A Generic Family of Optimal Sixteenth-Order Multiple-Root Finders and Their Dynamics Underlying Purely Imaginary Extraneous Fixed Points
Round 1
Reviewer 1 Report
The manuscript verses on a new multiple root finder of order sixteen. Up to the knowledge of the authors, is the first time that a method of this high order is published.
The proof of the convergence of the method is extensive and detailed, without losing any step of the process.
The method is tested on both numerical and dynamical performances, obtaining acceptable results.
Some questions about the dynamical study.
What software has been used for representing the dynamical planes? The numerical section is performed with Mathematica, as explained in line 211. However, the representation of dynamical planes (line 241) lacks of the stopping criteria and the applied software.
The following comment can be applied for the next examples and dynamical planes: Example 1 > Method W2A4; Example 2 > Methods W2A1, W2C2 and W2F2; Example 6 > Methods W2B3 and W2C1. In the manuscript the only explanation about this undesirable behavior is that these dynamical planes have more black points than the other ones. What is the reason why this behavior occurs? Maybe an study on the parameters planes, or on the extraneous points can give some light on this explanation.
Author Response
Your comments are greatly appreciated.
Based on your suggestions, all reponses to the required comments are in the attatched file.
Author Response File: 
Author Response.pdf
Reviewer 2 Report
In this article, the authors study a generic family of sixteenth-order multiple root finders. The paper seems to be interesting but it is very hard to read. A long list of formulas is often presented (this could be in annex). We cannot clearly see the goal of the paper and what is the novelty. I recommend a major revision in order to increase the readability of the paper.
Author Response
Your comments are greatly appreciated.
Based on your suggestions, all reponses to the required comments are in the attatched file.
Author Response File: Author Response.pdf
Round 2
Reviewer 1 Report
The comments have been answered by the authors. After an English language edition, the manuscript can be published in its present form.
Reviewer 2 Report
As the authors took into account my comments, I recommend the paper for publication.
