On Sinkhorn's DAD theorem and the self-consistency equation in COSMO-based activity coefficient models

In a 1966 paper, Sinkhorn proved that for any real square matrix A which has only positive entries there exists a uniquely determined real diagonal matrix D with positive diagonal entries such that B:=DAD is stochastic, i.e. all row sums of B are equal to 1. Moreover, Sinkhorn stated an iterative method for computing D. Nowadays, Sinkhorn's result and its variants are often referred to as DAD theorems. The purpose of this article is twofold. On the one hand, we give the link between Sinkhorn's DAD theorem and the self-consistency equation in COSMO-based activity coefficient models in chemical engineering. On the other hand, we give a new constructive proof of Sinkhorn's DAD theorem by using classical fixed-point theory. Hereby, the larger class of nonnegative matrices with positive diagonal is considered. Our proof uniformly provides convergence for a number of iterative methods for computing D. Some of them are used in practice although, to the best of our knowledge, a formal proof of convergence is missing.