## Journal of Applied Mathematics

• J. Appl. Math.
• Volume 2015, Special Issue (2015), Article ID 562529, 12 pages.

### LSMR Iterative Method for General Coupled Matrix Equations

#### Abstract

By extending the idea of LSMR method, we present an iterative method to solve the general coupled matrix equations ${\sum }_{k=1}^{q}{A}_{ik}{X}_{k}{B}_{ik}={C}_{i}$, $i=1,2,\dots ,p$, (including the generalized (coupled) Lyapunov and Sylvester matrix equations as special cases) over some constrained matrix groups $({X}_{1},{X}_{2},\dots ,{X}_{q})$, such as symmetric, generalized bisymmetric, and $(R,S)$-symmetric matrix groups. By this iterative method, for any initial matrix group $({X}_{1}^{(0)},{X}_{2}^{(0)},\dots ,{X}_{q}^{(0)})$, a solution group $({X}_{1}^{\mathrm{\ast}},{X}_{2}^{\mathrm{\ast}},\dots ,{X}_{q}^{\mathrm{\ast}})$ can be obtained within finite iteration steps in absence of round-off errors, and the minimum Frobenius norm solution or the minimum Frobenius norm least-squares solution group can be derived when an appropriate initial iterative matrix group is chosen. In addition, the optimal approximation solution group to a given matrix group $({\overline{X}}_{1},{\overline{X}}_{2},\dots ,{\overline{X}}_{q})$ in the Frobenius norm can be obtained by finding the least Frobenius norm solution group of new general coupled matrix equations. Finally, numerical examples are given to illustrate the effectiveness of the presented method.

#### Article information

Source
J. Appl. Math., Volume 2015, Special Issue (2015), Article ID 562529, 12 pages.

Dates
First available in Project Euclid: 15 April 2015

https://projecteuclid.org/euclid.jam/1429105060

Digital Object Identifier
doi:10.1155/2015/562529

Mathematical Reviews number (MathSciNet)
MR3332127

#### Citation

Toutounian, F.; Khojasteh Salkuyeh, D.; Mojarrab, M. LSMR Iterative Method for General Coupled Matrix Equations. J. Appl. Math. 2015, Special Issue (2015), Article ID 562529, 12 pages. doi:10.1155/2015/562529. https://projecteuclid.org/euclid.jam/1429105060

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