用修改的Lanczos方法解大型稀疏线性方程组

引言 解大型稀疏线性方程组Ax=b,已有许多方法,但这些方法基本上是针对对称正定矩阵或非零元分布较有规律的矩阵。对于一般的特大型稀疏矩阵的有效解法还在寻找过程中,其中一个途径是从Lanczos方法入手,采取各种变形。 我们受Paige和Saunders的算法SYMMLQ的启发,提出了两种解大型稀疏非对称     

用AOR方法求解大型稀疏最小二乘问题的收敛性

在许多实际问题中,我们都希望计算以下超定线性方程组 Ax=b (1)的最小二乘解.其中A为一大型疏m×n实矩阵,m>n,b为一给定的m维实向量.这里假定Rank(A)=n. 我们知道,(1)可叙述成,求唯一向量X∈R~n,使||b—AX||_2=min||b—Ay||_2对一切y∈R~n。由于Rank(A)=n,上述最小二乘问题等价于求一个n维向量X∈R~n和     

基于重启Lanczos过程的模型降阶方法

针对大规模的线性时不变系统,提出了基于重启Lanczos过程的模型降阶方法。首先,通过重启Lanczos过程分别得到原始系统的可控Gram矩阵的近似矩阵及可观Gram矩阵的近似矩阵。然后,根据原始系统的可控Gram矩阵及可观Gram矩阵所满足的Lyapunov方程构造映射Sylvester方程并求解,对解进行双正交化,得到降阶所需的变换矩阵,从而得到降阶系统。运用此方法对大规模线性时不变系统进行降阶,能够得到具有较高近似精度的稳定的降阶系统。最后,数值算例验证了此方法是行之有效的。     

分段线性规划的解法

其中A为m×n矩阵,x为n维向量,b为m维向量. 与通常线性规划所不同的是价值系数c_j(j=1,2,…,n)不是常数,而是分段常数.原问题要求x_j的值分成r_j段,分点     

大规模MIMO系统中基于Lanczos方法的低复杂度预编码

针对大规模MIMO系统中因基站天线数与用户数过大导致迫零(ZF)预编码矩阵求逆复杂度较高的问题,提出一种基于迭代子空间投影算法的Lanczos方法低复杂度预编码方案。根据大规模MIMO系统信道矩阵具有对角占优特性,将信道大矩阵求逆诺依曼级数的第1项作为迭代的初始值,从而加快算法的收敛速度,使得ZF预编码的复杂度从O(K~3)降低到O(K~2)。仿真结果表明,该算法以较快的收敛速度逼近传统ZF预编码方案的信道容量与误码率性能。     

免停滞的GMRES(m)算法研究

1.引 言 解大型线性方程组仍是当今数值计算中的一个重要问题[1—8],GMRES(m)算法是解大型非对称线性方程组的常用方法[1],其中A∈Rn×n为大型稀疏非奇异矩阵,x,b∈Rn.然而,当A为非正实阵时,GMRES(m)解问题(1.1)可能会停滞.为此我们在第二节将先给出GMRES(m)停     

一种最大向量平均个数的估计方法

提出一种估计n个d维向量中最大向量平均个数的方法。该方法通过分析单个向量与其他向量子集的支配关系,求出最大向量平均个数的解析式。证明解析式满足已知的递归关系,得到最大向量平均个数的近似估计。与已有方法相比,该方法可应用到估计k个其他向量支配的平均个数问题。     

基于非对称Lanczos 算法的线性分数阶系统模型降阶方法

采用非对称Lanczos 算法研究线性分数阶系统的模型降阶问题, 提出一种保持系统传递函数一定数量的分数阶矩的模型降阶方法. 根据Caputo 导数的运算法则给出线性分数阶系统的分数阶矩的计算方法; 利用非对称Lanczos 算法构造对应的非对称三对角矩阵; 根据非对称三对角矩阵的性质证明降阶系统与原系统具有相同的一定数量的分数阶矩; 给出降阶系统与原系统传递函数的误差估计, 为合理选择降阶系统的阶次提供理论依据. 数值实例的计算结果验证了所提出方法的有效性.

     

关于自逆变换

在R~n中考虑形如 S=I-uv~T (1)的变换.其中I是单位阵,u和v都是n维列向量,T表示转置.假定已知向量a和b,求变换S,使b=Sa,S~(-1)=S.那么,如果取 u=a-b (2)并选择v使     

矩阵特征值和特征向量的计算方法(上)

设 A 是 n×n 正方矩阵,一般而言其元素可以是复的。我们要解的问题就是要求出这样的λ值和这样的 n 维向量 v,使 Av=λv。每一个这样的λ值称为矩阵 A 的一个特征值,固有根或特征根,而对应的向量称为对应于该特征值的特征向量。     

一类特殊类型子空间上Ritz对的性质及其应用

§1.引言设A∈R~(M×N),定义增广矩阵 其中上标T表示转置。不失一般性,假设M≥N,设σ_i,i=1,2,…,N是A的奇异值,u_i和v_i分别是对应的左右奇异向量,奇异值按从小到大或从大到小的顺序排列,则     

Residuals of refined projection methods for large matrix eigenproblems

In terms of the deviation of the eigenvector ϕ from a projection subspace E, a priori error bounds are established for the residual norm of an approximate eigenpair obtained by a refined projection method. It is shown that the residual converges to zero as the deviation tends to zero. Finally, how to efficiently compute refined Ritz vectors is discussed in the refined symmetric Lanczos method.     

