Developing the CGLS algorithm for the least squares solutions of the general coupled matrix equations

2013 ◽  
Vol 37 (17) ◽  
pp. 2782-2798 ◽  
Author(s):  
Masoud Hajarian
Keyword(s):  
Least Squares ◽  
Matrix Equations ◽  
General Coupled Matrix Equations

scholarly journals An Iterative Algorithm for the Reflexive Solution of the General Coupled Matrix Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-15
Author(s):  
Zhongli Zhou ◽  
Guangxin Huang
Keyword(s):  
Iterative Algorithm ◽  
Matrix Group ◽  
Frobenius Norm ◽  
Matrix Equations ◽  
Matrix Solution ◽  
Initial Matrix ◽  
Special Cases ◽  
Reflexive Matrix ◽  
General Coupled Matrix Equations ◽  
Sylvester Matrix Equations

The general coupled matrix equations (including the generalized coupled Sylvester matrix equations as special cases) have numerous applications in control and system theory. In this paper, an iterative algorithm is constructed to solve the general coupled matrix equations over reflexive matrix solution. When the general coupled matrix equations are consistent over reflexive matrices, the reflexive solution can be determined automatically by the iterative algorithm within finite iterative steps in the absence of round-off errors. The least Frobenius norm reflexive solution of the general coupled matrix equations can be derived when an appropriate initial matrix is chosen. Furthermore, the unique optimal approximation reflexive solution to a given matrix group in Frobenius norm can be derived by finding the least-norm reflexive solution of the corresponding general coupled matrix equations. A numerical example is given to illustrate the effectiveness of the proposed iterative algorithm.

An Iterative algorithm for $\eta$-(anti)-Hermitian least-squares solutions of quaternion matrix equations

2015 ◽  
Vol 30 ◽  
Author(s):  
Fatemeh Beik ◽  
Salman Ahmadi-Asl
Keyword(s):  
Least Squares ◽  
Iterative Algorithm ◽  
Numerical Experiments ◽  
Iterative Algorithms ◽  
Iterative Approach ◽  
Matrix Equations ◽  
Quaternion Matrix ◽  
Hermitian Matrices ◽  
Convergence Properties ◽  
Least Squares Problems

Recently, some research has been devoted to finding the explicit forms of the η-Hermitian and η-anti-Hermitian solutions of several kinds of quaternion matrix equations and their associated least-squares problems in the literature. Although exploiting iterative algorithms is superior than utilizing the explicit forms in application, hitherto, an iterative approach has not been offered for finding η-(anti)-Hermitian solutions of quaternion matrix equations. The current paper deals with applying an efficient iterative manner for determining η-Hermitian and η-anti-Hermitian least-squares solutions corresponding to the quaternion matrix equation AXB + CY D = E. More precisely, first, this paper establishes some properties of the η-Hermitian and η-anti-Hermitian matrices. These properties allow for the demonstration of how the well-known conjugate gradient least- squares (CGLS) method can be developed for solving the mentioned problem over the η-Hermitian and η-anti-Hermitian matrices. In addition, the convergence properties of the proposed algorithm are discussed with details. In the circumstance that the coefficient matrices are ill-conditioned, it is suggested to use a preconditioner for accelerating the convergence behavior of the algorithm. Numerical experiments are reported to reveal the validity of the elaborated results and feasibility of the proposed iterative algorithm and its preconditioned version.

scholarly journals Vector least-squares solutions for coupled singular matrix equations

2007 ◽  
Vol 206 (2) ◽  
pp. 1051-1069 ◽  
Author(s):  
Adem Kılıçman ◽  
Zeyad Abdel Aziz Al Zhour
Keyword(s):  
Least Squares ◽  
Matrix Equations ◽  
Singular Matrix
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