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

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 14n3u, 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” \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\|A\|_M = \max_{ij} |a_{ij}|}$$\end{document}. 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(n3) flops.