Generalized Inverse Eigenvalue Problem for Centrohermitian Matrices

In this paper we first consider the existence and the general form of solution to the following generalized inverse eigenvalue problem(GIEP): given a set of n-dimension complex vectors {x j}m j=1 and a set of complex numbers {λ j}m j=1, find two n×n centrohermitian matrices A,B such that {x j}m j=1 and {λ j}m j=1 are the generalized eigenvectors and generalized eigenvalues of Ax=λBx, respectively. We then discuss the optimal approximation problem for the GIEP. More concretely, given two arbitrary matrices, , ∈C n×n, we find two matrices A and B such that the matrix (A*,B*) is closest to (,) in the Frobenius norm, where the matrix (A*,B*) is the solution to the GIEP. We show that the expression of the solution of the optimal approximation is unique and derive the expression for it.

Groundstate with Zero Eigenvalue for Generalized Sombrero-Shaped Potential in N-Dimensional Space

Based on an iterative method for solving the groundstate of Schrodinger equation, it is found that a kind of generalized Sombrero-shaped potentials in N-dimensional space has groundstates with zero eigenvalue. The restrictions on the parameters in the potential are discussed.

Solvability conditions for algebra inverse eigenvalue problem over set of anti-Hermitian generalized anti-Hamiltonian matrices

By using the characteristic properties of the anti-Hermitian generalized anti-Hamiltonian matrices, we prove some necessary and sufficient conditions of the solvability for algebra inverse eigenvalue problem of anti-Hermitian generalized anti-Hamiltonian matrices, and obtain a general expression of the solution to this problem. By using the properties of the orthogonal projection matrix, we also obtain the expression of the solution to optimal approximate problem of an n× n complex matrix under spectral restriction.

On the eigenvalues and eigendisplacement of the critical mode in horizontally layered media

Wave propagation in horizontally layered media is a classical problem in seismic-wave theory.In semi-infinite space,a nondispersive Rayleigh wave mode exists,and the eigendisplacement decays exponentially with depth.In a layered model with increasing layer velocity,the phase velocity of the Rayleigh wave varies between the S-wave velocity of the bottom half-space and that of the classical Rayleigh wave propagated in a supposed half-space formed by the parameters of the top layer.If the phase velocity is the same as the P-or S-wave velocity of the layer,which is called the critical mode or critical phase velocity of surface waves,the general solution of the wave equation is not a homogeneous(expressed by trigonometric functions)or inhomogeneous(expressed by exponential functions)plane wave,but one whose amplitude changes linearly with depth(expressed by a linear function).Theories based on a general solution containing only trigonometric or exponential functions do not apply to the critical mode,owing to the singularity at the critical phase velocity.In this study,based on the classical framework of generalized reflection and transmission coefficients,the propagation of surface waves in horizontally layered media was studied by introducing a solution for the linear function at the critical phase velocity.Therefore,the eigenvalues and eigenfunctions of the critical mode can be calculated by solving a singular problem.The eigendisplacement characteristics associated with the critical phase velocity were investigated for different layered models.In contrast to the normal mode,the eigendisplacement associated with the critical phase velocity exhibits different characteristics.If the phase velocity is equal to the S-wave velocity in the bottom half-space,the eigendisplacement remains constant with increasing depth.

Band structure calculation of scalar waves in two-dimensional phononic crystals based on generalized multipole technique

A multiple monopole （or multipole） method based on the generalized mul- tipole technique （GMT） is proposed to calculate the band structures of scalar waves in two-dimensional phononic crystals which are composed of arbitrarily shaped cylinders embedded in a host medium. In order to find the eigenvalues of the problem, besides the sources used to expand the wave field, an extra monopole source is introduced which acts as the external excitation. By varying the frequency of the excitation, the eigenvalues can be localized as the extreme points of an appropriately chosen function. By sweeping the frequency range of interest and sweeping the boundary of the irreducible first Brillouin zone, the band structure is obtained. Some numerical examples are presented to validate the proposed method.

