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least-squares solutions of Matrix Equations(AX = B,XC = D) for Hermitian Reflexive (Anti-Hermitian Reflexive) Matrices and Its Approximation

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Journal of Mathematical Research and Exposition 2011年第6期31卷 1108-1116页

作者： Shuo ZHOU Shi Tong YANG Wen WANG School of Mathematical Sciences Northeast Dianli University Jilin 132012 P. R. China

In this paper,the Hermitian reflexive（Anti-Hermitian reflexive）least-squares so-lutions of matrix equations（AX = B,XC = D）are *** special properties of partitioned matrices and Hermitian reflexive（Anti-Hermitian reflexive）matrices,the general expression of the solution is ***,the related optimal approximation problem to a given matrix over the solution set is considered.

关键词： matrix equations Hermitian reflexive matrix Anti-Hermitian reflexive matrix least-squares solution optimal approximation.

OPTIMAL APPROXIMATE solution OF THE MATRIX EQUATION AXB = C OVER SYMMETRIC MATRICES

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Journal of Computational Mathematics 2007年第5期25卷 543-552页

作者： Anping Liao Yuan Lei College of Mathematics and Econometrics Hunan University Changsha 410082 China

Let SE denote the least-squares symmetric solution set of the matrix equation A×B = C, where A, B and C are given matrices of suitable size. To find the optimal approximate solution in the set SE to a given matrix, we give a new feasible method based on the projection theorem, the generalized SVD and the canonical correction decomposition.

关键词： least-squares solution Optimal approximate solution Generalized singular value decomposition Canonical correlation decomposition.

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特殊矩阵反问题的最小二乘解及最佳逼近

高等学校计算数学学报 2008年第1期30卷 46-56页

作者：龚毅陈果良上海 201204 华东师范大学数学系华东师范大学数学系上海 200062 上海海关学院基础部

1引言近年来,关于矩阵的反问题国内外有诸多学者都做了研究工作.如,双对称(反对称)矩阵反问题、中心对称(反中心对称)矩阵反问题、对称次反对称(反对称次对称)矩阵反问题.这些矩阵在信息论、线性系统理论及数值分析等领域中有其广泛应用... 详细信息

1引言近年来,关于矩阵的反问题国内外有诸多学者都做了研究工作.如,双对称(反对称)矩阵反问题、中心对称(反中心对称)矩阵反问题、对称次反对称(反对称次对称)矩阵反问题.这些矩阵在信息论、线性系统理论及数值分析等领域中有其广泛应用.研究反问题的工具大多是奇异值分解(SVD)[6,8,9,5]、广义奇异值分解(GSVD)[4],且部分学者利用广义逆的方法[7]讨论上述的问题.本文的主要思想是找出这些矩阵的共同特点,利用矩

关键词： canonical correlation decomposition special matrix least-squares solution optimal approximation

A QR DECOMPOSITION BASED SOLVER FOR THE least squares PROBLEMS FROM THE MINIMAL RESIDUAL METHOD FOR THE SYLVESTER EQUATION

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Journal of Computational Mathematics 2007年第5期25卷 531-542页

作者： Zhongxiao Jia Yuquan Sun Department of Mathematical Sciences Tsinghua University Beijing 100084 China

Based on the generalized minimal residual （GMRES） principle, Hu and Reichel proposed a minimal residual algorithm for the Sylvester equation. The algorithm requires the solution of a structured least squares problem. They form the normal equations of the least squares problem and then solve it by a direct solver, so it is susceptible to instability. In this paper, by exploiting the special structure of the least squares problem and working on the problem directly, a numerically stable QR decomposition based algorithm is presented for the problem. The new algorithm is more stable than the normal equations algorithm of Hu and Reichel. Numerical experiments are reported to confirm the superior stability of the new algorithm.

关键词： least-squares solution Preconditioning Generalized singular value decomposition.

矩阵方程AXB+CYD=E对称最小范数最小二乘解的极小残差法

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高等学校计算数学学报 2010年第1期32卷 71-81页

作者：方玲廖安平雷渊后勤工程学院基础部湖南大学数学与计量经济学院

1引言本文用R;表示全体n×m实矩阵集合,用SR;表示全体n×n实对称矩阵集合,OR;表示全体n×n实正交矩阵集合.用I;表示n阶单位矩阵,用A*B表示矩阵A与B的Hadamard乘积.对任意矩阵A,B∈R;,定义内积〈A,B〉=tr（B;A）,其中

关键词： symmetric matrix least-squares solution least norm solution iteration method.