Geodesic
A numerical algorithm for solving the matrix equation $AX+X^\mathrm TB=C$
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 51 (2011) no. 5, pp. 739-747 Cet article a éte moissonné depuis la source Math-Net.Ru

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An algorithm of the Bartels–Stewart type for solving the matrix equation $AX+X^\mathrm TB=C$ is proposed. By applying the $\mathrm{QZ}$ algorithm, the original equation is reduced to an equation of the same type having triangular matrix coefficients $A$ and $B$. The resulting matrix equation is equivalent to a sequence of low-order systems of linear equations for the entries of the desired solution. Through numerical experiments, the situation where the conditions for unique solvability are “nearly” violated is simulated. The loss of the quality of the computed solution in this situation is analyzed. 
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     title = {A~numerical algorithm for solving the matrix equation $AX+X^\mathrm TB=C$},
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Yu. O. Vorontsov; Kh. D. Ikramov. A numerical algorithm for solving the matrix equation $AX+X^\mathrm TB=C$. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 51 (2011) no. 5, pp. 739-747. http://geodesic.mathdoc.fr/item/ZVMMF_2011_51_5_a0/

[1] Gantmakher F. R., Teoriya matrits, Nauka, M., 1966

[2] Ikramov Kh. D., Chislennoe reshenie matrichnykh uravnenii, Nauka, M., 1984

[3] Piao F., Zhang Q., Wang Z., “The solution to matrix equation $AX+X^\mathrm TB=C$”, J. Franklin Inst., 344:8 (2007), 1056–1062

[4] Ikramov Kh. D., “Ob usloviyakh odnoznachnoi razreshimosti matrichnogo uravneniya $AX+X^\mathrm TB=C$”, Dokl. RAN, 430:4 (2010), 444–447