The analysis by Bruma & Cuperman (1996)
clearly indicates that the accuracy of the vertical (*z*) integration
in problems of the type considered here,
*can be significantly improved upon formulating
the problem in terms of the value of the function to be integrated
(F) at z=0 and its derivatives at (equidistant) grid points above
z=0. * We here present some final results
obtained in the reference indicated above.

Denote by
the coordinates of a set of equidistant grid points along the
*z*-axis and by
.....
.....
the values of the function *F* at the corresponding grid points.
Let
and
;
then, about 500 grid points are involved.
The first ``moving" set is labeled *j*=0 and starts at the
photosphere,
(Notice that here *F* represents either
or

By the aid of the functions ( *z*=0,1,..,*n*)
it is possible to calculate their first order
derivatives, ;
then, in the resulting system of relationships, we eliminate the
functions
.....
as well as the derivative to obtain
a relation of the form

Thus, at a grid point we have a ten-grid-point-formula
for the function *F* in terms of the *F*-value at
and its derivatives at all other points. In particular, starting at
the photosphere where
the above symbolic formula reads

The explicit relation (20 (click here) ) is obtained by
the aid of the REDUCE-package and it reads

At this point the following remark is in order: at grid points , some simpler integration formulas can be used, namely:

As proposed and implemented by Bruma & Cuperman (1996), utilization of an appropriate iterative procedure leads to improvement of the zero-order results obtained at very low levels

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