3 Algorithm
For complex matrix equations (1), in which\(A,C\in C^{m\times n},\ B,D\in C^{n\times s}\ ,E,F\in C^{m\times s}\). the following Theorem 1, Corollary 1 and Corollary 2, we propose the following algorithms for solving the Hermitian minimum norm least square solution, the real symmetric minimum norm least square solution and the real dissymmetric minimum norm least square solution of (1).
Algorithm 1(solving the Hermitian minimum norm least square solution of (1) )
(1) Input\(\ A,\ B\ ,C,D,E,F\ H,K,G\),
(2) Form\(\text{\ A}^{R},C^{R},\text{\ B}_{c}^{R},D_{c}^{R},E_{c}^{R},\text{\ F}_{c}^{R},\)
(3) Calculat\(\ \par \begin{pmatrix}\text{ves}_{S}\left(X_{0}\right)\\ \text{ves}_{A}\left(X_{1}\right)\\ \end{pmatrix}={[\left(\frac{{{(B}_{c}^{R})}^{T}\otimes A^{R}}{\left(D_{c}^{R})^{T}\otimes C^{R}\right)}\right)HKG]}^{+}\left(\frac{E_{c}^{R}}{F_{c}^{R}}\right)\).
Algorithm 2 (solving the real symmetric minimum norm least square solution of (1))
(1) Input\(\ A,\ B\ ,C,D,E,F\ H,V,K_{n}\),
(2) Form \(A^{R},C^{R},B_{c}^{R},D_{c}^{R},E_{c}^{R},F_{c}^{R},\)
(3) Calculat\(\text{\ ves}_{S}\left(X_{0}\right)={[\left(\frac{{{(B}_{c}^{R})}^{T}\otimes A^{R}}{\left(D_{c}^{R})^{T}\otimes C^{R}\right)}\right)\text{HV}K_{n}]}^{+}\left(\frac{E_{c}^{R}}{F_{c}^{R}}\right)\).
Algorithm 3 (solving the real dissymmetric minimum norm least square solution of (1))
(1) Input\(\ A,\ B\ ,C,D,E,F\ H,V,L_{n}\),
(2) Form \(A^{R},C^{R},B_{c}^{R},D_{c}^{R},E_{c}^{R},F_{c}^{R}\),
(3) Calculat\(\ \text{ves}_{A}\left(X_{0}\right)={[\left(\frac{{{(B}_{c}^{R})}^{T}\otimes A^{R}}{\left(D_{c}^{R})^{T}\otimes C^{R}\right)}\right)\text{HV}L_{n}]}^{+}\left(\frac{E_{c}^{R}}{F_{c}^{R}}\right)\).