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.