### Abstract

We construct a family of high order iteration functions for finding polynomial roots of a known multiplicity s. This family is a generalization of a fundamental family of high order algorithms for simple roots that dates back to Schröder's 1870 paper. It starts with the well known variant of Newton's method B̂_{2}(x) = x - s • p(x)/p'(x) and the multiple root counterpart of Halley's method derived by Hansen and Patrick. Our approach demonstrates the relevance and power of algebraic combinatorial techniques in studying rational root-finding iteration functions.

Original language | English (US) |
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Pages (from-to) | 1897-1906 |

Number of pages | 10 |

Journal | Proceedings of the American Mathematical Society |

Volume | 138 |

Issue number | 6 |

DOIs | |

State | Published - Jun 1 2010 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

### Keywords

- Generating functions
- Iteration functions
- Multiple roots
- Recurrence relation
- Root-finding
- Symmetric functions

### Cite this

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**A combinatorial construction of high order algorithms for finding polynomial roots of known multiplicity.** / Jin, Yi; Kalantari, Bahman.

Research output: Contribution to journal › Article

TY - JOUR

T1 - A combinatorial construction of high order algorithms for finding polynomial roots of known multiplicity

AU - Jin, Yi

AU - Kalantari, Bahman

PY - 2010/6/1

Y1 - 2010/6/1

N2 - We construct a family of high order iteration functions for finding polynomial roots of a known multiplicity s. This family is a generalization of a fundamental family of high order algorithms for simple roots that dates back to Schröder's 1870 paper. It starts with the well known variant of Newton's method B̂2(x) = x - s • p(x)/p'(x) and the multiple root counterpart of Halley's method derived by Hansen and Patrick. Our approach demonstrates the relevance and power of algebraic combinatorial techniques in studying rational root-finding iteration functions.

AB - We construct a family of high order iteration functions for finding polynomial roots of a known multiplicity s. This family is a generalization of a fundamental family of high order algorithms for simple roots that dates back to Schröder's 1870 paper. It starts with the well known variant of Newton's method B̂2(x) = x - s • p(x)/p'(x) and the multiple root counterpart of Halley's method derived by Hansen and Patrick. Our approach demonstrates the relevance and power of algebraic combinatorial techniques in studying rational root-finding iteration functions.

KW - Generating functions

KW - Iteration functions

KW - Multiple roots

KW - Recurrence relation

KW - Root-finding

KW - Symmetric functions

UR - http://www.scopus.com/inward/record.url?scp=77951177786&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77951177786&partnerID=8YFLogxK

U2 - 10.1090/S0002-9939-10-10309-8

DO - 10.1090/S0002-9939-10-10309-8

M3 - Article

VL - 138

SP - 1897

EP - 1906

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

IS - 6

ER -