This paper is devoted to the exposition and illustration of a technique the authors have designated as the generalized variational method (GVM). The analysis is based on the variational approach and is an outgrowth of investigations in the hyper circle method. In essence, the GVM consists of considering the trial functions that appear symmetrically (quadratically) in a positive-semidefinite variational principle as independent functions. A proposition was proved to demonstrate generally that the approximate eigenvalue obtained from the GVM is at least as accurate as the geometric average of the associated approximate eigenvalues. Also, a conjecture was proposed that the accuracy of the generalized variational eigenvalue is comparable to that of a variational result employing a trial function incorporating the dimensionality of both associated trial functions. The application of the GVM to the perturbation-variational method yielded results that firmly establish the method. The generalized method completes the perturbation-variational method by providing the formerly missing even-order approximate results. For illustration, the GVM was employed to solve a bare reactor with a grey control sheet. Using Ray-leigh-Ritz optimized cosine series and optimized pyramid functions as associated solutions, the generalized variational eigenvalue accuracy indicated the effective combination of the dimensionalities of the associated trial functions.