The computational algorithm used in this work for the reconstruction of the three dimensional magnetic field above the photosphere consists of the several steps.
Using the observed photospheric field components 
(x, y, 0)
(i = x, y, z), by the aid of Eq. (6 (click here)),
one calculates the non-linear FFF
function 
. Now, inspection of Eq. (6 (click here))
reveals the existence of
mathematical singularities at points at which 
. While it can
be shown analytically that the actual indeterminacy can be removed by
techniques (see Cuperman et al. 1991b),
for numerical computational purposes, some suitable methods have to be used.
Thus, in our algorithm, 
is calculated at all points except in
some finite width bands on both sides of the curves 
; the
missing 
are then obtained by efficient interpolation
techniques. Thus, using 14-point derivative formulas (see Appendix
A), 
is calculated with a maximum relative error of
!
(This maximum error occurs at the largest r-values considered in this work,
, where 
becomes very small).
We perform a progressive vertical (z) integration for the extrapolation of the photospheric magnetic fields within the FFF model, using as boundary conditions the field values at the photosphere. (As mentioned above, these values are simulated by the exact analytical solution of the FFF model, Eqs. (8)-(13 (click here))).
First, from the (known) 
(x, y, z) - components we
compute the horizontal derivatives 
and
using the high order derivative formulas
given in Appendix A;
this is achieved with a relative accuracy of about 
;
Second, by the aid of Eqs. (4 (click here)) and (5 (click here))
we obtain the
vertical derivatives 
and
.
Third, using suitable 10-term extrapolation
formulas developed in Appendix B, we obtain the sought for results - the
three-component magnetic field in the half space z > 0.
The somewhat less
accurate results obtained at
very low vertical height 
are
corrected by a suitable iterative process; the maximum relative error at
height
is less than 
!