We have developed and implemented a computational algorithm for the reconstruction of the vector - chromospheric magnetic fields within the framework of the non-linear FFF theoretical model. This algorithm represents the extension, modification and improvement of previously proposed progressive vertical extrapolations (see, e.g. Wu et al. 1985; Cuperman et al. 1990) in the following sense:
1. In the calculation of the non-linear FFF-function 
(by the aid of Eq. (6 (click here))), the derivatives
and 
are computed by the aid of 14-term formulas, rather than by
2-3 term formulas as previously used.
Moreover, 14-versions of the 14-term formulas are used
at various points, in order to ensure about equal minimum
computational error. The maximum relative error in the
computation of these derivatives is smaller than 
.
As a consequence, 
is computed with a maximum relative error
smaller than 
(see Fig. 3 (click here)).
2. The same 14-term formula (and its 14 versions) is applied for the
calculation of the derivatives
and 
which enter the expressions of 
and
, Eqs. (4 (click here)) and (5 (click here)),
respectively.
Consequently, the same high computational precision is
obtained also for this purpose.
3. At points where 
, to avoid inherent
``mathematical" discontinuities, rather than calculating
by the aid of Eq. (6 (click here)) in conjunction with some
smoothing techniques, a suitable, different approach is used.
The same holds
at points where 
or 
, including the case in which 
.
Thus, at all points in the range
, the maximum
relative error in the computed function 
is smaller than
(See Fig. 3 (click here)).
4. The progressive vertical (z) extrapolation is based on a ``moving"
10-term formula, including information from ten consecutive grid
points, q=0,1,2,...,9. Thus, the extrapolated value
is expressed in terms of its derivatives
at grid points q=0,1,2,...,9 as well as the value of the function
itself at the point q=0; the various derivatives are calculated
according to Eqs. (4 (click here)) and (5 (click here)),
with 
,
and 
obtained as
indicated above.
5. A special correction is applied to the extrapolation results
obtained for the first few grid points above the photosphere
(especially at the first point, 
because in these cases the information required by the
``ten-term formula" is not available.
6. As a result ot the computational algorithm summarized above,
an exceptional good extrapolation accuracy is obtained:
at 
, the
maximum relative error in the extrapolated vector magnetic field
component 
is smaller than
Figure 10: Average relative error
(solid curve) and maximum relative
error (dotted curve) in the reconstruction of the magnetic field
components
and
as a function of the normalized height, 
Acknowledgements
The authors are grateful to G. Ai, J.J. Aly, P. Demoulin, C. Fang, J.C. Henoux, B. Leroy, J. Rayrole, T. Sakurai, M. Semel and H. Zhang for valuable discussions or/and for a critically reading of the manuscript.