### 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) |
---|---|

Title of host publication | Computer Algebra in Scientific Computing - 21st International Workshop, CASC 2019, Proceedings |

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

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) |
---|---|

Volume | 11661 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 21st International Workshop on Computer Algebra in Scientific Computing, CASC 2019 |
---|---|

Country | Russian Federation |

City | Moscow |

Period | 8/26/19 → 8/30/19 |

### 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)

## Fingerprint Dive into the research topics of 'Root-Finding with Implicit Deflation'. Together they form a unique fingerprint.

## 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