Review Reports

Round 1

Reviewer 1 Report

The manuscript: 

“Arbitrarily Accurate Analytical Approximations for the Error Function”, by R. M. Howard” (Ref. No.: mca-1504102-peer-review),

contains interesting material that potentially may be suitable for publication. However, it is not well-organized and requires a significant elaboration. In particular, it would be desirable to show the code for verification of the claims. Furthermore, one of the most applications of the error function erf(x) is the Faddeeva function w(x) (also known as the complex error function). It is necessary to include relations between erf(x) and w(x) to the manuscript. Perhaps, author also wants to mention the relation of erf(x) with Dawson’s integral as well.

It is also not clear if the proposed algorithms can also work with high accuracy when argument is complex z = x+i*y, where x and y are real and imaginary parts. And if yes, the covered domains with corresponding accuracies should be shown.

Apart from this the following should also be taken into account:

Abstract
1) The author should compare run-time algorithm with existing algorithms if possible. The results should be shown in the Abstract.

1. Introduction
1) The sentences: “Many other applications exist. In general, the error function is associated with a macro description of physical phenomena and the De-Moivre Laplace theorem is illustrative of the link between fundamental outcomes and a higher level model”. As it has been mentioned, the author should mention about complex error function w(z)=exp(-z^2)*(1-erf(-i*z)) and provide corresponding citations.

2. Spline Based Approximations for Error Function
1) Typically, spline based approximations of the error function are not accurate and rapid. The author should justify their applications.

2) The error function erf(x) at x> 6 becomes very close to 1. Is it really necessary to approximate it large argument? This should be justified and cited accordingly.

3) Figure 3 shows accuracy 10^-17. I am not sure if such a high accuracy is required. It may be more than enough to have an accuracy 10^-9. This should also be discussed.

3. Improved Approximations
1) Two more highly accurate approximations of the error function erf(z) are given in arXiv:1308.3399. Is it possible to show advantage of the proposed approximations with approximations (14) and (16) in terms of accuracy and computational speed?

4. Variable Sub-interval Approximations for Error Function
1) Table 5. Again, do we really need such a small relative error 10^-27?

2) Table 6. This shows 10^-50? Why such a high accuracy is needed?

5. Dynamic Constant plus Spline Approximation
1) The sentence: “The proof is detailed in Appendix 4”. Perhaps the proof should be shown within section 5 rather than in Appendix 4.

6. A Dynamical System to Yield Improved Approximations
1) Equation (65) should be cited. 

7. Applications
1) Is it possible to apply (or generalize) this approach for integral involving Gaussian function exp(-x^2)?

2) The sentence: “The proof is detailed in Appendix 6”. Perhaps the proof should be presented within section 7 since it looks important.

3) Approximation (126) resembles Salzer’s equation for the error function [Math. Tables Aids Comput. 5 (1951) 61-70]. It may be just a reformulation of the Salzer’s equation.

8. Conclusion
1) Conclusion should also include the computational run-time comparison if possible.

The manuscript requires a major mandatory revision.

Author Response

Thank you for your feedback on my manuscript. I have included the response to the comments made by both reviewers in the attached document. This will facilitate understanding the changes made to the original manuscript.

Author Response File: Author Response.pdf

Reviewer 2 Report

Since the error function is now available as the library function of most of the scientific program languages like Mathematica, Matlab, Maple, etc. the utility of its approximation within the finite domain is doubtful to me. 

So, the advantage of getting an approximation of the error function presented in this paper must be emphasized clearly in the revised version of the manuscript. 

The scale of the representation of relative error in the figures should be described appropriately.

The presentation of the paper deserves more attention so that the advantages and limitations of each method are clear for the reader/users.

Comments for author File: Comments.pdf

Author Response

Thank you for your feedback on my manuscript. I have included the response to the comments made by both reviewers in the attached document. This will facilitate understanding the changes made to the original manuscript.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

The manuscript:

"Arbitrarily Accurate Analytical Approximations for the Error Function", by Roy M. Howard (Ref. No.: mca-1504102-peer-review-v2),

has been improved after major revision. In particular, the author added the required material, provided clarifications and included citations. The manuscript is comprehensive and detailed. The author included Mathematica codes that can be used to verify the claims. Overall, I am satisfied with corrections. Therefore, it may be recommended for publication in the present form.