Submatrix Constrained Inverse Eigenvalue Problem involving Generalised Centrohermitian Matrices in Vibrating Structural Model Correction

Generalised centrohermitian and skew-centrohermitian matrices arise in a variety of applications in different fields. Based on the vibrating structure equation $M$$\ddot{x}$+$(D+G)$$\dot{x}$+$Kx$=$f(t)$ where $M$, $D$, $G$, $K$ are given matrices with appropriate sizes and x is a column vector, we design a new vibrating structure mode. This mode can be discretised as the left and right inverse eigenvalue problem of a certain structured matrix. When the structured matrix is generalised centrohermitian, we discuss its left and right inverse eigenvalue problem with a submatrix constraint, and then get necessary and sufficient conditions such that the problem is solvable. A general representation of the solutions is presented, and an analytical expression for the solution of the optimal approximation problem in the Frobenius norm is obtained. Finally, the corresponding algorithm to compute the unique optimal approximate solution is presented, and we provide an illustrative numerical example.     

Pseudo-Tournament Matrices and Their Eigenvalues

A tournament matrix and its corresponding directed graph both arise as a record of the outcomes of a round robin competition. An $n×n$ complex matrix $A$ is called $h$-pseudo-tournament if there exists a complex or real nonzero column vector $h$ such that $A+A^*=hh^*−I$. This class of matrices is a generalisation of well-studied tournament-like matrices such as $h$-hypertournament matrices, generalised tournament matrices, tournament matrices, and elliptic matrices. We discuss the eigen-properties of an $h$-pseudo-tournament matrix, and obtain new results when the matrix specialises to one of these tournament-like matrices. Further, several results derived in previous articles prove to be corollaries of those reached here.     

Scaling,sensitivity and stability in the numerical solution of quadratic eigenvalue problems

The most common way of solving the quadratic eigenvalue problem (QEP) (λ² M + λD + K)x = 0 is to convert it into a linear problem (λX + Y)z = 0 of twice the dimension and solve the linear problem by the QZ algorithm or a Krylov method. In doing so, it is important to understand the influence of the linearization process on the accuracy and stability of the computed solution. We discuss these issues for three particular linearizations: the standard companion linearization and two linearizations that preserve symmetry in the problem. For illustration we employ a model QEP describing the motion of a beam simply supported at both ends and damped at the midpoint. We show that the above linearizations lead to poor numerical results for the beam problem, but that a two‐parameter scaling proposed by Fan, Lin and Van Dooren cures the instabilities. We also show that half of the eigenvalues of the beam QEP are pure imaginary and are eigenvalues of the undamped problem. Our analysis makes use of recently developed theory explaining the sensitivity and stability of linearizations, the main conclusions of which are summarized. As well as arguing that scaling should routinely be used, we give guidance on how to choose a linearization and illustrate the practical value of condition numbers and backward errors. Copyright © 2007 John Wiley & Sons, Ltd.     

A Subspace-Projected Approximate Matrix Method for Systems of Linear Equations

Given two $n×n$ matrices $A$ and $A_0$ and a sequence of subspaces$\{0\}=\mathscr{V}_0⊂···⊂\mathscr{V}_n=\mathbb{R}^n$with dim$(\mathscr{V}_k)=\mathscr{k}$, the $k$-th subspace-projected approximated matrix $A_k$ is defined as $A_k=A+Π_k(A_0−A)Π_k$, where $Π_k$ is the orthogonal projection on $\mathscr{V}_{k}^⊥$. Consequently, $A_{k}v=Av$ and $v^{∗}A_{k}=v^{∗}A$ for all $v∈\mathscr{V}_{k}$. Thus $(A_{k})^{n}_{k≥0}$ is a sequence of matrices that gradually changes from $A_0$ into $A_n=A$. In principle, the definition of $\mathscr{V}_{k+1}$may depend on properties of $A_k$,which can be exploited to try to force $A_{k+1}$ to be closer to $A$ in some specific sense. By choosing $A_0$ as a simple approximation of $A$, this turns the subspace-approximated matrices into interesting preconditioners for linear algebra problems involving $A$. In the context of eigenvalue problems, they appeared in this role in Shepard et al. (2001), resulting in their Subspace Projected Approximate Matrix method. In this article, we investigate their use in solving linear systems of equations $Ax=b$. In particular, we seek conditions under which the solutions $x_k$ of the approximate systems $A_kx_k=b$ are computable at low computational cost, so the efficiency of the corresponding method is competitive with existing methods such as the Conjugate Gradient and the Minimal Residual methods. We also consider how well the sequence $(x_k)_{k≥0}$ approximates $x$, by performing some illustrative numerical tests.     

