Iterative Methods to Solve the Generalized Coupled Sylvester-Conjugate Matrix Equations for Obtaining the Centrally Symmetric (Centrally Antisymmetric) Matrix Solutions

The iterative method is presented for obtaining the centrally symmetric (centrally antisymmetric) matrix pair(X,Y)solutions of the generalized coupled Sylvester-conjugate matrix equationsA1X+B1Y=D1X¯E1+F1,A2Y+B2X=D2Y¯E2+F2. On the condition that the coupled matrix equations are consistent, we show that the solution pair(X*,Y*)can be obtained within finite iterative steps in the absence of round-off error for any initial value given centrally symmetric (centrally antisymmetric) matrix. Moreover, by choosing appropriate initial value, we can get the least Frobenius norm solution for the new generalized coupled Sylvester-conjugate linear matrix equations. Finally, some numerical examples are given to illustrate that the proposed iterative method is quite efficient.

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An Iterative Algorithm for the Generalized Reflexive Solutions of the Generalized Coupled Sylvester Matrix Equations

Journal of Applied Mathematics ◽

10.1155/2012/152805 ◽

2012 ◽

Vol 2012 ◽

pp. 1-28 ◽

Cited By ~ 2

Author(s):

Feng Yin ◽

Guang-Xin Huang

Keyword(s):

Iterative Algorithm ◽

Special Kind ◽

Frobenius Norm ◽

Matrix Equations ◽

Sylvester Matrix ◽

Special Cases ◽

The Matrix ◽

Solution Pair ◽

Sylvester Matrix Equations ◽

Matrix Pair

An iterative algorithm is constructed to solve the generalized coupled Sylvester matrix equations(AXB-CYD,EXF-GYH)=(M,N), which includes Sylvester and Lyapunov matrix equations as special cases, over generalized reflexive matricesXandY. When the matrix equations are consistent, for any initial generalized reflexive matrix pair[X1,Y1], the generalized reflexive solutions can be obtained by the iterative algorithm within finite iterative steps in the absence of round-off errors, and the least Frobenius norm generalized reflexive solutions can be obtained by choosing a special kind of initial matrix pair. The unique optimal approximation generalized reflexive solution pair[X̂,Ŷ]to a given matrix pair[X0,Y0]in Frobenius norm can be derived by finding the least-norm generalized reflexive solution pair[X̃*,Ỹ*]of a new corresponding generalized coupled Sylvester matrix equation pair(AX̃B-CỸD,EX̃F-GỸH)=(M̃,Ñ), whereM̃=M-AX0B+CY0D,Ñ=N-EX0F+GY0H. Several numerical examples are given to show the effectiveness of the presented iterative algorithm.

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Partially doubly symmetric solutions of general Sylvester matrix equations

Transactions of the Institute of Measurement and Control ◽

10.1177/0142331219875273 ◽

2019 ◽

Vol 42 (3) ◽

pp. 503-517

Author(s):

Masoud Hajarian

Keyword(s):

Control Systems ◽

Linear Matrix ◽

Frobenius Norm ◽

Matrix Equations ◽

Sylvester Matrix ◽

Sylvester Matrix Equations ◽

Linear Matrix Equations ◽

Scientific Fields ◽

Number Of Iterations ◽

Finite Number Of Iterations

The study of linear matrix equations is extremely important in many scientific fields such as control systems and stability analysis. In this work, we aim to design the Hestenes-Stiefel (HS) version of biconjugate residual (Bi-CR) algorithm for computing the (least Frobenius norm) partially doubly symmetric solution [Formula: see text] of the general Sylvester matrix equations [Formula: see text] for [Formula: see text]. We show that the proposed algorithm converges in a finite number of iterations. Finally, numerical results compare the proposed algorithm to alternative algorithms.

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Iterative method to solve the generalized coupled Sylvester-transpose linear matrix equations over reflexive or anti-reflexive matrix

Computers & Mathematics with Applications ◽

10.1016/j.camwa.2014.04.012 ◽

2014 ◽

Vol 67 (11) ◽

pp. 2071-2084 ◽

Cited By ~ 31

Author(s):

Yajun Xie ◽

Na Huang ◽

Changfeng Ma

Keyword(s):

Iterative Method ◽

Linear Matrix ◽

Matrix Equations ◽

Reflexive Matrix ◽

Linear Matrix Equations

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LSMR Iterative Method for General Coupled Matrix Equations

Journal of Applied Mathematics ◽

10.1155/2015/562529 ◽

2015 ◽

Vol 2015 ◽

pp. 1-12

Author(s):

F. Toutounian ◽

D. Khojasteh Salkuyeh ◽

M. Mojarrab

Keyword(s):

Iterative Method ◽

Symmetric Matrix ◽

Matrix Group ◽

Frobenius Norm ◽

Matrix Equations ◽

Approximation Solution ◽

Matrix Groups ◽

Special Cases ◽

Least Frobenius Norm Solution ◽

General Coupled Matrix Equations

By extending the idea of LSMR method, we present an iterative method to solve the general coupled matrix equations∑k=1qAikXkBik=Ci,i=1,2,…,p, (including the generalized (coupled) Lyapunov and Sylvester matrix equations as special cases) over some constrained matrix groups(X1,X2,…,Xq), such as symmetric, generalized bisymmetric, and(R,S)-symmetric matrix groups. By this iterative method, for any initial matrix group(X1(0),X2(0),…,Xq(0)), a solution group(X1*,X2*,…,Xq*)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(X¯1,X¯2,…,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.

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