Symmetric solutions of the coupled generalized Sylvester matrix equations via BCR algorithm

The symmetric solutions of linear matrix equations are extensively required in mathematics and engineering problems. The purpose of this paper is on deriving the biconjugate residual (BCR) algorithm for finding the least Frobenius norm symmetric solution pair (X,Y) of the coupled generalized Sylvester matrix equations{A1XB1+C1XD1+E1YF1=G1,A2XB2+C2XD2+E2YF2=G2.The convergence analysis shows that the BCR algorithm can compute the least Frobenius norm symmetric solution pair of the coupled generalized Sylvester matrix equations within a finite number of iterations in the absence of round-off errors. Finally, we give three numerical examples to illustrate the performance of the BCR algorithm.