Geodesic
Numerical algorithms for solving matrix equations $AX+BX^T=C$ and $AX+BX^*=C$
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 53 (2013) no. 6, pp. 843-852

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Conditions for the unique solvability of matrix equations of the form $AX+BX^T=C$ and $AX+BX^*=C$ are found. Numerical algorithms of the Bartels–Stewart type for solving such equations are described. Certain numerical tests with these algorithms are presented. In particular, the situation where the conditions for unique solvability are “almost” violated is modeled, and the deterioration of the quality of the computed solution in this situation is traced through. 
@article{ZVMMF_2013_53_6_a0,
     author = {Yu. O. Vorontsov and Khakim D. Ikramov},
     title = {Numerical algorithms for solving matrix equations $AX+BX^T=C$ and $AX+BX^*=C$},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {843--852},
     publisher = {mathdoc},
     volume = {53},
     number = {6},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2013_53_6_a0/}
}
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Yu. O. Vorontsov; Khakim D. Ikramov. Numerical algorithms for solving matrix equations $AX+BX^T=C$ and $AX+BX^*=C$. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 53 (2013) no. 6, pp. 843-852. http://geodesic.mathdoc.fr/item/ZVMMF_2013_53_6_a0/