In order to obtain an analytical least-squares solution we use the
simplified rotated echelle relation. The residual of a given line of index at position
(*x*_{i},*y*_{i}) and of wavelength , and the relative order
number *p*_{i}, is given by

One has to consider the case where the order number *m* is
also an unknown quantity since it has useful practical applications.
We rewrite, using

By determining the partial derivatives
of *R*_{i}^{2} with respect to each parameter (*m*,*a*,*b*,*c*) we obtain
a system of the form *A x* = *B* with *A* a matrix and *B* a
vector such as

with the quantities *S*_{x}, etc... being defined as

and similar formulae for *S*_{y}, ,
, , ,
, *S*_{y2}, , ,
.

The full system was solved using mathematical packages, and solutions for subsets with fixed parameters were determined with a view towards robust techniques for practical calibration procedures.

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