The method of algebraic function approximation in eigenvalue problems of closed lossless waveguides has been illustrated by means of two examples. The first example takes up a cylindrical guide with a concentric ferrite tube, magnetized axially. The second example consists in approximation of the eigenvalue (square of the propagation constant), by this technique in a cylindrical guide loaded coaxially by an isotropic dielectric rod. It is found that the type of singularities in the dispersion characteristics of the guiding system, can also be identified by the approach. It is shown that the method is both efficient and accurate numerically, in addition to the function theoretic insight it brings when used as a tool in analysis of the guiding system.