奇异模糊线性系统的扰动分析

当模糊线性系统的系数矩阵奇异时, 分析了模糊线性系统的右端向量和系数矩阵的扰动对模糊线性系统解的计算造成的影响,用矩阵的谱范数给出了相对误差的界.     

线性规划约束方程系数矩阵的敏感性分析

本文首先给出了用当前基解和系数矩阵A的行(或列)增向量直接计算新基解及其检验数的一个简单公式,进而讨论了关于线性规划系数矩阵A的一行(列)向量变化的敏感性分析问题。给出了保持最优基不变时ΔA的变化范围。     

一类广义半正定线性方程组的直接解法

1 引言 在具有等式约束的二次规划或椭圆型边值问题离散化分析中经常会遇到解线性方程组 (1)其中A∈R~(m×m)为对称正定矩阵,B∈R~(n×m)为行满秩矩阵,f∈R~m,g∈R~n为右端向量. 为了讨论的方便,首先引进, 定义1 若G∈R~(N×N),且对任何非零向量x∈R~N都有x~TGx>0(≥0),则称矩阵G     

求解大型非对称线性方程组的不完全广义最小向后扰动法

本文给出了求解大型非对称线性方程组的广义最小向后扰动法(GMBACK)的截断版本——不完全广义最小向后扰动法(IGMBACK).该方法基于Krylov向量的不完全正交化,从而在Krylov子空间上求出一个近似的或者拟最小向后扰动解.本文对新算法IGMBACK做了一些理论研究,包括算法的有限终止、解的存在性和唯一性等方面的研究;且给出了IGMBACK的执行.数值实验表明:IGMBACK通常比GMBACK和广义最小残量法(GMRES)更有效;且IGMBACK和GMBACK经常比GMRES收敛得更好.特殊地,如果系数矩阵是敏感矩阵,且方程组右侧的向量平行于系数矩阵的最小奇异值对应的左奇异向量时,重新开始的GMRES不一定收敛,而IGMBACK和GMBACK一般收敛,且比GMRES收敛得更好.     

调宽采样控制系统分析

一、系统描述考虑受控对象的动态特性由线性方程d/(dt)x(t)=Ax(t)+Bu(t)+d(t)(1.1)描述的控制系统,其中状态变量 x(t)是 p 维向量,控制变量 u(t)是 q 维向量,q≤p,A与 B 分别是 p×p 与 p×q 常系数矩阵。在(1.1)中,d(t)是系统的扰动,在本文中我们仅讨论阶跃扰动的情形,即 d(t)=d·1(t),d∈R~p 是常值向量。定义     

带内部耗散项的拟线性双曲型方程组的柯西问题

§1引言 对常系数常微分方程组的初值问题{其中x(t)是n维向量函数,A是n阶常数矩阵,由常微分方程理论知道,若A的特征值的实部均为非负,且关于实部为零的特征值所对应的初等因子是单重的,则(1.1)的解x(t)必是有界的。如果方程组(1.1)的右端还含未知函数的非线性项,即考虑初值问题     

基于向量变换的DGM(1,1,k~α)模型的病态性

系数矩阵谱条件数是度量灰色预测模型病态性的重要工具,而向量的数乘变换和旋转变换是降低系数矩阵谱条件数的有效方法.首先,利用向量的数乘变换和旋转变换研究DGM(1,1,k~α)模型的病态性,结果显示,DGM(1,1,k~α)模型的病态性主要受系数矩阵列向量的长度之比、夹角大小及时间幂的影响;其次,给出了基于向量变换的DGM(1,1,k~α)模型病态性的解决步骤;最后,通过一个算例验证了向量变换在解决矩阵病态性问题时的有效性和实用性.     

