Convergence analysis of the MCGNR algorithm for the least squares solution group of discrete-time periodic coupled matrix equations

The conjugate gradient normal equations residual minimizing (CGNR) algorithm is a popular tool for solving large nonsymmetric linear systems. In this study, we propose the matrix form of the CGNR (MCGNR) algorithm to find the least squares solution group of the discrete-time periodic coupled matrix equations

{ A 1 , t X t B 1 , t + C 1 , t X t + 1 D 1 , t + E 1 , t Y t F 1 , t = G 1 , t , A 2 , t X t B 2 , t + C 2 , t X t + 1 D 2 , t + E 2 , t Y t F 2 , t = G 2 , t , for t = 1 , 2 , …

with periodic matrices. We prove that the MCGNR algorithm converges in a finite number of steps in the absence of round-off errors, that is, this algorithm has the finite termination property. Also we show that the norms of the residual matrices of the MCGNR algorithm decrease monotonically during its iteration, that is, this algorithm has the residual reducing property. To show the efficiency of the MCGNR algorithm, two numerical examples are presented.