Gradient-based iterative algorithms for generalized coupled Sylvester-conjugate matrix equations

By applying the hierarchical identification principle, the gradient-based iterative algorithm is suggested to solve a class of complex matrix equations. With the real representation of a complex matrix as a tool, the sufficient and necessary conditions for the convergence factor are determined to guarantee that the iterative solutions given by the proposed algorithm converge to the exact solution for any initial matrices. Also, we solve the problem which is proposed by Wu et al. (2010). Finally, some numerical examples are provided to illustrate the effectiveness of the proposed algorithms and testify the conclusions suggested in this paper.     

The Updatable \overline{X} & S Control Charts

The tools for Statistical Process Control (SPC) should be continuously improved in order to continuously improve the product quality. This article proposes a scenario for continuously improving the & S control charts during the use of the charts. It makes use of the information collected from the out-of-control cases in a manufacturing process to update the charting parameters (i.e. the sample size, sampling interval and control limits) step-by-step. Consequently, the resultant control charts (called the updatable charts) become more and more effective to detect the mean shift δ_μ and standard deviation shift δ_σ for the particular process. The updatable charts are able to considerably reduce the average value of the loss function due to the occurrences of the out-of-control cases. Noteworthily, unlike the designs of the economic control charts, the designs of the updatable charts only require limited number of specifications that can be easily decided.     

求解矩阵方程AX=B几种算法的比较

探究了求解矩阵方程AX=B的广义共轭残量法（GCR）、正交极小化法（ORTHOMIN）、重开始的广义共轭残量法（GCR（k））、重开始的正交极小化法（ORTHOMIN（k））等四种算法的迭代思想,讨论了算法的收敛性和收敛速度;用数值实验比较四种算法的性能,得出了重开始的广义共轭残量法能更好地求解大规模矩阵方程的结论。     

An algorithm for solving the matrix equation X = FXF T + S

A now algorithm for solving the matrix equation X = FXF ^T + S, which is important in the control system design, is presented in this paper. The algorithm is based on the QR algorithm for finding the eigenvalues of a matrix and works efficiently for large dimensional problems. A simple example is given to illustrate the algorithm. The method is also applicable to other types of equations such as the Lyapunov equation A ^T X + XA + B = 0.     

Numerical solution of the matrix equation AX + XAT+ B = 0

A recursive algorithm is shown to solve the above equation accurately for large (n leq 146), lightly damped (zeta geq 10^{-3}) systems. About2.5n^{2}storage locations are required, and about2.5n^{3}multiplications are performed per recursion, ten recursions being typical.     

矩阵方程AX-EXF=BY的通解及其应用

给出矩阵方程AX—EXY=BY的一个完全解析的、具有显式表达式和完全自由度的参数解(X,Y)．这里假设矩阵束(E,A B)为R-能控的,F为任意的方阵．相比于现有结论,求解算法不要求矩阵A和F具有特殊的形式,且对它们的特征值没有任何的限制,此外,本文给出的通解还具有结构简洁的特点,作为一个应用,给出了广义系统正常Luenberger函数观测器的一种参数化的设计方法,算例证明了方法的有效性．     

Solving the generalized Sylvester matrix equation AV +BW = VF via Kronecker map

This note considers the solution to the generalized Sylvester matrix equation AV ＋ BW = VF with F being an arbitrary matrix, where V and W are the matrices to be determined. With the help of the Kronecker map, an explicit parametric solution to this matrix equation is established. The proposed solution possesses a very simple and neat form, and allows the matrix F to be undetermined.     

New matrix bounds and iterative algorithms for the discrete coupled algebraic Riccati equation

The discrete coupled algebraic Riccati equation (DCARE) has wide applications in control theory and linear system. In general, for the DCARE, one discusses every term of the coupled term, respectively. In this paper, we consider the coupled term as a whole, which is different from the recent results. When applying eigenvalue inequalities to discuss the coupled term, our method has less error. In terms of the properties of special matrices and eigenvalue inequalities, we propose several upper and lower matrix bounds for the solution of DCARE. Further, we discuss the iterative algorithms for the solution of the DCARE. In the fixed point iterative algorithms, the scope of Lipschitz factor is wider than the recent results. Finally, we offer corresponding numerical examples to illustrate the effectiveness of the derived results.     

Least-squares problem for the quaternion matrix equation AXB+CYD=E over different constrained matrices

For any A=A ₁+A ₂ j∈Q ^n×n and η∈<texlscub>i, j, k</texlscub>, denote A ^{η H} =?η A ^H η. If A ^{η H} =A, A is called an $\eta$-Hermitian matrix. If A ^{η H} =?A, A is called an η-anti-Hermitian matrix. Denote η-Hermitian matrices and η-anti-Hermitian matrices by η HQ ^n×n and η AQ ^n×n , respectively.

By using the complex representation of quaternion matrices, the Moore–Penrose generalized inverse and the Kronecker product of matrices, we derive the expressions of the least-squares solution with the least norm for the quaternion matrix equation AXB+CYD=E over X∈η HQ ^n×n and Y∈η AQ ^n×n .     

Finite iterative Hermitian R-conjugate solutions of the generalized coupled Sylvester-conjugate matrix equations

In this paper, an iterative algorithm for solving a generalized coupled Sylvester-conjugate matrix equations over Hermitian $R$-conjugate matrices given by $A_{1}V{B}_1+C_{1}W{D}_1=E_1\overline{V}{F}_1+G_1$ and $A_{2}V{B}_2+C_{2}W{D}_2=E_2\overline{V}{F}_2+G_2$ is presented. When these two matrix equations are consistent, the convergence theorem shows that a solution can be obtained within finite iterative steps in the absence of round-off error for any initial arbitrary Hermitian $R$-conjugate solution matrices $V_1$, $W_1$. Some lemmas and theorems are stated and proved where the iterative solutions are obtained. A numerical example is given to demonstrate the behavior of the proposed method and to support the theoretical results.