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) |
|---|---|
| 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
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On general convergence in extracting radicals via a fundamental family of iteration functions. / Jin, Yi; Kalantari, Bahman.
In: Journal of Computational and Applied Mathematics, Vol. 206, No. 2, 15.09.2007, p. 832-842.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 -