An iterative algorithm for a least squares solution of a matrix equation

In this article, we propose an iterative algorithm to compute the minimum norm least-squares solution of AXB+CYD=E, based on a matrix form of the algorithm LSQR for solving the least squares problem. We then apply this algorithm to compute the minimum norm least-squares centrosymmetric solution of min _X ‖AXB?E‖ _F . Numerical results are provided to verify the efficiency of the proposed method.     

Gradient based iterative solutions for general linear matrix equations

In this paper, we present a gradient based iterative algorithm for solving general linear matrix equations by extending the Jacobi iteration and by applying the hierarchical identification principle. Convergence analysis indicates that the iterative solutions always converge fast to the exact solutions for any initial values and small condition numbers of the associated matrices. Two numerical examples are provided to show that the proposed algorithm is effective.     

Efficient iterative solutions to general coupled matrix equations

Linear matrix equations are encountered in many systems and control applications.In this paper,we consider the general coupled matrix equations（including the generalized coupled Sylvester matrix equations as a special case）l t=1EstYtFst = Gs,s = 1,2,···,l over the generalized reflexive matrix group（Y1,Y2,···,Yl）.We derive an efcient gradient-iterative（GI） algorithm for fnding the generalized reflexive solution group of the general coupled matrix equations.Convergence analysis indicates that the algorithm always converges to the generalized reflexive solution group for any initial generalized reflexive matrix group（Y1（1）,Y2（1）,···,Yl（1））.Finally,numerical results are presented to test and illustrate the performance of the algorithm in terms of convergence,accuracy as well as the efciency.     

Hermitian tridiagonal solution with the least norm to quaternionic least squares problem

Quaternionic least squares (QLS) is an efficient method for solving approximate problems in quaternionic quantum theory. In view of the extensive applications of Hermitian tridiagonal matrices in physics, in this paper we list some properties of basis matrices and subvectors related to tridiagonal matrices, and give an iterative algorithm for finding Hermitian tridiagonal solution with the least norm to the quaternionic least squares problem by making the best use of structure of real representation matrices, we also propose a preconditioning strategy for the Algorithm LSQR-Q in Wang, Wei and Feng (2008) [14] and our algorithm. Numerical experiments are provided to verify the effectiveness of our method.     

Gradient-based maximal convergence rate iterative method for solving linear matrix equations

This paper is concerned with numerical solutions to general linear matrix equations including the well-known Lyapunov matrix equation and Sylvester matrix equation as special cases. Gradient based iterative algorithm is proposed to approximate the exact solution. A necessary and sufficient condition guaranteeing the convergence of the algorithm is presented. A sufficient condition that is easy to compute is also given. The optimal convergence factor such that the convergence rate of the algorithm is maximized is established. The proposed approach not only gives a complete understanding on gradient based iterative algorithm for solving linear matrix equations, but can also be served as a bridge between linear system theory and numerical computing. Numerical example shows the effectiveness of the proposed approach.     

Convergence characterisation of an iterative algorithm for periodic Lyapunov matrix equations

This paper is concerned with convergence characterisation of an iterative algorithm for a class of reverse discrete periodic Lyapunov matrix equation associated with discrete-time linear periodic systems. Firstly, a simple necessary condition is given for this algorithm to be convergent. Then, a necessary and sufficient condition is presented for the convergence of the algorithm in terms of the roots of polynomial equations. In addition, with the aid of the necessary condition explicit expressions of the optimal parameter such that the algorithm has the fastest convergence rate are provided for two special cases. The advantage of the proposed approaches is illustrated by numerical examples.     

加权范数下矩阵方程组的对称解及其最佳逼近

应用复合最速下降法,给出了求解矩阵方程组[(AXB=E,CXD=F)]加权范数下对称解及最佳逼近问题的迭代解法。对任意给定的初始矩阵,该迭代算法能够在有限步迭代计算之后得到矩阵方程组的对称解,并且在上述解集合中也可给出指定矩阵的最佳逼近矩阵。     

Matrix strategies for computing the least trimmed squares estimation of the general linear and SUR models

An algorithm for computing the exact least trimmed squares (LTS) estimator of the standard regression model has recently been proposed. The LTS algorithm is adapted to the general linear and seemingly unrelated regressions models with possible singular dispersion matrices. It searches through a regression tree to find the optimal estimates and has combinatorial complexity. The model is formulated as a generalized linear least squares problem. Efficient matrix techniques are employed to update the generalized residual sum of squares of a subset model. Specifically, the new algorithm utilizes previous computations to update a generalized QR decomposition by a single row. The sparse structure of the model is exploited. Theoretical measures of computational complexity are provided. Experimental results confirm the ability of the new algorithms to identify outlying observations.     

