O-对称矩阵的奇异值分解及其算法

本文研究了具有轴对称结构矩阵的奇异值分解,找出了这类矩阵奇异值分解与其子阵奇异值分解之间的定量关系.利用这些定量关系给出这类矩阵奇异值分解和Moore-Penrose逆的算法,据此可极大地节省求该类矩阵奇异值分解和Moore-Penrose逆时的计算量和存储量.     

关于矩阵的奇异值偏序

研究矩阵的奇异值偏序,给出了矩阵的奇异值偏序的等价刻画和性质,指出了相关文献关于矩阵*序刻画不真,利用强同时奇异值分解给出了矩阵*-序的刻画.     

酉延拓矩阵的奇异值分解及其广义逆

从普通奇异值分解出发,导出了酉延拓矩阵的奇异值和奇异向量与母矩阵的奇异值和奇异向量间的定量关系,同时对酉延拓矩阵的满秩分解及g逆,反射g逆,最小二乘g逆,最小范数g逆作了定量分析,得到了酉延拓矩阵的满秩分解矩阵F*和G*与母矩阵A的分解矩阵F和G之间的关系.最后给出了相应的快速求解算法,并举例说明该算法大大降低了分解的计算量和存储量,提高了计算效率.     

广义矩阵分布下多元回归模型的贝叶斯推断

考虑具有奇异矩阵椭球等高分布误差的多元线性回归模型的贝叶斯统计推断,在非信息先验下得到了系数矩阵关于Hausdorff测度的后验边缘分布和未来观察值的预测分布,并得到了一类特殊奇异矩阵椭球等高分布下误差协方差矩阵的后验边缘分布.对于具有奇异矩阵正态分布误差的多元线性回归模型,在广义正态-逆Wishart共轭先验下得到了类似的后验边缘分布和预测分布结果.在上述两种先验分布下,回归系数矩阵的后验边缘分布和预测分布是双奇异矩阵t分布,这种分布具有关于Hausdorff测度的精确密度.结果表明,在非信息先验下,回归系数矩阵的后验边缘分布和未来观察值的预测分布在奇异矩阵椭球等高分布类中具有稳健性.     

用随机奇异值分解算法求解矩阵恢复问题

本文研究了大型低秩矩阵恢复问题.利用随机奇异值分解（RSVD）算法,对稀疏矩阵做奇异值分解.该算法与Lanczos方法相比,在误差精度一致的同时运算时间大大降低,且该算法对相对低秩矩阵也有效.     

计算大规模矩阵最大最小奇异值和奇异向量的两个精化Lanczos算法

1.引言 在科学工程计算中经常需要计算大规模矩阵的少数最大或最小的奇异值及其所对应的奇异子空间。例如图像处理中要计算矩阵端部奇异值之比作为图像的分辨率,诸如此类的问题还存在于最小二乘问题、控制理论、量子化学中等等。然而大多实际问题中的矩阵是大型稀疏矩阵,且需要的是矩阵的部分奇异对。如果计算A的完全奇异值分解(SVD),则运算量和存储量极大,甚至不可能。因此必须寻求其它有效可靠的算法。 假设A的SVD为     

广义行(列)酉对称矩阵的Moore-penrose逆

通过对母矩阵进行奇异值分解的方法得到广义行(列)酉对称矩阵的奇异值分解进一步得到其Moore-penrose逆;用谱分解方法得到母矩阵的Moore-penrose逆,进一步得到广义行(列)酉对称矩阵的Moore-penrose逆.     

矩阵奇异值分解问题重分析的摄动法

本文提出了一般实矩阵奇异值分解问题重分析的摄动法.这是一种简捷、高效的快速重分析方法,对于提高各种需要反复进行矩阵奇异值分解的迭代分析问题的计算效率具有较重要的实用价值.文中导出了奇异值和左、右奇异向量的直到二阶摄动量的渐近估计算式.文末指出了将这种振动分析方法直接推广到一般复矩阵情况的途径.     