酉不变范数下极分解的扰动界

设A是m×n(m≥n)且秩为n的复矩阵.存在m×n矩阵Q满足Q*Q=I和n×n正定矩阵H使得A=QH,此分解称为A的极分解.本文给出了在任意酉不变范数下正定极因子H的扰动界,改进文[1,11]的结果;另外也首次提供了乘法扰动下酉极因子Q在任意酉不变范数下的扰动界.     

Q过程的μ-不变向量

设Q ={qij;i,j∈E}是可数集E上的全稳定q 矩阵 ,x ={xj;j∈E}是Q的有限 μ 不变向量 ,如Q零流出 ,则x是最小Q过程的μ 不变向量 ;一般地 ,Q不必零流出 ,但x满足infi∈Exi>0 ,则一定存在Q过程P(t) ,使x是P(t)的 μ 不变向量 .     

不变子空间与广义不变子空间——(Ⅱ)扰动定理

关于矩阵的不变子空间,自然会提出这样一个扰动问题:设Z_1∈C~(n×l)是A∈C~(n×n)的一个特征矩阵,若E∈C~(n×n)是一个扰动矩阵,问A+B是否存在特征矩阵Z_1,使得(Z_1)靠近R(Z_1)?关于矩阵对的广义不变子空间.也可以类似地提出问题。 对于这些问题,G.W.Stewart曾经讨论过,他的方法的关键是构造一种求解二次矩阵方程的迭代过程,用来逼近矩阵的一个不变子空间;而本文建议另一种迭代格式,用这种迭代逼近一个不变(或广义不变)子空间,具有二次收敛速度。     

The discrete linear restricted Chebyshev approximation

A numerically stable simplex algorithm for calculating the restricted Chebyshev solution of overdetermined systems of linear equations is described. In this algorithm minimum computer storage is required and no conditions are imposed on the coefficient matrix or on the right hand side of the system of equations. Also a new way of implementing a triangular decomposition method to the basis matrix is used. The ordinary Chebyshev solution, the one-sided Chebyshev solutions and the Chebyshev approximation by non-negative functions are obtained as special cases in this algorithm. Numerical results are given.     

Computing the conditioning of the components of a linear least‐squares solution

In this paper, we address the accuracy of the results for the overdetermined full rank linear least‐squares problem. We recall theoretical results obtained in (SIAM J. Matrix Anal. Appl. 2007; 29 (2):413–433) on conditioning of the least‐squares solution and the components of the solution when the matrix perturbations are measured in Frobenius or spectral norms. Then we define computable estimates for these condition numbers and we interpret them in terms of statistical quantities when the regression matrix and the right‐hand side are perturbed. In particular, we show that in the classical linear statistical model, the ratio of the variance of one component of the solution by the variance of the right‐hand side is exactly the condition number of this solution component when only perturbations on the right‐hand side are considered. We explain how to compute the variance–covariance matrix and the least‐squares conditioning using the libraries LAPACK (LAPACK Users' Guide (3rd edn). SIAM: Philadelphia, 1999) and ScaLAPACK (ScaLAPACK Users' Guide. SIAM: Philadelphia, 1997) and we give the corresponding computational cost. Finally we present a small historical numerical example that was used by Laplace (Théorie Analytique des Probabilités. Mme Ve Courcier, 1820; 497–530) for computing the mass of Jupiter and a physical application if the area of space geodesy. Copyright © 2008 John Wiley & Sons, Ltd.     

A simplex-type algorithm for continuous linear programs with constant coefficients

Mathematical Programming - We consider continuous linear programs over a continuous finite time horizon T, with a constant coefficient matrix, linear right hand side functions and linear cost...     

Matrix iterations and Cichon’s diagram

Using matrix iterations of ccc posets, we prove the consistency with ZFC of some cases where the cardinals on the right hand side of Cichon’s diagram take two or three arbitrary values (two regular values, the third one with uncountable cofinality). Also, mixing this with the techniques in J Symb Log 56(3):795–810, 1991, we can prove that it is consistent with ZFC to assign, at the same time, several arbitrary regular values on the left hand side of Cichon’s diagram.     

