Higher-order derivative-free families of Chebyshev–Halley type methods with or without memory for solving nonlinear equations

Argyros, Ioannis K.; Kansal, Munish; Kanwar, V.; Bajaj, Sugandha

doi:10.1016/j.amc.2017.07.051

Cited by 14 publications

(25 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Due to change (5), the parameters in (4) does not remain as before and we denote them by n τ and n α  . We call the iterations ( 1) and ( 4) the derivative presence (DP) and derivative-free (DF) variants respectively.…”

Section: The Optimal Derivative-free Three-point Iterationsmentioning

confidence: 99%

“…Theorem 3. The iteration (4) with n τ given by (7) and with n α  given by (11) have the order of convergence eight, if the following conditions hold:…”

Section: Application Of Sufficient Convergence Condition To Derive New Df Iterationsmentioning

confidence: 99%

“…In particular, the derivative-free methods are necessary when the derivative of the function f is unavailable or expensive to obtain. In the last decade, the derivative-free two and three-point methods with better convergence properties were developed (see [4]- [19] and references therein). It should be pointed out that most of these methods were proposed mainly for the concrete choice of parameters (see Table 1).…”

Section: Introductionmentioning

confidence: 99%

“…We shall find the coefficients , , A B C and , ω ∆ such that the iteration (4) with ( 7) and (32) has the order of convergence eight and state the following: Theorem 6. The iteration ( 4) with (7) and (32) has the order of convergence eight, if the following conditions hold:…”

mentioning

confidence: 99%

See 3 more Smart Citations

Constructive Theory of Designing Optimal Eighth-Order Derivative-Free Methods for Solving Nonlinear Equations

Жанлав¹,

Otgondorj²,

Mijiddorj³

2020

AJCM

View full text Add to dashboard Cite

This paper stresses the theoretical nature of constructing the optimal derivative-free iterations. We give necessary and sufficient conditions for derivative-free three-point iterations with the eighth-order of convergence. We also establish the connection of derivative-free and derivative presence three-point iterations. The use of the sufficient convergence conditions allows us to design wide class of optimal derivative-free iterations. The proposed family of iterations includes not only existing methods but also new methods with a higher order of convergence.

show abstract

Section: The Optimal Derivative-free Three-point Iterationsmentioning

confidence: 99%

“…Theorem 3. The iteration (4) with n τ given by (7) and with n α  given by (11) have the order of convergence eight, if the following conditions hold:…”

Section: Application Of Sufficient Convergence Condition To Derive New Df Iterationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

Constructive Theory of Designing Optimal Eighth-Order Derivative-Free Methods for Solving Nonlinear Equations

Жанлав¹,

Otgondorj²,

Mijiddorj³

2020

AJCM

View full text Add to dashboard Cite

show abstract

“…Finally, we compare them with schemes proposed by Argyros et al [1], out of them we consider expression (2.4) with (H 1 (τ ), γ = −0.01, α = 1) and (H 2 (τ ), γ = −0.01, α = 0), called by AM 1 and AM 2 , respectively. Actually, the methods AM 1 and AM 2 are designed only for simple roots according to their paper.…”

Section: Numerical Illustrationsmentioning

confidence: 99%

Higher-Order Families of Multiple Root Finding Methods Suitable for Non-Convergent Cases and Their Dynamics

Behl

Kanwar

Kim

2019

Mathematical Modelling and Analysis

Self Cite

View full text Add to dashboard Cite

In this paper, we present many new one-parameter families of classical Rall's method (modified Newton's method), Schröder's method, Halley's method and super-Halley method for the first time which will converge even though the guess is far away from the desired root or the derivative is small in the vicinity of the root and have the same error equations as those of their original methods respectively, for multiple roots. Further, we also propose an optimal family of iterative methods of fourth-order convergence and converging to a required root in a stable manner without divergence, oscillation or jumping problems. All the methods considered here are found to be more effective than the similar robust methods available in the literature. In their dynamical study, it has been observed that the proposed methods have equal or better stability and robustness as compared to the other methods.

show abstract

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

Rani

Soleymani

Kansal

et al. 2022

Math Methods in App Sciences

View full text Add to dashboard Cite

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.

show abstract

Higher-order derivative-free families of Chebyshev–Halley type methods with or without memory for solving nonlinear equations

Cited by 14 publications

References 17 publications

Constructive Theory of Designing Optimal Eighth-Order Derivative-Free Methods for Solving Nonlinear Equations

Constructive Theory of Designing Optimal Eighth-Order Derivative-Free Methods for Solving Nonlinear Equations

Higher-Order Families of Multiple Root Finding Methods Suitable for Non-Convergent Cases and Their Dynamics

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

Contact Info

Product

Resources

About