A New Perturbation Bound for the LDU Factorization of Diagonally Dominant Matrices

This work introduces a new perturbation bound for the $L$ factor of the LDU factorization of (row) diagonally dominant matrices computed via the column diagonal dominance pivoting strategy. This strategy yields $L$ and $U$ factors which are always well-conditioned and, so, the LDU factorization is guaranteed to be a rank-revealing decomposition. The new bound together with those for the $D$ and $U$ factors in [F. M. Dopico and P. Koev, Numer. Math., 119 (2011), pp. 337--371] establish that if 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 $L$ and $U$ and tiny relative entrywise variations of $D$ when column diagonal dominance pivoting is used. These results will allow us to prove in a follow-up work that such perturbations also lead to strong perturbation bounds for many other problems involving diagonally dominant matrices.

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