On the sensitivity of the LU factorization

This paper gives sensitivity analyses by two approaches forL andU in the factorizationA=LU for general perturbations inA which are sufficiently small in norm. By the matrix-vector equation approach, we derive the condition numbers for theL andU factors. By the matrix equation approach we derive corresponding condition estimates. We show how partial pivoting and complete pivoting affect the sensitivity of the LU factorization. The material presented here is a part of the first author's PhD thesis under the supervision of the second author. This research was supported by NSERC of Canada Grant OGP0009236.     

基于凸优函数的Newton类方法的收敛定理

本文研究了求解Banach空间上非线性算子方程f(x)=0的Newton类方法的收敛性.利用优函数原理,在A(x0)1f满足关于某一凸优函数的广义Lipschitz条件下,得到了Newton类方法的一个半局部收敛定理.同时,当f和A(x)及初始点x0给定时,针对广义Lipschitz条件构造了相应的优函数,推广了Newton类方法的相关结果.     

Spectral factorization of rectangular rational matrix functions with application to discrete Wiener-Hopf equations

The properties of a discrete Wiener-Hopf equation are closely related to the factorization of the symbol of the equation. We give a necessary and sufficient condition for existence of a canonical Wiener-Hopf factorization of a possibly nonregular rational matrix function W relative to a contour which is a positively oriented boundary of a region in the finite complex plane. The condition involves decomposition of the state space in a minimal realization of W and, if it is satisfied, we give explicit formulas for the factors. The results are generalized by means of centered realizations to arbitrary rational matrix functions. The proposed approach can be used to solve discrete Wiener-Hopf equations whose symbols are rational matrix functions which admit canonical factorization relative to the unit circle.     

Structured condition numbers and statistical condition estimation for the LDU factorization

In this article, we consider the structured condition numbers for LDU, factorization by using the modified matrix-vector approach and the differential calculus, which can be represented by sets of parameters. By setting the specific norms and weight parameters, we present the expressions of the structured normwise, mixed, componentwise condition numbers and the corresponding results for unstructured ones. In addition, we investigate the statistical estimation of condition numbers of LDU factorization using the probabilistic spectral norm estimator and the small-sample statistical condition estimation method, and devise three algorithms. Finally, we compare the structured condition numbers with the corresponding unstructured ones in numerical experiments.     

矩阵方程ATXA=D的条件数与向后扰动分析

讨论矩阵方程ATXA=D,该方程源于振动反问题和结构模型修正.本文利用Moore-Penrose广义逆的性质,给出该方程解的条件数的上、下界估计.同时,利用Schauder不动点理论给出该方程的向后扰动界,这些结果可用于该矩阵方程的数值计算.     

New perturbation bounds and condition numbers for the hyperbolic QR factorization

Using the modified matrix-vector equation approach, the technique of Lyapunov majorant function and the Banach fixed point theorem, we obtain some new rigorous perturbation bounds for R factor of the hyperbolic QR factorization under normwise perturbation. These bounds are always tighter than the one given in the literature. Moreover, the optimal first-order perturbation bounds and the normwise condition numbers for the hyperbolic QR factorization are also presented.     

Incremental Norm Estimation for Dense and Sparse Matrices

We present an incremental approach to 2-norm estimation for triangular matrices. Our investigation covers both dense and sparse matrices which can arise for example from a QR, a Cholesky or a LU factorization. If the explicit inverse of a triangular factor is available, as in the case of an implicit version of the LU factorization, we can relate our results to incremental condition estimation (ICE). Incremental norm estimation (INE) extends directly from the dense to the sparse case without needing the modifications that are necessary for the sparse version of ICE. INE can be applied to complement ICE, since the product of the two estimates gives an estimate for the matrix condition number. Furthermore, when applied to matrix inverses, INE can be used as the basis of a rank-revealing factorization.     

一些矩阵分解的敏度分析

１．引言 代数Ｒｉｃｃａｔｉ方程是线性系统理论与设计的核心课题之一．矩阵的Ｈｅｓｓｅｎｂｅｒｇ分解、Ｈａｍｉｌｔｏｎ矩阵的平方约化分解、辛矩阵的ＱＴ分解是数值求解代数Ｒｉｃｃａｔｉ 方程的基本工具．关于 Ｈｅｓｓｅｎｂｅｒｇ分解的研究工作有很多（参阅 ［４］及其参考文献）．最近, Ｓｕｎ［４］利用矩阵分裂算子研究了Ｈｅｓｓｅｎｂｅｒｇ分解因子的扰动分析,并根据所得的扰动上界定义了分解因子的条件数．本文第 ２节将运用局部展开方法引入 Ｈｅｓｓｅｎｂｅｒｇ分解因子的条件数．有趣的是所定义的条件数与Ｓｕｎ引…     

一类矩阵方程的正交对称解及其最佳逼近

1引言设Rn×m表示所有n×m实矩阵集合,I表示单位矩阵,AT表示矩阵A的转置矩阵, ORn×n={P|PTP=I)表示列正交矩阵集,SORn×n={P|PT=P,P2=I}表示对称正交对称矩阵集．如无特别说明,本文中的矩阵P均指这类对称正交对称矩阵．在Rn×m上定义内积为     

