  Context. Numerical solutions to transfer problems of polarized radiation in
solar and stellar atmospheres commonly rely on stationary iterative methods,
which often perform poorly when applied to large problems. In recent times,
stationary iterative methods have been replaced by state-of-the-art
preconditioned Krylov iterative methods for many applications. However, a
general description and a convergence analysis of Krylov methods in the
polarized radiative transfer context are still lacking. Aims. We describe the
practical application of preconditioned Krylov methods to linear transfer
problems of polarized radiation, possibly in a matrix-free context. The main
aim is to clarify the advantages and drawbacks of various Krylov accelerators
with respect to stationary iterative methods. Methods. We report the
convergence rate and the run time of various Krylov-accelerated techniques
combined with different formal solvers when applied to a 1D benchmark transfer
problem of polarized radiation. In particular, we analyze the GMRES, BICGSTAB,
and CGS Krylov methods, preconditioned with Jacobi, or (S)SOR. Results. Krylov
methods accelerate the convergence, reduce the run time, and improve the
robustness of standard stationary iterative methods. Jacobi-preconditioned
Krylov methods outperform SOR-preconditioned stationary iterations in all
respects. In particular, the Jacobi-GMRES method offers the best overall
performance for the problem setting in use. Conclusions. Krylov methods can be
more challenging to implement than stationary iterative methods. However, an
algebraic formulation of the radiative transfer problem allows one to apply and
study Krylov acceleration strategies with little effort. Furthermore, many
available numerical libraries implement matrix-free Krylov routines, enabling
an almost effortless transition to Krylov methods.
