Randomized Iterative Methods For Solving Linear Matrix Equations

Many practical problems in scientific calculations and engineering applications are transformed into solving linear matrix equations after modeling.Therefore,it is very important to design quick and effective methods to solve linear matrix equations,for which many scholars have done a lot of research workIn this paper,two kinds of randomized iteration methods are proposed to solve lin-ear matrix equations.One is to set up a sketched system by introducing low dimensional randomized matrices to the original system,and then take the projection point of cur-rent iteration on the solution space of sketched system as the next iteration.The other is to apply each column of the randomized matrices to act on the original system to form several sketched systems,and next iteration is obtained by weighted average of projec-tion points produced by respectively projecting current iteration onto solution spaces of these sketched systems.We not only induce stochastic iterative schemes from prospec-tive of geometric projection,but also investigate convergence and iteration complexity of proposed methods.In addition,some discrete distribution instances that enable itera-tion to converge are listed.With respect to solving the least squares problem and linear matrix equations with symmetric solutions,corresponding randomized variants are pre-sented,respectively.Finally,numerical experiments are used to verify the effectiveness of the proposed methods for solving linear matrix equations.