Approximation of a matrix with positive elements by a matrix of a unit rank

Most modern mathematical methods to solve the problems of the natural science, engineering, economics require the solution of linear problems of high dimension. To reduce the computational complexity a special structure of matrices corresponding to these problems is used. Block-low-rank matrices represent the approximation with good accuracy of dense matrices in a low-parametric format. Blocks of a small rank are given in the form of a product of matrices of smaller size. This can significantly save computer memory. The methods of approximate factorization of block-low-rank matrices can be used for approximate solution and preconditioning of the systems with dense matrices in the problems of aerodynamics, hydrodynamics and electrodynamics, in practical statistics and logistics as well. To build low-parametric representations of matrices based on low-rank approximations of separate blocks algebraic methods are widely used. This paper considers an efficient method of approximations of blocks of a matrix with positive elements by a matrix of a unit rank, i.e. in the form of a product of a column per row. The solution of the problem is sought among admissible representations minimizing the average modulus of logarithms of the ratio of approximate representation of the element to the exact value. The approximating problem is reduced to the problem of linear programming, for which the dual problem is the problem of developing the circulation of the minimum cost in a complete bipartite graph. To solve the problem obtained polynomial algorithm is given.