New extrapolation techniques are presented based on the inverse power method to facilitate solution of the multigroup neutron diffusion equation. Unlike the usual acceleration approaches, no estimate of the dominance ratio is required to calculate optimal extrapolation factors. At each outer iteration, the extrapolation factors that correspond to a stationary point of an appropriate functional have been calculated. This technique has been used successfully in the calculation of direct, direct/adjoint, and fixed-source eigenvalue problems for a multigroup formulation of the neutron diffusion equation discretized by finite elements. Numerical tests allow the performance of the variational method to be compared with that of the Chebyshev method.