ISSN 0278-6419, Moscow University Computational Mathematics and Cybernetics, 2018, Vol. 42, No. 2, pp. 51–54.
Allerton Press, Inc., 2018.
Original Russian Text
Kh.D. Ikramov, Yu.O. Vorontsov, 2018, published in Vestnik Moskovskogo Universiteta, Seriya 15: Vychislitel’naya Matematika i Kibernetika,
2018, No. 2, pp. 3–6.
Numerical Solution of a Semilinear Matrix Equation
of the Stein Type in the Normal Case
and Yu. O. Vorontsov
Faculty of Computational Mathematics and Cybernetics,
Moscow State University, Moscow, 119991 Russia
OOO Globus Media, Moscow, 115533 Russia
Received December 4, 2017
Abstract—It is known that the solution of the semilinear matrix equation X − AX
B = C can be
reduced to solving the classical Stein equation. The normal case means that the coeﬃcients on the
left-hand side of the resulting equation are normal matrices. A technique for solving the original
semilinear equation in the normal case is proposed. For equations of the order n = 3000, this allows
us to cut the time of computation almost in half, compared to Matlab’s library function dlyap, which
solves Stein equations in the Matlab package.
Keywords: discrete-time Sylvester equation, Stein equation, Schur’s form, normal matrix, Matlab’s
The matrix equation
X − AXB = C (1)
is called the discrete-time Sylvester equation or the Stein equation. We therefore call the semilinear
X − AX
B = C (2)
an equation of the Stein type. All four matrices A, B, C,andX can be rectangular matrices of the same
size, m × n.
It was shown in  that the solution of Eq. (2) can be reduced to solving the conventional Stein
Theorem 1. Equation (2) is uniquely solvable if and only if the equation
X − (AB
B)=C − AC
is uniquely solvable. In addition, both equations have the same solution, X.
Stein equations of modest size can be solved numerically by using one of two well-known orthogonal
methods: the algorithms of Bartels–Stewart (BS) and Golub–Nash–Van Loan (GNL). We brieﬂy
discuss these methods in Section 2. A more detailed description can be found in, e.g., . The latter
method applied to Stein equations is implemented by Matlab’s library function dlyap.
Theorem 1 oﬀers the following method for the numerical solution of equations of the Stein type:
1. Replace Eq. (2) by Eq. (3).
2. Apply the dlyap procedure to Eq. (3).