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。     

求解矩阵特征值和特征向量的PSO算法

提出一种基于粒子群优化算法的求解方法,将线性方程组的求解转化为无约束优化问题加以解决,采用粒子群优化算法求解矩阵特征值和特征向量。仿真实验结果表明,该方法求解精度高、收敛速度快,能够在10代左右收敛,可以有效获得任意矩阵的特征值和特征向量。     

基于动力学方程求解复矩阵特征值问题的并行实现

该文提出了一种利用动力学方程求解复特征值及其特征向量的并行实现方法。方法的原理为：首先将特征值问题通过优化技术转化为一个非线性动力学系统的求解问题，然后利用电路模拟中的波形松弛法并行计算这组动力学方程的解。该方法能够有效地确定复矩阵的全部特征值和特征向量。这是首次将波形松弛法引入大型矩阵的计算中，其并行算法已在IBM RS／6000 SuperPOWER2系统中有效地实现。     

解实对称矩阵特征值问题的Jacobi算法

求解n阶实对称阵A的特征值问题Ax=λx的Jacobi方法,是用一系列平面旋转变换化A为对角型,从而得到特征值和特征向量的.假设A_0=A,A_k=R_kA_(k-1)R_k~T,k=1,2,3,…,当k→∞时,A_k趋向于固定的对角型.简记某次变换为     

求解对称区间矩阵标准特征值的进化策略新算法

针对实对称区间矩阵的特征值问题,将区间不确定量看成是围绕区间中点的一种摄动,提出了一种基于区间扩张的对称区间矩阵特征值问题求解的进化策略算法。将区间矩阵中点作为平衡点,区间不确定量作为相应的扰动量,根据摄动公式求出区间矩阵的最大特征值和最小特征值,从而获得区间矩阵特征值问题的解。算例显示了该算法的有效性,其主要特点是收敛速度快、求解区间精度高。     

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.     

求实对称矩阵部分特征值的并行算法

提出了并行求解实对称稠密矩阵部分特征值的反幂法的预处理方法.该方法基于带状矩阵特征问题反幂法的信息传递复杂度低的特点,采用Householder变换并行算法约化大型实对称稠密矩阵为一定带宽的带状矩阵,针对带状矩阵用反幂法求解矩阵的在某一点的近似特征值;其中针对反幂法迭代中遇到的线性方程组,采用文献中的并行预处理共轭梯度算法求解.最后在Lenovo深腾1800集群上进行数值实验,并与预处理前反幂法的计算结果进行了比较,实验结果表明,经过预处理后的并行性远高于直接采用反幂法的并行性.     

求矩阵复特征值的双种群改进遗传算法

针对矩阵复特征值的特点,提出采用双种群改进遗传算法并行求解复特征值的近似值.该算法中双种群采用实数编码,在遗传过程中每个种群都根据适应度自动选择其交叉概率和变异概率,使个体对环境变化具有自适应调节能力.变异中采用了柯西变异,可以使个体很快跳出局部极小.仿真结果表明,此算法可以达到一定的精度,具有一定的通用性,并给求矩阵复特征值提供了一种快速的方法.     

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.     

基于增广矩阵束方法的平面天线阵列综合

针对平面阵列的稀布优化问题,提出了一种基于增广矩阵束方法的减少阵元数目、求解阵元位置和设计幅度激励的优化方法。首先对期望平面阵的方向图进行采样并由采样点数据构造增广矩阵,对此矩阵进行奇异值(SVD)分解,确定在误差允许范围内所需的最小阵元数目;然后基于广义特征值分解分别计算两组特征值,并根据类ESPRIT算法对特征值进行配对;最后在最小二乘准则条件下根据正确的特征值对求解平面阵列的阵元位置和激励。仿真结果表明该算法具有较高的计算效率和数值精度。     

五对角矩阵的特征值反问题

本文讨论了一类由五个特征值和相应特征向量构造实对称五对角矩阵的特征值反问题.研究了解的存在性以及存在解的充分必要条件,而且给出了算法和数值例子.     

