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\begin{document}

\setkomavar{signature}{Daniel Potts} % Unterschrift!
\setkomavar{subject}{2nd revision of
``Using NFFT~3 -- a software library for various nonequispaced fast
  Fourier transforms''
by J.~Keiner, S.~Kunis, and D.~Potts
}

\begin{letter}{Editor-in-Chief\\
Prof. Ian Gladwell\\
Department of Mathematics\\
Southern Methodist University\\
Dallas, TX 75275 USA}

\opening{Dear Professor Gladwell,}

we thank all three referees for their helpful comments and valuable
suggestions. In the following we report on the changes.

\textbf{Referee 1}
No changes.

\textbf{Referee 2}
The structure of the paper has been kept, in particular Section 3.2 shows the
typical workflow and details on the precomputation are still in the appendix.
The following suggestions of the referee have been incorporated, 1), 4), 5),
and 6).

2) ``Usually $\hat f$ is for the FT of f and not vice versa. If you have a
good reason for doing it like this, state it.'' and p.~3, sec 2.1 of the first
report: \\
The beginning of Section 2 now states that the FFT evaluates a
trigonometric polynomial at equispaced nodes and that this point of view is
most convenient for the generalisation to nonequispaced nodes.

3) ``P.3, line -10: what do you mean by no additional memory?''
The matrix entries are explicitly given and they are computed only when used.

A more detailed cost analysis and performance analysis has been
added, see the second issue of Referee 3.

\newpage

\textbf{Referee 3}
On ``The major issues''
\begin{enumerate}
\item The Sections 3.2 and 3.4 clearly state how the nodes have to be set.
\item A more detailed cost analysis and performance analysis has been
added. In particular, Algorithm 2 and 3 give clear upper bounds on the number
of arithmetic operations excluding the evaluation of the so called window
function which is detailed in Appendix A.3. Section 5.1 specifies these costs
under the tested default settings and gives a performance plot in Figure 5.1.
\item The size of the 3D problems is considered to be reasonable on standard hardware, since an
NFFT with $128^3$ Fourier coefficients, $128^3$ evaluation nodes, oversampling
factor $2$ and disjoint FFTW-in/output arrays takes approximately
\begin{equation*}
  (128^3 + 2 \times (2\times 128)^3 + 128^3) \times 16 =576 \text{MByte}.
\end{equation*}
A NFFT of size $256^3$ would already take over $4\text{GByte}$ with these
settings (the library can compute such a transform by using a smaller
oversampling factor etc. at increased cost).
\end{enumerate}

On ``The minor issues''
\begin{enumerate}
 \item[1,2.] Done, except  ``various $\rightarrow$ certain'', since
   ``Nonequispaced FFT'' is typically associated with (2.3) and ``various''
   refers to the fact that more general transforms are considered as well.
 \item[3.] Done, but we kept the version number when referring to the library.
 \item[4,5,6,7.] Done. However, we found no inconsistent vector notation and
   the terminology ``adjoint'' (or transposed) and ``inverse'' is in
   accordance with [19] 
\item[8.] We changed the section title. The statement on the complexity of the NDFT is correct.
\item[9.] Done, a detailed description of the precomputation is given in
  Appendix A.3, the naming system follows FFTW and does not use abbreviations
  like LAPACK.
\item[10.] It is more convenient to present first (up to Section 4) the
  theoretical part and finally the numerical results (Section 5 and 6).
  Test specifications are given in the second paragraph of Section 5.
\item[11.] Citations now follow the bibliographystyle \verb"acmtrans".
\end{enumerate}

In addition, the layout of tables has been changed to be compliant with
the ACM manuscript preparation guidelines.

\bigskip

Yours sincerely
\closing{\vspace{0.0cm}}
\end{letter}

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