Bound State Solutions of Schrodinger Equation for Generalized Morse Potential with Position-Dependent Mass

The effective mass one-dimensional Schroedinger equation for the generalized Morse potential is solved by using Nikiforov-Uvarov method. Energy eigenvalues and corresponding eigenfunctions are computed analytically. The results are also reduced to the constant mass case. Energy eigenvalues are computed numerically for some diatomic molecules. They are in agreement with the ones obtained before.

A MODIFIED INVERSE ITERATION FOR A LARGE SPARSE SPD GENERALIZED EIGENPROBLEM

In this paper, an algorithm based on a shifted inverse power iteration for computing generalized eigenvalues with corresponding eigenvectors of a large scale sparse symmetric positive definite matrix pencil is presented. It converges globally with a cubic asymptotic convergence rate, preserves sparsity of the original matrices and is fully parallelizable. The algebraic multilevel itera-tion method (AMLI) is used to improve the efficiency when symmetric positive definite linear equa-tions need to be solved.

Asymptotic Approximation of the Eigenvalues and the Eigenfunctions for the Orr-Sommerfeld Equation on Infinite Intervals

Asymptotic eigenvalues and eigenfunctions for the Orr-Sommerfeld equation in two-dimensional and three-dimensional incompressible flows on an infinite domain and on a semi-infinite domain are obtained. Two configurations are considered, one in which a short-wave limit approximation is used, and another in which a long-wave limit approximation is used. In the short-wave limit, Wentzel-Kramers-Brillouin (WKB) methods are utilized to estimate the eigenvalues, and the eigenfunctions are approximated in terms of Green’s functions. The procedure consists of transforming the Orr-Sommerfeld equation into a system of two second order ordinary differential equations for which the eigenvalues and the eigenfunctions can be approximated. In the long-wave limit approximation, solutions are expressed in terms of generalized hypergeometric functions. Our procedure works regardless of the values of the Reynolds number.

Generalized Irreducible α-Matrices and Its Applications

The class of generalized α-matrices is presented by Cvetkovi?, L. (2006), and proved to be a subclass of H-matrices. In this paper, we present a new class of matrices-generalized irreducible α-matrices, and prove that a generalized irreducible α-matrix is an H-matrix. Furthermore, using the generalized arithmetic-geometric mean inequality, we obtain two new classes of H-matrices. As applications of the obtained results, three regions including all the eigenvalues of a matrix are given.

Relativistic Treatment of Massive Klein-Gordon Particle by Modified Generalized Hulthen Potential

Approximate bound state solutions of spinless particles with a special case of equal scalar and vector modified generalized Hulthen potential has been obtained under the massive Klein-Gordon equation. The energy eigenvalues and the corresponding wave functions expressed in terms of a Jacobi polynomial are also obtained using the parametric generalization of the Nikiforov-Uvarov （NU） method. Under limiting cases our result are in agreement with the existing literature. Our results could be used to study the interactions and binding energies of the central potential for diatomic molecules in the relativistic framework which have many applications in physics and some others related disciplines.

A direct-variance-analysis method for generalized stochastic eigenvalue problem based on matrix perturbation theory

It has been extensively recognized that the engineering structures are becoming increasingly precise and complex,which makes the requirements of design and analysis more and more rigorous.Therefore the uncertainty effects are indispensable during the process of product development.Besides,iterative calculations,which are usually unaffordable in calculative efforts,are unavoidable if we want to achieve the best design.Taking uncertainty effects into consideration,matrix perturbation methodpermits quick sensitivity analysis and structural dynamic re-analysis,it can also overcome the difficulties in computational costs.Owing to the situations above,matrix perturbation method has been investigated by researchers worldwide recently.However,in the existing matrix perturbation methods,correlation coefficient matrix of random structural parameters,which is barely achievable in engineering practice,has to be given or to be assumed during the computational process.This has become the bottleneck of application for matrix perturbation method.In this paper,we aim to develop an executable approach,which contributes to the application of matrix perturbation method.In the present research,the first-order perturbation of structural vibration eigenvalues and eigenvectors is derived on the basis of the matrix perturbation theory when structural parameters such as stiffness and mass have changed.Combining the first-order perturbation of structural vibration eigenvalues and eigenvectors with the probability theory,the variance of structural random eigenvalue is derived from the perturbation of stiffness matrix,the perturbation of mass matrix and the eigenvector of baseline-structure directly.Hence the Direct-VarianceAnalysis(DVA)method is developed to assess the variation range of the structural random eigenvalues without correlation coefficient matrix being involved.The feasibility of the DVA method is verified with two numerical examples(one is trusssystem and the other is wing structure of MA700 commercial aircraft),in which the DVA method also shows superiority in computational efficiency when compared to the Monte-Carlo method.

