scholarly journals LSMR Iterative Method for General Coupled Matrix Equations

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.

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.

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

2014 ◽  
Vol 2014 ◽  
pp. 1-17 ◽  
Author(s):  
Yajun Xie ◽  
Changfeng Ma
Keyword(s):  
Iterative Method ◽  
Linear Matrix ◽  
Frobenius Norm ◽  
Matrix Equations ◽  
Initial Value ◽  
Least Frobenius Norm Solution ◽  
Centrally Symmetric ◽  
Solution Pair ◽  
Matrix Pair ◽  
Linear Matrix Equations

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.

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 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 Biconjugate residual algorithm for solving general Sylvester-transpose matrix equations

Filomat ◽  
2018 ◽  
Vol 32 (15) ◽  
pp. 5307-5318 ◽  
Author(s):  
Masoud Hajarian
Keyword(s):  
Finite Number ◽  
Efficient Algorithm ◽  
Frobenius Norm ◽  
Matrix Equations ◽  
Numerical Examples ◽  
Least Frobenius Norm Solution ◽  
Residual Algorithm ◽  
Sylvester Matrix Equations ◽  
Number Of Iterations ◽  
Finite Number Of Iterations

The present paper is concerned with the solution of the coupled generalized Sylvester-transpose matrix equations {A1XB1 + C1XD1 + E1XTF1 = M1, A2XB2 + C2XD2 + E2XTF2 = M2, including the well-known Lyapunov and Sylvester matrix equations. Based on a variant of biconjugate residual (BCR) algorithm, we construct and analyze an efficient algorithm to find the (least Frobenius norm) solution of the general Sylvester-transpose matrix equations within a finite number of iterations in the absence of round-off errors. Two numerical examples are given to examine the performance of the constructed algorithm.

scholarly journals A New Method for the Bisymmetric Minimum Norm Solution of the Consistent Matrix EquationsA1XB1=C1,A2XB2=C2

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Aijing Liu ◽  
Guoliang Chen ◽  
Xiangyun Zhang
Keyword(s):  
Iterative Method ◽  
New Method ◽  
Frobenius Norm ◽  
Minimum Norm ◽  
Matrix Equations ◽  
Minimum Norm Solution ◽  
New Iterative Method

We propose a new iterative method to find the bisymmetric minimum norm solution of a pair of consistent matrix equationsA1XB1=C1,A2XB2=C2. The algorithm can obtain the bisymmetric solution with minimum Frobenius norm in finite iteration steps in the absence of round-off errors. Our algorithm is faster and more stable than Algorithm 2.1 by Cai et al. (2010).

scholarly journals An iterative solution to coupled quaternion matrix equations

Filomat ◽  
2012 ◽  
Vol 26 (4) ◽  
pp. 809-826 ◽  
Author(s):  
Caiqin Song ◽  
Guoliang Chen ◽  
Xiangyun Zhang
Keyword(s):  
Iterative Algorithm ◽  
Special Kind ◽  
Iterative Solution ◽  
Inner Product ◽  
Complex Conjugate ◽  
Frobenius Norm ◽  
Matrix Equations ◽  
Quaternion Matrix ◽  
Approximation Solution ◽  
Quaternion Matrices

This note studies the iterative solution to the coupled quaternion matrix equations [?pi=1 T1i(Xi), ?pi=1 T2(Xi)... ?pi=1 Tp(Xi)] = [M1, M2,???, Mp], where Tsi,s = 1, 2,???, p; is a linear operator from Qmi,xni onto Qps?qs, Ms ? Qps?qs,s = 1, 2,???, p.i = 1, 2,???, p, by making use of a generalization of the classical complex conjugate graduate iterative algorithm. Based on the proposed iterative algorithm, the existence conditions of solution to the above coupled quaternion matrix equations can be determined. When the considered coupled quaternion matrix equations is consistent, it is proven by using a real inner product in quaternion space as a tool that a solution can be obtained within finite iterative steps for any initial quaternion matrices [X1(0),???,Xp (0)] in the absence of round-off errors and the least Frobenius norm solution can be derived by choosing a special kind of initial quaternion matrices. Furthermore, the optimal approximation solution to a given quaternion matrix can be derived. Finally, a numerical example is given to show the efficiency of the presented iterative method.

Iterative method to the coupled operator matrix equations with sub-matrix constraint and its application in control

2020 ◽  
pp. 014233122094756
Author(s):  
Caiqin Song
Keyword(s):  
Iterative Method ◽  
Iterative Algorithm ◽  
Numerical Solutions ◽  
Operator Matrix ◽  
Matrix Equations ◽  
Numerical Examples ◽  
Special Cases ◽  
Roundoff Errors ◽  
Norm Constraint ◽  
Exact Constraint

A finite iterative algorithm is presented for solving the numerical solutions to the coupled operator matrix equations in Zhang (2017b). In this paper, a new finite iterative algorithm is presented for solving the constraint solutions to the coupled operator matrix equations [Formula: see text], where the constraint solutions include symmetric solutions, bisymmetric solutions and reflexive solutions as special cases. If this system is consistent, for any initial constraint matrices, the exact constraint solutions can be obtained by the introduced algorithm within finite iterative steps in the absence of the roundoff errors. Also, if this system is not consistent, the least-norm constraint solutions can be obtained within the finite iteration steps in the absence of the roundoff errors. Furthermore, if a group of suitable matrices are given, the optimal approximation solutions can be derived. Finally, several numerical examples are given to show the effectiveness of the presented iterative algorithm.

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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