Monte Carlo evaluation of methods for pulse transfer function estimation

This paper describes the results of a Monte Carlo evaluation made of the methods proposed in current literature for the estimation of the pulse transfer function of a linear, time-invariant dynamic system with feedback. Considered are two basic methods for estimating the coefficients of a pulse transfer function, given only the normal operating input and output of the system obscured by noise and over a limited period of time. The most commonly proposed method is a linear method in which a set of simultaneous linear equations is formed from the sampled data and the coefficients obtained by a matrix inversion. The other method is an eigenvector method proposed by Levin which uses the eigenvector associated with the smallest eigenvalue of a matrix formed from the sampled data. This paper presents a set of examples designed to compare linear and eigenvector estimation methods and to verify experimentally the theoretical results and approximations given by Levin. The comparison shows that the eigenvector method generally gives estimates with equal or smaller rms errorsqrt{Variance+(Bias)^2}than the linear method. The eigenvector estimates had bias magnitudes which were consistently less than their standard deviations; the linear estimates did not, and thus had rms errors which often consisted largely of the bias. The approximate covariance matrix given by Levin for the coefficients estimated with the eigenvector method is found to be reasonably accurate.     

解大规模非对称矩阵特征问题的精化Arnoldi方法的一种变形

§1.引言 传统的投影类方法是计算大规模非对称矩阵特征问题Ax=λx部分特征对的主要方法,它们包括Arnoldi方法、块Arnoldi方法、同时迭代法、Davidson方法和Jacobi-Davidson方法,贾提出的精化投影类方法目前被公认为是另一类重要     

人脸识别中多目标最优不相关图像鉴别分析研究

考虑图像投影鉴别分析问题,为提高特征抽取的速度和识别率,利用图像矩阵直接构造图像散布矩阵,在具有统计不相关的条件下将Foley-Sammon鉴别分析(FSLDA)转化为两目标约束优化问题,并给出了有效投影向量的概念;根据多目标优化的最优性条件可将求取有效投影向量的问题归结为求广义特征方程的最大特征值对应的特征向量,并据此进行特征抽取,进而提出了两目标最优图像投影鉴别分析方法。与其他鉴别投影分析方法相比,该方法具有以下特点:(1)可直接由图像矩阵构建散布矩阵;(2)有效投影向量具有统计不相关性;(3)训练样本的类内散布矩阵不必为可逆的,也不需要求某种形式矩阵的逆。在ORL标准人脸库和NUST603人脸库上的试验结果表明,上述图像投影鉴别分析方法在识别性能上较以往的方法有一定的提高,尤其是特征抽取的速度有明显的提高。     

用Lanczos算法进行周期结构固有特性分析的研究

1．概述1．11程背景大型柔性空间结构（LargeFlexibleSpaceStructures,简称LFSS）是随着空间技术的发展而产生的一类特殊结构．六十年代以来,LFSS已成为卫星、太空站等的基本组成部分．这类柔性结构形式主要有三种：（1）梁式结构;（2）板式结构;（3）抛物天线及其他天线结构．LFSS是具有周期性的框架结构,其构成具有重复性,整体结构由相同构件按照一定规律彼此连接而成．该构件,称典型构件,由若干梁、杆、索、板等元件按一定方式连接而成．按结构是否封闭可将其分类为平移周期结构与旋转周期结构．平移周期结构由典型构件依…     

Descent Iterations for Improving Approximate Eigenpairs of Polynomial Eigenvalue Problems with General Complex Matrices

In this paper, we consider descent iterations with line search for improving an approximate eigenvalue and a corresponding approximate eigenvector of polynomial eigenvalue problems with general complex matrices, where an approximate eigenpair was obtained by some method. The polynomial eigenvalue problem is written as a system of complex nonlinear equations with nondifferentiable normalized condition. Convergence theorems for iterations are established. Finally, some numerical examples are presented to demonstrate the effectiveness of the iterative methods. Received April 9, 2001; revised October 2, 2001 Published online February 18, 2002     

The Symm-Wilkinson method for improving an approximate eigenvalue and its associated eigenvector

Here is discussed the Symm-Wilkinson method (called a relaxed algorithm in [4]) for improving an approximate simple eigenvalue of ann×n matrix and a corresponding approximate eigenvector which were obtained by some method. It is shown that their method is a Newton-like method applied to a system of nonlinear equations so that the process converges linearly under the usual assumptions. The Symm-Wilkinson method needs more multiplications than the standard Newton-like method applied to the same equations byn?1 at each step. Therefore, there does not seem to be any great advantage in using the former in place of the latter.     

两目标最优图像鉴别分析及其在人脸识别中的应用

考虑样本为图像矩阵的图像鉴别分析问题，将Foley—Sammon鉴别分析问题转化为一类带约束条件的两目标优化问题，给出了有效投影向量的概念．利用多目标优化的必要条件，得到有效投影向量应满足的条件，它为广义特征方程的最大特征值所对应的特征向量，从而得到了求有效投影向量集的方法，其中类内散布矩阵不必是非奇异的．实验结果表明：该方法节省了特征抽取的时间，并且识别性能要优于其他方法．     

An eigenvalue sensitivity study of a high temperature gas-cooled reactor system

This correspondence discusses the application of an eigenvalue sensitivity method [1] to a linear, time-invariant model of the high-temperature gas-cooled reactor (HTGR) manufactured by General Atomic. It is shown that for this type of system with distinct eigenvalues, a relation can be obtained between the solutions to the classical sensitivity. equation and the eigenvalue and eigenvector sensitivities. This relation indicates that the eigenvalue and eigenvector sensitivities are more fundamental indicators of parametric effects than the solutions of the sensitivity. equation. A Fortran program (SENSIT) was developed to implement the algorithms required for eigenvalue sensitivity calculations. This program was applied to a large scale (58 state) linear, time invariant model of the HTGR operating under closed-loop control.