A method for determining narrow upper and lower bounds for the fundamental eigenvalue of a nuclear reactor in either multigroup diffusion theory or transport theory has been developed. This method is based on the Barta-Polya theorem. The Barta-Polya theorem has been extended to yield bounds for multigroup diffusion theory eigenvalue problems. The trial function is a linear sum of known modes and unknown amplitude parameters. Determination of the optimum values of the parameters is a max-min problem of the type that occurs in optimum control and economics. To facilitate numerical computation, a coarse mesh is introduced. A computational method has been developed which quickly yields narrow eigenvalue bounds. Classical methods cannot determine optimum bounds since the function which is to be optimized is not differentiable at the max-min point. The bounds are determined using an iterative method. On each iteration the function is linearized about the mesh points, and linear programming is used to find the optimum solution to the approximate problem. Bounds have been found for a two-region, one-group diffusion theory problem and for a one-group transport theory problem. The bounds are superior to previous results and approach the exact solution as the number of terms in the trial function increases.
