Iterative Solutions of a Set of Matrix Equations by Using the Hierarchical Identification Principle

Abstract

This paper is concerned with iterative solution to a class of the real coupled matrix equations. By using the hierarchical identification principle, a gradient-based iterative algorithm is constructed to solve the real coupled matrix equations $A_{1}X{B}_1+A_{2}X{B}_2=F_1$ and $C_{1}X{D}_1+C_{2}X{D}_2=F_2$. The range of the convergence factor is derived to guarantee that the iterative algorithm is convergent for any initial value. The analysis indicates that if the coupled matrix equations have a unique solution, then the iterative solution converges fast to the exact one for any initial value under proper conditions. A numerical example is provided to illustrate the effectiveness of the proposed algorithm.