An efficient algebraic eigensolution technique is presented for the simultaneous determination of a set of the lowest frequencies and modal patterns of a structural system. The scheme utilizes a number of reduced generalized coordinates for collapsing the original eigensystem down to a much smaller size; a Stodola-Vianello iteration is employed for the convergence of the eigendata. This technique is then extended for obtaining frequencies and mode shapes in the intermediate to higher frequency ranges. A special application of this extension is given for the frequencies and mode shapes of a free-free structure. Selected examples in structural and continuum dynamics are given to illustrate this technique.