We introduce a generalization of the neutron diffusion equation, the solution of which is an accurate approximation to the transport scalar Flux. In this generalization we utilize auxiliary transport calculations of the system of interest to compute an accurate, pointwise diffusion coefficient. We have specified a procedure to generate and improve this auxiliary information in a systematic way, leading to improvement in the calculated diffusion scalar flux. This improvement is shown to be contingent upon satisfying the condition of positive calculated-diffusion coefficients, and we present an algorithm that ensures this positivity. Our generalized diffusion theory is also shown to be compatible with conventional diffusion theory in the sense that the same methods and codes can be used to calculate a solution for both. The accuracy of the method compared to reference SNtransport calculations is demonstrated for a wide variety of examples.