An efficient algorithm for the generalized (P,Q)-reflexive solution to a quaternion matrix equation and its optimal approximation |
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Authors: | Jing Jiang Ning Li |
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Institution: | 1. Department of Mathematics, Qilu Normal University, Jinan, 250013, P.R. China 2. School of Mathematics and Quantitative Economics, Shandong University of Finance and Economics, Jinan, 250002, P.R. China
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Abstract: | In the present paper, we propose an iterative algorithm for solving the generalized (P,Q)-reflexive solution to the quaternion matrix equation $\sum^{u}_{l=1}A_{l}XB_{l}+\sum^{v}_{s=1} C_{s}\overline{X}D_{s}=F$ . By this iterative algorithm, the solvability of the problem can be determined automatically. When the matrix equation is consistent over 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. |
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