拟Newton法在高阶矩阵中的应用——求解最大特征值及特征向量

将求解高阶矩阵的最大特征值及其对应的特征向量问题转化为高阶非线性方程组的求解问题。在此基础上,提出了求解矩阵最大特征值及其对应特征向量的拟Newton法,给出求解矩阵最大特征值及其单位化向量重新整理后的Broyden方法公式、BFS方法公式、DFP方法公式及其对应的Broyden算法,BFS算法,DFP算法。以层次分析法中高阶判断矩阵为例验证了该方法的可行性,说明了该方法相对收敛速度快的优势。     

求解矩阵特征值的改进PSO算法

为了改进粒子群算法在求解矩阵特征值时只能根据矩阵特征值范围逐一求解特征值的现状。提出了一种改进的粒子群算法。改进的粒子群算法采用寻找到一个特征值后,适当改变适应值函数的策略,使搜索区域远离已寻找到的特征值,继续寻找其他的特征值,如此反复,直到寻找到所有的特征值为止。利用四个不同类型的矩阵求解特征值进行仿真,实验结果也验证了算法的实用性和有效性。     

一种计算矩阵特征值特征向量的神经网络方法

当把Oja学习规则描述的连续型全反馈神经网络(Oja-N)用于求解矩阵特征值特征向量时,网络初始向量需位于单位超球面上,这给应用带来不便.由此,提出一种求解矩阵特征值特征向量的神经网络(1yNN)方法.在lyNN解析解基础上得到了以下结果:初始向量属于任意特征值对应特征向量张成的子空间,则网络平衡向量也将属于该空间;分析了lyNN收敛于矩阵最大特征值对应特征向量的初始向量取值条件;明确了lyNN收敛于矩阵不同特征值的特征子空间时,网络初始向量的最大取值空间;网络初始向量与已知特征向量垂直,则lyNN平衡解向量将垂直于该特征向量;证明了平衡解向量位于由非零初始向量确定的超球面上的结论.基于以上分析,设计了用lyNN求矩阵特征值特征向量的具体算法,实例演算验证了该算法的有效性.1yNN不出现有限溢,而基于Oja-N的方法在矩阵负定、初始向量位于单位超球面外时必出现有限溢,算法失效.与基于优化的方法相比,lyNN实现容易,计算量较小.     

求解job shop调度问题的量子粒子群优化算法*

针对粒子群优化算法搜索空间有限、容易出现早熟现象的缺陷,提出将量子粒子群优化算法用于求解作业车间调度问题。求解时,将每个调度按照一定的规则编码为一个矩阵,并以此矩阵作为算法中的粒子;然后根据调度目标确定目标函数,并按照量子粒子群优化算法的进化规则在调度空间内搜索最优解。仿真实例结果证明,该算法具有良好的全局收敛性能和快捷的收敛速度,调度效果优于遗传算法和粒子群优化算法。     

一种改进的求解TSP混合粒子群优化算法

为解决粒子群算法在求解组合优化问题中存在的早熟性收敛和收敛速度慢等问题,将粒子群算法与局部搜索优化算法结合,可抑制粒子群算法早熟收敛问题,提高粒子群算法的收敛速度。通过建立有效的局部搜索优化算法所需借助的参照优化边集,提高了局部搜索优化算法的求解质量和求解效率。新的混合粒子群算法高效收敛于中小规模旅行商问题的全局最优解,实验表明改进的混合粒子群算法是有效的。     

求解job-shop调度问题的量子粒子群优化算法

针对粒子群优化算法搜索空间有限、容易出现早熟现象的缺陷,提出将量子粒子群优化算法用于求解作业车间调度问题.求解时,将每个调度按照一定的规则编码为一个矩阵,并以此矩阵作为算法中的粒子;然后根据调度目标确定目标函数,并按照量子粒子群优化算法的进化规则在调度空间内搜索最优解.仿真实例结果证明,该算法具有良好的全局收敛性能和快捷的收敛速度,调度效果优于遗传算法和粒子群优化算法.     