矩阵方程组A1XB1=C1,A2XB2=C2的相容性与通解

本文推广了R.Penrose关于矩阵方程组AX=C,XB=D的工作,给出了矩阵方程组 A_1XB_1=C_1,A_2XB_2=C_2的相容性条件,给出了通解表达式以及唯一解的充要条件。     

Accuracy and Stability of the Null Space Method for Solving the Equality Constrained Least Squares Problem

The null space method is a standard method for solving the linear least squares problem subject to equality constraints (the LSE problem). We show that three variants of the method, including one used in LAPACK that is based on the generalized QR factorization, are numerically stable. We derive two perturbation bounds for the LSE problem: one of standard form that is not attainable, and a bound that yields the condition number of the LSE problem to within a small constant factor. By combining the backward error analysis and perturbation bounds we derive an approximate forward error bound suitable for practical computation. Numerical experiments are given to illustrate the sharpness of this bound.     

ON THE NECESSARY AND SUFFICIENT CONDITIONS TO SOLVE A HEAT EQUATION WITH GENERAL ADDITIVE GAUSSIAN NOISE

In this note, we consider stochastic heat equation with general additive Gaussian noise. Our aim is to derive some necessary and sufficient conditions on the Gaussian noise in order to solve the corresponding heat equation. We investigate this problem invoking two different methods, respectively, based on variance computations and on path-wise considerations in Besov spaces. We are going to see that, as anticipated, both approaches lead to the same necessary and sufficient condition on the noise. In addition, the path-wise approach brings out regularity results for the solution.     

ON GROWTH FACTORS OF THE LU FACTORIZATION

In this article, we derive upper bounds of different growth factors for the LU factorization, which are dominated by A11(k)-1A12(k),A21(k)A11(k)-1, where A11(k), A12(k), A21(k), A22(k) are sub-matrices of A. We also derive upper bounds of growth factors for the Cholesky factorization. Numerical examples are presented to verify our findings.     

The Bezoutian, state space realizations and Fisher’s information matrix of an ARMA process

In this paper we derive some properties of the Bezout matrix and relate the Fisher information matrix for a stationary ARMA process to the Bezoutian. Some properties are explained via realizations in state space form of the derivatives of the white noise process with respect to the parameters. A factorization of the Fisher information matrix as a product in factors which involve the Bezout matrix of the associated AR and MA polynomials is derived. From this factorization we can characterize singularity of the Fisher information matrix.     

LU分解的增长因子

In this article, we derive upper bounds of different growth factors for the LU factorization, which are dominated by A11(k)-1A12(k),A21(k)A11(k)-1, where A11(k), A12(k), A21(k), A22(k) are sub-matrices of A. We also derive upper bounds of growth factors for the Cholesky factorization. Numerical examples are presented to verify our findings.     

Factorization of the transition matrix for the general Jacobi system

The Jacobi system on a full‐line lattice is considered when it contains additional weight factors. A factorization formula is derived expressing the scattering from such a generalized Jacobi system in terms of the scattering from its fragments. This is performed by writing the transition matrix for the generalized Jacobi system as an ordered matrix product of the transition matrices corresponding to its fragments. The resulting factorization formula resembles the factorization formula for the Schrödinger equation on the full line. Copyright © 2016 John Wiley & Sons, Ltd.     

矩阵的WZ分解及其误差分析

在[1]中,对矩阵A进行WZ分解,其中矩阵W=(w_(i,j))和Z=(z_(i,j))形状如下: 1, i=j, w_(i,j)=任意,min(i,n-i+1)相似文献   

关于分块矩阵〔ABCD〕的g-逆的块独立性

本文证明了分块阵M=〔ABCD〕的g-逆有块独立性的充要条件是M适合秩可加性条件,M~+有一特定的表达式的充要条件也是M适合秩可加性条件.本文还给出了包含M的g-逆的子块的不变矩阵以及这些子块的定义方程.     

矩阵方程A~TXA=D的双对称最小二乘解

１．引 言 本文用 Ｒｎ×ｍ表示全体 ｎ×ｍ实矩阵集合，用 ＳＲｎ×ｎ（ＳＲ０ｎ×ｎ）表示全体 ｎ× ｎ实对称（实对称半正定）矩阵集合，ＯＲｎ×ｎ表示全体 ｎ× ｎ实正交矩阵集合，ＢＳＲｎ×ｎ表示全体ｎ×ｎ双对称实矩阵集合．这里，一个实对称矩阵Ａ＝（ａｉｊ）ｎ×ｎ被称为双对称矩阵，如果对所有的 　　　　　　　 　　　　　　　　　　　　　　　用Ａ×Ｂ表示矩阵 Ａ与 Ｂ的Ｈａｄａｍａｒｄ乘积，Ｉｋ表示 ｋ× ｋ阶单位矩阵，Ｏ表示零矩阵，Ｓｋ＝（ｅｋ，…，ｅ２，ｅ１）∈ Ｒｋ×ｋ，其中ｅｉ表示Ｉｋ的第ｉ列． 矩阵方程…     

有限域上n元n次方程只有零解的充要条件

本文研究了有限域上只有零解的n元n次方程的结构问题.利用对有限域上不可约多元多项式在其扩域中的分解特征的刻画,结合Chevalley定理,得到了有限域上n元n次方程只有零解的一个充要条件,并给出这类方程的一些新的具体构造.