A Padé approximate linearization algorithm for solving the quadratic eigenvalue problem with low‐rank damping

The low‐rank damping term appears commonly in quadratic eigenvalue problems arising from physical simulations. To exploit the low‐rank damping property, we propose a Padé approximate linearization (PAL) algorithm. The advantage of the PAL algorithm is that the dimension of the resulting linear eigenvalue problem is only n + ?m, which is generally substantially smaller than the dimension 2n of the linear eigenvalue problem produced by a direct linearization approach, where n is the dimension of the quadratic eigenvalue problem, and ? and m are the rank of the damping matrix and the order of a Padé approximant, respectively. Numerical examples show that by exploiting the low‐rank damping property, the PAL algorithm runs 33–47% faster than the direct linearization approach for solving modest size quadratic eigenvalue problems. Copyright © 2015 John Wiley & Sons, Ltd.     

A Problem of Ranking and Estimation With Poisson Processes

There are given k Poisson processes with parameters (rates of occurrence) λ₁, …, λ _k . Let λ₍₁₎ ≤ λ₍₂₎ ≤ … ≤ λ_(k) denote the ordered set of values of the parameters. A procedure is given for selecting the process corresponding to λ_(k) and estimating its parameter (λ_(k)). The given procedure controls the joint risk of improper selection and of large error in the estimate. Let θ > 1 and 0 < α, β < 1 be given numbers, and let δ denote the estimate of λ_(k). The joint probability that a correct selection is made and that |(δ/λ_(k)) ? 1| ≤ α is at least as large as β, for (λ_(k)/λ_(k?1)) ≥ θ. Two cases are considered, that is, when the processes are observed continuously in time, and when they are observed at successive intervals of time. Both the cases lead to the same theoretical results.     

Efficient computation of order and mode of corner singularities in 3D‐elasticity

A general numerical procedure is presented for the efficient computation of corner singularities, which appear in the case of non‐smooth domains in three‐dimensional linear elasticity. For obtaining the order and mode of singularity, a neighbourhood of the singular point is considered with only local boundary conditions. The weak formulation of the problem is approximated by a Galerkin–Petrov finite element method. A quadratic eigenvalue problem ( P +λ Q +λ² R ) u = 0 is obtained, with explicitly analytically defined matrices P , Q , R . Moreover, the three matrices are found to have optimal structure, so that P , R are symmetric and Q is skew symmetric, which can serve as an advantage in the following solution process. On this foundation a powerful iterative solution technique based on the Arnoldi method is submitted. For not too large systems this technique needs only one direct factorization of the banded matrix P for finding all eigenvalues in the interval ?e(λ)∈(?0.5,1.0) (no eigenpairs can be ‘lost’) as well as the corresponding eigenvectors, which is a great improvement in comparison with the normally used determinant method. For large systems a variant of the algorithm with an incomplete factorization of P is implemented to avoid the appearance of too much fill‐in. To illustrate the effectiveness of the present method several new numerical results are presented. In general, they show the dependence of the singular exponent on different geometrical parameters and the material properties. Copyright © 2001 John Wiley & Sons, Ltd.     

反自反矩阵的二次特征值反问题及其最佳逼近

二次特征值反问题是二次特征值问题的一个逆过程,在结构动力模型修正领域中应用非常广泛．本文由给定的部分特征值和特征向量,利用矩阵分块法、奇异值分解和Moore-Penrose广义逆,分析了二次特征值反问题反自反解的存在性,得出了解的一般表达式．然后讨论了任意给定矩阵在解集中最佳逼近解的存在性和唯一性．最后给出解的表达式和数值算法,由算例验证了结果的正确性．     

Calculation of eigenpair derivatives for asymmetric damped systems with distinct and repeated eigenvalues

An algorithm is derived for the computation of eigenpair derivatives of asymmetric quadratic eigenvalue problem with distinct and repeated eigenvalues. In the proposed method, the eigenvector derivatives of the damped systems are divided into a particular solution and a homogeneous solution. By introducing an additional normalization condition, we construct two extended systems of linear equations with nonsingular coefficient matrices to calculate the particular solution. The method is numerically stable, and the homogeneous solutions are computed by the second‐order derivatives of the eigenequations. Two numerical examples are used to illustrate the validity of the proposed method. Copyright © 2015 John Wiley & Sons, Ltd.     

