Nonlinear eigenvalue problem for second-order Hamiltonian systems

The nonlinear self-adjoint eigenvalue problem for a Hamiltonian system of two ordinary differential equations is examined under the assumption that the matrix of the system is a monotone function of the spectral parameter. Certain properties of eigenvalues that were previously established by the authors for Hamitonian systems of arbitrary order are now worked out in detail and made more precise for the above system. In particular, a single second-order ordinary differential equation is analyzed.     

Laplace特征值的一点注记

确定具有n个顶点e条边的图的Laplace的最大谱半径.     

Comparison theorems for self‐adjoint linear Hamiltonian eigenvalue problems

In this work we derive new comparison results for (finite) eigenvalues of two self‐adjoint linear Hamiltonian eigenvalue problems. The coefficient matrices depend on the spectral parameter nonlinearly and the spectral parameter is present also in the boundary conditions. We do not impose any controllability or strict normality assumptions. Our method is based on a generalization of the Sturmian comparison theorem for such systems. The results are new even for the Dirichlet boundary conditions, for linear Hamiltonian systems depending linearly on the spectral parameter, and for Sturm–Liouville eigenvalue problems with nonlinear dependence on the spectral parameter.     

A method for solving nonlinear spectral problems for a class of systems of differential algebraic equations

A method for calculating eigenvalues of a nonlinear spectral problem for one class of linear differential algebraic equations is proposed under the assumption of an analytical dependence on spectral parameter of the matrices appearing in the system of equations and the matrices determining boundary conditions.     

Acceleration of Convergence of Static and Dynamic Iterations

A new technique for acceleration of convergence of static and dynamic iterations for systems of linear equations and systems of linear differential equations is proposed. This technique is based on splitting the matrix of the system in such a way that the resulting iteration matrix has a minimal spectral radius for linear systems and a minimal spectral radius for some value of a parameter in Laplace transform domain for linear differential systems.This revised version was published online in October 2005 with corrections to the Cover Date.     

具有中心对称结构的非负张量的谱半径(英文)

The matrix with centrosymmetric structure is an important kind of structured matrices with many applications in various physics and engineering problems.The eigenvalue problem for a centrosymmetric matrix can be reduced to the other eigenvalue problems of lower order.We introduce the centrosymmetric structure to the tensor field and focus on the spectral radius of this particular tensor.Then we show that properties of centrosymmetric matrices hold true for tensors situation.     

由特征值唯一确定的实3×3Hankel矩阵(英文)

本文研究了由特征值唯一确定的3×3实Hankel矩阵.借助于M.Fielder[1]的结论并经过细致的讨论,得到3×3实Hankel矩阵由其特征值唯一确定的充分必要条件,刻画了3×3实Hankel矩阵的一种特征值性质.     

On optimality of two adaptive choices for the parameter of Dai–Liao method

Orthonormal matrices are a class of well-conditioned matrices with the least spectral condition number. Here, at first it is shown that a recently proposed choice for parameter of the Dai–Liao nonlinear conjugate gradient method makes the search direction matrix as close as possible to an orthonormal matrix in the Frobenius norm. Then, conducting a brief singular value analysis, it is shown that another recently proposed choice for the Dai–Liao parameter improves spectral condition number of the search direction matrix. Thus, theoretical justifications of the two choices for the Dai–Liao parameter are enhanced. Finally, some comparative numerical results are reported.     

基于信息压缩矩阵特征值映射的时变系统UD分解辨识算法

在系统辨识领域遗忘因子UD分解算法(一种通过对系统数据矩阵进行UD分解的在线辨识算法)具有对时变系统阶次和参数同步估计的优异性能,但传统的遗忘策略不能从根本上解决信息压缩矩阵数据过饱和问题,为了拓展现有UD分解算法在时变系统的适用范围,同时针对数据空间分布不均匀性,提出一种基于信息压缩矩阵特征值映射的UD分解辨识算法.从理论上分析辨识算法跟踪能力与参数估计矩阵有界性的对应关系,从而构造出一种基于信息压缩矩阵特征值映射的有界函数,特征值映射函数能够根据系统数据传递过程中信息量的大小动态调整遗忘因子,解决了参数辨识过程中数据过饱和及数据分布不均匀问题.仿真结果表明,相比于常规时变遗忘因子策略,带有特征值映射的UD分解算法能够更加准确跟踪系统参数的变化,且能够保证系统不是2N阶持续激励信号的情况下,也能对时变系统参数进行跟踪.     

Determination of the number of an eigenvalue of a singular nonlinear self-adjoint spectral problem for a linear Hamiltonian system of differential equations

We suggest a method for determining the number of an eigenvalue of a self-adjoint spectral problem nonlinear with respect to the spectral parameter, for some class of Hamiltonian systems of ordinary differential equations on the half-line. The standard boundary conditions are posed at zero, and the solution boundedness condition is posed at infinity. We assume that the matrix of the system is monotone with respect to the spectral parameter. The number of an eigenvalue is determined by the properties of the corresponding nontrivially solvable homogeneous boundary value problem. For the considered class of systems, it becomes possible to compute the numbers of eigenvalues lying in a given range of the spectral parameter without finding the eigenvalues themselves.     