Parameterized solution to a class of sylvester matrix equations

A class of formulas for converting linear matrix mappings into conventional linear mappings are presented. Using them, an easily computable numerical method for complete parameterized solutions of the Sylvester matrix equation AX-EXF=BY and its dual equation XA-FXE=YC are provided. It is also shown that the results obtained can be used easily for observer design. The method proposed in this paper is universally applicable to linear matrix equations.     

矩阵方程AXB+CYD=E最佳逼近自反解的迭代算法

利用复合最速下降法的迭代算法能够求出矩阵方程[AXB+CYD=E]的最佳逼近自反解,但其收敛速度很慢。针对这一问题,提出一种利用共轭方向法的迭代算法。对于任给初始自反矩阵[X1]和[Y1],无论矩阵方程[AXB+CYD=E]是否相容,该算法都可以经过有限次迭代计算出其最佳逼近自反解。两个数值例子表明该算法是可行的,且收敛速度更快。     

Iterative procedure for solution of sylvester generalized matrix equation

An iteration procedure for solving Sylvester generalized matrix equation is proposed in this paper. The sufficient conditions of stability of the iteration procedure for solving this equation are obtained. Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 183–186, May–June, 2000.     

A Numerical Approach To Generalized Periodic Sylvester Matrix Equation

The paper is dedicated to solving the generalized periodic discrete‐time Sylvester matrix equation, which is frequently encountered in control theory and applied mathematics. An iterative algorithm for this equation is presented. The proposed method is developed from a point of least squares method. The rationality of the method is testified by theoretical analysis. The presented approach is numerically reliable and requires less computation. Two numerical examples illustrate the effectiveness of the raised results.     

An iterative algorithm for solving a class of matrix equations

In this paper, an iterative algorithm is presented to solve the Sylvester and Lyapunov matrix equations. By this iterative algorithm, for any initial matrix X1, a solution X＊ can be obtained within finite iteration steps in the absence of roundoff errors. Some examples illustrate that this algorithm is very efficient and better than that of [ 1 ] and [2].     

Solutions to the generalized Sylvester matrix equations by a singular value decomposition

In this paper,solutions to the generalized Sylvester matrix equations AX-XF=BY and MXN-X=TY with A,M∈Rn×n,B,T∈Rn×n,F,N∈Rp×p and the matrices N,F being in companion form,are established by a singular value decomposition of a matrix with dimensions n×(n pr).The algorithm proposed in this paper for the euqation AX-XF=BY does not require the controllability of matrix pair(A,B)andthe restriction that A,F do not have common eigenvalues.Since singular value decomposition is adopted,the algorithm is numerically stable and may provide great convenience to the computation of the solution to these equations,and can perform important functions in many design problems in control systems theory.     

Solutions to the generalized Sylvester matrix equations by a singular value decomposition

In this paper, solutions to the generalized Sylvester matrix equations AX-XF=BY and MXN-X=TY with A,M ∈R ,B,T∈R, F,N∈R and the matrices N, F being in companion form, are established by a singular value decomposition of a matrix with dimensions n×(n + pr). The algorithm proposed in this paper for the euqation AX-XF = BY does not require the controllability of matrix pair (A,B) and the restriction that A,F don’t have common eigenvalues. Since singular value decomposition is adopted, the algorithm is numerically stable and may provide great convenience to the computation of the solution to these equations, and can perform important functions in many design problems in control systems theory.     

A new solution to the generalized Sylvester matrix equation

This note deals with the problem of solving the generalized Sylvester matrix equation AV-EVF=BW, with F being an arbitrary matrix, and provides complete general parametric expressions for the matrices V and W satisfying this equation. The primary feature of this solution is that the matrix F does not need to be in any canonical form, and may be even unknown a priori. The results provide great convenience to the computation and analysis of the solutions to this class of equations, and can perform important functions in many analysis and design problems in control systems theory.     

有理模型辨识的两类新方法——混合迭代与柔性最小二乘法

针对有理模型提出两类辨识方法.首先提出基于递阶辨识思想的混合辨识方法,将模型分解为分子和分母两个子模型,分别用最小二乘法辨识分子参数,用粒子群算法和智能多步长梯度迭代算法辨识分母参数.由于降低了模型维数,且信息向量与噪声不相关,相对于传统的偏差补偿最小二乘算法,混合迭代法可以提高辨识精度并降低计算量.然后,为消除模型结构已知的假设,且充分利用最新数据更新系统参数,提出柔性递推最小二乘辨识方法,将有理模型转化为时变参数系统,进而辨识出时变系统的参数.仿真例子验证了所提出方法的有效性.