Geodesic
Numerical algorithm for solving quadratic matrix equations of a certain class
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 54 (2014) no. 11, pp. 1707-1710 Cet article a éte moissonné depuis la source Math-Net.Ru

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A relationship is found between the solutions to the quadratic matrix equation $X^TDX+AX+X^TB+C=0$, where all the matrix coefficients are $n\times n$ matrices, and the neutral subspaces of the $2n\times 2n$ matrix $2n\times 2n$-матрицы $M=\begin{pmatrix}C&A\\B&D\end{pmatrix}$. This relationship is used to design an algorithm for solving matrix equations of the indicated type. Numerical results obtained with the help of the proposed algorithm are presented. 
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Yu. O. Vorontsov; Kh. D. Ikramov. Numerical algorithm for solving quadratic matrix equations of a certain class. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 54 (2014) no. 11, pp. 1707-1710. http://geodesic.mathdoc.fr/item/ZVMMF_2014_54_11_a1/

[1] Ikramov X. D., “O razreshimosti odnogo klassa kvadratichnykh matrichnykh uravnenii”, Dokl. AN, 455:2 (2014), 135–137 | DOI

[2] Khorn R., Dzhonson Ch., Matrichnyi analiz, Mir, M., 1989

[3] Ikramov X. D., “Razlozhenie Takagi simmetrichnoi unitarnoi matritsy kak konechnyi algoritm”, Zh. vychisl. matem. i matem. fiziki, 52:1 (2012), 4–7