QR factorization for row or column symmetric matrix

The problem of fast computing the QR factorization of row or column symmetric matrix is considered. We address two new algorithms based on a correspondence of Q and R matrices between the row or column symmetric matrix and its mother matrix. Theoretical analysis and numerical evidence show that, for a class of row or column symmetric matrices, the QR factorization using the mother matrix rather than the row or column symmetric matrix per se can save dramatically the CPU time and memory without loss of any numerical precision.     

A pivoting strategy for symmetric tridiagonal matrices

The LBL^T factorization of Bunch for solving linear systems involving a symmetric indefinite tridiagonal matrix T is a stable, efficient method. It computes a unit lower triangular matrix L and a block 1 × 1 and 2 × 2 matrix B such that T=LBL^T. Choosing the pivot size requires knowing a priori the largest element σ of T in magnitude. In some applications, it is required to factor T as it is formed without necessarily knowing σ. In this paper, we present a modification of the Bunch algorithm that can satisfy this requirement. We demonstrate that this modification exhibits the same bound on the growth factor as the Bunch algorithm and is likewise normwise backward stable. Copyright © 2005 John Wiley & Sons, Ltd.     

A simplified pivoting strategy for symmetric tridiagonal matrices

The pivoting strategy of Bunch and Marcia for solving systems involving symmetric indefinite tridiagonal matrices uses two different methods for solving 2 × 2 systems when a 2 × 2 pivot is chosen. In this paper, we eliminate this need for two methods by adding another criterion for choosing a 1 × 1 pivot. We demonstrate that all the results from the Bunch and Marcia pivoting strategy still hold. Copyright © 2006 John Wiley & Sons, Ltd.     

Twisted factorization of a banded matrix

The twisted factorization of a tridiagonal matrix T plays an important role in inverse iteration as featured in the MRRR algorithm. The twisted structure simplifies the computation of the eigenvector approximation and can also improve the accuracy. A tridiagonal twisted factorization is given by T=M _k Δ _k N _k where Δ _k is diagonal, M _k ,N _k have unit diagonals, and the k-th column of M _k and the k-th row of N _k correspond to the k-th column and row of the identity, that is . This paper gives a constructive proof for the existence of the twisted factorizations of a general banded matrix A. We show that for a given twist index k, there actually are two such factorizations. We also investigate the implications on inverse iteration and discuss the role of pivoting.      

A very fast algorithm for matrix factorization

We present a very fast algorithm for general matrix factorization of a data matrix for use in the statistical analysis of high-dimensional data via latent factors. Such data are prevalent across many application areas and generate an ever-increasing demand for methods of dimension reduction in order to undertake the statistical analysis of interest. Our algorithm uses a gradient-based approach which can be used with an arbitrary loss function provided the latter is differentiable. The speed and effectiveness of our algorithm for dimension reduction is demonstrated in the context of supervised classification of some real high-dimensional data sets from the bioinformatics literature.     

Unitary equivalence to a complex symmetric matrix: An algorithm

We present a necessary and sufficient condition for a 3×3 matrix to be unitarily equivalent to a symmetric matrix with complex entries, and an algorithm whereby an arbitrary 3×3 matrix can be tested. This test generalizes to a necessary and sufficient condition that applies to almost every n×n matrix. The test is constructive in that it explicitly exhibits the unitary equivalence to a complex symmetric matrix.     

关于周期对称三对角矩阵的广义特征值反问题

在给定部分特征值、部分特征向量及附加条件下提出了一类反问题,并给出了此问题解存在性的证明及求解的算法.     

A multilevel approach for nonnegative matrix factorization

Nonnegative matrix factorization (NMF), the problem of approximating a nonnegative matrix with the product of two low-rank nonnegative matrices, has been shown to be useful in many applications, such as text mining, image processing, and computational biology. In this paper, we explain how algorithms for NMF can be embedded into the framework of multilevel methods in order to accelerate their initial convergence. This technique can be applied in situations where data admit a good approximate representation in a lower dimensional space through linear transformations preserving nonnegativity. Several simple multilevel strategies are described and are experimentally shown to speed up significantly three popular NMF algorithms (alternating nonnegative least squares, multiplicative updates and hierarchical alternating least squares) on several standard image datasets.     

