Several new formulations of the response matrix doubling technique, which employ the combined use of a coarse and fine angular mesh, have been developed. The fine angular mesh is used to represent particle distributions that are highly anisotropic, while the coarse angular mesh is used for angular distributions that are more nearly isotropic. The fine and coarse mesh distributions are related by nonsquare response matrices. Calculations of transmitted and reflected currents for simple one-speed slab problems indicate that the new formulations can greatly improve the efficiency of response matrix calculations. Reflected currents calculated by using one of the new response matrix formulations were found to be 9 to 40 times more accurate than those obtained from conventional response matrix calculations using comparable computational effort. Improvements in transmitted current calculations were nearly as great. The new formulations also are applicable to more realistic calculations. The results of a multigroup calculation were quite encouraging. For energy-dependent problems, we can use a coarse and fine energy mesh as well as a coarse and fine angular mesh, so the potential for improvement appears to be even greater than for one-speed problems.