Estimating $T_{\rm eff}$, radius and luminosity of M-dwarfs using high resolution optical and NIR spectral features

We estimate effective temperature ($T_{\rm eff}$), stellar radius, and luminosity for a sample of 271 M-dwarf stars (M0V-M7V) observed as a part of CARMENES (Calar Alto high-Resolution search for M dwarfs with Exo-earths with Near-infrared and optical Echelle Spectrographs) radial-velocity planet survey. For the first time, using the simultaneously observed high resolution (R$\sim90000$) spectra in the optical (0.52 - 0.96 $\mu$m) and near-infrared (0.96 - 1.71 $\mu$m) bands, we derive empirical calibration relationships to estimate the fundamental parameters of these low-mass stars. We select a sample of nearby and bright M-dwarfs as our calibrators for which the physical parameters are acquired from high-precision interferometric measurements. To identify the most suitable indicators of $T_{\rm eff}$, radius, and luminosity (log $L/L_{\odot}$), we inspect a range of spectral features and assess them for reliable correlations. We perform multivariate linear regression and find that the combination of pseudo equivalent widths and equivalent width ratios of the Ca II at 0.854 $\mu$m and Ca II at 0.866 $\mu$m lines in the optical and the Mg I line at 1.57 $\mu$m in the NIR give the best fitting linear functional relations for the stellar parameters with root mean square errors (RMSE) of 99K, 0.06 $R_{\odot}$ and 0.22 dex respectively. We also explore and compare our results with literature values obtained using other different methods for the same sample of M dwarfs.