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.     

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.     

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。     

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。     

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。     

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。     

求解对称非线性矩阵特征值问题的一个三阶收敛的算法

考虑对称矩阵A(λ)∈R~(n×n),它的元素是λ的解析函数.求λ∈R,向量x≠0,使得求解(1.1)称为求解对称非线性矩阵特征值问题. 对于一般非线性矩阵特征值问题已经有了很多有效的方法.本文的目的是如何利用矩阵的对称性给出一个运算量与通常使用的二阶收敛方法的运算量相当的三阶收敛算法.     

用Lanczos方法解高阶稀疏矩阵广义特征值问题的某些经验

考虑n阶实对称矩阵偶K,M的广义特征值问题 Ky=ω~2My,(0.1)其中K是非负半定阵,M为对称正定矩阵。问题(0.1)的特征值分布为:0≤ω_1~2≤ω_2~2≤…≤ω_n~2。通常需要求解(0.1)的前k个特征解,即ω_1~2,ω_2~2,…,ω_k~2及其对应的特征向量     

基于两步对角化的对称稠密矩阵特征值问题快速求解算法

计算对称矩阵中的某些特定的特征值和特征向量问题是很多科学计算领域中都存在的重要课题。特别在电子结构的计算中,特征值计算成为计算瓶颈。以往在需要求解大部分特征值和特征向量的应用场合,一般使用直接求解的方式。为了更好地利用存储器性能优势,我们设计了对角化算法,对规约与逆变换过程进行拆分处理,通过对整个过程的重新设计,充分利用存储器结构上的优势,提升单核计算速度,同时改进并行效率。本文中我们重点讨论三对角矩阵到带状矩阵逆变换过程。本文中所提及到的算法应用于MESIA电子结构计算软件包之中,取得了一定的性能提升。     

微分方程在复杂网络社团发现中的研究

理解复杂网络的关键在于迅速精确地发现网络中的社团结构。基于图理论的谱聚类算法是一种有效并全局收敛的优秀社团发现算法,其计算量集中于特征值和特征向量的计算。结合常系数线性常微分方程的解与系数矩阵特征值的关系,提出了基于微分方程的谱聚类社团发现算法(AMCF和LMCF);这两种算法避免了矩阵的特征值和特征向量的复杂计算过程,为社团发现算法提供了新的思路。理论分析和实验验证了算法的有效性。     

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。     

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.     

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.     

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.     

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.     

A concise functional neural network computing the largest modulus eigenvalues and their corresponding eigenvectors of a real skew matrix

Quick extraction of the largest modulus eigenvalues of a real antisymmetric matrix is important for some engineering applications. As neural network runs in concurrent and asynchronous manner in essence, using it to complete this calculation can achieve high speed. This paper introduces a concise functional neural network (FNN), which can be equivalently transformed into a complex differential equation, to do this work. After obtaining the analytic solution of the equation, the convergence behaviors of this FNN are discussed. Simulation result indicates that with general initial complex values, the network will converge to the complex eigenvector which corresponds to the eigenvalue whose imaginary part is positive, and modulus is the largest of all eigenvalues. Comparing with other neural networks designed for the like aim, this network is applicable to real skew matrices.     

Partial solution of large symmetric generalized eigenvalue problems by nonlinear optimization of a modified Rayleigh quotient

A method is presented to solve numerically the lowest (or highest) eigenvalues and eigenvectors of the symmetric generalized eigenvalue problem. The technique proposed is iterative, does not transform the original matrices and yields eigencharacteristics in sequence, even for repeated eigenvalues. It is based on a nonlinear optimization of an unconstrained penalty function obtained from a generalization of the Rayleigh quotient. In addition, when the normality constraint is imposed, the eigenvectors are obtained by a sequence of solutions to linear equations, all with the same matrix. Examples demonstrate the validity of the method.