Sensitivity analysis of mixed integer programs: An application to environmental policy making

We consider a class of mixed integer programs in which the concave objective function and the constraint matrix are held fixed while some of the right hand side (RHS) coefficients are varied. An efficient iterative algorithm is developed for performing the above sensitivity analysis. A practical application of this class of programs is encountered in environmental policy making and accordingly it is used in illustrating the operation of the algorithm.     

Minimum-Euclidean-norm matrix correction for a pair of dual linear programming problems

For a pair of dual (possibly improper) linear programming problems, a family of matrix corrections is studied that ensure the existence of given solutions to these problems. The case of correcting the coefficient matrix and three cases of correcting an augmented coefficient matrix (obtained by adding the right-hand side vector of the primal problem, the right-hand-side vector of the dual problem, or both vectors) are considered. Necessary and sufficient conditions for the existence of a solution to the indicated problems, its uniqueness is proved, and the form of matrices for the solution with a minimum Euclidean norm is presented. Numerical examples are given.     

Privacy-preserving horizontally partitioned linear programs

We propose a simple privacy-preserving reformulation of a linear program whose equality constraint matrix is partitioned into groups of rows. Each group of matrix rows and its corresponding right hand side vector are owned by a distinct private entity that is unwilling to share or make public its row group or right hand side vector. By multiplying each privately held constraint group by an appropriately generated and privately held random matrix, the original linear program is transformed into an equivalent one that does not reveal any of the privately held data or make it public. The solution vector of the transformed secure linear program is publicly generated and is available to all entities.     

Right preconditioned MINRES for singular systems†

We consider solving large sparse symmetric singular linear systems. We first introduce an algorithm for right preconditioned minimum residual (MINRES) and prove that its iterates converge to the preconditioner weighted least squares solution without breakdown for an arbitrary right‐hand‐side vector and an arbitrary initial vector even if the linear system is singular and inconsistent. For the special case when the system is consistent, we prove that the iterates converge to a min‐norm solution with respect to the preconditioner if the initial vector is in the range space of the right preconditioned coefficient matrix. Furthermore, we propose a right preconditioned MINRES using symmetric successive over‐relaxation (SSOR) with Eisenstat's trick. Some numerical experiments on semidefinite systems in electromagnetic analysis and so forth indicate that the method is efficient and robust. Finally, we show that the residual norm can be further reduced by restarting the iterations.     

Fast deconvolution with approximated PSF by RSTLS with antireflective boundary conditions

The problem of reconstructing signals and images from degraded ones is considered in this paper. The latter problem is formulated as a linear system whose coefficient matrix models the unknown point spread function and the right hand side represents the observed image. Moreover, the coefficient matrix is very ill-conditioned, requiring an additional regularization term. Different boundary conditions can be proposed. In this paper antireflective boundary conditions are considered. Since both sides of the linear system have uncertainties and the coefficient matrix is highly structured, the Regularized Structured Total Least Squares approach seems to be the more appropriate one to compute an approximation of the true signal/image. With the latter approach the original problem is formulated as an highly nonconvex one, and seldom can the global minimum be computed. It is shown that Regularized Structured Total Least Squares problems for antireflective boundary conditions can be decomposed into single variable subproblems by a discrete sine transform. Such subproblems are then transformed into one-dimensional unimodal real-valued minimization problems which can be solved globally. Some numerical examples show the effectiveness of the proposed approach.     

Two-stage stochastic problems with correlated normal variables: computational experiences

Two-stage models are frequently used when making decisions under the influence of randomness. The case of normally distributed right hand side vector – with independent or correlated components – is treated here. The expected recourse function is computed by an enhanced Monte Carlo integration technique. Successive regression approximation technique is used for computing the optimal solution of the problem. Computational issues of the algorithm are discussed, improvements are proposed and numerical results are presented for random right hand side and a random matrix in the second stage problems.