连续时间LQ控制主要本征对的算法

本文首先提出了离散时间LQ控制的本征值方程当△t→0时怎样退化成为连续时间LQ控制的本征值方程.在建立了分离出的n阶连续时间的本征值方程,并保证了其本征值必定都在左半平面后,本文提出计算其最靠近于虚轴的若干个本征对,可以通过A_e=eA的矩阵变换.A_e的本征值全在单位圆之内.本征向量不变,至于本征值则只要做一次对数运算就可以求得原阵的本征值.A_e阵的最接近于单位圆的若干个本征对的算法,可以通过共轭子空间迭代解解决之.     

Numerical Determination of a Canonical Form of a Symplectic Matrix

We propose an algorithm that transforms a real symplectic matrix with a stable structure to a block diagonal form composed of three main blocks. The two extreme blocks of the same size are associated respectively with the eigenvalues outside and inside the unit circle. Moreover, these eigenvalues are symmetric with respect to the unit circle. The central block is in turn composed of several diagonal blocks whose eigenvalues are on the unit circle and satisfy a modification of the Krein-Gelfand-Lidskii criterion. The proposed algorithm also gives a qualitative criterion for structural stability.     

A structure-preserving doubling algorithm for quadratic eigenvalue problems arising from time-delay systems

We propose a structure-preserving doubling algorithm for a quadratic eigenvalue problem arising from the stability analysis of time-delay systems. We are particularly interested in the eigenvalues on the unit circle, which are difficult to estimate. The convergence and backward error of the algorithm are analyzed and three numerical examples are presented. Our experience shows that our algorithm is efficient in comparison to the few existing approaches for small to medium size problems.     

A spectral trichotomy method for symplectic matrices

This paper presents an algorithm for the numerical approximation of spectral projectors onto the invariant subspaces corresponding to the eigenvalues inside, on, and outside the unit circle of a symplectic matrix. The algorithm constructs iteratively three matrix sequences from which the projectors are obtained. The convergence depends essentially on the gap between the unit circle and the eigenvalues inside it. A larger gap leads to faster convergence. Theoretical and algorithmic aspects of the algorithm are developed. Numerical results are reported.     

A matricial computation of rational quadrature formulas on the unit circle

A matricial computation of quadrature formulas for orthogonal rational functions on the unit circle, is presented in this paper. The nodes of these quadrature formulas are the zeros of the para-orthogonal rational functions with poles in the exterior of the unit circle and the weights are given by the corresponding Christoffel numbers. We show how these nodes can be obtained as the eigenvalues of the operator Möbius transformations of Hessenberg matrices and also as the eigenvalues of the operator Möbius transformations of five-diagonal matrices, recently obtained. We illustrate the preceding results with some numerical examples.     

Circular trichotomy of the spectrum of regular matrix pencils

We introduce a method for approximating the right and left deflating subspaces of a regular matrix pencil corresponding to the eigenvalues inside, on and outside the unit circle. The method extends the iteration used in the context of spectral dichotomy, where the assumption on the absence of eigenvalues on the unit circle is removed. It constructs two matrix sequences whose null spaces and the null space of their sum lead to approximations of the deflating subspaces corresponding to the eigenvalues of modulus less than or equal to 1, equal to 1 and larger than or equal to 1. An orthogonalization process is then used to extract the desired delating subspaces. The resulting algorithm is an inverse free, easy to implement, and sufficiently fast. The derived convergence estimates reveal the key parameters, which determine the rate of convergence. The method is tested on several numerical examples.     

用矩阵符号函数解(广义)周期Sylvester方程

(广义)周期Sylvester方程来源于周期离散线性系统. 本文主要研究这类方程满足特征值分别位于开左半复平面和开右半复平面或位于单位圆周内和单位圆周外条件时用矩阵符号函数求解的数值方法.并通过数值例子说明我们的结论.     

关于一类无限维线性算子谱的刻画

利用有限维线性空间的理论,研究了一类无限阶Toeplitz矩阵的特征值问题,得到这类无限阶矩阵的特征值是连续变化的,并且其谱集合是由复平面上的单位圆盘{z∈C,|z|<1}被多项式函数f(z)=(?)a_iz~i作用后,所得到的像曲线内部的点组成.     

Robust D-stability analysis for linear discrete singular time-delay systems with structured and unstructured parameter uncertainties

In this paper, the robust D-stability problem (i.e. the robusteigenvalue-clustering in a specified circular region problem)of linear discrete singular time-delay systems with structured(elemental) and unstructured (norm-bounded) parameter uncertaintiesis investigated. Under the assumptions that the linear nominaldiscrete singular time-delay system is regular and impulse-free,and has all its finite eigenvalues lying inside a specifiedcircular region, a new sufficient condition is proposed to preservethe assumed properties when structured and unstructured parameteruncertainties are added into the linear nominal discrete singulartime-delay system. When all the finite eigenvalues are justrequired to locate inside the unit circle of the z-plane, theproposed criterion will become the stability robustness criterion.For the case that the linear discrete singular time-delay systemis only subject to structured parameter uncertainties, by anillustrative example, the presented sufficient condition isshown to be less conservative than the existing one reportedrecently in the literature.     

On manifolds of eigenfunctions and potentials generated by a family of periodic boundary-value problems

We consider a family of boundary-value problems with some potential as a parameter. We study the manifold of normalized eigenfunctions with even number of zeros in a period, and the manifold of potentials associated with double eigenvalues. In particular, we prove that the manifold of normalized eigenfunctions is a trivial fiber space over a unit circle and that the manifold of potentials with double eigenvalues is a homotopically trivial manifold trivially imbedded into the space of potentials.     

