iterative algorithm for the generalized $(p,q)$-reflexive solution of a quaternion matrix equation with $j$-conjugate of the unknowns
چکیده
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.