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.
Li, N. (2015). Iterative algorithm for the generalized $(P,Q)$-reflexive solution of a quaternion matrix equation with $j$-conjugate of the unknowns. Bulletin of the Iranian Mathematical Society, 41(1), 1-22.
MLA
N. Li. "Iterative algorithm for the generalized $(P,Q)$-reflexive solution of a quaternion matrix equation with $j$-conjugate of the unknowns". Bulletin of the Iranian Mathematical Society, 41, 1, 2015, 1-22.
HARVARD
Li, N. (2015). 'Iterative algorithm for the generalized $(P,Q)$-reflexive solution of a quaternion matrix equation with $j$-conjugate of the unknowns', Bulletin of the Iranian Mathematical Society, 41(1), pp. 1-22.
VANCOUVER
Li, N. Iterative algorithm for the generalized $(P,Q)$-reflexive solution of a quaternion matrix equation with $j$-conjugate of the unknowns. Bulletin of the Iranian Mathematical Society, 2015; 41(1): 1-22.