Geodesic
Numerical solution of the matrix equations $AX+X^TB=C$ and $X+AX^TB=C$ with rectangular coefficients
Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXV, Tome 405 (2012), pp. 54-58 
Y. O. Vorontsov; Kh. D. Ikramov. Numerical solution of the matrix equations $AX+X^TB=C$ and $X+AX^TB=C$ with rectangular coefficients. Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXV, Tome 405 (2012), pp. 54-58. http://geodesic.mathdoc.fr/item/ZNSL_2012_405_a4/
@article{ZNSL_2012_405_a4,
     author = {Y. O. Vorontsov and Kh. D. Ikramov},
     title = {Numerical solution of the matrix equations $AX+X^TB=C$ and $X+AX^TB=C$ with rectangular coefficients},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {54--58},
     year = {2012},
     volume = {405},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2012_405_a4/}
}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

It is shown how one should supplement the orthogonal algorithms previously proposed by the authors for the equations in the title of the article with square matrix coefficients so that these algorithms are able to solve equations with rectangular coefficients, provided that the latter satisfy the unique solvability.

[1] Yu. O. Vorontsov, Kh. D. Ikramov, “Chislennyi algoritm dlya resheniya matrichnogo uravneniya $AX+X^TB=C$”, Zh. vychisl. matem. matem. fiz., 51:5 (2011), 739–747 | MR | Zbl

[2] Kh. D. Ikramov, Yu. O. Vorontsov, “Matrichnoe uravnenie $X+AX^TB=C$: usloviya odnoznachnoi razreshimosti i algoritm chislennogo resheniya”, Dokl. RAN, 443:5 (2012), 545–548 | MR | Zbl

[3] R. Khorn, Ch. Dzhonson, Matrichnyi analiz, Mir, M., 1989 | MR