On the reflexive and anti-reflexive solutions of the generalised coupled Sylvester matrix equations. (English) Zbl 1196.65081

Dehghan, Mehdi; Hajarian, Masoud

Int. J. Syst. Sci. 41, No. 6, 607-625 (2010).

Authors’ abstract: The generalised coupled Sylvester matrix equations
\[ \begin{aligned} AXB + CYD &= J\\ EXF+GYH&=K,\end{aligned} \]
with unknown matrices \(X\) and \(Y\), have important applications in control and system theory. Also, it is well known that reflexive and anti-reflexive matrices have wide applications in many fields. In this article, we consider the generalised coupled Sylvester matrix equations over reflexive and anti-reflexive matrices. First we propose two new matrix equations equivalent to the generalised coupled Sylvester matrix equations over reflexive and anti-reflexive matrices, respectively. Then, two new iterative algorithms are proposed for solving these matrix equations. A convergence analysis of the proposed iterative algorithms is derived. Finally, some numerical examples are presented to illustrate the theoretical results of this article.

Reviewer: Sheng Chen (Harbin)

MSC:

65F30	Other matrix algorithms (MSC2010)
15A24	Matrix equations and identities
65F10	Iterative numerical methods for linear systems

Keywords:

generalised coupled Sylvester matrix equations; reflexive matrix; anti-reflexive matrix; iterative algorithm; Frobenius norm; convergence; numerical examples

References:

[1]	DOI: 10.1016/j.apnum.2006.07.005 · Zbl 1118.65028
[2]	DOI: 10.1016/0005-1098(95)00037-W · Zbl 0825.93992
[3]	DOI: 10.1137/S0895479895288759 · Zbl 0910.15005
[4]	Chen HC, Parallel Computations and Their Impact on Mechanics (AMD-Vol. 86) pp 101– (1987)
[5]	DOI: 10.1016/S0024-3795(87)90314-4 · Zbl 0631.15006
[6]	DOI: 10.1109/TAC.2005.852558 · Zbl 1365.65083
[7]	DOI: 10.1016/j.automatica.2005.03.026 · Zbl 1086.93063
[8]	DOI: 10.1109/TAC.2005.843856 · Zbl 1365.93551
[9]	Dullerud GE, A Course in Robust Control Theory–A Convex Approach (2000)
[10]	Golub GH, Matrix Computations,, 3. ed. (1996)
[11]	DOI: 10.1023/A:1014807923223 · Zbl 0992.65040
[12]	DOI: 10.1137/0613009 · Zbl 0746.65027
[13]	DOI: 10.1016/S0304-3975(00)00322-4 · Zbl 0972.68183
[14]	DOI: 10.1016/j.amc.2005.12.055 · Zbl 1148.65029
[15]	Peng YX, Applied Mathematics and Computations 160 pp 763– (2005) · Zbl 1068.65056
[16]	DOI: 10.1016/j.apnum.2007.01.025 · Zbl 1136.65046
[17]	DOI: 10.1016/0024-3795(91)90384-9 · Zbl 0736.65031
[18]	DOI: 10.1016/j.cam.2005.04.006 · Zbl 1088.15015
[19]	DOI: 10.1016/j.cam.2008.06.014 · Zbl 1161.65034