An iterative algorithm for the least squares bisymmetric solutions of the matrix equations A1XB1=C1,A2XB2=C2A1XB1=C1,A2XB2=C2

In this paper, an iterative algorithm is constructed to solve the minimum Frobenius norm residual problem: min‖(A1XB1A2XB2)−(C1C2)‖ over bisymmetric matrices. By this algorithm, for any initial bisymmetric matrix X0X0, a solution X∗X∗ can be obtained in finite iteration steps in the absence of roundoff errors, and the solution with least norm can be obtained by choosing a special kind of initial matrix. Furthermore, in the solution set of the above problem, the unique optimal approximation solution X̂ to a given matrix X¯ in the Frobenius norm can be derived by finding the least norm bisymmetric solution of a new corresponding minimum Frobenius norm problem. Given numerical examples show that the iterative algorithm is quite effective in actual computation.