Developing Kaczmarz method for solving Sylvester matrix equations

Abstract

The Kaczmarz method is one of the most simple and computationally efficient iterative methods for solving large-scale linear algebraic system of equations Ax=b, which at each iteration of it, only one row of coefficient matrix need to be known. The aim of this paper is to derive the matrix form of Kaczmarz method for finding the solution of Sylvester matrix equation. First, we present an iterative algorithm for finding the solution of the matrix equation AX=F by extending the Kaczmarz method and obtain the convergence of it for any initial matrix. Then, by applying the hierarchical approach, we obtain a new iterative algorithm for solving the Sylvester matrix equation AX+XB=C. Finally, three numerical examples are provided to illustrate the effectiveness of the proposed algorithm and compare with some similar gradient-based methods.