Generalized Inverse Eigenvalue Problem for (P,Q)-Conjugate Matrices and the Associated Approximation Problem

In this paper,the generalized inverse eigenvalue problem for the（P,Q）-conjugate matrices and the associated approximation problem are discussed by using generalized singular value decomposition（GSVD）.Moreover,the least residual problem of the above generalized inverse eigenvalue problem is studied by using the canonical correlation decomposition（CCD）.The solutions to these problems are derived.Some numerical examples are given to illustrate the main results.

A Note on the Hoffman-Wielandt Theorem for Generalized Eigenvalue Problems

Let {A, B} and {C, D} be diagonalizable pairs of order n, i.e., there exist invertible matrices P, Q and X, Ysuchthat A = P∧Q, B = PΩQ, C =XГY, D= X△Y, where∧ = diag（α1, α2, …, αn）, Ω= diag（βl, β2, …βn）,Г=diag（γ1,γ2,…,γn）, △=diag（δl,δ2,…,δn）.Let ρ（（α,β）, （γ,δ））=｜αδ-βγ｜/√｜α｜^2＋｜β｜^2√｜γ｜^2＋｜δ｜^2.In this paper, it will be proved that there is a permutation τ of {1,2,... ,n} such thatn∑i=1[ρ（（αi,βi）,（γτ（i）,δτ（i）））]^2≤n[1-1/κ^2（Y）κ^2（Q）（1-d2F（Z,W）/n）],where κ（Y） = ｜｜Y｜｜2｜｜Y^-1｜｜2,Z= （A,B）,W= （C, D） and dF（Z,W） = 1/√2｜｜Pz＊ -Pw＊｜｜F.

STURM SEQUENCE METHOD FOR GENERALIZED EIGENVALUE PROBLEM Ax=λBx

The generalized eigenvalue problem Ax=λBx, (1.1)

AN ADAPTIVE TRUST-REGION METHOD FOR GENERALIZED EIGENVALUES OF SYMMETRIC TENSORS

For symmetric tensors,computing generalized eigenvalues is equivalent to a homogenous polynomial optimization over the unit sphere.In this paper,we present an adaptive trustregion method for generalized eigenvalues of symmetric tensors.One of the features is that the trust-region radius is automatically updated by the adaptive technique to improve the algorithm performance.The other one is that a projection scheme is used to ensure the feasibility of all iteratives.Global convergence and local quadratic convergence of our algorithm are established,respectively.The preliminary numerical results show the efficiency of the proposed algorithm.

含虚拟同步机光伏并网电力系统小干扰稳定约束最优潮流算法

为改善含虚拟同步机(VSG)光伏(PV)并网电力系统小干扰稳定性,根据特征值及其阻尼比,筛选电力系统弱阻尼振荡模式和邻近失稳模式。建立上述危险模式对PV-VSG控制参数灵敏度,进而以潮流雅克比矩阵为桥梁,提出危险模式对同步发电机(SG)出力灵敏度的解析表达。以控制参数和SG出力调整量最小为目标函数,以危险模式阈值为约束,提出PV-VSG并网电力系统稳定约束最优潮流(SC-OPF)算法。算例结果分析了对危险模式的优化控制效果,验证了所提算法的有效性。