On singularities in the solution of three‐dimensional Stokes flow and incompressible elasticity problems with corners

In this paper, a numerical procedure is presented for the computation of corner singularities in the solution of three‐dimensional Stokes flow and incompressible elasticity problems near corners of various shape. For obtaining the order and mode of singularity, a neighbourhood of the singular point is considered with only local boundary conditions. The weak formulation of this problem is approximated using a mixed u , p Galerkin–Petrov finite element method. Additionally, a separation of variables is used to reduce the dimension of the original problem. As a result, the quadratic eigenvalue problem ( P +λ Q +λ² R ) d = 0 is obtained, where the saddle‐point‐type matrices P , Q , R are defined explicitly. For a numerical solution of the algebraic eigenvalue problem an iterative technique based on the Arnoldi method in combination with an Uzawa‐like scheme is used. This technique needs only one direct matrix factorization as well as few matrix–vector products for finding all eigenvalues in the interval ??(λ) ∈ (?0.5, 1.0), as well as the corresponding eigenvectors. Some benchmark tests show that this technique is robust and very accurate. Problems from practical importance are also analysed, for instance the surface‐breaking crack in an incompressible elastic material and the three‐dimensional viscous flow of a Newtonian fluid past a trihedral corner. Copyright © 2004 John Wiley & Sons, Ltd.     

广义特征值问题与两线性流形之间夹角的计算

高维欧氏空间中的两线性流形的夹角可用带二次等式约束的二次规划(QP-QEC)刻画。这样的夹角计算在统计学和数据分析中有许多重要应用,比如,两组随机变量的典型相关分析和核典型相关分析。本文用KKT条件探讨了更一般的QP-QEC与其对应的一般特征值问题之间的关系。在此基础上,借助一般特征值问题的解法,给出了这种夹角的算法。     

关于用多项式的单调序列逼近

本文证明,如果f∈C[0,1]不是多项式,则有n次多项式Pn(x)与Qn(x)使得在[0,1]上Qn(x)相似文献   

关于Fuzzy综合评价模型的研究

讨论了Fuzzy模型P=A(×)B=(p1,p2,…,pn)的算法规则(×)的三种情形,即Pj=max1≤i≤n[min(αi,rij)],pj=∑n)/(i=1αirij,Pj=∑n)/(i=1min(αi,rij), 以及二种评定法则:(1)简单评定法:当pk=max1≤j≤mpj时定为k级.(2)复合评定法:当∑k-1)/(j=1pj≤∑m)/(j=kpj且∑m)/(j=k+1pj≤∑k-1)/(j=1pj时定为k级,当∑k-1)/(j=1pj≥∑m)/(j=kpj且∑k-2)/(j=1pj≤∑m)/(j=k-1pj时定为(k-1)级,当∑m)/(j=k+1pj≥∑k-1)/(j=1pj且∑k-1)/(j=1pj≤∑m)/(j=kpj时定为(k+1)级.当模糊复杂度rij具有一定实际含义下,可能引发误判的三个定理.在三个定理条件下,一次性判定及联合判定均判定为k级,但当模糊复杂度rij有一定实际含义时,会发生不应判定为k级的现象.同时给出了减少误判的措施.     

Effects of sensor locations on air-coupled surface wave transmission measurements across a surface-breaking crack

Previous studies show that the surface wave transmission (SWT) method is effective to determine the depth of a surface-breaking crack in solid materials. However, nearfield wave scattering caused by the crack affects the reliability and consistency of surface wave transmission measurements. Prior studies on near-field scattering have focused on the case where crack depth h is greater than wavelength λ of surface waves (i.e., h/λ > 1). Near-field scattering of surface waves remains not completely understood in the range of h/λ for the SWT method (i.e., 0 ≤ h/λ ≤ 1/3), where the transmission coefficient is sensitive to crack depth change and monotonically decreases with increasing h/λ. In this study, the authors thoroughly investigated the near-field scattering of surface waves caused by a surface-breaking crack using experimental tests and numerical simulations for 0 ≤ h/λ ≤ 1/3. First, the effects of sensor locations on surface wave transmission coefficients across a surface-breaking crack are studied experimentally. Data are collected from Plexiglas and concrete specimens using air-coupled sensors. As a result, the variation of transmission coefficients is expressed in terms of the normalized crack depth (h/λ) as well as the normalized sensor location (x/λ). The validity of finite element models is also verified by comparing experimental results with numerical simulations (finite element method). Second, a series of parametric studies is performed using the verified finite element model to obtain more complete understanding of near-field scattering of surface waves propagating in various solid materials with different mechanical properties and geometric conditions. Finally, a guideline for selecting appropriate sensor arrangements to reliably obtain the crack depth using the SWT method is suggested.     

