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.1966年，辛克霍恩（Sinkhorn）在一篇论文中证明：对于任意仅含正元素的实方阵A，存在唯一确定的对角元均为正数的实对角矩阵D，使得B:=DAD为随机矩阵（stochastic matrix），即B的所有行和均等于1。此外，辛克霍恩还提出了一种求解D的迭代方法。如今，辛克霍恩的这一结论及其变体常被称为DAD定理（DAD theorems）。本文的研究目的分为两部分：一方面，我们建立了辛克霍恩DAD定理与化学工程中基于COSMO的活度系数模型内的自洽方程之间的关联；另一方面，我们借助经典不动点理论，给出了辛克霍恩DAD定理的全新构造性证明。本次研究将考虑更大的一类矩阵：对角元为正的非负矩阵。我们的证明统一推导了多种用于求解D的迭代方法的收敛性，其中部分方法已在实际中得到应用，但据我们所知，此前尚未有正式的收敛性证明。