A class of two-point sixth-order multiple-zero finders of modified double-Newton type and their dynamics

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

doi:10.1016/j.amc.2015.08.039

Cited by 53 publications

(72 citation statements)

References 38 publications

(35 reference statements)

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“…Other types of combinations have empirically shown poor convergence in the existing studies by [13,19,29,34] . A special attention will be paid to some selected cases 1A , 2A , 2B, 2C, 3A , 4A as well as all 5X and 5Y in order to pursue further properties of extraneous fixed points and relevant dynamics associated with their basins of attraction.…”

Section: Extraneous Fixed Pointsmentioning

confidence: 99%

“…The dynamics underlying basins of attraction was initiated by Stewart [41] and followed by works of Amat et al [2][3][4][5] , Scott et al [38] , Chun et al [10] , Chun-Neta [11] , Chicharro et al [8] , Cordero et al [15] , Neta et al [28,33] , Argyros-Magreñan [7] , Magreñan [27] , Magreñan et al [26] , Andreu et al [6] and Chun et al [12] . The only papers comparing basins of attraction for methods to obtain multiple roots are due to Neta et al [29] , Neta-Chun [30,34] , Chun-Neta [13,14] and Geum-Kim-Neta [19] .…”

Section: Extraneous Fixed Pointsmentioning

confidence: 99%

“…The top row for 5XB , 5XE , 5XF , in order from left to right, and the bottom row for 5XG , and 5YD , for the roots of the polynomial (z 4 − 1) 5 . [19] . The fastest method is 5YD (2588.39 s) followed by 5XF (3081.08 s).…”

Section: Examplementioning

confidence: 99%

“…The average number of iterations per point is least for GKN5YD (5.09) followed by GKN5XE (6.50) and GKN5XB (6.92) and the highest is for GKN5XF (51. 19). The least number of black points is for GKN5YD (652) and the highest for GKN5XA and GKN5XC .…”

Section: Examplementioning

confidence: 99%

“…Very recently Geum-Kim-Neta [19] have developed a class of twopoint sixth-order multiple-root finders by extending the classical modified double-Newton method with extensive analysis of their relevant dynamics behind the basins of attraction from the viewpoint of the extraneous fixed points. One member of the class is introduced as follows shown by (2.1) :…”

Section: Review Of Existing Sixth-order Multiple-root Findersmentioning

confidence: 99%

See 4 more Smart Citations

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

Geum

Kim

Neta

2016

Applied Mathematics and Computation

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Section: Extraneous Fixed Pointsmentioning

confidence: 99%

Section: Extraneous Fixed Pointsmentioning

confidence: 99%

Section: Examplementioning

confidence: 99%

Section: Examplementioning

confidence: 99%

Section: Review Of Existing Sixth-order Multiple-root Findersmentioning

confidence: 99%

See 3 more Smart Citations

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

Geum

Kim

Neta

2016

Applied Mathematics and Computation

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

Behl

Zafar

Junjua

2020

Math Methods in App Sciences

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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.

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An optimized Chebyshev–Halley type family of multiple solvers: Extensive analysis and applications

Rani

Soleymani

Kansal

et al. 2022

Math Methods in App Sciences

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In this manuscript, we introduce a higher‐order optimal family of Chebyshev–Halley type methods to solve a univariate nonlinear equation having multiple roots. The proposed scheme considers weight functions that are selected adequately to optimize the convergence order and demands only four functional evaluations at each iteration. An extensive convergence analysis is also provided that demonstrates the establishment of eighth‐order convergence of the developed scheme. As a result, the efficiency index of our scheme is optimal according to the Kung–Traub conjecture. Numerical experiments are performed on some real‐life as well as nonlinear academic problems. In addition, the dynamical study of iterative schemes reflects the good overview of their stability, convergence properties, and graphical aspects by drawing attraction basins in the complex plane. Numerical findings indicate that the proposed methods outperform the existing ones with similar characteristics.

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A class of two-point sixth-order multiple-zero finders of modified double-Newton type and their dynamics

Cited by 53 publications

References 38 publications

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

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

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

An optimized Chebyshev–Halley type family of multiple solvers: Extensive analysis and applications

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