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\title{Reply to referee report on:\\ PARAMETER ESTIMATION OF
MONOMIAL-EXPONENTIAL SUMS\\ by L. Fermo, C. van der Mee, and S. Seatzu}

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 First of all, we would like to express our appreciation for the effort invested
in writing the referee report. Nevertheless, we have a different assessment of
some of the issues raised by the referee.

The major differences between our paper and the five papers referred to by the
referee are the following:
\begin{itemize}
\item[1.] We choose to recover the parameters of the monomial-exponential sum
by using more data than parameters. Thus, using the matrix pencil method which
involves Hankel matrix systems, we arrive at rectangular rather than quadratic
matrix systems. As a result, we need to use the GSVD method.\footnote{In the
new version of our paper, we propose to explain this issue in more detail.} In
the five papers cited the number of data is always the same as the number of
parameters. In the application that has inspired our research (computing the
spectral data of the Zakharov-Shabat system and solving the Cauchy problem of
the nonlinear Schr\"o\-din\-ger equation), it is pretty easy to generate more
data than there are parameters, but using more data than parameters leads to
more accurate results.
\item[2.] We consider the case of multiple zeros (or zeros differing by a small
amount), where the corresponding Vandermonde matrix is either singular or
illconditioned. In the application that has inspired our research, multiple
zeros or single zeros close together are quite normal.
\item[3.] When rendering proofs of theoretical results, we rely on the theory
of linear higher order difference equations, where the so-called Casorati matrix
plays the role of the Wronskian matrix. One advantage of this approach, not
formulated in the paper, is that it leads to an immediate generalization to
systems of monomial-exponential sums, something of great interest to integrable
systems (such as the Manakov system generalizing the Zakharov-Shabat system
presented in the paper).
\end{itemize}
Thus in our opinion we have presented results going far beyond the contents of 
the five papers indicated by the referee.

Let us consider our major example: nonlinear PDE's of integrable type. The
referee may consider them ``very interesting," but this application actually
generates numerical problems not considered in previous papers and considered in
our paper: more data than parameters (to increase accuracy), multiple zeros or
close together simple zeros. We were baffled by the fact that none of the five
predecessor papers ever pointed out the relationship with the theory of
higher order linear difference equations, a fact that has led to more
complicated proofs of certain theoretical results in their papers. It is only
natural that we are asking for a reassment of the referee report.

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