Computation of eigenvalues of a real matrix

This paper presents a computational procedure for finding eigenvalues of a real matrix based on Alternate Quadrant Interlocking Factorization, a parallel direct method developed by Rao in 1994 for the solution of the general linear system Ax=b. The computational procedure is similar to LR algorithm as studied by Rutishauser in 1958 for finding eigenvalues of a general matrix. After a series of transformations the eigenvalues are obtained from simple 2×2 matrices derived from the main and cross diagonals of the limit matrix. A sufficient condition for the convergence of the computational procedure is proved. Numerical examples are given to demonstrate the method.     

Origin-shifted algorithm for matrix eigenvalues

In this paper an origin-shifted algorithm for matrix eigenvalues based on Frobenius-like form of matrix and the quasi-Routh array for polynomial stability is given. First, using Householder's transformations, a general matrix A is reduced to upper Hessenberg form. Secondly, with scaling strategy, the origin-shifted Hessenberg matrices are reduced to the Frobenius-like forms. Thirdly, using quasi-Routh array, the Frobenius-like matrices are determined whether they are stable. Finally, we get the approximate eigenvalues of A with the largest real-part. All the eigenvalues of A are obtained with matrix deflation. The algorithm is numerically stable. In the algorithm, we describe the errors of eigenvalues using two quantities, shifted-accuracy and satisfactory-threshold. The results of numerical tests compared with QR algorithm show that the origin-shifted algorithm is fiducial and efficient for all the eigenvalues of general matrix or for all the roots of polynomial.     

二维向量Sturm-Liouville问题的特征值

研究定义在有限区间[0,1]上的具有一般分离型边界条件的二维向量Sturm-Liouville问题。证明在一定条件下,该二维向量Sturm-Liouville问题只有有限个二重特征值。得到该二维向量Sturm-Liouville问题的某个特征值是二重特征值的必要条件。找到常数MQ使得大于该常数的特征值都是单的（一重）。     

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.     

An invariant subspace‐based approach to the random eigenvalue problem of systems with clustered spectrum

The repeated or closely spaced eigenvalues and corresponding eigenvectors of a matrix are usually very sensitive to a perturbation of the matrix, which makes capturing the behavior of these eigenpairs very difficult. Similar difficulty is encountered in solving the random eigenvalue problem when a matrix with random elements has a set of clustered eigenvalues in its mean. In addition, the methods to solve the random eigenvalue problem often differ in characterizing the problem, which leads to different interpretations of the solution. Thus, the solutions obtained from different methods become mathematically incomparable. These two issues, the difficulty of solving and the non‐unique characterization, are addressed here. A different approach is used where instead of tracking a few individual eigenpairs, the corresponding invariant subspace is tracked. The spectral stochastic finite element method is used for analysis, where the polynomial chaos expansion is used to represent the random eigenvalues and eigenvectors. However, the main concept of tracking the invariant subspace remains mostly independent of any such representation. The approach is successfully implemented in response prediction of a system with repeated natural frequencies. It is found that tracking only an invariant subspace could be sufficient to build a modal‐based reduced‐order model of the system. Copyright © 2012 John Wiley & Sons, Ltd.     

Multiplicative relative perturbation bounds of eigenvalues for diagonalizable matrices

In this paper, we present the general bounds of multiplicative relative perturbation for diagonalizable matrices, which are the improvement of recent results. The bounds gained here are sharper than those in related literatures. In addition, we obtain the multiplicative relative bounds for normal matrices, which are discussed first.     

Optimal Automatic Generation Control of Asynchronous Power Systems Using Output Feedback Control Strategy with Dynamic Participation of Wind Turbines

Abstract—This article presents the design of optimal output feedback automatic generation control regulators for an interconnected power system with dynamic participation of doubly fed induction generator based wind turbines. The power systems consist of plants with hydro-thermal turbines and are interconnected via parallel AC/DC links. Efforts have been made to propose optimal automatic generation control regulators based on feedback of output state variables, which are easily accessible and available for the measurement. The designed optimal output feedback automatic generation control regulators are implemented, and the system dynamic responses for various system states are obtained considering 1% load perturbation in one of the areas. The dynamic performance is compared with that obtained with optimal automatic generation control regulators designed using full state vector feedback. The pattern of closed-loop eigenvalues is also determined to test the system stability.     

A Method for Improving Multivariate Measurement Systems Assessment

Multivariate measurement systems analysis (MSA) is usually performed by designing suitable gauge repeatability and reproducibility (R&R) experiments, ignoring available data generated by the measurement system while used for inspection or process control. This article proposes an approach that, by using the data that are routinely available from the regular operation of the instrument, allows the measurement instrument's current precision to be compared against a benchmark. The proposed method may be appropriately used in an integrated and coordinated manner with the usual multivariate gauge study in the sense that it can be used to assess the stability or a possible deterioration in the precision of the measurement instrument while operational. Therefore, the complementary use of the proposed approach and the traditional multivariate gauge R&R studies can be a useful strategy for improving the overall quality of multivariate measurement systems. Furthermore, because it can be implemented at almost no additional cost, it may be effective in reducing the costs of a multivariate MSA performed with a certain frequency. Copyright © 2014 John Wiley & Sons, Ltd.     

基于MATLAB的磁悬浮系统奇异摄动参数计算

磁悬浮系统的稳定性是磁悬浮列车能否正式投入营运的关键因素。而实际的悬浮系统由于存在小电感奇异摄动参数,使得悬浮系统的动力学模型有相当高的阶数以及病态的数值特性,因此该文采用奇异摄动法处理这类问题。根据奇异摄动理论可知,要使系统稳定,摄动参数必存在一个稳定的上界。该文针对悬浮系统模型复杂和系统变参数的特点,借助MATLAB简化了磁悬浮系统奇异摄动参数稳定上界的计算。结果表明该方法的有效性和可行性,对磁悬浮系统的理论分析和工程研究有一定的参考价值。     

局部化的广义特征值最接近支持向量机

基于广义特征值的最接近支持向量机(Proximal Support Vector Machine via Generalized Eigenvalues,GEPSVM)是一种新的具有与SVM性能相当的两分类方法,通过求解广义特征值来获得两个彼此不平行的拟合两类样本的超平面.其决策是将测试样本归为距其最近的超平面所在的类.然而,该规则在某些情形会导致较差的分类结果.对此,在GEPSVM基础上,通过在类拟合超平面上寻找一个包含了所有训练样本投影的局部凸区域,来决定样本的类别.该局部方法不仅具有较GEPSVM更优的分类性能,同时还衍生出了求解超平面上凸壳的简单且易于核化的新算法.最后在人工和UCI数据集上获得了验证.