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 