Iterative algorithm for the generalized ‎$‎(P‎,‎Q)‎$‎-reflexive solution of a‎ ‎quaternion matrix equation with ‎$‎j‎$‎-conjugate of the unknowns

In the present paper‎, ‎we propose an iterative algorithm for‎ ‎solving the generalized $(P,Q)$-reflexive solution of the quaternion matrix‎ ‎equation $overset{u}{underset{l=1}{sum}}A_{l}XB_{l}+overset{v} ‎{underset{s=1}{sum}}C_{s}widetilde{X}D_{s}=F$‎. ‎By this iterative algorithm‎, ‎the solvability of the problem can be determined automatically‎. ‎When the‎ ‎matrix equation is consistent over a generalized $(P,Q)$-reflexive matrix $X$‎, ‎a‎ ‎generalized $(P,Q)$-reflexive solution can be obtained within finite iteration‎ ‎steps in the absence of roundoff errors‎, ‎and the least Frobenius norm‎ ‎generalized $(P,Q)$-reflexive solution can be obtained by choosing an‎ ‎appropriate initial iterative matrix‎. ‎Furthermore‎, ‎the optimal approximate‎ ‎generalized $(P,Q)$-reflexive solution to a given matrix $X_{0}$ can be‎ ‎derived by finding the least Frobenius norm generalized $(P,Q)$-reflexive‎ ‎solution of a new corresponding quaternion matrix equation‎. ‎Finally‎, ‎two‎ ‎numerical examples are given to illustrate the efficiency of the proposed methods‎.