New Relative Perturbation Bounds for Ldu Factorizations of Diagonally Dominant Matrices

This work introduces new relative perturbation bounds for the LDU factorization of (row) diagonally dominant matrices under structure-preserving componentwise perturbations. These bounds establish that if (row) diagonally dominant matrices are parameterized via their diagonally dominant parts and off-diagonal entries, then tiny relative componentwise perturbations of these parameters produce tiny relative normwise variations of the L and U factors and tiny relative entrywise variations of the factor D. These results improve previous bounds in an essential way, by including LDU factorizations computed via the column diagonal dominance pivoting strategy. This strategy is specific for (row) diagonally dominant matrices and has the key advantage of yielding L and U factors which are guaranteed to be well-conditioned and, so, the corresponding LDU factorization is guaranteed to be a rank-revealing decomposition. Since rank-revealing decompositions play a fundamental role in highly accurate matrix computations, the results presented in this paper have some important implications, because they will allow us to prove rigorously in a follow-up work that most of the standard tasks in numerical linear algebra can be performed with guaranteed high accuracy for the relevant class of diagonally dominant matrices.