3 Implementation of the algorithm

To implement DIAGONAL, we have decided to proceed in several phases. In this paper we will describe the first of them, and we will postpone the others for the near future. In the code used for comparisons with DELO we have included what we consider to be the three main characteristics of the proposed algorithm:

• The use of a diagonalization procedure to calculate the exponential in the formal solution (1).
• The possibility of treating gradients with depth for azimuth of the magnetic field.
• The linear approximation for the residual correction.

At a later stage, gradients with depth for the so-called $\alpha$ and $\xi$ angles will be treated analytically, and the analytical integration of the function f(z) may be performed (the reader is once more referred to Paper I for proper definitions of these quantities and their roles in the proposed algorithm). These later developments are not expected to bring great advances or new ideas, but just an improvement over what has yet been done.

3.1 Model atmospheres and spectral lines

The model atmosphere used has been limited to an optical depth range of $-4.0 < \log \tau < 1.0$, with a grid for the integration layers distributed linearly in $\log \tau$. Optical depth, $\tau$, at 5000 Å  is used at any moment as integration variable. Although several spectral lines have been tested (indeed Fei lines at 6302.5 Å, 6301.5 Å, 5225 Å  and 5250 Å), comparisons with DELO have been made for a virtual line at 5000 Å  and an electric dipole transition $7D1 \rightarrow 5D0$ (like the one for the Fei line at 5225 Å). Line profiles are calculated in a wavelength range of 300 bins around central wavelength, each bin sampling 2 mÅ.

We have considered not necessary for comparison purposes to use temperature and pressure parameters for description of the model atmosphere. Instead we have preferred to use more direct (in the sense of radiative transfer) parameters as Doppler width, line to continuum absorption coefficients ratio $\eta _0$, damping parameter a and a source function linear with optical depth. From these five parameters, all are kept constant with depth, except for the source function. Constancy or variability of these parameters add nothing to comparisons with DELO, as both codes handle them numerically in the same way. However it is important to mention that DIAGONAL may in the future incorporate analytical treatment of a linear variation with depth of $\eta _0$, once the integration of f(z) (see above) is implemented. Note that since Doppler width is a model--given parameter, microturbulence velocities are implicitly used. For the case of macroturbulence, its inclusion or exclusion in the set of model parameters will not change the results of comparisons. For the sake of simplicity, it is not used.

To these 5 parameters, one needs to add those ones related to the description of the magnetic field and the velocity in the l.o.s up to a total of 13 parameters. To describe magnetic field we use its intensity B, the inclination with the l.o.s $\gamma$and the azimuth $\xi$. The first two vary linearly with $\log \tau$ in our model atmospheres, while azimuth varies linearly with optical depth $\tau$, thus allowing analytical treatment. Finally, the velocity in the l.o.s is also assumed to vary linearly with $\log \tau$.