On Distributed Multiplication of Large-Scale Matrices

Abstract

Matrix multiplication is one of the main issues in matrix calculus. The multiplication of small-scale matrices does not cause any difficulties while multiplying of large-scale matrices usually faces many difficulties. In this case, block matrix multiplication is used. Since each block of the resultant matrix is formed independently of all the others, this allows the block matrix multiplication to be executed in parallel, significantly reducing the multiplication time. The article formulates the rules of optimal division of multiplied rectangular matrices into blocks depending on the number of processors used. The method of substantiation and assessment of the time complexity of the formation of the resulting matrix is also presented as well as further improvement of block matrix multiplication is proposed. In case of some kind of rectangular matrices, the process of improvement is reduced to the re-distribution of the elemental rows and columns of the original matrices into several block-rows and block-columns of the multiplied matrices. This leads to a decrease in the dimension of the largest block of the resultant matrix and, consequently, to an acceleration of the multiplication process.