基于复杂度追踪的模态参数识别方法对比研究

复杂度追踪(complexity pursuit, CP)是求解振动信号盲源分离(blind source separation, BSS)问题的一类经典方法。用复杂度追踪估计解混矩阵主要有基于源信号复杂度计算的梯度下降(complexity pursuit-gradient descent, CP-GD)算法和基于时间可预测度的广义特征值分解(temporal predictability-generalized eigenvalue decomposition, TP-GED)算法。当前,这两种算法的关联性与算法性能尚缺乏研究,因此对这两种算法的等价性和计算性能进行了研究。首先,给出CP-GD和TP-GED两种算法的具体理论及算法流程;其次,利用二、三自由度振动系统直观地展示并对比解混向量对应的源信号复杂度及可预测度的变化规律;最后,通过对多工况下多自由度系统的模态参数识别算例,对比研究两种算法的精度及计算量。研究结果表明:在低阻尼比及高信噪比条件下,两种方法得到的解混矩阵是相同的;考虑到计算信号复杂度和梯度下降较为耗时,CP-GD算法计算代价要高于TP-GED算法。

电压控制型虚拟同步发电机的高频谐振特性分析与抑制策略

虚拟同步发电机(VSG)模拟同步发电机的运行机制,引入了有功调频和无功调压控制环节,与传统电流控制型变流器不同,是一种电压控制型策略。文中立足电压控制型虚拟同步发电机控制结构和网源协调特性,构建小信号分析模型,分析在不同控制参数与电网参数场景下的振荡模态,进而掌握电压控制型虚拟同步发电机并网系统的高频谐振特性,并针对性提出高频谐振抑制策略。首先,分别建立电压控制型虚拟同步发电机、传统变流器各控制和电气环节的小信号分析模型,通过特征根分析方法,定量分析所有特征根的振荡模态和阻尼比,发现电压控制型虚拟同步发电机存在与传统变流器类似的高频振荡模态;同时分析虚拟同步发电机控制参数变化、电网强弱场景对其高频谐振阻尼作用机理。其次,基于电压控制型虚拟同步发电机控制策略,以有源阻尼方法为解决思路,提出基于电流内环前端通道引入虚拟阻抗控制策略的谐振抑制方法,抑制其高频谐振。最后,利用Matlab/Simulink仿真和RT-LAB控制器在环实验结果验证所提出的控制策略,结果表明虚拟阻抗控制环节起到有效正阻尼作用,相比于无源阻尼方法,在无需额外增加测点和外部控制量的情况下,电压控制型VSG网侧阻尼特性改善,暂态响应能力增强,高频失稳现象得到有效抑制。

基于SDW-MMSE的广义特征值稳健波束形成方法

最大输出信噪比(Signal-to-noise ratio,SNR)准则下,广义特征值(Generalized eigenvalue,GEV)波束形成存在复系数难以控制的问题,在复杂的声学环境中容易导致输出信号严重失真。针对复系数估计问题,本文提出一种基于最小均方误差(Minimum mean square error,MMSE)的复系数估计方法,并通过引入语音失真权重因子(Speech distortion weight,SDW),调节降噪效果和语音失真之间的权重关系,进而提出了基于SDW-MMSE的广义特征值稳健波束形成方法。通过最大似然法估计目标信号和噪音信号的功率谱,进而求解主广义特征向量。进一步基于SDW-MMSE估计复系数,将复系数与主广义特征向量相结合,从而得到基于SDW-MMSE的广义特征值稳健波束形成滤波向量。仿真实验结果表明,本文提出的波束形成方法可有效消除相干噪声和非相干噪声,具有输出信噪比高、语音失真少等稳健性能。

基于双边Lanczos算法的声子晶体复模态重分析

针对声子晶体拓扑结构修改时模态计算效率低的问题,提出基于双边Lanczos算法的复模态重分析。与全分析不同,本方法利用声子晶体原始结构的模态分析结果,通过双边Lanczos算法构建投影向量矩阵,将广义特征值方程映射进子空间来压缩矩阵规模,求解方程后再利用近似模态关系得到最终解。通过对尺寸和形状发生修改的声子晶体进行分析,验证了方法所求结果具有高精度,与全分析相比缩减了约35%的计算时间,在处理拓扑修改变化量大和计算规模大的声子晶体模态分析问题上有很大潜力。