Numerical issues in computing the antitriangular factorization of symmetric indefinite matrices

An algorithm for computing the antitriangular factorization of symmetric matrices, relying only on orthogonal transformations, was recently proposed. The computed antitriangular form straightforwardly reveals the inertia of the matrix. A block version of the latter algorithm was described in a different paper, where it was noticed that the algorithm sometimes fails to compute the correct inertia of the matrix.In this paper we analyze a possible cause of the failure of detecting the inertia and propose a procedure to recover it. Furthermore, we propose a different algorithm to compute the antitriangular factorization of a symmetric matrix that handles most of the singularities of the matrix at the very end of the algorithm.Numerical results are also given showing the reliability of the proposed algorithm.