矩阵方程AXB+CYD=E的M对称解的迭代算法

在共轭梯度思想的启发下,结合线性投影算子,给出迭代算法求解了线性矩阵方程AXB+CYD=E的M对称解[X,Y]及其最佳逼近.当矩阵方程AXB+CYD=E有M对称解时,应用迭代算法,在有限的误差范围内,对任意初始M对称矩阵对[X_,Y_1],经过有限步迭代可得到矩阵方程的M对称解;选取合适的初始迭代矩阵,还可得到极小范数M对称解.而且,对任意给定的矩阵对[X,Y],矩阵方程AXB+CYD=E的最佳逼近可以通过迭代求解新的矩阵方程AXB+CYD=E的极小范数M对称解得到.文中的数值例子证实了该算法的有效性.     

矩阵方程AXB+CXD=F对称解的迭代算法

在共轭梯度思想的启发下,本文给出了迭代算法求解约束矩阵方程AXB+CXD=F的对称解及其最佳逼近.应用迭代算法,矩阵方程AXB+CXD=F的相容性可以在迭代过程中自动判断.当矩阵方程AXB+CXD=F有对称解时,在有限的误差范围内,对任意初始对称矩阵X1,运用迭代算法,经过有限步可得到矩阵方程的对称解;选取合适的初始迭代矩阵,还可以迭代出极小范数对称解.而且,对任意给定的矩阵X0,矩阵方程AXB+CXD=F的最佳逼近对称解可以通过迭代求解新的矩阵方程A(X)B+C(X)D=(F)的极小范数对称解得到.文中的数值例子证实了该算法的有效性.     

求解矩阵方程组A1XB1+C1XD1=E1,A2XB2+C2XD2=E2的迭代算法北大核心CSCD

应用共轭梯度方法和线性投影算子,给出迭代算法求解了线性矩阵方程组A1XB1＋C1XD1=E1,A2XB2＋C2XD2=E2在任意线性子空间上的约束解及其最佳逼近.可以证明,当矩阵方程组A1XB1＋C1XD1=E1,A2XB2＋C2XD2=E2相容时,所给迭代算法经过有限步迭代可得到矩阵方程组的约束解,极小范数解和最佳逼近.文中的数值例子证实了该算法的有效性.     

矩阵方程AX＋XB=C的双对称解及其最佳逼近

提出一种求解线性矩阵方程AX＋XB=C双对称解的迭代法.该算法能够自动地判断解的情况,并在方程相容时得到方程的双对称解,在方程不相容时得到方程的最小二乘双对称解.对任意的初始矩阵,在没有舍入误差的情况下,经过有限步迭代得到问题的一个双对称解.若取特殊的初始矩阵,则可以得到问题的极小范数双对称解,从而巧妙地解决了对给定矩...     

矩阵方程AXB=C的反对称解和极小范数反对称解

建立了求矩阵方程AXB=C反对称解的迭代方法.使用该方法不仅能够判断反对称解的存在性,而且在有反对称解时,能够在有限步迭代计算之后得到反对称解.选取特殊的初始矩阵,可求得极小范数反对称解.     

一类Lyapunov矩阵方程的双反对称解的迭代算法

基于求线性代数方程组的共轭梯度法的思想,建立一种求Lyapunov矩阵方程的双反对称解的迭代算法,对任意给定的初始双反对称矩阵,算法能够在有限步迭代计算后得到矩阵方程的极小范数双反对称解,同时在上述解集中也可得出指定矩阵的最佳逼近双反称矩阵.数值算例表明,迭代算法是有效的.     

关于矩阵方程XB=C的结构解

很多应用中导出矩阵方程XB=G,本文考虑此方程的结构解.首先考虑自伴矩阵解及反自伴矩阵解,接下来考虑广义对称解及广义反对称解,最后讨论更广泛的矩阵方程AXB=C的酉矩阵解.所得结果推广了Sun,Tisseur,Trench等人的-些结果.     

