Accurate and efficient LDU decomposition of diagonally dominant M-matrices

Abstract

NA † Abstract. An efficient method for the computation to high relative accuracy of the LDU decomposition of an n × n row diagonally dominant M-matrix is presented, assuming that the off-diagonal entries and row sums are given. This method costs an additional O(n2) elementary operations over the cost of Gaussian elimination, and leads to a lower triangular, column diagonally dominant matrix and an upper triangular, row diagonally dominant matrix. Comparisons with other methods in the literature are commented and illustrated. 1. Introduction. Recent advances in Numerical Linear Algebra have shown that certain classes of matrices allow computation of certain matrix functions to high relative accuracy, independently of the size of the classical condition number. Some of these classes of matrices are defined by special sign or other structure and require knowledge of some natural parameters to high relative accuracy. In most of those cases, accurate spectral computation (eigenvalues, singular values) is assured once we have an accurate matrix factorization with a suitable pivoting. For instance, the bidiagonal decomposition in the case of totally nonnegative matrices (see also (5), (10)) or an LDU factorization after a symmetric pivoting in the case of diagonally dominant matrices (cf. (4), (13), (14)). Let us focus now on the problem considered in this paper. An algorithm published in (2) computes with high relative accuracy the LDU factorization of an n × n row diagonally dominant M-matrix, if the off-diagonal entries and the row sums are given. The trick is to modify Gaussian elimination to compute the off-diagonal entries and the row sums of each Schur complement without performing subtractions. In addition, symmetric complete pivoting was used in (4) in order to obtain well conditioned L and U factors (U is even row diagonally dominant). This factorization is a special case of a