# The Sinkhorn-Knopp algorithm : convergence and applications

Knight, P.A. (2008) The Sinkhorn-Knopp algorithm : convergence and applications. SIAM Journal on Matrix Analysis and Applications, 30 (1). pp. 261-275. ISSN 0895-4798

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## Abstract

As long as a square nonnegative matrix A contains sufficient nonzero elements, then the Sinkhorn-Knopp algorithm can be used to balance the matrix, that is, to find a diagonal scaling of A that is doubly stochastic. It is known that the convergence is linear, and an upper bound has been given for the rate of convergence for positive matrices. In this paper we give an explicit expression for the rate of convergence for fully indecomposable matrices. We describe how balancing algorithms can be used to give a measure of web page significance. We compare the measure with some well known alternatives, including PageRank. We show that, with an appropriate modi. cation, the Sinkhorn-Knopp algorithm is a natural candidate for computing the measure on enormous data sets.