nth Root extraction: Double iteration process and Newton's method

The present investigation deals with the critical study of the works of Lancaster and Traub, who have developed $n$th root extraction methods of a real number. It is found that their developed methods are equivalent and the particular cases of Halley's and Householder's methods. Again the methods presented by them are easily obtained from simple modifications of Newton's method, which is the extension of Heron's square root iteration formula. Further, the rate of convergency of their reported methods are studied.

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Determining radius of convergence of Newton’s method using radius of curvature

MATEMATIKA ◽

10.11113/matematika.v33.n1.836 ◽

2017 ◽

Vol 33 (1) ◽

pp. 43

Author(s):

Ridwan Pandiya ◽

Ismail Mohd

Keyword(s):

Newton’S Method ◽

Newton's Method ◽

Iteration Process ◽

Initial Guess ◽

Local Extremum ◽

Radius Of Convergence ◽

Radius Of Curvature

In this paper, we propose a method on how to manage the convergence of Newton’s method if its iteration process encounters a local extremum. This idea establishes the osculating circle at a local extremum. It then uses the radius of the osculating circle also known as the radius of the curvature as an additional number of the local extremum. It then takes that additional number and combines it with the local extremum. This is then used as an initial guess in finding a root near to that local extremum. This paper will provide several examples which demonstrate that our idea is successful and they perform to fulfill the aim of this paper.

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Optimal Power Flow Using Newton’s Method

International Journal of Scientific Research ◽

10.15373/22778179/feb2014/54 ◽

2012 ◽

Vol 3 (2) ◽

pp. 167-169

Author(s):

F.M.PATEL F.M.PATEL ◽

◽

N. B. PANCHAL N. B. PANCHAL

Keyword(s):

Newton’S Method ◽

Newton's Method ◽

Power Flow ◽

Optimal Power Flow ◽

Optimal Power

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Newton's Method for Fractional Combinatorial Optimization,

10.21236/ada323687 ◽

1992 ◽

Cited By ~ 4

Author(s):

Thomas Radzik

Keyword(s):

Combinatorial Optimization ◽

Newton’S Method ◽

Newton's Method ◽

Fractional Combinatorial Optimization

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An Efficient Sixth-Order Convergent Newton-Type Iterative Method for Nonlinear Equations

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.220-223.2585 ◽

2012 ◽

Vol 220-223 ◽

pp. 2585-2588

Author(s):

Zhong Yong Hu ◽

Fang Liang ◽

Lian Zhong Li ◽

Rui Chen

Keyword(s):

Iterative Method ◽

Nonlinear Equations ◽

Newton’S Method ◽

Newton's Method ◽

Efficiency Index ◽

Sixth Order ◽

Type Method ◽

Second Derivatives ◽

Better Than

In this paper, we present a modified sixth order convergent Newton-type method for solving nonlinear equations. It is free from second derivatives, and requires three evaluations of the functions and two evaluations of derivatives per iteration. Hence the efficiency index of the presented method is 1.43097 which is better than that of classical Newton’s method 1.41421. Several results are given to illustrate the advantage and efficiency the algorithm.

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