In this paper, strong relative perturbation bounds are developed for a number of linear algebra problems involving diagonally dominant matrices. The key point is to parameterize diagonally dominant matrices using their off-diagonal entries and diagonally dominant parts and to consider small relative componentwise perturbations of these parameters. This allows us to obtain new relative perturbation bounds for the inverse, the solution to linear systems, the symmetric indefinite eigenvalue problem, the singular value problem, and the nonsymmetric eigenvalue problem. These bounds are much stronger than traditional perturbation results, since they are independent of either the standard condition number or the magnitude of eigenvalues/singular values. Together with previously derived perturbation bounds for the LDU factorization and the symmetric positive definite eigenvalue problem, this paper presents a complete and detailed account of relative structured perturbation theory for diagonally dominant matrices.; This research was partially supported by the Ministerio de Economía y Competitividad of Spain under grant MTM2012-32542.
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.; Research supported in part by Ministerio de Economía y Competitividad
of Spain under grant MTM2012-32542.