The alpha-eigenvalue problem describes the criticality and fundamental neutron flux mode of a nuclear system. Traditionally, the alpha-eigenvalue problem has been solved using methods that focus on supercritical systems with large, positive eigenvalues. These methods, however, struggle for very subcritical problems where the negative eigenvalue can lead to negative absorption, potentially causing the methods to diverge. We present a Rayleigh quotient fixed- point method that is applied to demonstrably primitive discretizations of the one-, two- and three-dimensional Cartesian geometry, multigroup in energy, neutron transport equation. By analyzing the structure of the discretized neutron transport matrix equations, we derive an alpha-eigenvalue update readily implementable in existing neutron transport codes. The derived eigenvalue update is optimal in the least-squares sense and positive eigenvector updates are guaranteed from the Froebenius-Perron Theorem for primitive matrices.
