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

Research output: Contribution to journalArticle

2 Citations (Scopus)

### Abstract

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.

Original language English (US) 832-842 11 Journal of Computational and Applied Mathematics 206 2 https://doi.org/10.1016/j.cam.2006.08.035 Published - Sep 15 2007

### Fingerprint

Iteration Function
Newton-Raphson method
Algebra
Visualization
Newton Methods
Polynomials
Polynomial Roots
Root-finding
High-order Methods
Computer algebra system
Maple
Roots of Unity
Trivial
Family
Partial
Computing

### All Science Journal Classification (ASJC) codes

• Computational Mathematics
• Applied Mathematics

### Cite this

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title = "On general convergence in extracting radicals via a fundamental family of iteration functions",
abstract = "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.",
author = "Yi Jin and Bahman Kalantari",
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language = "English (US)",
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In: Journal of Computational and Applied Mathematics, Vol. 206, No. 2, 15.09.2007, p. 832-842.

Research output: Contribution to journalArticle

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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

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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.

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