A uniform object-oriented solution to the eigenvalue problem for real symmetric and Hermitian matrices

We present a system of classes, SHMatrix, to deal in a unified way with the computation of eigenvalues and eigenvectors in real symmetric and Hermitian matrices. Thus, two descendant classes, one for the real symmetric and other for the Hermitian cases, override the abstract methods defined in a base class. The use of the inheritance relationship and polymorphism allows handling objects of any descendant class using a single reference of the base class. The system of classes is intended to be the core element of more sophisticated methods to deal with large eigenvalue problems, as those arising in the variational treatment of realistic quantum mechanical problems. The present system of classes allows computing a subset of all the possible eigenvalues and, optionally, the corresponding eigenvectors. Comparison with well established solutions for analogous eigenvalue problems, as those included in LAPACK, shows that the present solution is competitive against them.

Program summary

Program title: SHMatrixCatalogue identifier: AEHZ_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEHZ_v1_0.htmlProgram obtainable from: CPC Program Library, Queen?s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 2616No. of bytes in distributed program, including test data, etc.: 127 312Distribution format: tar.gzProgramming language: Standard ANSI C++.Computer: PCs and workstations.Operating system: Linux, Windows.Classification: 4.8.Nature of problem: The treatment of problems involving eigensystems is a central topic in the quantum mechanical field. Here, the use of the variational approach leads to the computation of eigenvalues and eigenvectors of real symmetric and Hermitian Hamiltonian matrices. Realistic models with several degrees of freedom leads to large (sometimes very large) matrices. Different techniques, such as divide and conquer, can be used to factorize the matrices in order to apply a parallel computing approach. However, it is still interesting to have a core procedure able to tackle the computation of eigenvalues and eigenvectors once the matrix has been factorized to pieces of enough small size. Several available software packages, such as LAPACK, tackled this problem under the traditional imperative programming paradigm. In order to ease the modelling of complex quantum mechanical models it could be interesting to apply an object-oriented approach to the treatment of the eigenproblem. This approach offers the advantage of a single, uniform treatment for the real symmetric and Hermitian cases.Solution method: To reach the above goals, we have developed a system of classes: SHMatrix. SHMatrix is composed by an abstract base class and two descendant classes, one for real symmetric matrices and the other for the Hermitian case. The object-oriented characteristics of inheritance and polymorphism allows handling both cases using a single reference of the base class. The basic computing strategy applied in SHMatrix allows computing subsets of eigenvalues and (optionally) eigenvectors. The tests performed show that SHMatrix is competitive, and more efficient for large matrices, than the equivalent routines of the LAPACK package.Running time: The examples included in the distribution take only a couple of seconds to run.     

The projective power method for the algebraic eigenvalue problem

A new method for computing several largest eigenvalues of a matrix has some common features with the power method but uses orthogonal projections instead of the customary ways of deflation. The rate of convergence is basically the same as for the power method but the fast refinement of the approximations to eigenvalues and eigenvectors in the cases of real symmetric and Hermitian matrices can be done even without the inverse iterations.     

The equivalence of the constrained Rayleigh quotient and Newton methods for matrix polynomials expressed in different polynomial bases along with the confluent case

We show that using the constrained Rayleigh quotient method to find the eigenvalues of matrix polynomials in different polynomial bases is equivalent to applying the Newton method to certain functions. We find those functions explicitly for a variety of polynomial bases including monomial, orthogonal, Newton, Lagrange and Bernstein bases. In order to do so, we provide explicit symbolic formulas for the right and left eigenvectors of the generalized companion matrix pencils for matrix polynomials expressed in those bases. Using the properties of the Newton basis, we also find two different formulas for the companion matrix pencil corresponding to the Hermite interpolation. We give pairs of explicit LU factors associated with these pencils. Additionally, we explicitly find the right and left eigenvectors for each of these pencils.     

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.     

基于LDL~T分解求实对称矩阵特征值的递归算法

基于线性代数与矩阵理论,给出利用LDLT分解计算实对称矩阵特征值的递归算法。该算法可求出实对称矩阵在给定区间内的特征值的个数,并可计算满足精度要求的特征值。理论分析和实际测试证明该算法是有效的。     

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。     

用Excel求实对称阵的全部特征值和特征向量

给出了用Excel求实对称矩阵的全部特征值和特征向量的方法,该方法简单、直观,不需要设计程序,也不需要专门的数学软件,不仅为课堂教学,也为数值计算提供了方便。