Abstract. 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.Key words. accurate computations, column diagonal dominance pivoting, diagonally dominant matrices, diagonally dominant parts, LDU factorization, rank-revealing decomposition, relative perturbation theory AMS subject classifications. 65F05, 65F15, 15A18, 15A23, 15B99 DOI. 10.1137/13093858X1. Introduction. Perturbation analysis is a classical topic in matrix theory and numerical linear algebra [23,24,35] which still attracts a lot of attention. In recent years, considerable effort has been devoted to deriving sharper perturbation bounds when structured perturbations of important classes of structured matrices are considered (see, as a sample, [1,3,4,7,12,14,18,21,22,26,27,28,29,30,31,33,34,37,38,40]). In this paper, we present a new perturbation bound for the L factor of the LDU factorization of diagonally dominant matrices under a class of componentwise structure-preserving perturbations which are important in numerical computations [14,39,40]. Here, A = LDU is an LDU factorization of A if L is a unit lower triangular matrix, D is a diagonal matrix, and U is a unit upper triangular matrix.Solution of this problem is motivated by several facts. First, apart from its classical applications [20], the LDU factorization has been applied recently to computing accurate rank-revealing decompositions (RRD) [11] of many classes of structure matrices, which are used to perform matrix computations with high relative accuracy [5,11,13,15,17]. In this context, an LDU factorization is an RRD if L and U are well-conditioned. A key point in computing an LDU factorization as an RRD is that the standard partial pivoting strategy does not produce, in general, well-conditioned factors L and U , and that neither complete nor rook pivoting guarantees that L and
Abstract. 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.
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