Philip A. Knight

The Sinkhorn-Knopp Algorithm: Convergence and Applications

As long as a square nonnegative matrix

A Symmetry Preserving Algorithm for Matrix Scaling

A

On the periodically perturbed logistic equation

contains sufficient nonzero elements, the Sinkhorn-Knopp algorithm can be used to balance the matrix, that is, to find a diagonal scaling of

Matrix Powers in Finite Precision Arithmetic

A

Rank-driven Markov processes

that is doubly stochastic. We relate balancing to problems in traffic flow and describe how balancing algorithms can be used to give a two sided measure of nodes in a graph. We show that with an appropriate modification, the Sinkhorn-Knopp algorithm is a natural candidate for computing the measure on enormous data sets.

Fast rectangular matrix multiplication and QR decomposition

We present an iterative algorithm which asymptotically scales the

A fast algorithm for matrix balancing

\infty

A First Course in Network Theory

-norm of each row and each column of a matrix to one. This scaling algorithm preserves symmetry of the original matrix and shows fast linear convergence with an asymptotic rate of 1/2. We discuss extensions of the algorithm to the 1-norm, and by inference to other norms. For the 1-norm case, we show again that convergence is linear, with the rate dependent on the spectrum of the scaled matrix. We demonstrate experimentally that the scaling algorithm improves the conditioning of the matrix and that it helps direct solvers by reducing the need for pivoting. In particular, for symmetric matrices the theoretical and experimental results highlight the potential of the proposed algorithm over existing alternatives.

Bak-Sneppen-type models and rank-driven processes.

We study the logistic equation modified by a periodic time dependence. The perturbation introduces bifurcation delays which can be calculated explicitly and we qualitatively explain how the bifurcation diagram deforms as the perturbation is increased.

The Hertz contact problem, coupled Volterra integral equations and a linear complementarity problem

If