We report preliminary results of convergence properties for nonperturbative resolvent approximations to Anderson_like models of magnetic ions in metals. Our study is initially focused on the spin_ 1/2 Anderson model for magnetic impurities, but the methods studied can include multiplet and crystal_field effects which are needed for more accurate descriptions of real systems. We will compare the nonperturbative Lanczos method (tridiagonalization) and similar truncation schemes to exact ground_state energies for the impurity model and assess the efficacy of these nonperturbative approaches to understanding the Anderson lattice, heavy fermions, and other strongly interacting electronic systems.