The aim of this paper is to develop an efficient numerical method to compute the eigenvalues of the stability analysis of a problem describing the motion of a fluid within a cylindrical container heated non-homogeneously from below. An axisymmetric stationary motion settles in, at certain values of the external parameters appearing in the set of partial differential equations modeling the problem. This basic solution is computed by discretizing the equations with a Chebyshev collocation method. Its linear stability is formulated with a generalized eigenvalue problem. The numerical approach (generalized Arnoldi method) uses the idea of preconditioning the eigenvalue problem with a modified Cayley transformation before applying the Arnoldi method. Previous works have dealt with transformations requiring regularity to one of the submatrices. In this article we extend those results to the case in which that submatrix is singular. This method allows a fast computation of the critical eigenvalues which determine whether the steady flow is stable or unstable. The algorithm based on this method is compared to the QZ method and is found to be computationally more efficient. The reliability of the computed eigenvalues in terms of stability is confirmed via pseudospectra calculations.