The distance to instability of a matrix A is a robust measure for the stability of the corresponding dynamical system x = Ax, known to be far more reliable than checking the eigenvalues of A. In this paper, a new algorithm for computing such a distance is sketched. Built on existing approaches, its computationally most expensive part involves a usually modest number of shift-and-invert Amoldi iterations. This makes it possible to address large sparse matrices, such as those arising from discretized partial differential equations.