### 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 language | English (US) |
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Title of host publication | Computer Algebra in Scientific Computing - 21st International Workshop, CASC 2019, Proceedings |

Editors | Evgenii V. Vorozhtsov, Timur M. Sadykov, Werner M. Seiler, Wolfram Koepf, Matthew England |

Publisher | Springer-Verlag |

Pages | 236-245 |

Number of pages | 10 |

ISBN (Print) | 9783030268305 |

DOIs | |

State | Published - Jan 1 2019 |

Event | 21st International Workshop on Computer Algebra in Scientific Computing, CASC 2019 - Moscow, Russian Federation Duration: Aug 26 2019 → Aug 30 2019 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 11661 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 21st International Workshop on Computer Algebra in Scientific Computing, CASC 2019 |
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Country | Russian Federation |

City | Moscow |

Period | 8/26/19 → 8/30/19 |

### Fingerprint

### 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

*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.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*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

}

TY - GEN

T1 - Root-Finding with Implicit Deflation

AU - Imbach, Rémi

AU - Pan, Victor Y.

AU - Yap, Chee

AU - Kotsireas, Ilias S.

AU - Zaderman, Vitaly

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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.

AB - 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.

KW - Deflation

KW - Efficiency

KW - Ehrlich’s iterations

KW - Functional iterations

KW - Maps of the variable

KW - Newton’s iterations

KW - Polynomial roots

KW - Taming wild roots

KW - Weierstrass’s iterations

UR - http://www.scopus.com/inward/record.url?scp=85071504563&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85071504563&partnerID=8YFLogxK

U2 - 10.1007/978-3-030-26831-2_16

DO - 10.1007/978-3-030-26831-2_16

M3 - Conference contribution

SN - 9783030268305

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 236

EP - 245

BT - Computer Algebra in Scientific Computing - 21st International Workshop, CASC 2019, Proceedings

A2 - Vorozhtsov, Evgenii V.

A2 - Sadykov, Timur M.

A2 - Seiler, Werner M.

A2 - Koepf, Wolfram

A2 - England, Matthew

PB - Springer-Verlag

ER -