanEmail author
  • Chee Yap
  • Ilias S. Kotsireas
  • Vitaly Zaderman
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11661)

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.

Keywords

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

2000 Math. Subject Classification:

26C10 30C15 65H05 

Notes

Acknowledgements

The research of R. Inbach, V. Y. Pan, C. Yap, and V. Zaderman was supported by NSF Grant CCF–1563942. The research of V. Y. Pan and V. Zaderman was also supported by NSF Grants CCF 1116736 and PSC CUNY Award 69813 00 48. The research of Ilias Kotsireas was supported by an NSERC grant.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Rémi Imbach
    • 1
  • Victor Y. Pan
    • 2
    • 3
    Email author
  • Chee Yap
    • 1
  • Ilias S. Kotsireas
    • 4
  • Vitaly Zaderman
    • 3
  1. 1.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA
  2. 2.Department of Computer ScienceLehman College of the City University of New YorkBronxUSA
  3. 3.Ph.D. Programs in Mathematics and Computer ScienceThe Graduate Center of the City University of New YorkNew YorkUSA
  4. 4.Department of Physics and Computer ScienceWilfrid Laurier UniversityWaterlooCanada

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