Perturbation theory for the LDU factorization and accurate computations for diagonally dominant matrices

Abstract

We present a structured perturbation theory for the LDU factorization of (row) diagonally dominant matrices and we use this theory to prove that a recent algorithm of Ye (Math Comp 77(264):2195-2230, 2008 computes the L , D and U factors of these matrices with relative errors less than 14n 3 u, where u is the unit roundoff and n × n is the size of the matrix. The relative errors for D are componentwise and for L and U are normwise with respect the "max norm" A M = max i j |a i j |. These error bounds guarantee that for any diagonally dominant matrix A we can compute accurately its singular value decomposition and the solution of the linear system Ax = b for most vectors b, independently of the magnitude of the traditional condition number of A and in O(n 3 ) flops. Mathematics Subject Classification (2000)

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.…”

“…We note that using A D and v as parameters is crucial in the development of high relative accuracy algorithms for such matrices in [45]. Study of such parametric perturbations for the LDU factorization is also essential for the error analysis presented in [14] for the algorithm in [45]. Additionally, as mentioned in the introduction, the diagonally dominant parts v and off-diagonal entries often represent physical parameters in applications and it is natural to consider perturbed problems with small variations in these parameters.…”

“…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.…”

“…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.…”