### Abstract:

In this paper we generalize an Approximate Lambda Iteration (ALI) technique developed for scalar transfer problems to a vectorial transfer problem for polarized radiation. Scalar ALI techniques are based on a suitable decomposition of the Lambda operator governing the integral form of the transfer equation. Lambda operators for scalar transfer equations are diagonally dominant, offering thus the possibility to use iterative methods of the Jacobi type where the iteration process is based on the diagonal of the Lambda operator (Olson et al., 1986, JQSRT 35, 431). Here we consider resonance polarization, created by the scattering of an anisotropic radiation field, for spectral lines formed with complete frequency redistribution in a 1D axisymmetric medium. The problem can be formulated as an integral equation for a 2-component vector (Rees, 1978PASJ...30..455R) or, as shown by Ivanov (1995A&A...303..621I), as an integral equation for a (2x2) matrix source function which involves the same generalized Lambda operator as the vector integral equation. We find that this equation holds also in the presence of a weak turbulent magnetic field. The generalized Lambda operator is a (2x2) matrix operator. The element {11} describes the propagation of the intensity and is identical to the Lambda operator of non-polarized problems. The element {22} describes the propagation of the polarization. The off-diagonal terms weakly couple the intensity and the polarization. We propose a block Jacobi iterative method and show that its convergence properties are controlled by the propagator for the intensity. We also show that convergence can be accelerated by an Ng acceleration method applied to each element of the source matrix. We extend to polarized transfer a convergence criterion introduced by Auer et al. (1994A&A...292..599A) based on the grid truncation error of the converged solution.