An iterative algorithm for the least squares bisymmetric solutions of the matrix equations

In this paper, an iterative algorithm is constructed to solve the minimum Frobenius norm residual problem: min ‖ ( A 1 X B 1 A 2 X B 2 ) − ( C 1 C 2 ) ‖ over bisymmetric matrices. By this algorithm, for any initial bisymmetric matrix X 0 , a solution 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 X ̂ 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.