Iterative least-squares solutions of coupled Sylvester matrix equations

Abstract

In this paper, we present a general family of iterative methods to solve linear equations, which includes the well-known Jacobi and Gauss–Seidel iterations as its special cases. The methods are extended to solve coupled Sylvester matrix equations. In our approach, we regard the unknown matrices to be solved as the system parameters to be identified, and propose a least-squares iterative algorithm by applying a hierarchical identification principle and by introducing the block-matrix inner product (the star product for short). We prove that the iterative solution consistently converges to the exact solution for any initial value. The algorithms proposed require less storage capacity than the existing numerical ones. Finally, the algorithms are tested on computer and the results verify the theoretical findings.