求解矩阵特征值及特征向量的新方法

提出一种基于进化策略求解矩阵特征值及特征向量的新方法。该方法在进化过程中通过重组、突变、选择对个体进行训练学习,向最优解逼近。当达到预先给定的误差时,程序终止,得到最优解。实验结果表明,与传统方法相比,该方法的收敛速度较快,求解精度提高了10倍。该算法能够快速有效地获得任意矩阵对应的特征值及特征向量。     

矩阵特征值估计的粒子群优化算法

利用Gersehgorin圆盘定理与矩阵特征值的性质,将特征值的求解问题转化为最优化问题.借助粒子群优化算法与二分法思想,精确地估计了实(复)方矩阵的全体特征值,并与Matlab软件中基于QR算法设计的特征值求解函数eig的计算结果作对比,绝对误差达到10-7数量级以上.同时,也解决了特征值分离度的估计问题.     

基于PSO的Fisher准则下小样本最佳鉴别变换

小样本条件下,Fisher准则中类内散布矩阵一般是奇异的,无法直接求解.本文提出利用粒子群优化理论,在无需求类内散布矩阵逆的情况下求解Fisher准则下小样本最佳鉴别变换的方法.讨论了通过粒子群优化算法的位置-速度搜索模型获取最佳鉴别投影向量的方法和步骤.实验对比类内散布矩阵非奇异时,采用计算特征向量方法和本文方法的差异.分析验证小样本条件下类内散布矩阵奇异时,通过本文方法进行最佳鉴别变换的分类效果.实验证实本文算法的有效性.     

基于参数方程处理等式约束优化的粒子群算法

针对目前已有的粒子群优化算法求解有等式约束优化问题时对收敛速度和解的精度的影响,提出了一种新的基于参数方程的粒子群优化算法.它是粒子群在初始化和选代进化过程中使用求解参数方程的方法处理等式约束设计出的粒子群优化算法.数值实验结果表明,新算法是有效的.它不仅提高了收敛速度和解的精度,而且是一种通用的智能算法.     

Eigenfactor solution of the matrix Riccati equation--A continuous square root algorithm

This paper introduces a new algorithm for solving the matrix Riccati equation. Differential equations for the eigenvalues and eigenvectors of the solution matrix are developed in which their derivatives are expressed in terms of the eigenvalues and eigenvectors themselves and not as functions of the solution matrix. The solution of these equations yields, then, the time behavior of the eigenvalues and eigenvectors of the solution matrix. A reconstruction of the matrix itself at any desired time is immediately obtained through a trivial similarity transformation. This algorithm serves two purposes. First, being a square root solution, it entails all the advantages of square root algorithms such as nonnegative definiteness and accuracy. Secondly, it furnishes the eigenvalues and eigenvectors of the solution matrix continuously without resorting to the complicated route of solving the equation directly and then decomposing the solution matrix into its eigenvalues and eigenvectors. The algorithm which handles cases of distinct as well as multiple eigenvalues is tested on several examples. Through these examples it is seen that the algorithm is indeed more accurate than the ordinary one. Moreover, it is seen that the algorithm works in cases where the ordinary algorithm fails and even in cases where the closed-form solution cannot be computed as a result of numerical difficulties.     

自洽场方法中广义本征值方程求解及其C++程序设计

本文研究自洽场方法中广义本征值方程求解的算法,并设计相应的C 程序来实现该算法。首先对重叠矩阵进行分解,并将广义本征值方程化为标准的本征值方程,再利用Householder变换将上一步变换所得的矩阵化为对称三对角矩阵,进而用QL方法求解这个三对角矩阵的本征值和本征矢量,从而得到自洽场方法中广义本征值方程的本征值和本征矢量。     

Robust adaptive beamforming method using principal eigenpairs with modification of PASTd

Adaptive arrays suffer from performance degradation in the presence of steering vector errors. The doubly constrained robust Capon beamformer (DCRCB) can deal with the problem, utilizing all the eigenvalues and eigenvectors of the covariance matrix, which leads to high computational complexity. This paper presents a robust beamforming method which is computationally efficient, exploiting principal eigenpairs only. The eigenpairs can be estimated based on the projection approximation subspace tracking with deflation (PASTd). The original PASTd algorithm, which does not provide orthonormal eigenvectors in general, is modified so that the orthonormalization of eigenvectors can be efficiently made using the structure of the modified algorithm. The proposed beamforming method significantly reduces the computational load, particularly when the number of the directional signals is much less than that of sensor elements, and substantially has the same performance as the conventional one utilizing all the eigenpairs.     

