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 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 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.

Solving the coupled Sylvester-like matrix equations via a new finite iterative algorithm

2017 ◽  
Vol 34 (5) ◽  
pp. 1446-1467 ◽  
Author(s):  
Masoud Hajarian
Keyword(s):  
Finite Number ◽  
Iterative Algorithm ◽  
Design Methodology ◽  
Matrix Form ◽  
Matrix Equations ◽  
Content Type ◽  
Numerical Examples ◽  
The Matrix ◽  
Number Of Iterations ◽  
Finite Number Of Iterations

Purpose The purpose of this paper is to obtain an iterative algorithm to find the solution of the coupled Sylvester-like matrix equations. Design/methodology/approach In this work, the matrix form of the conjugate direction (CD) algorithm to find the solution X of the coupled Sylvester-like matrix equations: {A1XB1+M1f1(X)N1=F1,A2XB2+M2f2(X)N2=F2,with fi(X) = X, fi(X) = X¯, fi(X) = XT and fi(X) = XH for i = 1; 2 has been established. Findings It is proven that the algorithm converges to the solution within a finite number of iterations in the absence of round-off errors. Finally, four numerical examples were used to test the proficiency and convergence of the established algorithm. Originality/value The numerical examples have led the author to believe that the generalized CD (GCD) algorithm is efficient and it converges more rapidly in comparison with the CGNR and CGNE algorithms.

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 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 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 Numerical Solution of Duffing Equation by Using an Improved Taylor Matrix Method

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Berna Bülbül ◽  
Mehmet Sezer
Keyword(s):  
Matrix Method ◽  
Numerical Approach ◽  
Duffing Equation ◽  
Matrix Equations ◽  
Algebraic Equations ◽  
Solution Function ◽  
Numerical Examples ◽  
Collocation Points ◽  
Nonlinear Algebraic Equations ◽  
The Matrix

We have suggested a numerical approach, which is based on an improved Taylor matrix method, for solving Duffing differential equations. The method is based on the approximation by the truncated Taylor series about center zero. Duffing equation and conditions are transformed into the matrix equations, which corresponds to a system of nonlinear algebraic equations with the unknown coefficients, via collocation points. Combining these matrix equations and then solving the system yield the unknown coefficients of the solution function. Numerical examples are included to demonstrate the validity and the applicability of the technique. The results show the efficiency and the accuracy of the present work. Also, the method can be easily applied to engineering and science problems.

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.

Sign in / Sign up

Export Citation Format

Share Document