Comments on the paper

[YehorYudinIPP]

Dear Andrew,

I have read your JOSS paper and have several comments and suggestions on it, mainly regarding the citing.

Firstly, it would be appreciated if the paper had DOI or other indicators for the cited papers.

Secondly, the paper would benefit from citing the Golub-Welsch algorithm implementation by Laurie and Gautschi, providing a more explicit description of this algorithm, as well as commenting on the Dijkstra algorithm.

Thirdly, it would be nice if the paper explicitly included the expressions of the solved integrals and necessary parameters like the integral domain and function calls to compute the integrals. This includes a concealed expression cited as the equation (32) for |𝐴32(𝑥1, 𝑥2)| from Hewett et al. (2019).

All the best,
Yehor

[AndrewGibbs]

Dear Yehor,

Thank you for these suggestions, the paper has certainly improved as a result. Please let me know if you are able to access the latest version easily?

1. Where possible, I have added DOIs. This now includes a DOI for the Popov 1979 paper, which was not included in the original submission. However, for older papers, websites, and books, this is not possible - the information is not on MathSciNet or Google Scholar. In these cases, I have included journal/publisher/URL information. I hope this is sufficient?
2. I have added references in the acknowledgements section for the Dijstkra and Golub-Welsch algorithms. The links in the preceeding sentence are to the Wikipedia articles, to quickly demonstrate the size of these algorithms when written in full! I have not included a more explicit description of either algorithm; given that these are well-studied algorithms and there are many thorough explanations available, this would take up a considerable amount of unnecessary space and I feel it would detract from the message of the paper (which has a fairly small word limit!). For added context: There are many other root-finding algorithms (Newton, Bisection, Companion matrices) which are called from the PathFinder algorithm and described in the corresponding JCP paper, but are not mentioned in this JOSS submission about the code, because for those subroutines, the implementation was my own.
3. Thank you for this excellent suggestion. I have included expressions of the Cusp and Swallowtail integrals, and the integral used in Hewett et al, as you suggest. For the latter two of these, I have also included an example PathFinder call used to compute them at a given point $(x_1,x_2)$. I feel these more advanced examples have made the scope and syntax clearer, thanks again.

I will not close this issue until you are satisfied with the changes made.

Best wishes,

Andrew