scholarly journals An Iterative Algorithm for the Generalized Reflexive Solution of the Matrix EquationsAXB=E,CXD=F

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Deqin Chen ◽  
Feng Yin ◽  
Guang-Xin Huang
Keyword(s):  
Approximate Solution ◽  
Iterative Algorithm ◽  
Matrix Equation ◽  
Iterative Solution ◽  
Linear Matrix ◽  
Optimal Approximate Solution ◽  
Linear Matrix Equation ◽  
Numerical Examples ◽  
Reflexive Matrix ◽  
The Matrix

An iterative algorithm is constructed to solve the linear matrix equation pairAXB=E, CXD=Fover generalized reflexive matrixX. When the matrix equation pairAXB=E, CXD=Fis consistent over generalized reflexive matrixX, for any generalized reflexive initial iterative matrixX1, the generalized reflexive solution can be obtained by the iterative algorithm within finite iterative steps in the absence of round-off errors. The unique least-norm generalized reflexive iterative solution of the matrix equation pair can be derived when an appropriate initial iterative matrix is chosen. Furthermore, the optimal approximate solution ofAXB=E, CXD=Ffor a given generalized reflexive matrixX0can be derived by finding the least-norm generalized reflexive solution of a new corresponding matrix equation pairAX̃B=Ẽ, CX̃D=F̃withẼ=E-AX0B, F̃=F-CX0D. Finally, several numerical examples are given to illustrate that our iterative algorithm is effective.

scholarly journals An Iterative Algorithm for the Least Squares Generalized Reflexive Solutions of the Matrix Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
Feng Yin ◽  
Guang-Xin Huang
Keyword(s):  
Approximate Solution ◽  
Iterative Algorithm ◽  
Applied Mathematics ◽  
Frobenius Norm ◽  
Matrix Equations ◽  
Optimal Approximate Solution ◽  
Numerical Examples ◽  
Reflexive Matrix ◽  
The Matrix ◽  
Residual Problem

The generalized coupled Sylvester systems play a fundamental role in wide applications in several areas, such as stability theory, control theory, perturbation analysis, and some other fields of pure and applied mathematics. The iterative method is an important way to solve the generalized coupled Sylvester systems. In this paper, an iterative algorithm is constructed to solve the minimum Frobenius norm residual problem: min over generalized reflexive matrix . For any initial generalized reflexive matrix , by the iterative algorithm, the generalized reflexive solution can be obtained within finite iterative steps in the absence of round-off errors, and the unique least-norm generalized reflexive solution can also be derived when an appropriate initial iterative matrix is chosen. Furthermore, the unique optimal approximate solution to a given matrix in Frobenius norm can be derived by finding the least-norm generalized reflexive solution of a new corresponding minimum Frobenius norm residual problem: with , . Finally, several numerical examples are given to illustrate that our iterative algorithm is effective.

scholarly journals Finite iterative algorithms for the generalized reflexive and anti-reflexive solutions of the linear matrix equation AXB = C

Filomat ◽  
2017 ◽  
Vol 31 (7) ◽  
pp. 2151-2162 ◽  
Author(s):  
Xiang Wang ◽  
Xiao-Bin Tang ◽  
Xin-Geng Gao ◽  
Wu-Hua Wu
Keyword(s):  
Iterative Method ◽  
Matrix Equation ◽  
Iterative Algorithms ◽  
Iterative Solution ◽  
Necessary Condition ◽  
Linear Matrix ◽  
Linear Matrix Equation ◽  
Reflexive Matrix ◽  
Roundoff Errors ◽  
Sufficient And Necessary Condition

In this paper, an iterative method is presented to solve the linear matrix equation AXB = C over the generalized reflexive (or anti-reflexive) matrix X (A ? Rpxn, B ? Rmxq, C ? Rpxq, X ? Rnxm). By the iterative method, the solvability of the equation AXB = C over the generalized reflexive (or anti-reflexive) matrix can be determined automatically. When the equation AXB = C is consistent over the generalized reflexive (or anti-reflexive) matrix X, for any generalized reflexive (or anti-reflexive) initial iterative matrix X1, the generalized reflexive (anti-reflexive) solution can be obtained within finite iterative steps in the absence of roundoff errors. The unique least-norm generalized reflexive (or anti-reflexive) iterative solution of AXB = C can be derived when an appropriate initial iterative matrix is chosen. A sufficient and necessary condition for whether the equation AXB = C is inconsistent is given. Furthermore, the optimal approximate solution of AXB = C for a given matrix X0 can be derived by finding the least-norm generalized reflexive (or anti-reflexive) solution of a new corresponding matrix equation AXB = C. Finally, several numerical examples are given to support the theoretical results of this paper.

scholarly journals An Efficient Algorithm for the Reflexive Solution of the Quaternion Matrix EquationAXB+CXHD=F

