scholarly journals An accelerated Jacobi-gradient based iterative algorithm for solving sylvester matrix equations

Filomat ◽  
2017 ◽  
Vol 31 (8) ◽  
pp. 2381-2390 ◽  
Author(s):  
Zhaolu Tian ◽  
Maoyi Tian ◽  
Chuanqing Gu ◽  
Xiaoning Hao
Keyword(s):  
Theoretical Analysis ◽  
Iterative Algorithm ◽  
Matrix Equations ◽  
Initial Value ◽  
True Solution ◽  
Numerical Examples ◽  
Sylvester Matrix ◽  
Gradient Based ◽  
Sylvester Matrix Equations

In this paper, an accelerated Jacobi-gradient based iterative (AJGI) algorithm for solving Sylvester matrix equations is presented, which is based on the algorithms proposed by Ding and Chen [6], Niu et al. [18] and Xie et al. [25]. Theoretical analysis shows that the new algorithm will converge to the true solution for any initial value under certain assumptions. Finally, three numerical examples are given to verify the eficiency of the accelerated algorithm proposed in this paper.

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

2012 ◽  
Vol 2012 ◽  
pp. 1-28 ◽  
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.

scholarly journals Algorithms for Solving Nonhomogeneous Generalized Sylvester Matrix Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-5
Author(s):  
Ehab A. El-Sayed ◽  
Eid E. El Behady
Keyword(s):  
Explicit Solution ◽  
Second Order ◽  
New Method ◽  
Matrix Equations ◽  
Numerical Examples ◽  
First Order ◽  
Hessenberg Form ◽  
Sylvester Matrix ◽  
The Matrix ◽  
Sylvester Matrix Equations

This paper considers a new method to solve the first-order and second-order nonhomogeneous generalized Sylvester matrix equations AV+BW= EVF+R and MVF2+DV F+KV=BW+R, respectively, where A,E,M,D,K,B, and F are the arbitrary real known matrices and V and W are the matrices to be determined. An explicit solution for these equations is proposed, based on the orthogonal reduction of the matrix F to an upper Hessenberg form H. The technique is very simple and does not require the eigenvalues of matrix F to be known. The proposed method is illustrated by numerical examples.

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