矩阵的奇异值与特征值的关系探究及应用

讨论矩阵的奇异值与特征值的关系,并给出奇异值在最优化理论分析中的一个应用.     

广义奇异值的扰动

本文给出了一对矩阵的广义奇异值扰动的一致上界,并由之可导出普通奇异值扰动的经典定理。     

Perturbation of the SVD in the presence of small singular values

This paper gives SVD perturbation bounds and expansions that are of use when an m × n, m ? n matrix A has small singular values. The first part of the paper gives subspace bounds that are closely related to those of Wedin but are stated so as to isolate the effect of any small singular values to the left singular subspace. In the second part first and second order approximations are given for perturbed singular values. The subspace bounds are used to show that all approximations retain accuracy when applied to small singular values. The paper concludes by deriving a subspace bound for multiplicative perturbations and using that bound to give a simple approximation to a singular value perturbed by a multiplicative perturbation.     

奇异值的极值性质(英文)

考虑了矩阵奇异值的相关问题,研究了其极值性质,并得出了一些特殊情形下的结果.最后,考察了其在多元统计分析中的应用.     

Using semiseparable matrices to compute the SVD of a general matrix product/quotient

In this work we reduce the computation of the singular values of a general product/quotient of matrices to the computation of the singular values of an upper triangular semiseparable matrix. Compared to the reduction into a bidiagonal matrix the reduction into semiseparable form exhibits a nested subspace iteration. Hence, when there are large gaps between the singular values, these gaps manifest themselves already during the reduction algorithm in contrast to the bidiagonal case.     

Note on the Characteristic Values and the Three Types of Singular Values of a Complex Matrix

In this paper we prove several inequalities involving the characteristic values, the singular values, the real singular values and the imaginary singular values of a complex matrix.     

矩阵的广义奇异值的若干不等式

本文引进广义奇异值概念，获得了若干广义奇异值不等式及交错定理．     

Sensitivity Analysis of Zero Singular Values and Multiple Singular Values

Some results of the author are used to discuss the sensitivity of zero singular values and multiple singular values of a real matrix analytically dependent on several parameters.     

Singular value expansion for the Green function of Helmholtz operator

We consider the integral operator defined on a circular disk, and with kernel the Green function of the Helmholtz operator. We present an analytic framework for the explicit computation of the singular system of this kernel. In particular, the main formulas of this framework are given by a characteristic equation for the singular values and explicit expressions for the corresponding singular functions. We provide also a property of the singular values, that gives an important information for the numerical evaluation of the singular system. Finally, we present a simple numerical experiment, where the singular system computed by a simple implementation of these analytic formulas is compared with the singular system obtained by a discretization of the Green function of the Helmholtz operator.     

An efficient and accurate parallel algorithm for the singular value problem of bidiagonal matrices

Summary. In this paper we propose an algorithm based on Laguerre's iteration, rank two divide-and-conquer technique and a hybrid strategy for computing singular values of bidiagonal matrices. The algorithm is fully parallel in nature and evaluates singular values to tiny relative error if necessary. It is competitive with QR algorithm in serial mode in speed and advantageous in computing partial singular values. Error analysis and numerical results are presented. Received March 15, 1993 / Revised version received June 7, 1994     

Sensitivity analysis for the generalized singular value decomposition

In this paper, we discuss the sensitivity of multiple nonzero finite generalized singular values and the corresponding generalized singular matrix set of a real matrix pair analytically dependent on several parameters. From our results, the partial derivatives of multiple nonzero singular values and their left and right singular vector matrices are obtained.Copyright © 2012 John Wiley & Sons, Ltd.     

NEW ESTIMATES FOR SINGULAR VALUES OF A MATRIX

New estimates are provided for singular values of a matrix in this paper. These results generalize and improve corresponding estimates for singular values in [4]-[6].