Modeling of through-focus aerial image with aberration and imaginary mask edge effects in optical lithography simulation

Aerial image through focus in the presence of aberrations and electromagnetic edge effects modeled by adding ±π/2 phase at pattern edges is expanded by a quadratic equation with respect to focus. The quadratic equation is expressed by four coefficients that are adequately independent of both mask layout and the variations in the optical setting in projection printing, thus saving the computation cost of the quadratic fit for each individual layout edge position in a new mask pattern or variation from a nominal optical setting. The error of this method is less than 1% for any typical integrated circuit features. This accuracy holds when the defocus is less than one Rayleigh unit (0.5λ/NA(2), where λ is a wavelength and NA is the numerical aperture) and the root mean square of the existing aberration is less than 0.02λ, which encompasses current lithography practice. More importantly, the method is a foundation for future first-cut accurate algebraic imaging models that have sufficient speed for assessing the desired or undesired changes in the through-focus images of millions of features as the optical system conditions change. These optical system changes occur naturally across the image field, and aberration levels are even programmed in tuning modern tooling to compensate for electromagnetic mask edge effects.     

The penetration depth of HgBa2CuO4+δ with 0.07 ≤ δ ≤ 0.35

The magnetic penetration depth λ(T) of three HgBa₂CuO_4+δ samples with 0.16 < δ ≤ 0.27 has been determined from the reversible magnetization. The obtained λ follows a BCS-like correlation of 1/λ² ∝ 1?(T/T_c)² over whole measured temperature range in an underdoped sample with T_c ～ 90 K, but deviates significantly from similar fits in an overdoped sample with the same T_c and an optimum doped sample, whose 1/λ² 's depends on T nearly linearly below T_c/2. This asymmetry between the underdoped and overdoped samples suggests that the T-dependence of 1/λ² is affected by doping in a complicated way.     

热释电材料的一般解

利用Stroh公式,给出了热释电材料热弹性问题的一般解,此解适用于四对复共轭压电本征值和一对热本征值互不相等的情况.然后,讨论了压电本征值问题出现重根时退化热释电材料的通解,并给出相应通解的形式.当热本征值与一个或多个压电本征值相等时,给出了退化热释电材料热弹性问题特解的具体形式.最后,通过共线界面裂纹问题证明了退化热释电材料和非退化热释电材料一般解中的任意函数f_α(z_α)形式相同,只是其中的某些系数有所变化.     

Determining the refractive index and average thickness of AsSe semiconducting glass films from wavelength measurements only

The dispersive refractive index n(λ) and thickness d of chalcogenide glass thin films are usually calculated from measurements of both optical transmission and wavelength values. Many factors can influence the transmission values, leading to large errors in the values obtained for n(λ) and d. Anovel optical method is used to derive n(λ) and d for AsSe semiconducting glass thin films deposited by thermal evaporation in the spectral region where k(2) ? n(2), using only wavelength values. This entails obtaining two transmission spectra: one at normal incidence and another at oblique incidence. The procedure yields values for the refractive index and average thickness of thermally evaporated chalcogenide films to an accuracy better than 3%.     

一类双曲型积分微分问题有限元逼近的超收敛估计

本文研究双曲型积分微分方程的半离散有限元逼近格式的超收敛估计.基于一种新的初值近似,得到了有限元解与精确解的Ritz-Volterra投影的Ws,p(Ω)模的如下超收敛估计k＞1,s=0,2≤p≤∞时,超收敛1阶;k＞1,s=1,2≤p＜∞时,超收敛2阶;k＞1,s=1,p=∞时,几乎超收敛2阶;k=1,s=1,2≤p ≤∞时,超收敛1阶.