The three-dimensional (3D) reconstruction of magnetic configurations above the photosphere is consid- ered within the framework of the nonlinear force-free-field (FFF) model. The physical- computational algorithm pro- posed and tested incorporates, for the first time, the fol- lowing basic features: 1) Both photospheric vector field, B (z, y, O) and chromospheric line of sight field component, B,(z, y, z) data are utilized; this reduces significantly the degree of ill-posedness characterizing the Cauchy problem corresponding to the case when only B(z,y,O) - values are used as boundary conditions. 2) A high-order, very efficient computational algorithm is developed and used: horizontal derivatives are evaluated by 14 - terms formu- las in 14 different forms, selected such as to provide op- timal computational accuracy; the vertical integration is achieved by the use of moving 10 - term formulas ex- pressed in terms of 10 derivatives and the first Bi(z, y, z) values (i = z,y,z). 3) At neutral points, where inher- ent computational singularities in the values of the FFF- function a arise, rather than using smoothing techniques based on four-neighbouring- values averages, suitable pro- cedures ensuring continuity are developed and used. The overall result of the incorporation of these novel features is an improvement by orders of magnitude of the accu- racy with which the chromospheric fields are reconstructed in the case in which one uses (i) only B(z,y,O) - values as boundary conditions and (ii) relative simple compu- tational formulas and smoothing techniques; at Z = 20, ABi/Bi < 10-3 ! The elimination/minimization of mea- surement errors as well as the fitting of the corrected date to FFF-model-States is also discussed.