基于自协调函数的信道容量和最大熵的计算

信道容量和最大熵的计算是信息论中的经典问题.讨论了利用自协调函数理论计算信道容量,尤其是带约束的信道容量的方法.将最大熵的计算作为信道容量计算的特殊情况.作为应用,在证明了单位成本信道容量函数的单峰性的基础上,提出了其相应的多项式时间算法.     

Calculating the minimal/maximal eigenvalue of symmetric parameterized matrices using projection

In applications of linear algebra including nuclear physics and structural dynamics, there is a need to deal with uncertainty in the matrices. We focus on matrices that depend on a set of parameters ω and we are interested in the minimal eigenvalue of a matrix pencil ( A , B ) with A , B symmetric and B positive definite. If ω can be interpreted as the realization of random variables, one may be interested in statistical moments of the minimal eigenvalue. In order to obtain statistical moments, we need a fast evaluation of the eigenvalue as a function of ω . Because this is costly for large matrices, we are looking for a small parameterized eigenvalue problem whose minimal eigenvalue makes a small error with the minimal eigenvalue of the large eigenvalue problem. The advantage, in comparison with a global polynomial approximation (on which, e.g., the polynomial chaos approximation relies), is that we do not suffer from the possible nonsmoothness of the minimal eigenvalue. The small‐scale eigenvalue problem is obtained by projection of the large‐scale problem. Our main contribution is that, for constructing the subspace, we use multiple eigenvectors and derivatives of eigenvectors. We provide theoretical results and document numerical experiments regarding the beneficial effect of adding multiple eigenvectors and derivatives.     

Structured preconditioners for nonsingular matrices of block two-by-two structures

For the large sparse block two-by-two real nonsingular matrices, we establish a general framework of practical and efficient structured preconditioners through matrix transformation and matrix approximations. For the specific versions such as modified block Jacobi-type, modified block Gauss-Seidel-type, and modified block unsymmetric (symmetric) Gauss-Seidel-type preconditioners, we precisely describe their concrete expressions and deliberately analyze eigenvalue distributions and positive definiteness of the preconditioned matrices. Also, we show that when these structured preconditioners are employed to precondition the Krylov subspace methods such as GMRES and restarted GMRES, fast and effective iteration solvers can be obtained for the large sparse systems of linear equations with block two-by-two coefficient matrices. In particular, these structured preconditioners can lead to efficient and high-quality preconditioning matrices for some typical matrices from the real-world applications.

     

子空间上的对称正定及对称半正定阵的左右特征值反问题

文[1][2][3]中讨论AX＝B的对称阵逆特征值问题，文[4][5][6]中讨论了半正定阵的逆特征值问题。本文讨论了空间了子空间上的对称正定及对称半正定阵的左右特征值反问题，给出了解存在的充分条件及解的表达式。     

Asymptotic distributions for the elementary symmetric functions of two matrices under the assumption of linearity

The asymptotic distributions of the elementary symmetric functions (esf's) of the characteristic roots of a noncentral multivariate beta matrix and of the generalized correlation matrix (noncentral under the assumption of linearity) are derived.     

Extremal eigenvalue intervals of symmetric tridiagonal interval matrices

Computing the extremal eigenvalue bounds of interval matrices is non‐deterministic polynomial‐time (NP)‐hard. We investigate bounds on real eigenvalues of real symmetric tridiagonal interval matrices and prove that for a given real symmetric tridiagonal interval matrices, we can achieve its exact range of the smallest and largest eigenvalues just by computing extremal eigenvalues of four symmetric tridiagonal matrices.     

The inverse eigenvalue problem for symmetric quasi anti-bidiagonal matrices

In this paper we construct the symmetric quasi anti-bidiagonal matrix that its eigenvalues are given, and show that the problem is also equivalent to the inverse eigenvalue problem for a certain symmetric tridiagonal matrix which has the same eigenvalues. Not only elements of the tridiagonal matrix come from quasi anti-bidiagonal matrix, but also the places of elements exchange based on some conditions.     

Adjacency preserving functions on symmetric elements and linear preservers of non-zero decomposable symmetric tensors

Let A be a non-empty set and m be a positive integer. Let ≡ be the equivalence relation defined on A ^m such that (x ₁, …, x _m ) ≡ (y ₁, …, y _m ) if there exists a permutation σ on {1, …, m} such that y _σ(i) = x _i for all i. Let A ^(m) denote the set of all equivalence classes determined by ≡. Two elements X and Y in A ^(m) are said to be adjacent if (x ₁, …, x _m?1, a) ∈ X and (x ₁, …, x _m?1, b) ∈ Y for some x ₁, …, x _m?1 ∈ A and some distinct elements a, b ∈ A. We study the structure of functions from A ^(m) to B ⁽ⁿ⁾ that send adjacent elements to adjacent elements when A has at least n + 2 elements and its application to linear preservers of non-zero decomposable symmetric tensors.     

A characterization of the type of the Cauchy-Hua measure on real symmetric matrices

The Cauchy-Hua measure is the probability on the set ofn×n real symmetric matrices of densitys C _n/det(I + s ₂)_{(n + 1)/2} . For any probability measure on the set ofn×n real symmetric matrices, we define (if verifies an additional condition) the image of by a 2n×2n real symplectic matrix, and we introduce the type of . Then, the type of the Cauchy-Hua measure is characterized from its invariance by the symplectic group.     

