For a class of augmented systems for the computation of singular points, a Gauss-Newton like iteration is developed which only needs a represention of the Moore-Penrose pseudoinverse of a matrix which only has the size of the Jacobian of the original system. A technique based on deflated block elimination is presented to determine this pseudoinverse even in the case of large systems. To illustrate the applicability and efficiency of the methods, several numerical examples are included. @[Ð