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 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.

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 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 Modified CGLS iterative algorithm for solving the generalized Sylvester-conjugate matrix equation

Filomat ◽  
2020 ◽  
Vol 34 (4) ◽  
pp. 1329-1346
Author(s):  
Caiqin Song ◽  
Qing-Wen Wang
Keyword(s):  
Iterative Method ◽  
Least Squares ◽  
Iterative Algorithm ◽  
Matrix Equation ◽  
Inner Product ◽  
Frobenius Norm ◽  
Finite Convergence ◽  
Numerical Examples ◽  
Roundoff Errors ◽  
New Iterative Method

By introducing the real inner product, this paper offers an modified conjugate gradient least squares iterative algorithm (MCGLS)for solving the generalized Sylvester-conjugate matrix equation. The properties of this algorithm are discussed and the finite convergence of this algorithm is proven. This new iterative method can obtain the symmetric least squares Frobenius norm solution within finite iteration steps in the absence of roundoff errors. Finally, two numerical examples are offered to illustrate the effectiveness of the proposed algorithm.

scholarly journals A New Iterative Algorithm for Solving a Class of Matrix Nearness Problem

2012 ◽  
Vol 2012 ◽  
pp. 1-6
Author(s):  
Xuefeng Duan ◽  
Chunmei Li
Keyword(s):  
Iterative Algorithm ◽  
Projection Algorithm ◽  
Frobenius Norm ◽  
Matrix Equations ◽  
Von Neumann ◽  
Numerical Examples ◽  
Matrix Nearness Problem ◽  
Least Frobenius Norm Solution ◽  
Alternating Projection ◽  
The Matrix

Based on the alternating projection algorithm, which was proposed by Von Neumann to treat the problem of finding the projection of a given point onto the intersection of two closed subspaces, we propose a new iterative algorithm to solve the matrix nearness problem associated with the matrix equations AXB=E, CXD=F, which arises frequently in experimental design. If we choose the initial iterative matrix X0=0, the least Frobenius norm solution of these matrix equations is obtained. Numerical examples show that the new algorithm is feasible and effective.

scholarly journals Real Representation Approach to Quaternion Matrix Equation Involving ϕ-Hermicity

2019 ◽  
Vol 2019 ◽  
pp. 1-8 ◽  
Author(s):  
Xin Liu ◽  
Huajun Huang ◽  
Zhuo-Heng He
Keyword(s):  
Matrix Equation ◽  
Sufficient Conditions ◽  
Complete Characterization ◽  
Quaternion Matrix ◽  
Real Representation ◽  
Quaternion Matrix Equation ◽  
The Matrix ◽  
Hermitian Solution ◽  
Necessary And Sufficient

For a quaternion matrix A, we denote by Aϕ the matrix obtained by applying ϕ entrywise to the transposed matrix AT, where ϕ is a nonstandard involution of quaternions. A is said to be ϕ-Hermitian or ϕ-skew-Hermitian if A=Aϕ or A=−Aϕ, respectively. In this paper, we give a complete characterization of the nonstandard involutions ϕ of quaternions and their conjugacy properties; then we establish a new real representation of a quaternion matrix. Based on this, we derive some necessary and sufficient conditions for the existence of a ϕ-Hermitian solution or ϕ-skew-Hermitian solution to the quaternion matrix equation AX=B. Moreover, we give solutions of the quaternion equation when it is solvable.

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