  The calculation of the emerging radiation from a model atmosphere requires
knowledge of the emissivity and absorption coefficients, which are proportional
to the atomic level population densities of the levels involved in each
transition. Due to the intricate interdependency of the radiation field and the
physical state of the atoms, iterative methods are required in order to
calculate the atomic level population densities. A variety of different methods
have been proposed to solve this problem, which is known as the Non-Local
Thermodynamical Equilibrium (NLTE) problem.
  In this study we have developed a Jacobian-Free Newton-Krylov method (JFNK)
to solve multi-level NLTE radiative transfer problems. Using the Rybicki &
Hummer (1992) method as a reference (Rybicki, G. B. & Hummer, D. G. 1992, A&A,
262, 209), our results show that our JFNK solver can achieve up to a factor two
speed up when using local approximate operators / preconditioners, while also
achieving a lower residual error in the statistical equilibrium equations.
Another advantage of this method is that the addition of charge conservation
and partial redistribution effects should be straight forward.
  Our method can help accelerating the calculation of the emerging spectra from
numerical models and also the reconstruction of chromospheric datasets through
NLTE inversions.
