scholarly journals Generalized Reflexive and Generalized Antireflexive Solutions to a System of Matrix Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Yong Lin ◽  
Qing-Wen Wang
Keyword(s):  
Iterative Algorithms ◽  
Optimal Approximation ◽  
Matrix Equations ◽  
Initial Value ◽  
Numerical Examples ◽  
System Of Matrix Equations ◽  
Iterative Solutions ◽  
The Given ◽  
True Values

Two efficient iterative algorithms are presented to solve a system of matrix equationsA1X1B1+A2X2B2=E,C1X1D1+C2X2D2=Fover generalized reflexive and generalized antireflexive matrices. By the algorithms, the least norm generalized reflexive (antireflexive) solutions and the unique optimal approximation generalized reflexive (antireflexive) solutions to the system can be obtained, too. For any initial value, it is proved that the iterative solutions obtained by the proposed algorithms converge to their true values. The given numerical examples demonstrate that the iterative algorithms are efficient.

scholarly journals (Anti-)Hermitian Generalized (Anti-)Hamiltonian Solution to a System of Matrix Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
Juan Yu ◽  
Qing-Wen Wang ◽  
Chang-Zhou Dong
Keyword(s):  
Least Squares ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Optimal Approximation ◽  
Matrix Equations ◽  
Approximation Solution ◽  
Numerical Examples ◽  
System Of Matrix Equations ◽  
Necessary And Sufficient ◽  
The Given

We mainly solve three problems. Firstly, by the decomposition of the (anti-)Hermitian generalized (anti-)Hamiltonian matrices, the necessary and sufficient conditions for the existence of and the expression for the (anti-)Hermitian generalized (anti-)Hamiltonian solutions to the system of matrix equationsAX=B,XC=Dare derived, respectively. Secondly, the optimal approximation solutionmin⁡X∈K⁡∥X^-X∥is obtained, whereKis the (anti-)Hermitian generalized (anti-)Hamiltonian solution set of the above system andX^is the given matrix. Thirdly, the least squares (anti-)Hermitian generalized (anti-)Hamiltonian solutions are considered. In addition, algorithms about computing the least squares (anti-)Hermitian generalized (anti-)Hamiltonian solution and the corresponding numerical examples are presented.

scholarly journals Numerical Algorithms of the Discrete Coupled Algebraic Riccati Equation Arising in Optimal Control Systems

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Li Wang
Keyword(s):  
Optimal Control ◽  
Robust Control ◽  
Control Systems ◽  
Riccati Equation ◽  
Numerical Algorithms ◽  
Iterative Algorithms ◽  
Iterative Solution ◽  
Algebraic Riccati Equation ◽  
Initial Value ◽  
Numerical Examples

The discrete coupled algebraic Riccati equation (DCARE) has wide applications in robust control, optimal control, and so on. In this paper, we present two iterative algorithms for solving the DCARE. The two iterative algorithms contain both the iterative solution in the last iterative step and the iterative solution in the current iterative step. And, for different initial value, the iterative sequences are increasing and bounded in one algorithm and decreasing and bounded in another. They are all monotonous and convergent. Numerical examples demonstrate the convergence effect of the presented algorithms.

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 Iterative Solutions of a Set of Matrix Equations by Using the Hierarchical Identification Principle

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Huamin Zhang
Keyword(s):  
Unique Solution ◽  
Iterative Algorithm ◽  
Iterative Solution ◽  
Matrix Equations ◽  
Convergence Factor ◽  
Initial Value ◽  
Hierarchical Identification ◽  
The Real ◽  
Gradient Based ◽  
Iterative Solutions

This paper is concerned with iterative solution to a class of the real coupled matrix equations. By using the hierarchical identification principle, a gradient-based iterative algorithm is constructed to solve the real coupled matrix equationsA1XB1+A2XB2=F1andC1XD1+C2XD2=F2. The range of the convergence factor is derived to guarantee that the iterative algorithm is convergent for any initial value. The analysis indicates that if the coupled matrix equations have a unique solution, then the iterative solution converges fast to the exact one for any initial value under proper conditions. A numerical example is provided to illustrate the effectiveness of the proposed algorithm.

scholarly journals The{P,Q,k+1}-Reflexive Solution to System of Matrix EquationsAX=C,XB=D

