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

Abstract

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

Relative Perturbation Theory for Diagonally Dominant Matrices

“…The perturbation bounds obtained in [4,33] amplify δA by factors (1 − 2γ) −1 or (1 − γ) −1 , which can be considered as condition numbers of the corresponding problems and are very large if γ ≈ 1/2 or γ ≈ 1. In contrast, the bounds derived in [10,14,46] and in this work are free of condition numbers for the class of perturbations we consider.…”

“…In this section, we give an overview of diagonally dominant matrices and present some results proved recently in [10,14] that will be used in the subsequent sections. More information on diagonally dominant matrices can be found in [10, section 2] and [14, section 2], and the references therein.…”

“…Several of our results are based on the perturbation bounds for the LDU factorization recently obtained in [10,14]. We first recall that if the LU , or LDU , factorization of a nonsingular matrix exists, then it is unique.…”

“…In [14], a structured perturbation theory is presented for the LDU factorization of diagonally dominant matrices that provides simple and strong bounds on the entrywise relative variations for the diagonal matrix D and the normwise relative variations for the factors L and U . This result has been recently improved in an essential way in [10] by allowing the use of a certain pivoting strategy which guarantees that the factors L and U are always well-conditioned. Computationally, a new algorithm is presented in [45] that accurately computes the LDU factorization of diagonally dominant matrices with entrywise accurate factor D and normwise accurate factors L and U , which is a significant improvement over the classical results in [20,44].…”

“…We shall rely heavily on the LDU perturbation results from [10,14]. Indeed, most of the new bounds in this paper are derived starting from the perturbation bounds i , for i = max, 1, ∞, F, 2.…”