扰动的Couette－Poiseuille流的特征值的计算

本文考虑的问题是二维粘性渠流。对0到2000之间的雷诺数,计算了平稳扰动的Couette-Poiseuille流的下游特征值,其特征方程类似于Orr-Sommerfeld方程。所用的方法是谱方法和初值方法(复合矩阵方法).就几种有趣的流量,给出了相应的特征值的计算结果。这些特征值确定了扰动的衰减率。     

Matrix Riccati equations and matrix Sturm-Liouville problems

The asymptotic behavior of determinants of unitary solutions of matrix Riccati differential equations containing a large parameter is determined. The result leads to theorems on existence and asymptotic distribution of eigenvalues of indefinite matrix Sturm-Liouville problems.     

Oscillation theorems for symplectic difference systems

We consider symplectic difference systems involving a spectral parameter, together with the Dirichlet boundary conditions. The main result of the paper is a discrete version of the so-called oscillation theorem which relates the number of finite eigenvalues less than a given number to the number of focal points of the principal solution of the symplectic system. In two recent papers the same problem was treated and an essential ingredient was to establish the concept of the multiplicity of a focal point. But there was still a rather restrictive condition needed, which is eliminated here by using the concept of finite eigenvalues (or zeros) from the theory of matrix pencils.     

Boundary-value problem for a class of second-order parameter-dependent dynamic equations on a time scale

In this study, we consider a boundary value problem generated by a second-order dynamic equation on a time scale and boundary conditions depending on the spectral parameter. We give some properties of the solutions and obtain a formulation of the number of eigenvalues of the problem.     

HSS METHOD WITH A COMPLEX PARAMETER FOR THE SOLUTION OF COMPLEX LINEAR SYSTEM

In this paper, a complex parameter is employed in the Hermitian and skew-Hermitian splitting (HSS) method (Bai, Golub and Ng: SIAM J. Matrix Anal. Appl., 24(2003), 603-626) for solving the complex linear system $Ax=f$. The convergence of the resulting method is proved when the spectrum of the matrix $A$ lie in the right upper (or lower) part of the complex plane. We also derive an upper bound of the spectral radius of the HSS iteration matrix, and an estimated optimal parameter $α$(denoted by $α_{est}$) of this upper bound is presented. Numerical experiments on two modified model problems show that the HSS method with $α_{est}$has a smaller spectral radius than that with the real parameter which minimizes the corresponding upper bound. In particular, for the 'dominant' imaginary part of the matrix $A$, this improvement is considerable. We also test the GMRES method preconditioned by the HSS preconditioning matrix with our parameter $α_{est}$.     

二次自伴矩阵多项式阵特征值界的数值计算

在参数不确定性线性系统的鲁棒控制研究中,常用到的一个指标就是使不确定性系统在输出反馈或状态反馈控制下的闭环系统在H∞-范数界γ的条件下的二次稳定.是否二次稳定,一般要验证能否找到一个正常数,ε使相应的R iccati方程有正定解.而R iccati方程一般情况下求解相当困难.本文通过具体的分析,提出了一种在给定正定矩阵的条件下,找使此正定阵是R iccati方程的解相对应的正常数ε的可能范围的方法,即求解二次自伴矩阵多项式阵特征值界的方法.文中详细给出了所用理论及算法.给出了求正常数ε范围的一个实例.     

广义特征值极小扰动问题的一类黎曼共轭梯度法

研究含参数$l$非方矩阵对广义特征值极小扰动问题所导出的一类复乘积流形约束矩阵最小二乘问题.与已有工作不同,本文直接针对复问题模型,结合复乘积流形的几何性质和欧式空间上的改进Fletcher-Reeves共轭梯度法,设计一类适用于问题模型的黎曼非线性共轭梯度求解算法,并给出全局收敛性分析.数值实验和数值比较表明该算法比参数$l=1$的已有算法收敛速度更快,与参数$l=n$的已有算法能得到相同精度的解.与部分其它流形优化相比与已有的黎曼Dai非线性共轭梯度法具有相当的迭代效率,与黎曼二阶算法相比单步迭代成本较低、总体迭代时间较少,与部分非流形优化算法相比在迭代效率上有明显优势.     

Inverse spectral problems for differential pencils with boundary conditions dependent on the spectral parameter

In this paper, we discuss two inverse problems for differential pencils with boundary conditions dependent on the spectral parameter. We will prove the Hochstadt–Lieberman type theorem of 1 – 3 except for arbitrary one eigenvalue and the Borg type theorem of 1 – 3 except for at most arbitrary two eigenvalues, respectively. The new results are generalizations of the related results. Copyright © 2016 John Wiley & Sons, Ltd.     

Numerical solution of vector Sturm–Liouville problems with Dirichlet conditions and nonlinear dependence on the spectral parameter

A numerical-analytical iterative method is proposed for solving generalized self-adjoint regular vector Sturm–Liouville problems with Dirichlet boundary conditions. The method is based on eigenvalue (spectral) correction. The matrix coefficients of the equations are assumed to be nonlinear functions of the spectral parameter. For a relatively close initial approximation, the method is shown to have second-order convergence with respect to a small parameter. Test examples are considered, and the model problem of transverse vibrations of a hinged rod with a variable cross section is solved taking into account its rotational inertia.