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Chang-Zhou Dong ◽  
Qing-Wen Wang
Keyword(s):  
General Solution ◽  
Least Squares ◽  
Approximation Problem ◽  
Optimal Approximation ◽  
Matrix Equations ◽  
Solvability Conditions ◽  
Least Squares Solution ◽  
System Of Matrix Equations ◽  
Conjugate Transpose

LetP∈Cm×mandQ∈Cn×nbe Hermitian and{k+1}-potent matrices; that is,Pk+1=P=P⁎andQk+1=Q=Q⁎,where·⁎stands for the conjugate transpose of a matrix. A matrixX∈Cm×nis called{P,Q,k+1}-reflexive (antireflexive) ifPXQ=X (PXQ=-X). In this paper, the system of matrix equationsAX=CandXB=Dsubject to{P,Q,k+1}-reflexive and antireflexive constraints is studied by converting into two simpler cases:k=1andk=2.We give the solvability conditions and the general solution to this system; in addition, the least squares solution is derived; finally, the associated optimal approximation problem for a given matrix is considered.

scholarly journals On the HermitianR-Conjugate Solution of a System of Matrix Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Chang-Zhou Dong ◽  
Qing-Wen Wang ◽  
Yu-Ping Zhang
Keyword(s):  
Least Squares ◽  
Sufficient Conditions ◽  
Approximation Problem ◽  
Matrix Equations ◽  
Complex Matrix ◽  
Solvability Conditions ◽  
Numerical Examples ◽  
System Of Matrix Equations ◽  
Necessary And Sufficient ◽  
Conjugate Solution

LetRbe annbynnontrivial real symmetric involution matrix, that is,R=R−1=RT≠In. Ann×ncomplex matrixAis termedR-conjugate ifA¯=RAR, whereA¯denotes the conjugate ofA. We give necessary and sufficient conditions for the existence of the HermitianR-conjugate solution to the system of complex matrix equationsAX=C and XB=Dand present an expression of the HermitianR-conjugate solution to this system when the solvability conditions are satisfied. In addition, the solution to an optimal approximation problem is obtained. Furthermore, the least squares HermitianR-conjugate solution with the least norm to this system mentioned above is considered. The representation of such solution is also derived. Finally, an algorithm and numerical examples are given.

Difference Schemes for Nonlinear BVPS on the Semiaxis

2007 ◽  
Vol 7 (1) ◽  
pp. 25-47 ◽  
Author(s):  
I.P. Gavrilyuk ◽  
M. Hermann ◽  
M.V. Kutniv ◽  
V.L. Makarov
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Boundary Value Problem ◽  
Difference Scheme ◽  
Adaptive Algorithm ◽  
Order Differential Equation ◽  
Constructive Method ◽  
Numerical Examples ◽  
Exact Difference ◽  
The Given

Abstract The scalar boundary value problem (BVP) for a nonlinear second order differential equation on the semiaxis is considered. Under some natural assumptions it is shown that on an arbitrary finite grid there exists a unique three-point exact difference scheme (EDS), i.e., a difference scheme whose solution coincides with the projection of the exact solution of the given differential equation onto the underlying grid. A constructive method is proposed to derive from the EDS a so-called truncated difference scheme (n-TDS) of rank n, where n is a freely selectable natural number. The n-TDS is the basis for a new adaptive algorithm which has all the advantages known from the modern IVP-solvers. Numerical examples are given which illustrate the theorems presented in the paper and demonstrate the reliability of the new algorithm.

scholarly journals Two matrix methods for solution of nonlinear and linear Lane-Emden type equations with mixed condition by operational matrix

2015 ◽  
Vol 4 (3) ◽  
pp. 420 ◽  
Author(s):  
Behrooz Basirat ◽  
Mohammad Amin Shahdadi
Keyword(s):  
Bernstein Polynomials ◽  
Operational Matrix ◽  
Numerical Procedure ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Mixed Condition ◽  
Matrix Methods ◽  
Numerical Examples ◽  
Linear Algebraic Equations ◽  
The Given

<p>The aim of this article is to present an efficient numerical procedure for solving Lane-Emden type equations. We present two practical matrix method for solving Lane-Emden type equations with mixed conditions by Bernstein polynomials operational matrices (BPOMs) on interval [<em>a; b</em>]. This methods transforms Lane-Emden type equations and the given conditions into matrix equation which corresponds to a system of linear algebraic equations. We also give some numerical examples to demonstrate the efficiency and validity of the operational matrices for solving Lane-Emden type equations (LEEs).</p>

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