The analysis of $\delta$ Eri is based on the spectral synthesis method.
The starting values for the atmospheric parameters are discussed in Section~\ref{para}.
The metallicity of the atmospheric models used for the analysis was set to zero for all calculations.
The starting value for the macroturbulence was $\zeta$=0~\kms.

The main tool we used, SME, is very flexible and allows fitting of many spectral parameters at the same time. Basically it finds an optimal fit to the data by minimizing the $\chi^2$, i.e. minimizing the difference between synthetic spectrum and observed data. This is done by selecting the desired spectral regions and choosing a set of free parameters. The code changes these free parameters and re-calculates the synthesis until convergence is achieved. If the initial parameters are not known well it is better not to set more than two free parameters, in order to avoid correlations between the tuned parameters.

We started with the optical wavelength region, finding the best-fit values for \Teff\ as well as for the turbulent and radial velocities. The latter was set to be free throughout all calculation runs, since it does not correlate with other fundamental parameters. For \logg\ we relied on a recent determination using the parallax and kept it fixed. Macroscopic line broadening was assumed to be represented by a Gaussian profile parameterized by the macroturbulent velocity only. Thus, the derived value of $\zeta$ includes rotational and instrumental broadening. A preliminary synthesis calculation resulted in \Teff=5026\,K, $\xi$=1\,\kms and $\zeta$=3.5\,\kms. The radial velocity was found to be $v_{\rm rad}=-3.5$\,\kms.

To estimate the effect which can be produced by choosing different initial parameters, we ran SME several times, where the effective temperature was perturbed by $\pm$200~K and individual abundances of Fe and Ca by $\pm$0.2~dex. For a more precise determination of \Teff\ we included four Fe~II lines free of blends with $\log gf$ values taken from \citet{2009A&A...497..611M}. %(see Table~\ref{tab:FeIIopt}). {\bf Table to be added!}%
The derived average \Teff=5035 $\pm$32\,K was used for all further calculations in the optical as well as in the IR (RVS) region.

The final step was to derive the individual abundances of Ca and Fe from fitting the optical and the RVS regions separately. For the optical region, we derive $\log\epsilon$(Fe I)=7.56 $\pm$0.03\,dex and $\log\epsilon$(Ca I)=6.37 $\pm$0.03\,dex.
For the RVS region, we derive $\log\epsilon$(Fe I)=7.57$\pm$0.03\,dex and $\log\epsilon$(Ca II)=6.39 $\pm$0.01\,dex.
Thus, the abundances derived from the two different spectral regions are in a good agreement.
Relative to the solar abundances used for the line-list calibration (Section~\ref{linelist}), our analysis results in [Fe/H]=+0.13 $\pm$0.03\,dex and [Ca/H]=+0.07 $\pm$0.03\,dex. These values agree with the lower end of the overabundance range ([M/H]= +0.1 ... +0.25) found for $\delta$~Eri in previous studies (see Table~\ref{tab:para}). As a final step, we performed a consistency check by assuming the metallicity of +0.13 dex. It showed no changes for the optical part of the spectrum between solar abundances of Ca and Fe if compared to the abundances, derived with non-solar metallicity. However, the same calculations performed for the infra-red part of the spectrum yielded the difference for Ca and Fe of +0.07 and +0.05 dex respectively. These numbers show that the atmosphere model changes its structure when we are moving towards the infra-red part of the spectrum. This fact is not taken into account by SME because it does an interpolation between the models with the closest parameters. And finally, the comparison of the respective abundances, derived in the optical and IR, yields for the solar metallicity $\epsilon(Ca)_{optical} - \epsilon(Ca)_{IR} = -0.015$ dex and $\epsilon(Fe)_{optical} - \epsilon(Fe)_{IR} = -0.001$ dex.
All results, derived in both regions can be found in Table~\ref{tab:results}.

\begin{table}[h]
   \caption{\label{tab:results} Summary of the results derived for $\delta$ Eri from optical and infra-red spectral regions.}
\centering
\begin{tabular}{l r r r r}
\hline\hline
$T_{eff}$, K & \logg & log $\epsilon$(Fe) & log $\epsilon$(Ca) \\
\hline
5035 $\pm$ 32 & 3.78 & 7.56 $\pm$ 0.03 & 6.37 $\pm$ 0.03 & Optical \\
              &      & 7.57 $\pm$ 0.03 & 6.39 $\pm$ 0.01 &   IR    \\
\hline
\end{tabular}
\end{table}