矩阵方程AXB+CX~TD=E自反最佳逼近解的迭代算法

本文研究了Sylvester矩阵方程AXB+CXTD=E自反(或反自反)最佳逼近解.利用所提出的共轭方向法的迭代算法,获得了一个结果:不论矩阵方程AXB+CXTD=E是否相容,对于任给初始自反(或反自反)矩阵X1,在有限迭代步内,该算法都能够计算出该矩阵方程的自反(或反自反)最佳逼近解.最后,三个数值例子验证了该算法是有效性的.     

Sylvester矩阵方程极小范数最小二乘解的迭代解法

研究了Sylvester矩阵方程最小二乘解以及极小范数最小二乘解的迭代解法,首先利用递阶辨识原理,得到了求解矩阵方程AX+YB=C的极小范数最小二乘解的一种迭代算法,进而,将这种算法推广到一般线性矩阵方程A_iX_iB_i=C的情形,最后,数值例子验证了算法的有效性.     

矩阵方程AXAT=C的对称斜反对称解

设A∈Rm×n,C∈Rm×m给定,利用矩阵的广义奇异值分解和对称斜反对称矩阵的性质,得到了矩阵方程(1)AXAT=C存在对称斜反对称解的充要条件和通解表达式;证明了若方程(1)有解,则一定存在唯一极小范数解,并给出了极小范数解的具体表达式和求解步骤.     

LSQR iterative method for generalized coupled Sylvester matrix equations

An iterative method is proposed to solve generalized coupled Sylvester matrix equations, based on a matrix form of the least-squares QR-factorization (LSQR) algorithm. By this iterative method on the selection of special initial matrices, we can obtain the minimum Frobenius norm solutions or the minimum Frobenius norm least-squares solutions over some constrained matrices, such as symmetric, generalized bisymmetric and (R, S)-symmetric matrices. Meanwhile, the optimal approximate solutions to the given matrices can be derived by solving the corresponding new generalized coupled Sylvester matrix equations. Finally, numerical examples are given to illustrate the effectiveness of the present method.     

Circulant preconditioners for discrete ill-posed Toeplitz systems

Circulant preconditioners are commonly used to accelerate the rate of convergence of iterative methods when solving linear systems of equations with a Toeplitz matrix. Block extensions that can be applied when the system has a block Toeplitz matrix with Toeplitz blocks also have been developed. This paper is concerned with preconditioning of linear systems of equations with a symmetric block Toeplitz matrix with symmetric Toeplitz blocks that stem from the discretization of a linear ill-posed problem. The right-hand side of the linear systems represents available data and is assumed to be contaminated by error. These kinds of linear systems arise, e.g., in image deblurring problems. It is important that the preconditioner does not affect the invariant subspace associated with the smallest eigenvalues of the block Toeplitz matrix to avoid severe propagation of the error in the right-hand side. A perturbation result indicates how the dimension of the subspace associated with the smallest eigenvalues should be chosen and allows the determination of a suitable preconditioner when an estimate of the error in the right-hand side is available. This estimate also is used to decide how many iterations to carry out by a minimum residual iterative method. Applications to image restoration are presented.     

Minimum residual methods for augmented systems

For large systems of linear equations, iterative methods provide attractive solution techniques. We describe the applicability and convergence of iterative methods of Krylov subspace type for an important class of symmetric and indefinite matrix problems, namely augmented (or KKT) systems. Specifically, we consider preconditioned minimum residual methods and discuss indefinite versus positive definite preconditioning. For a natural choice of starting vector we prove that when the definite and indenfinite preconditioners are related in the obvious way, MINRES (which is applicable in the case of positive definite preconditioning) and full GMRES (which is applicable in the case of indefinite preconditioning) give residual vectors with identical Euclidean norm at each iteration. Moreover, we show that the convergence of both methods is related to a system of normal equations for which the LSQR algorithm can be employed. As a side result, we give a rare example of a non-trivial normal(1) matrix where the corresponding inner product is explicitly known: a conjugate gradient method therefore exists and can be employed in this case. This work was supported by British Council/German Academic Exchange Service Research Collaboration Project 465 and NATO Collaborative Research Grant CRG 960782     