用神经网络计算矩阵特征值与特征向量

该文研究用神经网格求解一般实对称矩阵的全部特征向量的问题。详细讨论了网络的平均态度合的结构并建立了平衡态集合的构造定理。通过求解简单的一维微分方程求出了网络的解析表达式。这一表达式是由对称矩阵的特征值与特征向量表达的、因而非常清晰利用解的解析表达式分析了网络的解的全局渐近行为。提出了用一些单位向量作为网络初始值计算对称矩阵的全部特征值与特征向量的具体算法。     

基于KL变换的模糊C-均值聚类彩色图像分割

根据图像色彩特征空间的正交特性，以及构成特征空间的特征向量和特征值之间的统计特性，提出了一种新的彩色图像指定区域分割算法。首先在指定区域选取采样像素，通过KL变换计算采样像素的协方差矩阵、特征值、特征向量；由特征向量构成指定区域的色彩特征空间，然后对原色彩空间中的向量进行空间变换和权重变换；最后用模糊C-均值聚类方法聚类变换后的向量，得到分割结果。文中给出了静物图像的聚类分割结果，体现了算法对于指定区域细节分割的准确性。     

一种求解复Hermite矩阵特征值的方法

介绍几种求解矩阵特征值和特征向量的经典算法及各自优缺点,通过理论推导,提出了一种性能稳健的方法,可以求解信号处理中常见的复Hermite阵.将对复Hermite矩阵求特征值和特征向量的问题转化为求解实对称阵的特征值和特征向量,而实对称阵的求解采用一种改进的三对角Householder法.最后把结果与Matlab仿真结果比较,可以看出该方法有很高的精确度.     

Spectral clustering based on matrix perturbation theory

This paper exposes some intrinsic characteristics of the spectral clustering method by using the tools from the matrix perturbation theory. We construct a weight ma- trix of a graph and study its eigenvalues and eigenvectors. It shows that the num- ber of clusters is equal to the number of eigenvalues that are larger than 1, and the number of points in each of the clusters can be approximated by the associated eigenvalue. It also shows that the eigenvector of the weight matrix can be used directly to perform clustering; that is, the directional angle between the two-row vectors of the matrix derived from the eigenvectors is a suitable distance measure for clustering. As a result, an unsupervised spectral clustering algorithm based on weight matrix (USCAWM) is developed. The experimental results on a number of artificial and real-world data sets show the correctness of the theoretical analysis.     

A guided genetic algorithm for diagonalization of symmetric and Hermitian matrices

The eigenvalues and eigenvectors of a matrix have many applications in engineering and science, such us studying and solving structural problems in both the treatment of signal or image processing, and the study of quantum mechanics. One of the most important aspects of an algorithm is the speed of execution, especially when it is used in large arrays. For this reason, in this paper the authors propose a new methodology using a genetic algorithm to compute all the eigenvectors and eigenvalues in real symmetric and Hermitian matrices. The algorithm uses a general-purpose library developed by the authors for genetic algorithms (GALGA). The speed of execution and the influence of population size have been studied. Moreover, the algorithm has been tested in different matrices and population sizes by comparing the speed of execution to the number of the eigenvectors. This new methodology is faster than the previous algorithm developed by the authors and all eigenvectors can be obtained with it. In addition, the performance using the Coope matrix has been tested contrasting the results with another technique published in the scientific literature.     

Higher order eigensensitivity analysis of damped systems with repeated eigenvalues

A simplified method for the computation of first-, second- and higher-order derivatives of eigenvalues and eigenvectors associated with repeated eigenvalues is presented. Adjacent eigenvectors and orthonormal conditions are used to compose an algebraic equation. The algebraic equation which is developed can be used to compute derivatives of eigenvalues and eigenvectors simultaneously. Since the coefficient matrix in the proposed algebraic equation is non-singular, symmetric and based on N-space, it is numerically stable and very efficient compared to previous methods. To verify the efficiency of the proposed method, the finite element model of the cantilever beam and a mechanical system in the case of a non-proportionally damped system are considered.     

Inverse iteration method for finding eigenvectors

An iteration method which is not sensitive to small errors in the eigenvalues is developed for finding eigenvectors. The method finds the eigenvector of both the normal and transposed matrix. These eigenvectors can then be used in the Rayleigh quotient to improve the value for the eigenvalue.