A symmetry exploiting Lanczos method for symmetric Toeplitz matrices

Several methods for computing the smallest eigenvalues of a symmetric and positive definite Toeplitz matrix T have been studied in the literature. Most of them share the disadvantage that they do not reflect symmetry properties of the corresponding eigenvector. In this note we present a Lanczos method which approximates simultaneously the odd and the even spectrum of T at the same cost as the classical Lanczos approach.     

A property of eigenvalue bounds for a class of symmetric tridiagonal interval matrices

The eigenvalue bounds of interval matrices are often required in some mechanical and engineering fields. In this paper, we consider an interval eigenvalue problem with symmetric tridiagonal matrices. A theoretical result is obtained that under certain assumptions the upper and lower bounds of interval eigenvalues of the problem must be achieved just at some vertex matrices of the interval matrix. Then a sufficient condition is provided to guarantee the assumption to be satisfied. The conclusion is illustrated also by a numerical example. Copyright © 2010 John Wiley & Sons, Ltd.     

Graphical characterization of positive definite non symmetric quasi-Cartan matrices

It is known that each positive definite quasi-Cartan matrix $A$ is $\mathbb{Z}$-equivalent to a Cartan matrix $A_{\Delta }$ called Dynkin type of $A$, the matrix $A_{\Delta }$ is uniquely determined up to conjugation by permutation matrices. However, in most of the cases, it is not possible to determine the Dynkin type of a given connected quasi-Cartan matrix by simple inspection. In this paper, we give a graph theoretical characterization of non-symmetric connected quasi-Cartan matrices. For this purpose, a special assemblage of blocks is introduced. This result complements the approach proposed by Barot (1999, 2001), for ${\mathbb{A}}_n$, ${\mathbb{D}}_n$ and ${\mathbb{E}}_m$ with $m=6,7,8$.     

On self-concordant barrier functions for conic hulls and fractional programming

Given a self-concordant barrier function for a convex set , we determine a self-concordant barrier function for the conic hull of . As our main result, we derive an “optimal” barrier for based on the barrier function for . Important applications of this result include the conic reformulation of a convex problem, and the solution of fractional programs by interior-point methods. The problem of minimizing a convex-concave fraction over some convex set can be solved by applying an interior-point method directly to the original nonconvex problem, or by applying an interior-point method to an equivalent convex reformulation of the original problem. Our main result allows to analyze the second approach showing that the rate of convergence is of the same order in both cases.     

Hermite interpolation and its numerical differentiation formula involving symmetric functions

By utilizing symmetric functions,this paper presents explicit representations for Hermite interpolation and its numerical differentiation formula.And the corresponding error estimates are also provided.     

实对称带状矩阵逆特征值问题

研究了一类实对称带状矩阵逆特征值问题：给定三个互异实数λ,μ和v及三个非零实向量x,y和z,分别构造实对称五对角矩阵T和实对称九对角矩阵A,使其都具有特征对(λ,x),(μ,y)和(v,z)．给出了此类问题的两种提法,研究了问题的可解性以及存在惟一解的充分必要条件,最后给出了数值算法和数值例子．     

Bush‐type Hadamard matrices and symmetric designs

Abstact: A symmetric 2‐(100, 45, 20) design is constructed that admits a tactical decomposition into 10 point and block classes of size 10 such that every point is in either 0 or 5 blocks from a given block class, and every block contains either 0 or 5 points from a given point class. This design yields a Bush‐type Hadamard matrix of order 100 that leads to two new infinite classes of symmetric designs with parameters and where m is an arbitrary positive integer. Similarly, a Bush‐type Hadamard matrix of order 36 is constructed and used for the construction of an infinite family of designs with parameters and a second infinite family of designs with parameters where m is any positive integer. © 2000 John Wiley & Sons, Inc. J Combin Designs 9: 72–78, 2001     

线性流形上对称正交反对称矩阵反问题的最小二乘解

设P是n阶对称正交矩阵,如果n阶矩阵A满足AT=A和(PA)T=-PA,则称A为对称正交反对称矩阵,所有n阶对称正交反对称矩阵的全体记为SARnp.令S={A∈SARnp f(A)=‖AX-B‖=m in,X,B〗∈Rn×m本文讨论了下面两个问题问题Ⅰ给定C∈Rn×p,D∈Rp×p,求A∈S使得CTAC=D问题Ⅱ已知A~∈Rn×n,求A∧∈SE使得‖A~-A∧‖=m inA∈SE‖A~-A‖其中SE是问题Ⅰ的解集合.文中给出了问题Ⅰ有解的充要条件及其通解表达式.进而,指出了集合SE非空时,问题Ⅱ存在唯一解,并给出了解的表达式,从而得到了求解A∧的数值算法.     

On distinguishing trees by their chromatic symmetric functions

Let T be an unrooted tree. The chromatic symmetric functionXT, introduced by Stanley, is a sum of monomial symmetric functions corresponding to proper colorings of T. The subtree polynomialST, first considered under a different name by Chaudhary and Gordon, is the bivariate generating function for subtrees of T by their numbers of edges and leaves. We prove that ST=〈Φ,XT〉, where 〈⋅,⋅〉 is the Hall inner product on symmetric functions and Φ is a certain symmetric function that does not depend on T. Thus the chromatic symmetric function is a stronger isomorphism invariant than the subtree polynomial. As a corollary, the path and degree sequences of a tree can be obtained from its chromatic symmetric function. As another application, we exhibit two infinite families of trees (spiders and some caterpillars), and one family of unicyclic graphs (squids) whose members are determined completely by their chromatic symmetric functions.