2013 ◽  
Vol 2013 ◽  
pp. 1-14 ◽  
Author(s):  
Ning Li ◽  
Qing-Wen Wang ◽  
Jing Jiang
Keyword(s):  
Iterative Algorithm ◽  
Matrix Equation ◽  
Efficient Algorithm ◽  
Frobenius Norm ◽  
Quaternion Matrix ◽  
Numerical Examples ◽  
Reflexive Matrix ◽  
Roundoff Errors ◽  
Quaternion Matrix Equation ◽  
The Matrix

We propose an iterative algorithm for solving the reflexive solution of the quaternion matrix equationAXB+CXHD=F. When the matrix equation is consistent over reflexive matrixX, a reflexive solution can be obtained within finite iteration steps in the absence of roundoff errors. By the proposed iterative algorithm, the least Frobenius norm reflexive solution of the matrix equation can be derived when an appropriate initial iterative matrix is chosen. Furthermore, the optimal approximate reflexive solution to a given reflexive matrixX0can be derived by finding the least Frobenius norm reflexive solution of a new corresponding quaternion matrix equation. Finally, two numerical examples are given to illustrate the efficiency of the proposed methods.

scholarly journals The least-squares solutions of the matrix equation $ A^{\ast}XB+B^{\ast}X^{\ast}A = D $ and its optimal approximation

2021 ◽  
Vol 7 (3) ◽  
pp. 3680-3691
Author(s):  
Huiting Zhang ◽  
◽  
Yuying Yuan ◽  
Sisi Li ◽  
Yongxin Yuan ◽  
...  
Keyword(s):  
Least Squares ◽  
Matrix Equation ◽  
Canonical Correlation ◽  
Approximation Problem ◽  
Optimal Approximation ◽  
Linear Matrix ◽  
General Representation ◽  
Linear Matrix Equation ◽  
The Matrix

<abstract><p>In this paper, the least-squares solutions to the linear matrix equation $ A^{\ast}XB+B^{\ast}X^{\ast}A = D $ are discussed. By using the canonical correlation decomposition (CCD) of a pair of matrices, the general representation of the least-squares solutions to the matrix equation is derived. Moreover, the expression of the solution to the corresponding weighted optimal approximation problem is obtained.</p></abstract>

scholarly journals The Relaxed Gradient Based Iterative Algorithm for the Symmetric (Skew Symmetric) Solution of the Sylvester EquationAX+XB=C

2017 ◽  
Vol 2017 ◽  
pp. 1-8 ◽  
Author(s):  
Xiaodan Zhang ◽  
Xingping Sheng
Keyword(s):  
Iterative Methods ◽  
Iterative Algorithm ◽  
Matrix Equation ◽  
Symmetric Solution ◽  
Symmetric Matrix ◽  
Iterative Algorithms ◽  
Iterative Solution ◽  
Sylvester Matrix Equation ◽  
Numerical Examples ◽  
Gradient Based

In this paper, we present two different relaxed gradient based iterative (RGI) algorithms for solving the symmetric and skew symmetric solution of the Sylvester matrix equationAX+XB=C. By using these two iterative methods, it is proved that the iterative solution converges to its true symmetric (skew symmetric) solution under some appropriate assumptions when any initial symmetric (skew symmetric) matrixX0is taken. Finally, two numerical examples are given to illustrate that the introduced iterative algorithms are more efficient.

scholarly journals Iterative Algorithm for Solving a Class of Quaternion Matrix Equation over the Generalized(P,Q)-Reflexive Matrices

2013 ◽  
Vol 2013 ◽  
pp. 1-15 ◽  
Author(s):  
Ning Li ◽  
Qing-Wen Wang
Keyword(s):  
System Theory ◽  
Iterative Algorithm ◽  
Matrix Equation ◽  
Frobenius Norm ◽  
Matrix Equations ◽  
Quaternion Matrix ◽  
Special Cases ◽  
Reflexive Matrix ◽  
Quaternion Matrix Equation ◽  
The Matrix

The matrix equation∑l=1uAlXBl+∑s=1vCsXTDs=F,which includes some frequently investigated matrix equations as its special cases, plays important roles in the system theory. In this paper, we propose an iterative algorithm for solving the quaternion matrix equation∑l=1uAlXBl+∑s=1vCsXTDs=Fover generalized(P,Q)-reflexive matrices. The proposed iterative algorithm automatically determines the solvability of the quaternion matrix equation over generalized(P,Q)-reflexive matrices. When the matrix equation is consistent over generalized(P,Q)-reflexive matrices, the sequence{X(k)}generated by the introduced algorithm converges to a generalized(P,Q)-reflexive solution of the quaternion matrix equation. And the sequence{X(k)}converges to the least Frobenius norm generalized(P,Q)-reflexive solution of the quaternion matrix equation when an appropriate initial iterative matrix is chosen. Furthermore, the optimal approximate generalized(P,Q)-reflexive solution for a given generalized(P,Q)-reflexive matrixX0can be derived. The numerical results indicate that the iterative algorithm is quite efficient.

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