In the preceding chapters, we have seen a number of situations in which group inverses of singular and irreducible M-matrices have facilitated the analysis of various phenomena. If these group inverses are to be useful in applied settings, then we need efficient and reliable methods to compute them. In this chapter, we consider some of the algorithmic and stability issues associated with computing the group inverse of irreducible singular M-matrices. Section 8.1 shows how the QR factorisation of an irreducible singular M-matrix yields a method for computing its group inverse, and discusses the use of the Cholesky factorisation for finding the group inverse of the Laplacian matrix of a connected weighted graph. In section 8.2, we consider two algorithms for computing the Drazin inverse of a singular matrix, and their specific implementation to the case of singular irreducible M-matrices. Next, in section 8.3 we pick up on a theme introduced in section 6.4, and present a method by which the group inverse of a matrix of the form I − T , where T is irreducible and stochastic, can be computed in a parallel fashion. Finally, in section 8.4, we present some results on the sensitivity and conditioning of the group inverse for various types of singular, irreducible M-matrices.