A numerical algorithm for solving the matrix equation $AX+X^\mathrm TB=C$

An algorithm of the Bartels–Stewart type for solving the matrix equation $AX+X^\mathrm TB=C$ is proposed. By applying the $\mathrm{QZ}$ algorithm, the original equation is reduced to an equation of the same type having triangular matrix coefficients $A$ and $B$. The resulting matrix equation is equivalent to a sequence of low-order systems of linear equations for the entries of the desired solution. Through numerical experiments, the situation where the conditions for unique solvability are “nearly” violated is simulated. The loss of the quality of the computed solution in this situation is analyzed.