A sixth-order family of three-point modified Newton-like multiple-root finders and the dynamics behind their extraneous fixed points

Geum, Young Hee; Kim, Young Ik; Neta, Beny

doi:10.1016/j.amc.2016.02.029

Cited by 43 publications

(59 citation statements)

References 39 publications

(56 reference statements)

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…We take the three‐point sixth‐order method presented by Thukral, given by expression and denote it by TM. We take another three‐point family of sixth‐order methods proposed by Geum et al; out of those, we consider the following method:

\begin{matrix} alignleft \\ align-1 y_{n} & align-2 = x_{n} - m \frac{f (x_{n})}{f^{'} (x_{n})}, m \geq 1, \\ align-1 w_{n} & align-2 = x_{n} - m [1 + h_{n} + 2 h_{n}^{2}] \frac{f (x_{n})}{f^{'} (x_{n})}, \\ align-1 x_{n + 1} & align-2 = x_{n} - m [1 + h_{n} + 2 h_{n}^{2} + (1 + 2 h_{n}) l_{n}] \frac{f (x_{n})}{f^{'} (x_{n})}, \end{matrix}

h_{n} = {[\frac{f (y_{n})}{f (x_{n})}]}^{\frac{1}{m}}, l_{n} = {[\frac{f (z_{n})}{f (x_{n})}]}^{\frac{1}{m}},

denoted by GK.…”

Section: Numerical Results and Applicationsmentioning

confidence: 99%

“…We also compare our methods with the existing three‐ and two‐point schemes presented by Zafar et al, Thukral, and Geum et al on a concrete variety of standard numerical examples and three real‐life problems. The proposed family has an advantage over the scheme of Geum et al that it provides two choices: ie, the selection of weight functions at first as well at the second step of the two‐point scheme.…”

Section: Discussionmentioning

confidence: 99%

“…Geum et al have also presented a new family of three‐point sixth‐order methods for multiple roots, which is given below:

\begin{matrix} alignleft \\ align-1 y_{n} & align-2 = x_{n} - m \frac{f (x_{n})}{f^{'} (x_{n})}, m \geq 1, \\ align-1 w_{n} & align-2 = x_{n} - m L (h_{n}) \frac{f (x_{n})}{f^{'} (x_{n})}, \\ align-1 x_{n + 1} & align-2 = x_{n} - m K (h_{n}, l_{n}) \frac{f (x_{n})}{f^{'} (x_{n})}, \end{matrix}

where

h_{n} = \sqrt[m]{\frac{f false(y_{n} false)}{f false(x_{n} false)}}

and

l_{n} = \sqrt[m]{\frac{f false(w_{n} false)}{f false(x_{n} false)}}

.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A new two‐point scheme for multiple roots of nonlinear equations

Behl

Zafar

Junjua

2020

Math Methods in App Sciences

View full text Add to dashboard Cite

In most of the earlier research for multiple zeros, in order to obtain a new iteration function from the existing scheme, the usual practice is to make no change at the first substep. In this paper, we explore the idea that what are the advantages if the flexibility of choice is also given at the first substep. Therefore, we present a new two‐point sixth‐order scheme for multiple roots (m>1). The main advantages of our scheme over the existing schemes are flexibility at both substeps, simple body structure, smaller residual error, smaller error difference between two consecutive iterations, and smaller asymptotic error constant. The development of the scheme is based on midpoint formula and weight functions of two variables. We compare our methods with the existing methods of the same order with real‐life applications as well as standard test problems. From the numerical results, we find that our methods can be considered as better alternates for the existing methods of the same order. Finally, dynamical study of the proposed schemes is presented that confirms the theoretical results.

show abstract

\begin{matrix} alignleft \\ align-1 y_{n} & align-2 = x_{n} - m \frac{f (x_{n})}{f^{'} (x_{n})}, m \geq 1, \\ align-1 w_{n} & align-2 = x_{n} - m [1 + h_{n} + 2 h_{n}^{2}] \frac{f (x_{n})}{f^{'} (x_{n})}, \\ align-1 x_{n + 1} & align-2 = x_{n} - m [1 + h_{n} + 2 h_{n}^{2} + (1 + 2 h_{n}) l_{n}] \frac{f (x_{n})}{f^{'} (x_{n})}, \end{matrix}

h_{n} = {[\frac{f (y_{n})}{f (x_{n})}]}^{\frac{1}{m}}, l_{n} = {[\frac{f (z_{n})}{f (x_{n})}]}^{\frac{1}{m}},

denoted by GK.…”

Section: Numerical Results and Applicationsmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

“…Geum et al have also presented a new family of three‐point sixth‐order methods for multiple roots, which is given below:

\begin{matrix} alignleft \\ align-1 y_{n} & align-2 = x_{n} - m \frac{f (x_{n})}{f^{'} (x_{n})}, m \geq 1, \\ align-1 w_{n} & align-2 = x_{n} - m L (h_{n}) \frac{f (x_{n})}{f^{'} (x_{n})}, \\ align-1 x_{n + 1} & align-2 = x_{n} - m K (h_{n}, l_{n}) \frac{f (x_{n})}{f^{'} (x_{n})}, \end{matrix}

where

h_{n} = \sqrt[m]{\frac{f false(y_{n} false)}{f false(x_{n} false)}}

and

l_{n} = \sqrt[m]{\frac{f false(w_{n} false)}{f false(x_{n} false)}}

.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A new two‐point scheme for multiple roots of nonlinear equations

Behl

Zafar

Junjua

2020

Math Methods in App Sciences

View full text Add to dashboard Cite

show abstract

“…As a preliminary task, we first describe the following lemma regarding the negative real roots of a quadratic equation, which would play a role in determining the desired purely imaginary extraneous fixed points in connection with the prototype quadratic polynomial f (z) = z 2 − 1. The following lemma holds according to the analysis of [34]. To begin the detailed study regarding the purely imaginary extraneous fixed points, we now consider Case 2 described by (3.14) to discuss another selection of 10 free parameters…”

Section: Purely Imaginary Extraneous Fixed Pointsmentioning

confidence: 99%

An optimal family of eighth-order simple-root finders with weight functions dependent on function-to-function ratios and their dynamics underlying extraneous fixed points

Lee

Kim

Neta

2017

Journal of Computational and Applied Mathematics

Self Cite

View full text Add to dashboard Cite

b s t r a c tWe extend in this paper an optimal family of three-step eighth-order methods developed by Džunić et al. (2011) with higher-order weight functions employed in the second and third sub-steps and investigate their dynamics under the relevant extraneous fixed points among which purely imaginary ones are specially treated for the analysis of the rich dynamics. Their theoretical and computational properties are fully investigated along with a main theorem describing the order of convergence and the asymptotic error constant as well as proper choices of special cases. A wide variety of relevant numerical examples are illustrated to confirm the underlying theoretical development. In addition, this paper investigates the dynamics of selected existing optimal eighth-order iterative maps with the help of illustrative basins of attraction for various polynomials.

show abstract

“…In recent work, one can find a visual comparison, by plotting the basins of attraction for the methods. The idea of using basins of attraction appeared first in Stewart [59] and followed by the works of Amat et al [2], [3], and [4], Scott et al [57], Chicharro et al [8], Chun et al [9], [10], [11], [12], Cordero et al [21], Neta et al [48], [49], Argyros and Magreñan, [5], Magreñan, [40] and Geum et al [24], [25], [26] and [27]. In later works ( [11], [12], [13], [14], [15]), we have introduced a more quantitative comparison, by listing the average number of iterations per point, the CPU time and the number of points requiring 40 iterations.…”

Section: Introductionmentioning

confidence: 99%