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.

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