An algorithm for the asymptotic solution of boundary value problems involving vibrations of thin cylindrical shells by means of symbolic computation is presented. The algorithm is based on the method of asymptotic integration of the vibration equations of thin shells, developed by Goldenveizer, Lidsky and Tovstik. A linear shell theory of the Kirchhoff-Love type is employed. The equations describing the vibrations of thin shells contain several parameters, the main of which is the small parameter of the shell thickness. Formal asymptotic solutions in different domains of the space of the parameters are obtained by using a computational geometry approach. Computer algebra methods are employed to study the characteristic equation that involves the construction of the convex hull of a set of points.  The study is limited to the cases for which the asymptotic representation of the solution is the same in the entire domain of integration, and solutions are linearly independent (no turning points, no multiple roots). Axisymmetric as well as non-axisymmetric vibrations are considered. The constructed solutions are used for studying the free vibration spectra of the shells.  The numerical results obtained by applying this algorithm to the particular problem of low frequency vibrations of thin cylindrical shells are in good agreement with the results obtained by finite element analysis, as well as with asymptotic results obtained by authors using other solution techniques.