Two- or three-stage least squares?

Two elements enter the choice between 2 and 3SLS for full-system estimation: statistical efficiency and computational cost. 2SLS always has the computational edge, but 3SLS can be more efficient, a relative advantage that increases with the strength of the interrelations among the error terms. A measure of these interrelations is thus helpful in making the choice, and, when there are only two equations, this has suggested using a high pairwise error correlation as an indicator of when to use 3SLS. In larger systems of equations, however, these pairwise correlations can remain small even though more general interrelations give 3SLS the relative advantage. More general indicators are therefore needed, and this paper suggests three such and demonstrates their efficacy.Professor of Economics, Boston College, and Principal Research Associate, Center for Computational Research in Economics and Management Science, MIT. All computation was done on the TROLL system at MIT. My thanks go to Josh Charap for his able research assistance. This research was sponsored in part by the National Science Foundation under grant #IST-8420614.     

Eigensolution of specially structured matrices with hyper‐symmetry

Recently, four canonical forms have been developed and applied to the dynamics and stability analysis of symmetric frames. In this paper, hyper‐symmetric matrices and specially structured matrices are defined and efficient methods are proposed for the eigensolution of such matrices. Applications are extended to hyper‐graphs and specially structured graphs. Simple methods are developed for calculating the eigenvalues of the Laplacian matrices of such graphs. The developments presented in this paper can also be considered as generalization of Form II and Form III symmetry, previously defined by the authors. Copyright © 2006 John Wiley & Sons, Ltd.     

基于特征根比分布的协作频谱感知算法研究

介绍两种基于大维随机矩阵理论(Random Matrix Theory,RMT)的频谱感知方法,综合对比讨论并验证其感知性能。这些方法以采样协方差矩阵的最大最小特征根之比作为判决统计量与相应的判决门限相比较来确定频谱是否空闲。通过与能量检测(Energy Detection,ED)算法仿真对比,它们克服了能量检测中需要预知噪声信息的缺点,并具有不受噪声不确定度影响的优点,有较高检测性能和优越性。     

Parallel block tridiagonalization of real symmetric matrices

Two parallel block tridiagonalization algorithms and implementations for dense real symmetric matrices are presented. Block tridiagonalization is a critical pre-processing step for the block tridiagonal divide-and-conquer algorithm for computing eigensystems and is useful for many algorithms desiring the efficiencies of block structure in matrices. For an “effectively” sparse matrix, which frequently results from applications with strong locality properties, a heuristic parallel algorithm is used to transform it into a block tridiagonal matrix such that the eigenvalue errors remain bounded by some prescribed accuracy tolerance. For a dense matrix without any usable structure, orthogonal transformations are used to reduce it to block tridiagonal form using mostly level 3 BLAS operations. Numerical experiments show that block tridiagonal structure obtained from this algorithm directly affects the computational complexity of the parallel block tridiagonal divide-and-conquer eigensolver. Reduction to block tridiagonal form provides significantly lower execution times, as well as memory traffic and communication cost, over the traditional reduction to tridiagonal form for eigensystem computations.     

进化策略算法在矩阵特征值求解中的应用

工程实践中多种振动问题的求解常常归纳为求矩阵特征值问题,另外一些稳定性分析问题及相关分析问题也可以转化为求矩阵特征值问题.为了有效求解此类问题,提出了一种新的求解矩阵特征值的进化策略算法,该算法可用于求解任意矩阵的特征值.实验结果表明,这种基于进化策略算法求解矩阵特征值的方法,与传统方法相比,表现出求解精度高,收敛速度快等优点.     

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。     

联合听觉掩蔽效应的子空间语音增强算法

在经典子空间语音增强算法中,因语音特征值估计偏差会造成语音失真和音乐噪声。针对该问题,提出一种联合听觉掩蔽效应的语音增强算法。该算法联合掩蔽阈值自适应调节噪声特征值的抑制系数,并利用维纳滤波对音乐噪声的抑制性,对该特征值并行修正,最终还原出纯净的语音。实验结果证明,该算法在白噪声和有色噪声的背景下,与经典子空间的语音增强算法相比,能提高信噪比,减少语音失真和音乐噪声。     

Necessary and sufficient conditions for global optimality of eigenvalue optimization problems

The necessary and sufficient conditions for global optimality are derived for an eigenvalue optimization problem. We consider the generalized eigenvalue problem where real symmetric matrices on both sides are linear functions of design variables. In this case, a minimization problem with eigenvalue constraints can be formulated as Semi-Definite Programming (SDP). From the Karush-Kuhn-Tucker conditions of SDP, the necessary and sufficient conditions are derived for arbitrary multiplicity of the lowest eigenvalues for the case where important lower bound constraints are considered for the design variables. Received May 18, 2000     

箭头矩阵在最小最大特征值条件下的完成问题

引入并讨论了对称箭头矩阵完成问题:在事先给定的对称箭头矩阵中嵌入一行一列使之成为新的对称箭头矩阵,并且具有指定最小最大特征值.利用箭头矩阵特征多项式之间的递归关系,给出并证明了这个问题存在惟一解的充要条件,以及解的一般公式与计算方法.同时还给出了存在非负解及均匀箭柄解的充要条件.利用该问题解决了逆特征值问题:求一个对称箭头矩阵,使它的各阶顺序主子阵具有给定的最小最大特征值.并给出该逆特征值问题解的计算方法.数值计算表明,该算法更有效.     

Hybrid shape parameter strategy for the RBF approximation of vibrating systems

Shape parameters play an important role in radial basis function (RBF) approximations. Therefore, the choice of them is an active field in numerical analysis research. In this paper, first we review the available strategies in the literature for selecting shape parameters. Then, we introduce an alternative approach called hybrid strategy for scaling the RBFs. This strategy is constructed based on the advantages of the older ones and discards their disadvantages. The efficiency of the new strategy is demonstrated by comparing the effects of different strategies on approximating the eigenvalues of ordinary and partial differential equations with different boundary conditions.