### Abstract

Newton's method is well-known to be generally convergent for solving x^{n} - c = 0. In this paper, we first extend this result to the next two members of an infinite family of high order methods referred to here as the Basic Family which starts with Newton's method. While computing roots of unity numerically is a trivial task, studying the general convergence of the Basic Family in this simple case is an important first step toward the understanding of the global behavior of this fundamental family. With the aid of polynomiography, techniques for the visualization of polynomial root-finding, we further conjecture the general convergence of all members of the Basic Family when extracting radicals. Using the computer algebra system Maple, we obtain some partial results toward the proof of our conjecture.

Original language | English (US) |
---|---|

Pages (from-to) | 832-842 |

Number of pages | 11 |

Journal | Journal of Computational and Applied Mathematics |

Volume | 206 |

Issue number | 2 |

DOIs | |

State | Published - Sep 15 2007 |

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### All Science Journal Classification (ASJC) codes

- Computational Mathematics
- Applied Mathematics

### Cite this

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*Journal of Computational and Applied Mathematics*, vol. 206, no. 2, pp. 832-842. https://doi.org/10.1016/j.cam.2006.08.035

**On general convergence in extracting radicals via a fundamental family of iteration functions.** / Jin, Yi; Kalantari, Bahman.

Research output: Contribution to journal › Article

TY - JOUR

T1 - On general convergence in extracting radicals via a fundamental family of iteration functions

AU - Jin, Yi

AU - Kalantari, Bahman

PY - 2007/9/15

Y1 - 2007/9/15

N2 - Newton's method is well-known to be generally convergent for solving xn - c = 0. In this paper, we first extend this result to the next two members of an infinite family of high order methods referred to here as the Basic Family which starts with Newton's method. While computing roots of unity numerically is a trivial task, studying the general convergence of the Basic Family in this simple case is an important first step toward the understanding of the global behavior of this fundamental family. With the aid of polynomiography, techniques for the visualization of polynomial root-finding, we further conjecture the general convergence of all members of the Basic Family when extracting radicals. Using the computer algebra system Maple, we obtain some partial results toward the proof of our conjecture.

AB - Newton's method is well-known to be generally convergent for solving xn - c = 0. In this paper, we first extend this result to the next two members of an infinite family of high order methods referred to here as the Basic Family which starts with Newton's method. While computing roots of unity numerically is a trivial task, studying the general convergence of the Basic Family in this simple case is an important first step toward the understanding of the global behavior of this fundamental family. With the aid of polynomiography, techniques for the visualization of polynomial root-finding, we further conjecture the general convergence of all members of the Basic Family when extracting radicals. Using the computer algebra system Maple, we obtain some partial results toward the proof of our conjecture.

UR - http://www.scopus.com/inward/record.url?scp=34249819938&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34249819938&partnerID=8YFLogxK

U2 - 10.1016/j.cam.2006.08.035

DO - 10.1016/j.cam.2006.08.035

M3 - Article

VL - 206

SP - 832

EP - 842

JO - Journal of Computational and Applied Mathematics

JF - Journal of Computational and Applied Mathematics

SN - 0377-0427

IS - 2

ER -