A convenient and practical method of numerical determination of the fundamental eigenfunction and eigenvalue in a class of linear eigenvalue problems has been developed and applied in two and three dimensions. The method is based on use of a network and of difference equations, but departs from previous methods in that it is not iterative. Rather, a polynomial operator is applied to a trial function just once, to accomplish a determinable degree of reduction in all eigenfunctions other than the fundamental that are contained in the trial function. In the case of the diffusion equation, the polynomial operator is a Tschebyscheff polynomial of a simple averaging operator. It is shown that this operator, when of degreem, is ``better'' than any other polynomial operator of this degree and much ``better'' thanmiterations of a simple averaging operator—``better'' in the sense of accomplishing to a greater degree the elimination ofallunwanted eigenfunctions. Techniques for the use of computing equipment for application of the polynomial operator are discussed. By orthogonalization, the method can be applied to modes other than the fundamental.