Randomized Monte Carlo algorithms for matrix iterations and solving large systems of linear equations

Abstract Randomized scalable vector algorithms for calculation of matrix iterations and solving extremely large linear algebraic equations are developed. Among applications presented in this paper are randomized iterative methods for large linear systems of algebraic equations governed by M-matrices. The crucial idea of the randomized method is that the iterations are performed by sampling random columns only, thus avoiding not only matrix-matrix but also matrix-vector multiplications. The suggested vector randomized methods are highly efficient for solving linear equations of high dimension, the computational cost depends only linearly on the dimension.