A criterion for the convergence of the Gauss-Seidel method

The solution of linear equations by iterative methods requires for convergence that the absolute magnitudes of all the eigenvalues of the iteration matrix should be less than unity. The test for convergence however is often difficult to apply because of the computation required. In this paper a method for determining the convergence of the Gauss-Seidel iteration is proposed. The method involves the numerical integration of initial value differential equations in the complex plane around the unit circle. The Gauss-Seidel method converges if the number of roots inside the unit circle is equal to the order of the iteration matrix.     

Numerical methods for palindromic eigenvalue problems: Computing the anti‐triangular Schur form

We present structure‐preserving numerical methods for the eigenvalue problem of complex palindromic pencils. Such problems arise in control theory, as well as from palindromic linearizations of higher degree palindromic matrix polynomials. A key ingredient of these methods is the development of an appropriate condensed form—the anti‐triangular Schur form. Ill‐conditioned problems with eigenvalues near the unit circle, in particular near ±1, are discussed. We show how a combination of unstructured methods followed by a structured refinement can be used to solve such problems accurately. Copyright © 2008 John Wiley & Sons, Ltd.     

Inverse scattering transform for the two-component Gerdjikov–Ivanov equation with nonzero boundary conditions: Dark-dark solitons

The two-component Gerdjikov–Ivanov equation with nonzero boundary conditions is studied by the inverse scattering transform. A fundamental set of analytic eigenfunctions is obtained with the aid of the associated adjoint problem. Three symmetry conditions are discussed to curb the scattering data. The behavior of the Jost functions and the scattering matrix at the branch points is discussed. The inverse scattering problem is formulated by a matrix Riemann–Hilbert problem. The trace formula in terms of the scattering data and the so-called asymptotic phase difference for the potential are obtained. The solitons classification is described in detail. When the discrete eigenvalues lie on the circle, the dark-dark soliton is obtained for the first time in this work. And the discrete eigenvalues off the circle generate the dark-bright, bright-bright, breather-breather, M(-type)-W(-type) solitons, and their interactions.     

The characteristic polynomial of a random permutation matrix

We establish a central limit theorem for the logarithm of the characteristic polynomial of a random permutation matrix. We relate this result to a central limit theorem of Wieand for the counting function for the eigenvalues lying in some interval on the unit circle.     

On the linear representations of circle expanding maps

In this paper, it is proved that the two real analytic expanding endomorphisms of the unit circle are equivalent if and only if they have equal eigenvalues along corresponding cycles. A sufficient and necessary condition that a real analytic solution of Cvitanovic-Feigenbaum equation induces a real analytic expanding map is given. Supported by the NSFC.     

关于稳定矩阵分解定理的一个简单证明

一个矩阵称为稳定的,如果这个矩阵的特征值全包含在单位开圆盘内.利用Parker关于复方阵的分解定理给出了稳定矩阵分解定理的一个简单证明,并对奇异值全部严格小于1的矩阵给出了类似的结论.     

覆盖单位圆的序贯方法

本文应用数论方法解决一个实际中遇到的单位圆的覆盖问题。用单位圆上的均匀布点方法估计覆盖面积S的均值、方差及其分布函数,结果显示Beta分布可以较好的拟合S/π的分布。为了增大覆盖面积,推荐采用序贯方法安排随机圆,模拟结果显示序贯方法非常有效。本文还指出了序贯方法在加权单位圆的覆盖问题中效果也显著。     

On an eigenvalue for the Laplace operator in a disk with Dirichlet boundary condition on a small part of the boundary in a critical case

A boundary-value problem of finding eigenvalues is considered for the negative Laplace operator in a disk with Neumann boundary condition on almost all the circle except for a small arc of vanishing length, where the Dirichlet boundary condition is imposed. A complete asymptotic expansion with respect to a parameter (the length of the small arc) is constructed for an eigenvalue of this problem that converges to a double eigenvalue of the Neumann problem.     

On improperly posed Caucfay problems and their approximate solution

It is well known that, in general, the Cauchy problem for theLaplace equation does not allow a solution and therefore isill-posed in both the Hadamard and the Tikhonov senses. Thepresent work focuses on the question whether the problem hasany meaningful approximate solution for arbitrary boundary conditions.Firstly, it is shown that it is possible to construct an analyticfunction which assumes some prescribed value on part of theboundary of a simply-connected domain. This problem is thenshown to be equivalent to the Cauchy problem under consideration,the solution to which can thus be invariably approximated toany degree of accuracy on the unit circle centred at the originwhen both the potential and the flux are specified as square-integrablefunctions over half the unit circle boundary. The uniquenessof the exact solution to the problem is also established. Theseresults are actually true for any simply-connected domain whichcan be conformally mapped onto the unit circle so that the partof its boundary with prescribed potential and flux correspondsto one-half of the unit circle boundary. Finally, the feasibilityof a boundary element formulation for a generic type of ill-posedboundary value problems is briefly discussed.     

Eigenanalysis of some preconditioned Helmholtz problems

Summary. In this work we calculate the eigenvalues obtained by preconditioning the discrete Helmholtz operator with Sommerfeld-like boundary conditions on a rectilinear domain, by a related operator with boundary conditions that permit the use of fast solvers. The main innovation is that the eigenvalues for two and three-dimensional domains can be calculated exactly by solving a set of one-dimensional eigenvalue problems. This permits analysis of quite large problems. For grids fine enough to resolve the solution for a given wave number, preconditioning using Neumann boundary conditions yields eigenvalues that are uniformly bounded, located in the first quadrant, and outside the unit circle. In contrast, Dirichlet boundary conditions yield eigenvalues that approach zero as the product of wave number with the mesh size is decreased. These eigenvalue properties yield the first insight into the behavior of iterative methods such as GMRES applied to these preconditioned problems. Received March 24, 1998 / Revised version received September 28, 1998