An iterative method for the least squares symmetric solution of matrix equation AXB = C

This paper an iterative method is presented to solve the minimum Frobenius norm residual problem: with unknown symmetric matrix . By the iterative method, for any initial symmetric matrix , a solution can be obtained within finite iteration steps in the absence of roundoff errors, and the solution with least Frobenius norm can be obtained by choosing a special kind of initial symmetric matrix. In addition, in the solution set of the minimum Frobenius norm residual problem, the unique optimal approximation solution to a given matrix in Frobenius norm can be expressed as , where is the least norm symmetric solution of the new minimum residual problem: with . Given numerical examples are show that the iterative method is quite efficient.Research supported by Scientific Research Fund of Hunan Provincial Education Department of China (05C797), by China Postdoctoral Science Foundation (2004035645) and by National Natural Science Foundation of China (10571047).     

求矩阵方程AXB+CXD=F的中心对称最小二乘解的迭代算法

该文建立了求矩阵方程AXB+CXD=F的中心对称最小二乘解的迭代算法.使用该算法不仅可以判断该矩阵方程的中心对称解的存在性,而且无论中心对称解是否存在,都能够在有限步迭代计算之后得到中心对称最小二乘解.选取特殊的初始矩阵时,可求得极小范数中心对称最小二乘解.同时,也能给出指定矩阵的最佳逼近中心对称矩阵.     

LSQR iterative common symmetric solutions to matrix equations AXB = E and CXD = F

A new matrix based iterative method is presented to compute common symmetric solution or common symmetric least-squares solution of the pair of matrix equations AXB = E and CXD = F. By this iterative method, for any initial matrix X₀, a solution X^∗ can be obtained within finite iteration steps if exact arithmetic was used, and the solution X^∗ with the minimum Frobenius norm can be obtained by choosing a special kind of initial matrix. In addition, the unique nearest common symmetric solution or common symmetric least-squares solution to given matrix in Frobenius norm can be obtained by first finding the minimum Frobenius norm common symmetric solution or common symmetric least-squares solution of the new pair of matrix equations. The given numerical examples show that the matrix based iterative method proposed in this paper has faster convergence than the iterative methods proposed in [1] and [2] to solve the same problems.     

双变量矩阵方程组对称最小二乘解的迭代算法

本文研究了求双变量线性矩阵方程组的对称最小二乘解的问题.利用求解线性代数方程组的共轭梯度法的基本思想,通过对有关矩阵和系数的变形与近似处理,建立了一种迭代算法.拓宽了共轭梯度法的适用范围.算例表明,迭代算法是有效的.     

Preconditioning and iterative solution of symmetric indefinite linear systems arising from interior point methods for linear programming

We propose to compute the search direction at each interior-point iteration for a linear program via a reduced augmented system that typically has a much smaller dimension than the original augmented system. This reduced system is potentially less susceptible to the ill-conditioning effect of the elements in the (1,1) block of the augmented matrix. A preconditioner is then designed by approximating the block structure of the inverse of the transformed matrix to further improve the spectral properties of the transformed system. The resulting preconditioned system is likely to become better conditioned toward the end of the interior-point algorithm. Capitalizing on the special spectral properties of the transformed matrix, we further proposed a two-phase iterative algorithm that starts by solving the normal equations with PCG in each IPM iteration, and then switches to solve the preconditioned reduced augmented system with symmetric quasi-minimal residual (SQMR) method when it is advantageous to do so. The experimental results have demonstrated that our proposed method is competitive with direct methods in solving large-scale LP problems and a set of highly degenerate LP problems. Research supported in parts by NUS Research Grant R146-000-076-112 and SMA IUP Research Grant.