Two Dynamic Remarks on the Chebyshev–Halley Family of Iterative Methods for Solving Nonlinear Equations

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

In the end of page 2 when it says spatial should say special

In the end of page 3 a space is missing after multiplier

After section 4 authors must write a section with conclusions

Author Response

We have corrected the two typos indicated by the referee and we have added a section with conclusions.

Reviewer 2 Report

Comments and Suggestions for Authors

The paper focuses on the dynamic study of the Chebyshev-Halley family of iterative methods for solving nonlinear equations, with objectives including characterizing the existence of extraneous attracting fixed points and studying the free critical points of the methods. It identified situations where the methods in the Chebyshev-Halley family have a bad behavior from the root-finding point of view. The paper provides insights into the behavior of the Chebyshev-Halley family of iterative methods for solving nonlinear equations, which can be valuable for researchers and practitioners in the field of numerical analysis and computational mathematics.

 

1. Given the existence of extraneous attracting fixed points, how would researchers using the Chebyshev-Halley family of iterative methods avoid them? What insight can we gain from the parameter plan analysis into a proper and efficient way to choose parameters?
2. As the author said, the difficulty from a high number of free critical points for the methods in the family discourages their use in the parameter plane analysis. What would be a solution to that? The limitations of the parameter plane as a graphical tool are not addressed in detail.
3. The paper does not provide a comparison with other existing iterative methods for solving nonlinear equations.
4. What conclusion can be drawn from the study? The manuscript lacks a conclusion section in the end.

 

Minors:

1. There are some typos that need to be corrected, for example “new evaluations of f(x) an its derivatives”, “Halley’s method must be exclude”, “because it has not attracting extraneous fixed points”, etc.
2. The x and y axis of some figures (e.g., the parameter plan) need to have their labels, and the some figures need legends to clarify what the color indicates.
3. In figure captions, instead of saying “on the left” or ”on the right”, use A and B.

Comments on the Quality of English Language

Some typos and grammar errors need to be corrected.

Author Response

See the attached document

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

The paper is very well-written and organized. This paper presented a dynamic study of the well-known Chebyshev-Halley family of iterative methods for solving nonlinear equations.

The author should improve the English language and remove several grammatical mistakes. However, the overall quality is good.

 

The author is requested to add a separate conclusion paragraph stating the achieved outcomes of this work. and future plans. In this conclusion paragraph, the author should include the following statement as it gives more value to the paper: This work can be extended to nonlinear ordinary differential equations [1] and can provide promising results in solving these problems using an iterative scheme.

[1] M. Ben-Romdhane, H. Temimi, M. Baccouch, An iterative finite difference method for approximating the two-branched solution of Bratu's problem, Applied Numerical Mathematics, Vol. 139, pp. 62-76, 2019.

Author Response

A new section with the conclusions of the paper, together with the suggested reference, has been included.

Reviewer 4 Report

Comments and Suggestions for Authors

The paper investigates the Chebyshev-Halley family of iterative methods for solving nonlinear equations. Namely, the existence of extraneous attracting fixed points is analyzed when the methods of this family are applied to polynomial equations and the free critical points of the family methods are studied as a previous step to determine the existence of attracting cycles.

Сomments:

1) In formula (4) it should be clarified that the coefficients of the polynomial p(z) are constant or variable quantities.

2) Before the formulation of Theorem 1 it is written that the result of Nayak and Pal will be generalized, but after the formulation of Theorem 1 it is written that the proof mimics the one given by Nayak and Pal. It seems to me that it would be nice to provide a complete proof of Theorem 1 in the article, since a generalization or analogy is not always an exact match.

3) There is no Conclusion section at the end of the article.

4) The list of references is not formatted according to the rules of the journal Mathematics.

Author Response

1) In formula (4) it should be clarified that the coefficients of the polynomial p(z) are constant or variable quantities.

Done.

2) Before the formulation of Theorem 1 it is written that the result of Nayak and Pal will be generalized, but after the formulation of Theorem 1 it is written that the proof mimics the one given by Nayak and Pal. It seems to me that it would be nice to provide a complete proof of Theorem 1 in the article, since a generalization or analogy is not always an exact match.

We have included the main ideas of the proof of Nayak and Pal. The original proof of these two authors uses the solutions of equation L_p(z)=-2, that is the corresponding to the case \alpha=0 in our equation L_p(z)=2/(2\alpha-1). The main ideas are the same. Only this technical detail is different, but a generalization to all the methods in the family is posible.

3) There is no Conclusion section at the end of the article.

We have included a section with conclusions in the revised version.

4) The list of references is not formatted according to the rules of the journal Mathematics.

It's true. We have the compromise of adapting the manuscript to the journal style I fit is accepted.

 

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The authors perform a dynamical study of the Chebyshev-Halley family of methods on polynomial equations. They used dynamical tools as dynamical and parameter planes and found superattracting fixed points different from the roots and a rich dynamical behavior, including attracting cycles. The work is limited to nonlinear equations, but can be extended to nonlinear systems of equations.

Reviewer 2 Report

Comments and Suggestions for Authors

The authors have addressed all my questions. 

Reviewer 4 Report

Comments and Suggestions for Authors

I see that the author has addressed all my comments accordingly. The manuscript is revised and improved. Thus, I recommend this article for publication in the journal "Axioms".