Accurate computation of the smallest eigenvalue of a diagonally dominant M-matrix

If each off-diagonal entry and the sum of each row of a diagonally dominant M-matrix are known to certain relative accuracy, then its smallest eigenva lue and the entries of its inverse are known to the same order relative acc uracy independent of any condition numbers. In this paper, we devise algori thms that compute these quantities with relative errors in the magnitude of the machine precision. Rounding error analysis and numerical examples are presented to demonstrate the numerical behaviour of the algorithms.