Computing eigenvectors of block tridiagonal matrices based on twisted block factorizations

New methods for computing eigenvectors of symmetric block tridiagonal matrices based on twisted block factorizations are explored. The relation of the block where two twisted factorizations meet to an eigenvector of the block tridiagonal matrix is reviewed. Based on this, several new algorithmic strategies for computing the eigenvector efficiently are motivated and designed. The underlying idea is to determine a good starting vector for an inverse iteration process from the twisted block factorizations such that a good eigenvector approximation can be computed with a single step of inverse iteration.     

r-对称循环矩阵及逆矩阵三角分解的快速算法

根据r-对称循环矩阵的特殊结构给出了求这类矩阵本身及其逆矩阵三角分解的快速算法,算法的运算量均为O(n2),一般矩阵及逆矩阵三角分解的运算量均为O(n3).     

基于判别超图和非负矩阵分解的人脸识别方法

非负矩阵分解是一种流行的数据表示方法,已广泛应用于图像处理和模式识别等问题.但是非负矩阵分解忽略了数据的几何结构. 而现有的基于简单图的学习方法只考虑了图像的成对信息,并且对计算相似度时的参数选择非常敏感. 超图学习方法可以有效地解决这些问题. 超图利用超边将多个顶点相连接用以表示图像的高维结构信息. 然而, 现有的大部分超图学习方法都是无判别的学习方法.为了提高识别效果, 提出了基于具有判别信息的超图和非负矩阵分解方法的新模型, 利用交替方向法进行迭代求解新模型, 并结合最近邻方法进行人脸识别. 在几个常用标准人脸图像数据库上进行实验, 实验结果表明提出的方法是有效的.     

Real valued iterative methods for solving complex symmetric linear systems

Complex valued systems of equations with a matrix R + 1S where R and S are real valued arise in many applications. A preconditioned iterative solution method is presented when R and S are symmetric positive semi‐definite and at least one of R, S is positive definite. The condition number of the preconditioned matrix is bounded above by 2, so only very few iterations are required. Applications when solving matrix polynomial equation systems, linear systems of ordinary differential equations, and using time‐stepping integration schemes based on Padé approximation for parabolic and hyperbolic problems are also discussed. Numerical comparisons show that the proposed real valued method is much faster than the iterative complex symmetric QMR method. Copyright © 2000 John Wiley & Sons, Ltd.     

Several variants of the Hermitian and skew‐Hermitian splitting method for a class of complex symmetric linear systems

This paper is concerned with several variants of the Hermitian and skew‐Hermitian splitting iteration method to solve a class of complex symmetric linear systems. Theoretical analysis shows that several Hermitian and skew‐Hermitian splitting based iteration methods are unconditionally convergent. Numerical experiments from an n‐degree‐of‐freedom linear system are reported to illustrate the efficiency of the proposed methods. Copyright © 2014 John Wiley & Sons, Ltd.     

On the inverse of a general tridiagonal matrix

In the current paper a new efficient computational algorithm to find the inverse of a general tridiagonal matrix is presented. The algorithm is suited for implementation using computer algebra systems such as MAPLE, MATHEMATICA, MATLAB and MACSYMA. Symbolic and numeric examples are given.     

求解一类复对称线性系统的改进的SNS和SSS迭代法

本文在Bai的基础上提出改进的斜正规分裂(MSNS)和斜尺度化分裂(MSSS)迭代法,用以求解一类应用广泛的复对称线性系统,并证实MSNS和MSSS迭代法是无条件收敛的.通过利用一些Krylov子空间方法,本文给出相对应的非精确版本的MSNS(MSSS)方法.数值实验说明了所给方法的有效性.     

Preconditioned GSOR iterative method for a class of complex symmetric system of linear equations

In this paper, we present a preconditioned variant of the generalized successive overrelaxation (GSOR) iterative method for solving a broad class of complex symmetric linear systems. We study conditions under which the spectral radius of the iteration matrix of the preconditioned GSOR method is smaller than that of the GSOR method and determine the optimal values of iteration parameters. Numerical experiments are given to verify the validity of the presented theoretical results and the effectiveness of the preconditioned GSOR method. Copyright © 2015 John Wiley & Sons, Ltd.     

Cauchy型矩阵及其逆矩阵的快速三角分解

Abstract, In this paper,algorithms for determining the triangular factorization of Cauchy typematrices and their inverses are derived by using O(n2) operations.     

A property of eigenvalue bounds for a class of symmetric tridiagonal interval matrices

The eigenvalue bounds of interval matrices are often required in some mechanical and engineering fields. In this paper, we consider an interval eigenvalue problem with symmetric tridiagonal matrices. A theoretical result is obtained that under certain assumptions the upper and lower bounds of interval eigenvalues of the problem must be achieved just at some vertex matrices of the interval matrix. Then a sufficient condition is provided to guarantee the assumption to be satisfied. The conclusion is illustrated also by a numerical example. Copyright © 2010 John Wiley & Sons, Ltd.